DE102019210264B4 - Method and device for heating dielectric objects with a predeterminable heat distribution by means of high-frequency radiation - Google Patents

Abstract

Method for heating dielectric objects with a predeterminable heat distribution by means of high-frequency radiation, in which - an object group of several objects is introduced into an irradiation zone and irradiated with coherent high-frequency radiation via an RF transmission unit with at least two separate RF transmission channels, - with the object group being heated prior to heating At least one differential measurement of scattering parameters is carried out, in which high-frequency radiation backscattered from the irradiation zone is detected with a different number of objects in the object group in the irradiation zone at irradiation locations of the RF transmission channels, - from the differential measurement of scattering parameters, heating matrices for the objects in the object group are estimated, - from the Heating matrices Operating parameters for the RF transmission channels are determined, which lead to a predetermined heat distribution in the object group, and the RF transmission channels then for heating the object group with de n operating parameters determined from the heating matrices.

Description

Technical field of application

The present invention relates to a method and a device for heating dielectric objects with a predeterminable heat distribution by means of high-frequency radiation, in which the objects are introduced into an irradiation zone and irradiated with coherent high-frequency radiation via a high-frequency transmission unit with at least two separate high-frequency transmission channels.

The heating of dielectric objects with the help of high frequency (HF) electromagnetic radiation, especially microwave radiation, has a long history. The use of microwaves for heating materials is not limited to food, but has spread in the industrial processing of a wide variety of materials, for example in the ceramic, rubber and plastic industries and in many specialized processes in the chemical industry.

In many applications of microwave heating, the objects to be heated, also referred to below as loads or items to be heated, are located in a closed, electromagnetically insulated (mostly metallic) cavity that is fed by one or more high-frequency electromagnetic radiation sources. One of the main problems with such a configuration is that the multiple reflections of the electromagnetic (EM) waves form standing wave patterns on the walls of the cavity and the surface of the material to be heated, which can significantly affect the uniformity of the heating. Since the energy input, which is converted into heat, is proportional to the squared amplitude of the electric field, the uneven field distribution leads directly to uneven heating or temperature distribution during the heating process. This is mostly undesirable, especially if the distribution is nondeterministic or uncontrollable, and can considerably impair the quality of the material to be processed or the material to be heated. Even in non-isolated areas of application of microwave radiation, however, irregularities in the power distribution can occur. Configurations with several coherent radiation sources can, for example, also generate interference patterns in open space, which disturb the uniformity of the heating. Furthermore, even in open scenarios, geometric properties of the load or its surroundings can lead to scatter phenomena, which could lead to uneven power distribution.

However, the unevenness of the heating can also be desirable in some applications. For example, it would be advantageous if two or more foods or dishes of different types could be cooked or prepared in one appliance at the same time in gastronomy. If these foods differ in the heat absorption capacity, the desired degree of cooking, in the amount, etc., it is necessary to apply different electromagnetic power to different dielectric volumes. Another example in which the uneven distribution of the EM power can be advantageous is the microwave-assisted breaking up of inhomogeneous rock by focusing the EM power. In all these cases it is necessary to focus the electromagnetic power on predefined sub-areas of the load, to heat some parts of the load more than the rest, or generally to impress a desired EM power profile into the load.

State of the art

For both problems, both uniform and focused heating, there are two categories of solutions: deterministic and so-called “blind” methods. Deterministic methods are based on information regarding the electromagnetic field distribution inside the load, which has been obtained through measurements or simulations.

For example, the US 2013/0186887 A1 a deterministic method in which the relationship between excitation or operating parameters of the EM sources and the power patterns they produce in the load is determined experimentally by means of infrared measurements. However, the hardware required for such a sensor system is often very complex and expensive. In some cases it is also incompatible with the requirements for the type of use. Finally, some of these methods can only capture surface temperature as a measure of EM power consumption, which is not always meaningful for bulky objects.

Another sub-category of deterministic methods is based on electromagnetic simulations such as the US 9459346 B2 or pre-optimized structures to predict the power distribution in the load or to optimize it in advance. The main disadvantage of such methods is that they only work for one limited number of very strictly predetermined loads are applicable. Furthermore, the correspondence between the EM simulations and the physical reality for complex and relatively large structures (especially in multi-resonant cavities) is not always sufficiently precise.

Blind methods dispense with such information and either use the same type and method of excitation for all possible loads or rely on external measurements.

One of the most important techniques of most of the blind methods for uniformity optimization is the stochastic mixing or "swirling" of the electromagnetic modes within a cavity. This attempts to generate a statistically homogeneous EM field over time. The mixture can be achieved by mechanical means (cf. US 7 030 347 B2 ) or by modulating the frequency (cf.e.g. U.S. 5,961,871 A ) and the phase. Mechanical approaches have the disadvantage that they are often based on complex hardware that has to be maintained frequently and intensively. Furthermore, such hardware can only adapt to different heating goods to a limited extent, if at all. Frequency and phase modulation approaches overcome these disadvantages to a certain point, but often have poorer performance than mechanical approaches, in part because of the limited range of modes that can be excited. Furthermore, a stochastic modulation of the frequency and phase of the EM field requires frequent switching between a large number of excitation parameters, which places high demands on the electromagnetic power source.

The range of previous blind methods is much more limited for focusing the EM power within the material to be heated. The most common and simplest approach is to electromagnetically isolate the application area of the EM power with suitable hardware, as is the case, for example, in the US 8 420 991 B2 is shown. The resulting sub-areas are fed separately with EM power, so that the ratio of the services in said areas can be controlled. In many applications, however, such strict insulation of the material to be heated is not possible, either because of physical or geometric restrictions (e.g. dishes that consist of different foods and have to be prepared in a common container, biological tissue in vivo, etc.).

An alternative way of electromagnetically highlighting a part of the material to be heated is to introduce a contrast medium into this part which has a high electromagnetic contrast. By means of differential reflection measurements before and after the introduction of the contrast agent, suitable excitation parameters for the focusing can then be determined, as shown in FIG G. Bellizzi et al. "Blind focusing of electromagnetic fields in hyperthermia exploiting target contrast variations." IEEE Transactions on Biomedical Engineering 62.1 (2015): 208-217 is described. However, this approach is not suitable for all applications of electromagnetic heating, because the introduction of a contrast medium is often either impossible, impractical or undesirable, as is the case with food heating, for example.

the EP 3 073 803 A1 describes a method for cooking with microwaves and a microwave cooking device with which the efficiency of a microwave cooking process is optimized by minimizing the reflected microwave power. The determination of the complex scattering parameters of the cooking chamber makes it possible to adjust the relative phase positions accordingly. Furthermore, the amplitudes of the microwave radiation can be optimized accordingly on the basis of the detected forward-running and reverse-running microwave signals.

the US 2016/0 330 803 A1 describes a method for controlling the irradiation of an object in an irradiation room by means of at least two transmitting antennas, with the radiated power, amplitude, phase and frequency of the microwave radiation being adapted for the transmitting antennas via determined scatter matrices. This can be done to minimize the reflected microwave power or to enter a specific target power.

The object of the present invention is to provide a method and a device for heating dielectric objects by means of high-frequency radiation, which allow a group of objects to be heated with a predeterminable heat distribution or even an object to be heated evenly without temperature measurements on the objects or additional hardware expenditure.

Presentation of the invention

The object is achieved with the method and the device according to claims 1, 2 and 19. Advantageous refinements of the method are the subject matter of the dependent claims or can be found in the following description and the exemplary embodiments.

In the proposed method, differential parameter measurements, i.e. S-parameter measurements that take place at different times, are carried out on the object or group of objects to be heated, and heating matrices for the objects of the group of objects or partial areas of the object are estimated from these measurements. With the help of these heating matrices, operating or excitation parameters (amplitude, phase, frequency) for the RF transmission channels of the RF transmission unit used for heating can then be determined, which lead to the desired heat distribution in the object or objects. Depending on the application, two process variants can be used.

In the first variant of the method, an object group made up of several objects is introduced into an irradiation zone and irradiated with coherent high-frequency radiation via an HF transmission unit with several separate HF transmission channels. The RF transmission channels each feed one or more antennas via which the high-frequency radiation is radiated into the irradiation zone. In this context, high-frequency radiation is understood to mean a range between 1 kHZ and 300 THz. High-frequency radiation in the range between 3 MHz and 300 GHz is preferably used. In this first variant of the method, at least one differential measurement of scattering parameters is carried out before the object group is heated, in which high-frequency radiation backscattered from the irradiation zone is detected with a suitable measuring device at the irradiation locations of the RF transmission channels for a different number of objects in the object group in the irradiation zone. A differential measurement of scattering parameters should be understood to mean two separate measurements of scattering parameters at different points in time. Heating matrices for the objects in the object group are then estimated from the differential measurement of scattering parameters and operating or excitation parameters for the RF transmission channels are determined from the heating matrices, which lead to a predetermined heat distribution in the object group. The RF transmission channels are then operated to heat the object group with the operating or excitation parameters determined from the heating matrices.

In the second variant of the method, a uniform distribution of heat within a dielectric object is aimed for, as is desired, for example, when thawing frozen food. In this variant of the method, in turn, an object or a group of objects is introduced into an irradiation zone and irradiated with coherent high-frequency radiation over a heating period via an HF transmission unit with several separate HF transmission channels. In turn, one or more differential scatter parameter measurements are carried out in which, in this case, high-frequency radiation backscattered from the irradiation zone is detected before the start and at a time during the heating period and / or at different times during the heating period at the irradiation locations of the RF transmission channels. From each differential measurement of scattering parameters, heating matrices are then estimated for areas that are more strongly heated relative to other areas of the object or the object group, so-called hotspots. Operating parameters for the RF transmission channels are then determined from the heating matrices, which lead to a uniform distribution of heat in the object or the group of objects. The HF transmission channels are then operated, i.e. after each new differential measurement of scattering parameters, for heating or further heating of the object group with the operating parameters determined from the heating matrices.

In the proposed method variants, by estimating the heating matrices from the differential scatter parameter measurements, information is obtained about which power patterns or power distributions and thus heat distributions are impressed on the load with any (within the operating parameter limits) excitations of the RF transmission channels used for the irradiation. In the method, the relationship between the operating parameters of the RF transmission channels and the heat or absorbed power distribution is estimated by external measurements, ie by measurements that take place neither within the irradiation zone nor within the material to be heated. Instead, the scattering parameters of the irradiation zone are measured at the irradiation locations or irradiation gates before and after the introduction of predetermined objects or object groups or at different times during the heating. The heating matrices for these objects or object groups or partial areas of the objects can be determined by suitable evaluation of these measurements. With the help of these heating matrices, the electromagnetic power can then be determined that is required for any excitation of the system, that is to say any operating parameters, in each of these objects or object groups or sub-areas in Heat is converted. The relationship between input (excitation) and result (consumed power) is thus quantified in the form of heating matrices in order to use this relationship for optimizing the uniformity or focusing the heating or any predeterminable heat distribution.

Correspondingly, the device for carrying out the method has an RF transmission unit with an RF source and power amplifier and at least two RF transmission channels, via which objects introduced into the irradiation zone can be irradiated with coherent high-frequency radiation, and a measuring device with which the RF- Transmission channels in the irradiation zone complex scattering parameters of the loaded irradiation zone can be measured. The RF transmission channels can have one or more transmission antennas per channel. A control and evaluation device is available for controlling the RF transmission unit and measuring device, which is designed to carry out the method in accordance with one or more of the method variants and configurations described in the present patent application.

The proposed method is based on differential scatter matrix measurements (S), i.e. measurements of the scatter parameters (S parameters) at the irradiation locations of the RF transmission channels in the irradiation zone, also referred to below as EMAZ (Electromagnetic Application Zone), before and after the change in load (first alternative method) or change in the temperature of the load (second alternative method) of the irradiation zone, and on the evaluation of the corresponding change in the S-parameters.

In contrast to the above publication by G. Bellizzi et al. The method described for focusing microwaves is assigned a physical meaning to the change in the S parameters in the proposed method, in that heating matrices are calculated or estimated from this change. These heating matrices can then not only be used for focusing, but also for general optimization of the shape of the power distribution (among other things for optimizing uniformity). The proposed method is based on the change in the S matrix product Δ (S ^H S), and not on the matrix product ΔS ^H ΔS of the changes.

The proposed method requires an irradiation zone for the high-frequency radiation or energy, in which the material to be heated is located, and a power amplifier with an HF source that can generate electromagnetic oscillations in a certain frequency range at several (N ≥ 2) channels or outputs. This radiation zone (EMAZ) can be a fully or partially electromagnetically isolated cavity (e.g. a cavity whose walls are made of metal), or an open or partially open region within which most of the RF radiation and power is concentrated (e.g. the imaging chamber of a magnetic resonance tomograph, the area of application of a microwave-assisted quarry device, etc.). The oscillations of the power amplifier must be able to be generated coherently, i.e. they must be able to have the same frequency and a constant phase difference. The frequency, amplitude and phase of the vibrations must be adjustable and controllable by the amplifier. The amplifier feeds the EMAZ, in which the material to be heated is located, via the HF transmission channels with high-frequency electromagnetic radiation at N inlets or irradiation locations. Accordingly, the amplifier has N channels. At each inlet, one or more antennas are arranged as part of the respective RF transmission channel, which radiate into the EMAZ. There is also a measuring device that can measure the scatter parameters (S matrix) of the EMAZ at the N ≥ 2 irradiation locations within the frequency bandwidth of the amplifier.

In the first variant of the proposed method, the material to be heated consists of L ≥ 2 separable parts or objects that can be introduced separately into the EMAZ. For example, the material to be heated can be placed on metallic or other types of carriers that can be inserted separately into the EMAZ. The material to be heated can also be located in separate containers which are at least partially transparent to the high-frequency electromagnetic radiation within the operating frequency spectrum.

These containers can be inserted into the EMAZ or removed from the EMAZ separately and independently of one another.

1. (a) Abschätzung der Erwärmungsmatrizen des Erwärmungsguts.
2. (b) Bestimmung der Anregungen der Kanäle des Verstärkers anhand einer Optimierung der Erwärmung.
3. (c) Anwendung der obigen Anregungen zur Erwärmung der Last.

The first variant of the process can be broken down into three phases:

1. (a) Estimation of the heating matrices of the material to be heated.
2. (b) Determination of the excitations of the channels of the amplifier on the basis of an optimization of the heating.
3. (c) Apply the suggestions above to heat the load.

To estimate the heating matrices of the material to be heated (first phase), different subsets or objects of the material to be heated are introduced into the EMAZ and the S parameters of the partially loaded EMAZ are measured at the irradiation locations. In addition, the S-parameters of the fully loaded EMAZ are measured, i.e. the EMAZ when all the material to be heated has been introduced. The difference between these S-parameter measurements is evaluated using the mathematical method described below in order to calculate the heating matrices of the different subsets or objects of the material to be heated. This step can be repeated for any subsets of the material to be heated, as long as its shape, consistency and the loading options of the EMAZ allow, until the heating matrices of all subsets or objects that make up the material to be heated have been calculated.

The second method variant of the proposed method is suitable for items to be heated that have to be heated as uniformly as possible, their dielectric properties preferably depending strongly on the local temperature. Frequently occurring application examples that fall under this category relate to the thawing of frozen goods (e.g. food, ready meals, frozen blood samples or transplant organs, etc.). If the field distribution within the food to be heated (or defrosted) is uneven, there are "hotspots", i.e. localized areas whose temperature rises faster than average compared to the other areas. If the losses of the material also increase with increasing temperature, there is the so-called “runaway effect”, during which such hotspots consume more and more power and get hotter even faster, which in turn continuously worsens the uniformity of the heating. So this is a positive feedback loop, the undesirable results of which are constantly self-reinforcing. To counteract this catastrophic effect, the excitation parameters must be readjusted so that as little power as possible is introduced into the hotspots that are constantly forming. In the second variant of the method, S-parameter measurements are carried out repeatedly, preferably regularly, in order to detect the occurrence of these hotspots, to calculate their heating matrices and to counteract their further heating. As a result, the uniformity of the heating or thawing is constantly monitored and regulated. The occurrence and the heating matrices of the hotspots are detected or determined by comparing successive S-parameter measurements. If the S-parameters of the EMAZ have changed significantly within a time interval while the excitation has remained the same, this indicates that part of the food to be heated or thawed has changed its properties due to its warming. On the basis of this change, suitable suggestions are calculated according to the second variant of the method, which heat the affected “hot” areas the least.

1. (a) Messung der S-Parameter der EMAZ vor dem Start des Erwärmungsprozesses.
2. (b) Auswahl eines Anregungsparametersatzes.
3. (c) Anregung der EMAZ mit gewähltem Parametersatz.
4. (d) Erneute Messung der S-Parameter, Abschätzung der Erwärmungsmatrizen eventueller Hotspots aus der Differenz der S-Parametermessungen und Berechnung neuer Anregungsparameter anhand der Erwärmungsmatrizen.
5. (e) Falls der Erwärmungsprozess noch nicht zu beenden ist, gehe zu Schritt (c).

The second variant of the process can be broken down into the following phases:

1. (a) Measurement of the S-parameters of the EMAZ before the start of the warming process.
2. (b) Selection of a set of excitation parameters.
3. (c) Excitation of the EMAZ with the selected parameter set.
4. (d) Renewed measurement of the S-parameters, estimation of the heating matrices of any hotspots from the difference in the S-parameter measurements and calculation of new excitation parameters based on the heating matrices.
5. (e) If the heating process has not yet ended, go to step (c).

The estimation of the heating matrices from the difference between the S-parameter measurements is again carried out using a mathematical method that will be described in more detail later. In this way, based on the change in the S-parameters, suitable excitation parameters are determined in order to avoid the formation of hotspots during the heating process.

In the proposed method, heating matrices of partial areas of a load that are to be heated by means of coherent electromagnetic power are determined from differential scatter parameter measurements. In the first variant of the method, these sub-areas are determined by the user and / or the geometric or physical boundary conditions. The user can introduce these partial areas into the irradiation zone or remove them therefrom. From the estimated according to the procedure Heating matrices for these sub-areas can be used to optimize the profile of the electromagnetic power distribution, which is impressed on the load by the high-frequency radiation. Accordingly, different applications of coherent high-frequency heating can be implemented, for example uniform heating or focusing of the power in one or more sub-areas. The second variant of the method is about heating objects as uniformly and gently as possible, preferably objects in which an abrupt phase transition takes place during the heating process. An example of this is frozen food, in which the phase transition from solid to liquid should be as homogeneous as possible and at the same time. The aim of the method is to detect the formation of so-called hotspots, to determine their heating matrices and to counteract the formation of hotspots. In this case, the sub-areas whose heating matrices are determined by the proposed method are the hotspots.

The proposed method is what is known as a blind method, in which no sensor data on physical parameters such as temperature or electromagnetic field strength in the load or EMAZ are required. The method is based only on measurements at the irradiation locations or inlets of the high-frequency radiation or the outputs of the amplifier channels. These are scatter parameter measurements by the EMAZ. This means that there is no need for cumbersome, expensive and difficult-to-maintain sensors. The heating process and usability are not affected by inserting or removing sensors in the load or EMAZ. In the first variant of the method, the determination of the heating matrices of the individual objects or partial areas of the load enables a wide range of heating options, from homogeneous heating to selective focusing of the electromagnetic power. Between these extreme power distributions (homogeneous / focused), any other distributions can also be impressed on the load. In the case of uniformity optimization, no moving parts are required to generate a statistically homogeneous electromagnetic field. Nor is electronically generated mode noise due to random frequency or phase variations required, which would place high demands on the switching speed of the power amplifier and would not be adaptable to the different loading scenarios. Use of contrast media is also not necessary with the proposed method. In one embodiment of the second variant of the method, a phase transition of the load can be tracked or detected quickly and reliably on the basis of the calculation and monitoring of the mean absorption rate of the load. The advantage of the method described for this purpose lies in the fact that the average of the absorption rate can be derived directly from the S parameters without performing time-consuming (and often insufficient) parameter runs. The proposed method and the associated device can be used in areas of application in which objects or groups of objects are to be heated in a defined manner by high-frequency radiation. One example is the use in microwave or combination ovens for cooking, preparing and defrosting food in large kitchens and restaurants. These ovens have to heat the food evenly and efficiently, among other things in order to comply with the hygienic regulations, to take better account of the properties of the individual ingredients, to better preserve the nutritional value of the food, to accelerate the cooking process. On the other hand, it is of great advantage if any part of the food can be heated more than the rest, because this allows different dishes or foods to be cooked at the same time or similar foods to be prepared in different degrees of cooking. Another example relates to the application in industry, especially in food processing (e.g. pasteurization, drying processes, etc.), in chemical engineering (e.g. microwave-assisted synthesis and processing of chemical substances, preparation processes for pharmaceutical products) or in material processing (e.g. polymerization, heating of liquids and solids, including in the ceramic and steel industry, sintering). Another possible use is in medical technology, for example for selective or uniform heating of tissues, for thawing frozen goods or transplant organs or for optimizing high-frequency electromagnetic fields in magnetic resonance tomography.

Figure list

• 1 eine schematische Darstellung eines Beispiels der vorgeschlagenen Vorrichtung für die Erwärmung eines oder mehrerer Objekte;
• 2 eine beispielhafte Aufteilung des Erwärmungsguts in vier Teilbereiche in der EMAZ;
• 3 ein Beispiel für eine teilbeladene EMAZ, deren ursprüngliche Beladung in 2 dargestellt ist;
• 4 ein weiteres Beispiel für eine teilbeladene EMAZ, deren ursprüngliche Beladung in 2 dargestellt ist;
• 5 ein Beispiel für die Wahl der Betriebs- bzw. Anregungsparameter nach einem Zeitschlitzverfahren; und
• 6 ein Beispiel für die zeitliche Entwicklung der Wärmeverteilung während der Erwärmung eines Erwärmungsguts.

The proposed method variants and the associated device are described in more detail below using exemplary embodiments in conjunction with the drawings. Here show:

• 1 a schematic representation of an example of the proposed device for heating one or more objects;
• 2 an exemplary division of the material to be heated into four sub-areas in the EMAZ;
• 3 an example of a partially loaded EMAZ whose original load was in 2 is shown;
• 4th Another example of a partially loaded EMAZ whose original load was in 2 is shown;
• 5 an example for the selection of the operating or excitation parameters according to a time slot method; and
• 6th an example of the development over time of the heat distribution during the heating of a material to be heated.

Ways of Carrying Out the Invention

• • Eine elektromagnetische Bestrahlungszone (EMAZ) 1, die in 1 beispielhaft als elektromagnetisch isolierte Kammer ausgebildet ist. Die Erfindung ist jedoch nicht auf eine derartige Ausgestaltung beschränkt. Die EMAZ kann auch offen oder halboffen ausgebildet, nichtisoliert oder nur teilweise isoliert sein.
• • Einen kohärenten hochfrequenten Leistungsverstärker 4 mit HF-Quelle, der N ≥ 2 separate elektromagnetische Schwingungen auf N (also gleich vielen) Kanälen gleichzeitig generieren kann, deren Frequenzen, Amplituden und relative Phasendifferenzen durch den Nutzer oder eine selbstständige Steuerungssoftware innerhalb bekannter Toleranzen festgelegt bzw. variiert werden können. Die Frequenzen der Ausgangssignale des Leistungsverstärkers 4 müssen nicht zwangsläufig dieselbe Frequenz haben. Es muss jedoch die Möglichkeit bestehen, einige oder alle Ausgangssignale kohärent zu generieren, d.h. bei exakt derselben stabilen Frequenz und mit fester relativer Phasendifferenz zueinander.
• • Zwei oder mehr Antennen 2, die hochfrequente elektromagnetische Strahlung in die EMAZ einspeisen können und Teil der HF-Sendekanäle der Vorrichtung sind. An jedem Kanal des Leistungsverstärkers 4 können hierzu eine oder mehrere Antennen angeschlossen sein, auch wenn in 1 beispielhaft nur eine Antenne pro Kanal angezeigt wird.
• • Eine Messeinrichtung 5, die die komplexen Streuparameter bzw. Streumatrizen der EMAZ mit vordefinierter Frequenzauflösung innerhalb der Betriebsbandbreite zu willkürlichen Zeitpunkten vor, während und nach dem Erwärmungsprozess messen kann. Die Messeinrichtung kann entweder in die oben genannte Signalquelle eingebaut oder separat davon angeordnet sein.
• • Eine Steuer- und Auswerteeinheit 9, die an vier Datenströmen angebunden ist: einen benutzerdefinierten Eingangsdatenstrom 7, einen Befehlsdatenstrom 8, einen Rückmeldungsstrom 6 aus dem Leistungsverstärker 4 und der Messeinrichtung 5, und (optional) einen Rückmeldungsstrom 10 aus der EMAZ 1. Der Rückmeldungsstrom 6 enthält Daten der S-Parametermessungen aus der Messeinrichtung 5. Der Eingangsdatenstrom 7 enthält Anweisungen vom Benutzer (beispielsweise, aber nicht ausschließlich, Leistungspegel, Dauer der Erwärmung, Frequenzauflösung, Anzahl der Zyklen, Erwärmungsprofil usw.). Der Befehlsdatenstrom 8 beinhaltet die Anregungsparameter für die unterschiedlichen Kanäle des Verstärkers (u.a. Frequenz, Amplitude, Phasendifferenz, Dauer des Signals). Der Rückmeldungsstrom 10 kann Sensordaten aus der EMAZ und/oder dem Erwärmungsgut enthalten (z.B. Temperatur, Feuchtegrad usw.), falls solche Sensoren vorhanden sind. Die Steuer- und Auswerteeinheit 9 beinhaltet Software, die unterschiedliche mathematische Verfahren A und B zur Abschätzung der Erwärmungsmatrizen und Bestimmung der Betriebs- bzw. Anregungsparameter ausführt. Das Verfahren A schätzt anhand differentieller S-Parametermessungen (d.h. S-Parametermessungen, die an unterschiedlichen Zeitpunkten stattfinden) die Erwärmungsmatrizen, im Folgenden auch als Q-Matrizen bezeichnet, eines oder mehrerer Teilbereiche des Erwärmungsguts ab. Das Verfahren B bestimmt anhand der Q-Matrizen, die mit Verfahren A berechnet bzw. abgeschätzt wurden, die passenden Anregungen, um eine erwünschte Wärme- bzw. Leistungsverteilung im Erwärmungsgut zu erzeugen. Die Steuer- und Auswerteeinheit 9 ist ein zentrales Element in einer Feedbackschleife, die den Erwärmungsprozess kontinuierlich überwacht und steuert.

A device embodied by way of example for carrying out the proposed method is shown in FIG 1 shown schematically. The device has the following components:

• • An electromagnetic radiation zone (EMAZ) 1 , in the 1 is designed for example as an electromagnetically isolated chamber. However, the invention is not limited to such an embodiment. The EMAZ can also be open or semi-open, non-insulated or only partially insulated.
• • A coherent high frequency power amplifier 4th with HF source that can generate N ≥ 2 separate electromagnetic oscillations on N (i.e. the same number) channels at the same time, the frequencies, amplitudes and relative phase differences of which can be defined or varied by the user or independent control software within known tolerances. The frequencies of the output signals from the power amplifier 4th do not necessarily have to have the same frequency. However, it must be possible to generate some or all of the output signals coherently, ie at exactly the same stable frequency and with a fixed relative phase difference to one another.
• • Two or more antennas 2 that can feed high-frequency electromagnetic radiation into the EMAZ and are part of the device's RF transmission channels. On each channel of the power amplifier 4th one or more antennas can be connected for this purpose, even if in 1 for example only one antenna per channel is displayed.
• • A measuring device 5 that can measure the complex scatter parameters or scatter matrices of the EMAZ with a predefined frequency resolution within the operating bandwidth at arbitrary times before, during and after the heating process. The measuring device can either be built into the above-mentioned signal source or arranged separately from it.
• • A control and evaluation unit 9 connected to four data streams: a user-defined input data stream 7th , a command stream 8th , a feedback stream 6th from the power amplifier 4th and the measuring device 5 , and (optional) a feedback stream 10 from the EMAZ 1 . The feedback stream 6th contains data of the S-parameter measurements from the measuring device 5 . The input data stream 7th contains instructions from the user (for example, but not limited to, power level, duration of heating, frequency resolution, number of cycles, heating profile, etc.). The command stream 8th contains the excitation parameters for the different channels of the amplifier (including frequency, amplitude, phase difference, duration of the signal). The feedback stream 10 can contain sensor data from the EMAZ and / or the material to be heated (e.g. temperature, degree of humidity, etc.), if such sensors are available. The control and evaluation unit 9 contains software that executes different mathematical methods A and B for estimating the heating matrices and determining the operating and excitation parameters. Method A uses differential S-parameter measurements (ie S-parameter measurements that take place at different times) to estimate the heating matrices, also referred to below as Q matrices, of one or more partial areas of the material to be heated. Method B uses the Q matrices that were calculated or estimated using method A to determine the appropriate stimuli in order to generate a desired heat or power distribution in the material to be heated. The control and evaluation unit 9 is a central element in a feedback loop that continuously monitors and controls the heating process.

In 1 is also a load or good to be heated 3 to recognize that has been introduced into the EMAZ and can consist of one or more dielectric objects.

In the following, two exemplary embodiments of the proposed method are explained in more detail, in the context of which the mathematical methods A and B are also described in more detail.

First execution

1. 1. Abschätzung der Erwärmungsmatrizen der L Teilbereiche.
2. 2. Bestimmung der Anregungen der Kanäle des Verstärkers anhand von Optimierung des Leistungsmusters.
3. 3. Anwendung der obigen Anregungen zur Erwärmung der Last.

The first exemplary embodiment relates to heating scenarios in which the user can divide the material to be heated into partial quantities and introduce it separately into the EMAZ or remove it from the EMAZ. This can occur, for example, in a microwave oven or combi oven with several racks, in which the dishes or food can be placed on different carriers (e.g. baking trays). So one assumes that the material to be heated consists of L parts, which are in the 2 until 4th are marked as G ₁ , G ₂ , ..., G _l , ..., G _L and are separated into the EMAZ 1 introduced or by the EMAZ 1 can be removed. The process consists of three phases:

1. 1. Estimation of the heating matrices of the L sub-areas.
2. 2. Determination of the excitations of the channels of the amplifier on the basis of optimization of the power pattern.
3. 3. Applying the suggestions above to heat the load.

1. 1. Der Benutzer führt das gesamte Erwärmungsgut (G₁, G₂, ..., G_L) in EMAZ 1 ein und das System misst die N × N S-Matrix der EMAZ bei mehreren Frequenzen ƒ₁,ƒ₂, ..., ƒ_{N ƒ} mithilfe der Messeinrichtung 5. Die Daten werden als N_f komplexe Matrizen S₀(ƒ_n),n = 1, ...,N_ƒ, mittels Datenstrom 6 in den Speicher von Steuer- und Auswerteeinheit 9 übertragen. In 2 ist eine beispielhafte Aufteilung des Erwärmungsguts 3 in vier Teilbereiche G₁, G₂, G₃, G₄ dargestellt.
2. 2. Für l = 1 bis L
   1. a. Der Benutzer entfernt den Teil C_l des Erwärmungsguts. Dabei kann entweder G_l samt seinen Träger entfernt werden (siehe z.B. 3) oder nur das Erwärmungsgut entfernt werden (siehe z.B. 4). In letzterem Fall ist es in der Regel praktischer, zuerst G_l samt seinen Träger zu entfernen und danach einen anderen leeren, identischen Träger einzuführen. Wenn die Geometrie und das Material des Trägers das EM-Feld innerhalb der EMAZ stark beeinflusst, kann diese alternative Vorgehensweise Vorteile haben.
   2. b. Das System misst die N × N S-Matrix der EMAZ bei denselben Frequenzen ƒ_n, n = 1, ..., N_ƒ mithilfe der Messeinheit 5. Die Daten werden als N_ƒ komplexe Matrizen S_l(ƒ_n), n = 1, ...,N_ƒ, mittels Datenstrom 6 in den Speicher von Steuer- und Auswerteeinheit 9 übertragen.
   3. c. Steuer- und Auswerteeinheit 9 berechnet die Q-Matrix von Teilbereich G_l mithilfe von Verfahren A.
   4. d. Der Benutzer führt Teilmenge G_l wieder in die EMAZ ein.

Phase 1 (recognition phase):

1. 1. The user enters the entire item to be _{heated (G 1} , G ₂ , ..., G _{L) in EMAZ} 1 one and the system measures the N × N S matrix of the EMAZ at several frequencies ƒ ₁ , ƒ ₂ , ..., ƒ _{N ƒ} using the measuring device 5 . The data are provided as N _f complex matrices S ₀ (ƒ _n ), n = 1, ..., N _ƒ , by means of a data stream 6th into the memory of the control and evaluation unit 9 transfer. In 2 is an exemplary division of the material to be heated 3 shown in four sub-areas G ₁ , G ₂ , G ₃ , G ₄ .
2. 2. For l = 1 to L
   1. a. The user removes the part C _{l of} the material to be heated. Either G _l and its carrier can be removed (see e.g. 3 ) or only the material to be heated must be removed (see e.g. 4th ). In the latter case, it is usually more practical to first _{remove G 1} and its carrier and then to insert another empty, identical carrier. If the geometry and material of the carrier have a strong influence on the EM field within the EMAZ, this alternative approach can have advantages.
   2. b. The system measures the N × N S matrix of the EMAZ at the same frequencies ƒ _n , n = 1, ..., N _ƒ using the measuring unit 5 . The data are provided as N _ƒ complex matrices S _l (ƒ _n ), n = 1, ..., N _ƒ , by means of a data stream 6th into the memory of the control and evaluation unit 9 transfer.
   3. c. Control and evaluation unit 9 calculates the Q-matrix of sub-area G _l using method A.
   4. d. The user re-introduces subset G _l into the EMAZ.

The following explains the form of Method A in the case of performing the method for the first time. The aim of this algorithm is to determine the heating matrix (Q matrix) of a sub-area G _l . This matrix is an N × N complex symmetric matrix Q _l , where N represents the number of channels. If the excitation of the channels is given by the complex vector a = [a ₁ a ₂ ... a _N] ^T (where | a _n | ^{2 represents} the power of channel n and arg a _n its phase), then by definition equals the electromagnetic power converted into heat within G _l $$p_l=a^H{Q}_{l}a.$$

wobei E_k(r) das elektrische Feld, das von Kanal k erzeugt wird, wenn nur Kanal k sendet, und σ(r) die äquivalente elektrische Leitfähigkeit im Erwärmungsgut darstellen.The elements of a Q-matrix can be defined by the following formula: $$q_{ij}^l=\frac{1}{2}{\displaystyle \int {\displaystyle \int {\displaystyle {\int }_{G_l}\sigma \left(r\right){\mathrm{E.}}_i^{\ast }\left(r\right)\cdot {\mathrm{E.}}_j\left(r\right)dV}}}$$

where E _k (r) is the electric field that is generated by channel k if only channel k is transmitting, and σ (r) represents the equivalent electrical conductivity in the material to be heated.

wobei p_w = a^HQ_wa die parasitären Verluste der EMAZ sind. Solche Verluste können bspw. an den metallischen Wänden der EMAZ entstehen, falls die Bestrahlungszone durch metallische Wände von ihrer Umgebung teilweise oder vollständig isoliert wird. Ferner können parasitäre Verluste auch an anderen metallischen Gegenständen (z.B. Antennen, Lastträger) entstehen, die nicht zum Erwärmungsgut gehören. Falls die EMAZ nicht vollständig (wenn überhaupt) von ihrer Umgebung isoliert ist, kann man schließlich unter parasitären Verlusten auch die EM-Strahlungsleckage aus der Bestrahlungszone in ihre Umgebung einstufen.Using the law of conservation of energy, the total power consumed in the EMAZ is equal to the power fed in || a || 2 = a ^H a minus the reflected power || S ₀ a || ² = a ^H S ₀ ^H S ₀ a. Note that the reflected power is calculated by the S matrix S ₀ , ie by the measurement that was carried out when the EMAZ was fully loaded. thats why $$p_1+p_2+\cdots +p_{\mathrm{L.}}+p_w=a^{H}a-a^H{\mathrm{S.}}_0^H{\mathrm{S.}}_{0}a=a^H\left(\mathrm{I.}-{\mathrm{S.}}_0^H{\mathrm{S.}}_0\right)a,$$

where p _w = a ^H Q _w a are the parasitic losses of the EMAZ. Such losses can arise, for example, on the metallic walls of the EMAZ if the irradiation zone is partially or completely isolated from its surroundings by metallic walls. Furthermore, parasitic losses can also occur on other metallic objects (e.g. antennas, load carriers) that are not part of the material to be heated. If the EMAZ is not completely (if at all) isolated from its surroundings, one can finally classify the EM radiation leakage from the irradiation zone into its surroundings under parasitic losses.

für alle möglichen Vektoren a ∈ C^N . Daraus lässt sich folgende Matrixgleichung ableiten: $$Q_w+{\displaystyle \sum _{k=1}^L{Q}_k}=I-S_0^H{S}_0$$

Hence the equation results $$a^H\left(Q_w+{\displaystyle \sum _{k=1}^{\mathrm{L.}}{Q}_k}\right)a=a^H\left(\mathrm{I.}-{\mathrm{S.}}_0^H{\mathrm{S.}}_0\right)a$$

for all possible vectors a ∈ C ^N. The following matrix equation can be derived from this: $$Q_w+{\displaystyle \sum _{k=1}^{\mathrm{L.}}{Q}_k}=\mathrm{I.}-{\mathrm{S.}}_0^H{\mathrm{S.}}_0$$

If the EMAZ is not fully loaded, but a part (e.g. G _l ) is missing, the following equation can be derived using a similar procedure: $$Q_w+{\displaystyle \sum _{k\ne 1}{Q}_k}\simeq \mathrm{I.}-{\mathrm{S.}}_l^H{\mathrm{S.}}_l$$

It should be noted that in equation (3) the S parameter _{measurement S I of} the EMAZ without the l -th part of the food G _l and the Q matrices Q _{k of} the EMAZ appear with a full load. An implicit assumption in equation (3) is that the Q matrices of all subsets of the material to be heated (Q _k , k ≠ l) that have remained in the EMAZ and the wall losses are the same as in the case of a full load (see equation (2 )) have stayed. This is only an approximation, because removing part of the material to be heated will slightly change the electromagnetic fields within the EMAZ, which in turn leads to a change in the Q matrices of the sub-areas G _l according to equation (1). Therefore equation (3) is only an approximation.

If one subtracts equations (2) and (3) from one another, one obtains the following estimate for the Q-matrix of the part G _{l of} the material to be heated: $$Q_l\simeq {\mathrm{S.}}_l^H{\mathrm{S.}}_l-{\mathrm{S.}}_0^H{\mathrm{S.}}_0=\mathrm{\Delta }\left({\mathrm{S.}}^H\mathrm{S.}\right)$$

A good approximation for the heating matrix of a sub-area of the material to be heated is therefore the change in the S-parameter ^{product S H S after said sub-area has been removed from the EMAZ.} This has the following simple physical interpretation: the change in SHS is equal to the change in reflected power. It is clear that the power consumed must decrease and the reflected power increase by the same amount if part of the material to be heated is removed. The change in the S-parameter product Δ (S ^H S) must therefore be equal to the power Q _l that was consumed in the sub-area G _l.

unnatürliche negative Eigenwerte aufweisen. Solche Eigenwerte sind unnatürlich, weil alle Eigenwerte von Q-Matrizen Leistungswerte repräsentieren, wie aus der Gleichung v^HQ̃_lv = v^H(λv) = λ||v||²ersichtlich ist, wo v der Eigenvektor von Q̃_l ist, der dem Eigenwert λ entspricht. Der Eigenwert ist die Leistung, die der Eigenvektor in Teilbereich G_l generiert, normiert durch die Eingangsleistung des Eigenvektors. Leistungswerte können nur positiv sein, was demnach auch für die Eigenwerte gelten muss. Um dieses Problem zu beheben, kann man die geschätzten Q-Matrizen Q̃_l durch die beste positiv semidefinite Approximation ersetzen. Diese Approximation lässt sich durch folgende Formel berechnen: $${\widehat{\mathbf{Q}}}_l=\frac{1}{2}\left(U+V\right)\mathrm{\Sigma }{\mathbf{V}}^H$$

wo Q̃_l = UΣV^H die Singulärwertzerlegung von Q̃_l. Matrix Q̂_l ist demnach die beste Approximation von Q̃_l, die „natürliche Eigenschaften“, also nichtnegative Eigenwerte hat. Das Verfahren, das durch das Hinzufügen von Approximation (5) entsteht, wird im Folgenden als A1 Variation von Verfahren A bezeichnet.The method according to equation (4) is referred to below with A0. Since redistribution effects of the electromagnetic field pattern when subsets are removed can lead to errors in the estimation of the Q-matrices, it is possible that the estimated Q-matrices $${\tilde{Q}}_l\equiv {\mathrm{S.}}_l^H{\mathrm{S.}}_l-{\mathrm{S.}}_0^H{\mathrm{S.}}_0$$

have unnatural negative eigenvalues. Such eigenvalues are unnatural because all eigenvalues of Q-matrices represent power ^{values, as from the equation v H} Q̃ _l v = v ^H (λv) = λ || v || ^{2 it} can be seen where v is the eigenvector of Q̃ _l , which corresponds to the eigenvalue λ. The eigenvalue is the power that the eigenvector generated in sub-area G _l , normalized by the input power of the eigenvector. Performance values can only be positive, which must therefore also apply to the eigenvalues. To solve this problem, the estimated Q-matrices Q̃ _{l can be} replaced by the best positive semidefinite approximation. This approximation can be calculated using the following formula: $${\widehat{\mathbf{Q}}}_l=\frac{1}{2}\left(U+V\right)\mathrm{\Sigma }{\mathbf{V}}^H$$

where Q̃ _l = UΣV ^{H is} the singular value decomposition of Q̃ _l . Matrix Q̂ _l is therefore the best approximation of Q̃ _l , which has “natural properties”, that is, nonnegative eigenvalues. The procedure created by adding approximation ( 5 ) is referred to below as the A1 variation of method A.

• Finde Matrizen Q̂_l, l = 1,2, ..., L die:
  1. 1. Positiv semidefinit sind, d.h. Q̂_l ≥ 0.
  2. 2. Die Gleichung ${\tilde{Q}}_w+{\displaystyle {\sum }_{k=1}^L{\widehat{\mathbf{Q}}}_l}=I-S_0^H{S}_0$
     
     erfüllen.
  3. 3. Folgende Norm minimieren: ${{\displaystyle {\sum }_{l=1}^L\Vert {\widehat{\mathbf{Q}}}_l-{\tilde{Q}}_l\Vert }}^2.$
     

There is another physical condition which, if met, could improve the estimate of the Q-matrices, namely the law of conservation of energy, which is embodied by equation (2). Since the Q matrices estimated by (4) contain a certain degree of inaccuracy, it is to be expected that equation (2) will not be met _{by the estimated matrices Q̃ l.} However, one could _{search for new matrices Q̂ l} , which on the one hand lie as close as possible to the original estimates Q̃ _l , on the other hand are positive semidefinite and satisfy equation (2). The following optimization problem can thus be formulated:

• Find matrices Q̂ _l , l = 1,2, ..., L which:
  1. 1. Are positive semidefinite, ie Q̂ _l ≥ 0.
  2. 2. The equation ${\tilde{Q}}_w+{\displaystyle {\sum }_{k=1}^{\mathrm{L.}}{\widehat{\mathbf{Q}}}_l}=\mathrm{I.}-{\mathrm{S.}}_0^H{\mathrm{S.}}_0$
     
     fulfill.
  3. 3. Minimize the following norm: ${{\displaystyle {\sum }_{l=1}^{\mathrm{L.}}\Vert {\widehat{\mathbf{Q}}}_l-{\tilde{Q}}_l\Vert }}^2.$
     

berechnet, wobei S_e die S-Parameter der leeren EMAZ repräsentieren.Note that you step in 2 cannot use the exact Q matrix of parasitic losses Q̃ _w because it is unknown. However, one can use an estimate Q̃ _{w of} said Q-matrix instead, which can be determined, for example, by calculating the losses of the empty EMAZ by means of a formula ${\tilde{Q}}_w=\mathrm{I.}-{\mathrm{S.}}_e^H{\mathrm{S.}}_e$

calculated, where S _{e represent} the S parameters of the empty EMAZ.

The above optimization problem is convex and can be solved by means of corresponding numerical methods, which are carried out by software on the control and evaluation unit 9 are implemented. We call this variation of method A, which requires the solution of the above convex optimization problem, A2.

Using the law of conservation of energy, phase 1 can also be modified in such a way that one only has to carry out L - 1 instead of L steps. When the Q-matrices of the first L - 1 sub-areas have been calculated, the warming matrix of the L-th sub-area can be calculated using the law of conservation of energy as follows: $${\tilde{Q}}_{\mathrm{L.}}=\mathrm{I.}-{\mathrm{S.}}_0^H{\mathrm{S.}}_0-{\displaystyle \sum _{l=1}^{\mathrm{L.}-1}{\tilde{Q}}_l-{\tilde{Q}}_w}$$

1. 1. Miss die S-Parameter der leeren EMAZ S_e bei Frequenzen f₁, f_z, ...,f_{N f} und berechne ${\tilde{Q}}_w=I-S_e^H{S}_e$
   
2. 2. Führe den ersten Teil der Last G₁ in die EMAZ ein und miss erneut die S-Parameter S₂ der EMAZ.
3. 3. Ohne den Teil G₁ zu entfernen, führe den zweiten Teil der Last G₂ in die EMAZ ein und miss die S-Parameter S₀ der vollbeladenen EMAZ.
4. 4. Berechne Q-Matrix von G₂ durch ${\tilde{Q}}_2=S_2^{\mathrm{<figure-callout\; id="2"\; label="H\; S"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">H</figure-callout>}}{S}_{}{}_2-S_0^{\mathrm{<figure-callout\; id="0"\; label="H\; S"\; filenames="DE102019210264B4\_0153.png"\; state="\{\{state\}\}">H</figure-callout>}}{S}_{}{}_0.$
   
5. 5. Berechne Q-Matrix von G₁ durch ${\tilde{Q}}_1=I-S_0^{\mathrm{<figure-callout\; id="0"\; label="H\; S"\; filenames="DE102019210264B4\_0153.png"\; state="\{\{state\}\}">H</figure-callout>}}{S}_{}{}_0-{\tilde{Q}}_2-{\tilde{Q}}_w.$
   

The last term of equation (6) can either be omitted if the parasitic losses are small, or _{estimated by measuring the S parameters S e of} the empty EMAZ, as described above. For the case L = 2 as an example, i.e. subdivision of the material to be heated into two areas, the phase 1 simplify as follows:

1. 1. Measure the S parameters of the empty EMAZ S _e at frequencies f ₁ , f _z , ..., f _{N f} and calculate ${\tilde{Q}}_w=\mathrm{I.}-{\mathrm{S.}}_e^H{\mathrm{S.}}_e$
   
2. 2. Introduce the first part of the load G ₁ into the EMAZ and measure the S-parameters S _{2 of} the EMAZ again.
3. 3. Without _{removing part G 1} , introduce the second part of load G ₂ into the EMAZ and measure the S-parameters S _{0 of} the fully loaded EMAZ.
4. 4. Compute Q-matrix of G ₂ by ${\tilde{Q}}_2={\mathrm{S.}}_2^H{\mathrm{S.}}_2-{\mathrm{S.}}_0^H{\mathrm{S.}}_0.$
   
5. 5. Compute Q-matrix of G ₁ by ${\tilde{Q}}_1=\mathrm{I.}-{\mathrm{S.}}_0^H{\mathrm{S.}}_0-{\tilde{Q}}_2-{\tilde{Q}}_w.$
   

Phase 2 (optimization phase):

In this phase of the first exemplary embodiment of the method, the user or the system must place one or more requirements on the electromagnetic power distribution and use an optimization method to select the channel excitations so that the generated power pattern meets the requirements. Different requirements can be made that require a corresponding optimization process. Two of the most frequently used requirements are described below as examples.

1) Focusing the electromagnetic power:

wobei p_k die Leistung, die im Teilbereich G_k verbraucht wird und V_k das entsprechende Volumen besagten Teilbereichs darstellen. In der obigen Definition ist als zu fokussierender Bereich der l -te Teilbereich G_l angenommen. Das Akronym MPDR steht für „Mean Power Density Ratio“, also Verhältnis der durchschnittlichen Leistungsdichten. Zur Maximierung von MDPR kann ein Algorithmus eingesetzt werden, wie er beispielsweise in der DE 10 2016 202 234 B3 in den Absätzen [0049]-[0051] beschrieben ist.This requirement is about concentrating most of the power in a sub-area G _{l of} the material to be heated. Since the volume of said sub-area can be much smaller than that of the rest of the material to be heated, it makes more sense to talk about power density, i.e. about power normalized by the volume in which it is consumed. A criterion for the quality of the focusing that is often used is the power contrast, i.e. the ratio of the average power density in the focused area to the average power density in the rest of the material to be heated: $$\text{MPDR =}\frac{{\displaystyle {\sum }_{k\ne l}{V}_k}}{V_l}\cdot \frac{p_l}{{\displaystyle {\sum }_{k\ne l}{p}_k}}$$

where p _{k is} the power that is consumed in sub-area G _k and V _{k represents} the corresponding volume of said sub-area. In the above definition, the l-th sub-area G _{l is} assumed to be the area to be focused. The acronym MPDR stands for "Mean Power Density Ratio". To maximize MDPR, an algorithm such as that described in the DE 10 2016 202 234 B3 in paragraphs [0049] - [0051].

If one omits the constant prefactor in equation (7), it is sufficient to maximize the following quotient: $$\frac{a^H{Q}_{l}a}{a^H\left({\displaystyle {\sum }_{k\ne l}{Q}_k}\right)a}$$

der dem größten Eigenwert λ entspricht. Dieses mathematische Problem lässt sich leicht anhand bekannter numerischer Verfahren lösen.This problem can be presented again as a generalized eigenvalue problem. Accordingly, the sought-after excitation vector is the eigenvector of the equation $$Q_{l}a=\lambda \left({\displaystyle \sum _{k\ne l}{Q}_k}\right)a$$

which corresponds to the greatest eigenvalue λ. This mathematical problem can be easily solved using well-known numerical methods.

Note that the above method generates an optimal focusing vector a _max (ƒ) per frequency. One can therefore apply all excitation vectors at the corresponding frequencies f _n , n = 1, ..., N _ƒ one after the other. Since each frequency has a different potential for focusing, it is possible that using different frequencies will increase the average focus quality. Furthermore, this can lead to an improvement in the uniformity of the heating within the individual sub-areas, since different frequencies generate different power patterns.

2) Optimizing the evenness:

wobei $\overline{p}={\displaystyle {\sum }_{l=1}^L{p}_l}/{\displaystyle {\sum }_{l=1}^L{V}_l}$

die durchschnittliche Leistungsdichte darstellt. Die Bezeichnung „RSD“ steht für „Relative Standard Deviation“, d.h. relative Standardabweichung. Je niedriger der RSD-Wert ist, desto höher die Gleichmäßigkeit der Erwärmung.Another, very often asked requirement for electromagnetic heating is that the material to be heated is heated as evenly as possible. In this context, the excitation of the channels can be optimized so that the electromagnetic power that is consumed in the material to be heated is distributed as evenly as possible. A measure of the unevenness is the relative standard deviation of the power density values of the sub-areas of the material to be heated (p ₁ / V ₁ , p ₂ / V ₂ , ..., p _L / V _L ), i.e. their standard deviation normalized by the corresponding mean value (i.e. average power density ): $$\text{RSD}=\frac{\sqrt{\frac{1}{\mathrm{L.}}{\displaystyle {\sum }_{l=1}^{\mathrm{L.}}{\left(\frac{p_l}{v_l}-\overline{p}\right)}^2}}}{\overline{p}}$$

whereby $\overline{p}={\displaystyle {\sum }_{l=1}^{\mathrm{L.}}{p}_l}/{\displaystyle {\sum }_{l=1}^{\mathrm{L.}}{V}_l}$

represents the average power density. The term “RSD” stands for “Relative Standard Deviation”, ie relative standard deviation. The lower the RSD value, the higher the uniformity of heating.

An effective optimization method to minimize this value is described below. According to this method there is a double periodic loop of excitation vectors. The inner loop applies N excitation _{vectors at the same frequency f n} one after the other, while the outer loop _{applies such N vector sets at N f} different frequencies one after the other (a so-called double time slot method, see also 5 ).

hintereinander zu verwenden, um ein sogenanntes „Modestirring“, also ein Modenmischen, durchzuführen. Die Anregung mit mehreren Parametern hat den Vorteil, dass für jeden Anregungsparametersatz ein anderes Leistungsmuster im Erwärmungsgut hervorgerufen wird. Durch die zeitliche Überlagerung aller so erzeugten Muster kann die Gleichmäßigkeit der Erwärmung verbessert werden.The principle of the method is based on several sets or vectors of excitation _{while staying at a frequency f n} $a_1^n,a_2^n,a_3^n,...$

to be used one after the other in order to carry out a so-called “mode stirrring”, ie a mode mixing. The excitation with several parameters has the advantage that a different performance pattern is produced in the material to be heated for each excitation parameter set. The uniformity of the heating can be improved by the temporal superposition of all patterns generated in this way.

Während des l -ten Intervalls werden die Kanäle durch den komplexen Anregungsvektor a_l angeregt. Die dadurch erzeugte Leistungsdichte im k -ten Teilbereich ist $p_k^l=a_l^H{Q}_k{a}_l/V_k.$

Im Folgenden werden wir Q_k/V_k mit Q̈_k kennzeichnen. Die zeitliche Überlagerung der unterschiedlichen Leistungen hat als Endeffekt, dass die äquivalente Leistungsdichte am Ende von t_n Sekunden wie folgt aussieht: $$p_k=\frac{1}{t_n}{\displaystyle \sum _{l=1}^{N_T}{\tau }_{nl}{a}_l^H{\overline{\mathbf{Q}}}_k{a}_l}$$

Assume that the signal source dwells at a frequency f = f _n for a period of t _n seconds. This time period is subdivided into N _T time intervals, each of which has a time duration of τ _n1 , τ _n2 , τ _η3 , ..., τ _{nN T} Seconds (see 5 ). It is of course true ${\displaystyle {\sum }_{l=1}^{N_T}{\tau }_{nl}}=t_n.$

During the l th interval, the channels are excited _{by the complex excitation vector a l.} The power density generated in this way is in the k -th sub-area $p_k^l=a_l^H{Q}_k{a}_l/V_k.$

In the following we will denote Q _{k /} V _k with Q̈ _k . The final effect of the temporal superposition of the different powers is that the equivalent power density at the end of t _n seconds looks like this: $$p_k=\frac{1}{t_n}{\displaystyle \sum _{l=1}^{N_T}{\tau }_{nl}{a}_l^H{\overline{\mathbf{Q}}}_k{a}_l}$$

inkorporieren. Wenn man also a_l durch $a_l\sqrt{{\tau }_{nl}/t_n}$

ersetzt, wird obige Formel wie folgt vereinfacht: $$p_k={\displaystyle \sum _{l=1}^{N_T}{a}_l^H{\overline{\mathbf{Q}}}_k{a}_l}$$

In order to simplify the description of the method, the prefactor τ _nl / t _n for each individual power pattern can be converted into the amplitudes of the excitation vectors by multiplying by the factor $\sqrt{{\tau }_{nl}/t_n}$

incorporate. So if you go through _{a l} $a_l\sqrt{{\tau }_{nl}/t_n}$

replaced, the above formula is simplified as follows: $$p_k={\displaystyle \sum _{l=1}^{N_T}{a}_l^H{\overline{\mathbf{Q}}}_k{a}_l}$$

wobei $c_{ij}={\displaystyle {\sum }_{l=1}^{N_T}{a}_{lj}{a}_{li}^{\ast }}.$

Wenn man mit A = [a_ij] die Matrix der Anregungen darstellt, dessen i -te Zeile den Anregungsvektor während des i -ten zeitlichen Intervalls ist, dann ist C = A^HA, wo C = [c_ij]. Aus diesem Verhältnis zwischen C und A schließt man, dass die Elemente der Matrix C nicht beliebig sein können, sondern eine hermitesche und positiv semi-definite Matrix bilden müssen. Ferner sieht man aus (8), dass die eigentlichen Parameter, von denen das Leistungsmuster p_k abhängt, die Elemente von C und nicht von A sind. Unabhängig von der Anzahl N_T der Zeitschlitze wird C eine N x N hermitesche Matrix sein. Deswegen ist es möglich, ohne Verlust der Allgemeinheit die Anzahl der Zeitschlitze während des Verweilens bei einer Frequenz gleich der Anzahl der Kanäle zu wählen, d.h. N_T = N. Mehr Freiheitsgrade für A würden die Anzahl der Freiheitsgrade für C nicht erhöhen.If the complex amplitude of the j th channel during the l th time interval is α _lj , then the above formula takes the form: $$p_k={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N\left({\displaystyle \sum _{l=1}^{N_T}{a}_{lj}{a}_{li}^{\ast }}\right)}}{\overline{q}}_{ij}^k={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N{c}_{ij}{\overline{q}}_{ij}^k}}$$

whereby $c_{ij}={\displaystyle {\sum }_{l=1}^{N_T}{a}_{lj}{a}_{li}^{\ast }}.$

If the matrix of the excitations is represented with A = [a _ij] , the i-th row of which is the excitation vector during the i-th time interval, then C = A ^H A, where C = [c _ij] . From this relationship between C and A one concludes that the elements of the matrix C cannot be arbitrary, but must form a Hermitian and positive semi-definite matrix. It can also be seen from (8) that the actual parameters on which the performance pattern p _k depends are the elements of C and not of A. Regardless of the number N _T of time slots, C will be an N x N Hermitian matrix. Therefore it is possible without Loss of generality to choose the number of timeslots while staying at a frequency equal to the number of channels, ie N _T = N. More degrees of freedom for A would not increase the number of degrees of freedom for C.

wobei $\overline{p}=\frac{1}{L}{\displaystyle {\sum }_{k=1}^L{p}_k}$

der durchschnittliche Wert aller Teilleistungsdichten p_k ist.In the following, the method aims at the skillful selection of the matrix C in order _{to achieve as few deviating values p k} as possible. The measure of uniformity that is used is the normalized standard deviation of the values p _k , previously referred to as RSD, ie $${\widehat{\sigma }}_p=\frac{1}{\sqrt{\mathrm{L.}}}\cdot \frac{\sqrt{{\displaystyle {\sum }_{k=1}^{\mathrm{L.}}{\left(p_k-\overline{p}\right)}^2}}}{\overline{p}}$$

whereby $\overline{p}=\frac{1}{\mathrm{L.}}{\displaystyle {\sum }_{k=1}^{\mathrm{L.}}{p}_k}$

is the average value of all partial power densities p _k .

wo C_m positiv-definite hermitesche Matrizen. Eine mögliche Basis des oben genannten Vektorraumes kann mithilfe folgender Matrizen aufgebaut werden: $$E_k\left[e_{ij}^k\right],\text{ wo }{e}_{ij}^k=\{\begin{array}{rr}\hfill \mathrm{1,}& \hfill \text{wenn }i=j=k\\ \hfill \mathrm{0,}& \hfill \text{sonst}\end{array}$$

$$F_{kl}=\left[{ƒ}_{ij}^k\right],\text{ wo }{ƒ}_{ij}^{kl}=\{\begin{array}{rr}\hfill \mathrm{1,}& \hfill \text{wenn }i=k,j=l,oderi=l,j=k\\ \hfill \mathrm{0,}& \hfill \text{sonst}\end{array}$$

$$G_{kl}=\left[g_{ij}^{kl}\right],\text{ wo }{g}_{ij}^{kl}=\{\begin{array}{rr}\hfill \sqrt{-1},& \hfill \text{wenn }i=k,j=l\\ \hfill -\sqrt{-1}& \hfill \text{wenn }i=l,j=k\\ \hfill \mathrm{0,}& \hfill \text{sonst}\end{array}$$

In order _{to minimize σ p} , the matrix C must be chosen appropriately. The latter belongs to the space of the Hermitian matrices (ie C ^H = C); this space is N ² dimensional. That means that there are N ² real degrees of freedom γ _m and basis “vectors” (basis matrices) C _m , so that $$\mathbf{C.}={\displaystyle {\sum }_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\gamma }_m{\mathrm{C.}}_m}$$

where C _m positive-definite Hermitian matrices. A possible basis of the above-mentioned vector space can be constructed with the help of the following matrices: $${\mathrm{E.}}_k\left[e_{ij}^k\right],\text{Where}{e}_{ij}^k=\{\begin{array}{rr}\hfill \mathrm{1,}& \hfill \text{if}i=j=\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE102019210264B4\_0153.png"\; state="\{\{state\}\}">k</figure-callout>}\\ \hfill \mathrm{0,}& \hfill \text{otherwise}\end{array}$$

$${\mathrm{F.}}_{kl}=\left[{ƒ}_{ij}^k\right],\text{Where}{ƒ}_{ij}^{kl}=\{\begin{array}{rr}\hfill \mathrm{1,}& \hfill \text{if}i=k,j=l,Oderi=l,j=\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE102019210264B4\_0153.png"\; state="\{\{state\}\}">k</figure-callout>}\\ \hfill \mathrm{0,}& \hfill \text{otherwise}\end{array}$$

$$G_{kl}=\left[G_{ij}^{kl}\right],\text{Where}{G}_{ij}^{kl}=\{\begin{array}{rr}\hfill \sqrt{-1},& \hfill \text{if}i=k,j=l\\ \hfill -\sqrt{-1}& \hfill \text{if}i=l,j=\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE102019210264B4\_0153.png"\; state="\{\{state\}\}">k</figure-callout>}\\ \hfill \mathrm{0,}& \hfill \text{otherwise}\end{array}$$

A basis with N ² elements, which forms a basis of the space to which C belongs, consists of the following matrices: $$\begin{array}{l}·\left\{{\mathrm{E.}}_k\right\},k=\mathrm{1.2,}...,N\\ ·\left\{{\mathrm{F.}}_{kl}+{\mathrm{E.}}_k+{\mathrm{E.}}_l\right\},k=\mathrm{1.2,}...,N,l=k+\mathrm{1.2,}...,N\\ ·\left\{{\mathbf{G}}_{kl}+{\mathrm{E.}}_k+{\mathrm{E.}}_l\right\},k=\mathrm{1.2,}...,N,l=k+\mathrm{1.2,}...,N\end{array}$$

If you number the above matrices continuously from m = 1 to m = N ² , you get the basis {C _m }, m = 1,2, ..., N ² .

muss man darauf achten, dass die Koeffizienten γ_m nicht beliebig sein können, sondern solche Werte haben müssen, dass C ≥ 0, d.h. C muss positiv semi-definit sein (sie muss ausschließlich nichtnegative Eigenwerte besitzen). Das geht aus der Definition von C = A^HA hervor.When decomposing the matrix C in the above basis $\left(\mathrm{C.}={\displaystyle {\sum }_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\gamma }_m{\mathrm{C.}}_m}\right)$

one must take care that the coefficients γ _m cannot be arbitrary, but must have such values that C ≥ 0, ie C must be positive semi-definite (it must only have non-negative eigenvalues). This is evident from the definition of C = A ^{H A.}

die Elemente der Basis-Matrix C_m. Daher nimmt Gleichung (8) folgende Form an: $$p_k={\displaystyle {\sum }_{m=1}^{N^2}{\gamma }_m}{\displaystyle {\sum }_{i,j}{c}_{ij}^m{\overline{q}}_{ij}^k=}{\displaystyle {\sum }_{m=1}^{N^2}{\gamma }_m{p}_k^m}$$

wobei $p_k^1,p_k^2,\mathrm{...},p_k^m,\mathrm{...},p_k^{N^2}$

elementare Basismuster darstellen, die wie folgt definiert sind: $$p_k^m={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N{c}_{ij}^m{\overline{q}}_{ij}^k}}$$

According to the above decomposition, the individual elements of C can be written as follows: $c_{ij}={\displaystyle {\sum }_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\gamma }_m{c}_{ij}^m,\text{Where}{c}_{ij}^m}$

the elements of the basic matrix C _m . Hence equation (8) takes the form: $$p_k={\displaystyle {\sum }_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\gamma }_m}{\displaystyle {\sum }_{i,j}{c}_{ij}^m{\overline{q}}_{ij}^k=}{\displaystyle {\sum }_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\gamma }_m{p}_k^m}$$

whereby ${\mathrm{<figure-callout\; id="1"\; label="p\; k"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">p</figure-callout>}}_k^{},{\mathrm{<figure-callout\; id="2"\; label="p\; k"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">p</figure-callout>}}_k^{},\mathrm{...},p_k^m,\mathrm{...},{\mathrm{<figure-callout\; id="2"\; label="p\; k\; N"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">p</figure-callout>}}_{k{N}^{}}^{{}^2}$

represent elementary basic patterns, which are defined as follows: $$p_k^m={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N{c}_{ij}^m{\overline{q}}_{ij}^k}}$$

Decomposition (11) of performance pattern p _k has the direct consequence that the maximum number of degrees of freedom that one has available through this method to shape the performance pattern is N ² , regardless of how finely one subdivides the time cycle, ie no matter how big N _T is.

berechnet werden. Damit der Ausdruck vereinfacht wird, werden vorher die Elementarmuster $p_k^m$

so normiert, dass ihre Summe 1 Watt/m³ ist, und neue Koeffizienten definiert: $$p_k={\displaystyle \sum _{m=1}^{N^2}{\gamma }_m{p}_k^m={\displaystyle \sum _{m=1}^{N^2}{\widehat{\gamma }}_m}{\widehat{p}}_k^m}$$

wobei ${\widehat{p}}_k^m=p_k^m/{\displaystyle {\sum }_{k=1}^L{p}_k^m}und{\widehat{\gamma }}_m{\displaystyle {\sum }_{k=1}^L{p}_k^m}.$

Anhand dieser Definitionen ist das Maß der Ungleichmäßigkeit durch folgende Formel gegeben: $${\widehat{\sigma }}_p\left(\widehat{\gamma }\right)=\sqrt{L\frac{{\widehat{\gamma }}^T{\mathbf{P}}_t\widehat{\gamma }}{{\left({\displaystyle {\sum }_m{\widehat{\gamma }}_m}\right)}^2}-1}$$

wobei ${\mathbf{P}}_t=\left[{\pi }_{mn}^{\text{t}}\right]\text{ eine }{N}^2\times N^2$

symmetrische reelle Matrix mit Elementen ${\pi }_{mn}^t={\displaystyle {\sum }_{k=1}^L{\widehat{p}}_k^m}{\widehat{p}}_k^n,$

Wenn man die Summe der unbekannten Koeffizienten auf 1 normiert, d.h. Σ_m γ̂_m = 1, dann ist die Minimierung der Ungleichmäßigkeit laut Formel (12) äquivalent zur Minimierung der quadratischen Form f (γ̂) = γ̂^TP_tγ̂. Daher läuft Verfahren B auf folgendes Optimierungsproblem hinaus:

• • Finde einen reellen N² -dimensionalen Vektor γ, der folgenden Bedingungen genügt:
  • [i] ${\displaystyle {\sum }_{m=1}^{N^2}{\widehat{\gamma }}_m}=1.$
    
  • [ii] Die Matrix $\mathbf{C}={\displaystyle {\sum }_{m=1}^{N^2}{\gamma }_m{\mathbf{C}}_m}$
    
    ist positiv semi-definit.
  • [iii] Er minimiert die Funktion f (γ̂) = γ̂^TP_tγ̂.

The optimization goal, ie the minimization of the unevenness, which is _{expressed by σ̂ p} , now depends on the choice of the coefficients γ ₁ , γ ₂ , ..., γ _{m '} ..., γ _N 2, which are converted into a real one Vector γ = [γ ₁ γ ₂ ... γ _N ² ] ^T can summarize. The analytical dependence of the optimization function σ̂ _p on γ can be determined using the decomposition $p_k={\displaystyle {\sum }_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\gamma }_m}{p}_k^m$

be calculated. In order to simplify the expression, the elementary patterns are made beforehand $p_k^m$

normalized so that their sum 1 Watt / m ³ , and new coefficients are defined: $$p_k={\displaystyle \sum _{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\gamma }_m{p}_k^m={\displaystyle \sum _{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\widehat{\gamma }}_m}{\widehat{p}}_k^m}$$

whereby ${\widehat{p}}_k^m=p_k^m/{\displaystyle {\sum }_{k=1}^{\mathrm{L.}}{p}_k^m}und{\widehat{\gamma }}_m{\displaystyle {\sum }_{k=1}^{\mathrm{L.}}{p}_k^m}.$

Using these definitions, the degree of unevenness is given by the following formula: $${\widehat{\sigma }}_p\left(\widehat{\gamma }\right)=\sqrt{\mathrm{L.}\frac{{\widehat{\gamma }}^T{\mathbf{P.}}_t\widehat{\gamma }}{{\left({\displaystyle {\sum }_m{\widehat{\gamma }}_m}\right)}^2}-1}$$

whereby ${\mathbf{P.}}_t=\left[{\pi }_{mn}^{\text{t}}\right]\text{one}{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">N</figure-callout>}}^2\times {\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">N</figure-callout>}}^2$

symmetric real matrix with elements ${\pi }_{mn}^t={\displaystyle {\sum }_{k=1}^{\mathrm{L.}}{\widehat{p}}_k^m}{\widehat{p}}_k^n,$

If the sum of the unknown coefficients is normalized to 1, ie Σ _m γ̂ _m = 1, then minimizing the non-uniformity according to formula (12) is equivalent to minimizing the quadratic form f (γ̂) = γ̂ ^T P _t γ̂. Method B therefore results in the following optimization problem:

• • Find a real N ² -dimensional vector γ that satisfies the following conditions:
  • [i] ${\displaystyle {\sum }_{m=1}^{N^2}{\widehat{\gamma }}_m}=1.$
    
  • [ii] The matrix $\mathbf{C.}={\displaystyle {\sum }_{m=1}^{N^2}{\gamma }_m{\mathbf{C.}}_m}$
    
    is positive semi-definite.
  • [iii] It minimizes the function f (γ̂) = γ̂ ^T P _t γ̂.

The above-mentioned optimization problem falls into the category of semi-definite programming (SDP) and can be numerically solved using known mathematical methods for convex optimization, as found in specialist books (e.g. Boyd, Stephen, and Lieven Vandenberghe. Convex optimization. Cambridge University Press, 2004) is known.

The numerical solution of the above optimization problem calculates the optimal coefficients γ _{k of} the decomposition of matrix C, which minimizes the unevenness. From the C matrix, which is calculated by formula (10), one can find the matrix of the optimal channel excitations A by solving the Find matrix equation C = AHA. This equation has an infinite number of solutions, which, however, lead to the same power distribution in the material to be heated, because the latter depends exclusively on the matrix C. The simplest of these solutions is obtained by diagonalizing the C-matrix. According to the spectral theorem for Hermitian matrices, this matrix can be decomposed as follows: C = VDV ^H , where V is a unitary matrix with the eigenvectors of C as columns and D is a diagonal matrix that contains the eigenvalues of C. This decomposition can be determined using standard numerical methods. So a solution to the equation C = AHA looks like this: $$\mathbf{A.}={\mathbf{<figure-callout\; id="1"\; label="VD"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">VD</figure-callout>}}^{1/2}{\mathbf{V}}^H$$

The optimization problem is thus solved; that is, for each time segment τ _nl , l = 1, 2, ..., N _T , the excitation of the jth channel is given by the complex entry a _{lj of} the A matrix. By additive superimposition of these N _T patterns, the desired, equivalent power pattern p _{k is} generated, which has maximum uniformity. It must be emphasized again that the minimally sufficient number of different excitation vectors or time intervals to achieve maximum uniformity is N _T = N, since formula (13) shows that A is quadratic. This means that as many excitation vectors as channels are required in order to achieve maximum uniformity using the above method.

• • Input: Ein Satz von L Erwärmungsmatrizen Q₁, Q₂, ..., Q_L, bei einer bestimmten Frequenz f_n.
• • Output: Ein Satz von N Anregungsvektoren a₁, a₂, ..., a_N, die zeitlich nacheinander für τ_n1, τ_n2, ..., τ_nN Sekunden als Anregungsparameter des Verstärkers während des Verweilens bei Frequenz f_n angewandt werden sollen (vgl. 2).
• • Schritte:
  1. 1. Berechne die Basismatrizen ${\mathbf{C}}_m=\left[c_{ij}^m\right].$
     
  2. 2. Mithilfe der normierten Q-Matrizen ${\mathbf{Q}}_k/V_k=\left[{\overline{q}}_{ij}^k\right]$
     
     berechne die N² Basismuster durch $$p_k^m={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N{c}_{ij}^m{\overline{q}}_{ij}^k}},m=\mathrm{1,2,}\mathrm{...},{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">N</figure-callout>}}^2$$
     
  3. 3. Normiere die Basismuster auf Summe von 1 Watt: ${\widehat{p}}_k^m=p_k^m/{\displaystyle {\sum }_{k=1}^L{p}_k^m}.$
     
  4. 4. Berechne die Matrix ${\mathbf{P}}_t=\left[{\pi }_{mn}^t\right],\text{ wo }{\pi }_{mn}^t={\displaystyle {\sum }_{k=1}^L{p}_k^m/{\displaystyle {\sum }_{k=1}^L{\widehat{p}}_k^m{\widehat{p}}_k^n.}}$
     
  5. 5. Minimiere (bspw. durch konvexe Optimierung) die Funktion f (γ̂) = γ̂^TP_tγ̂ unter folgenden Bedingungen:
     • ■ ${\displaystyle {\sum }_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\widehat{\gamma }}_m}=1$
       
       ■ Die Matrix $\mathbf{C}={\displaystyle {\sum }_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\gamma }_m{\mathbf{C}}_m.}$
       
       ist positiv semi-definit.
  6. 6. Berechne nichtnormierte Koeffizienten durch ${\gamma }_m={\widehat{\gamma }}_m/{\displaystyle {\sum }_{k=1}^L{p}_k^m}.$
     
  7. 7. Berechne C-Matrix durch $\mathbf{C}={\displaystyle {\sum }_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\gamma }_m{\mathbf{C}}_m.}$
     
  8. 8. Führe die Eigenwertzerlegung von C durch: C = VDV^H.
  9. 9. Berechne die Anregungsmatrix durch A = VD^1/2V^H. Die optimalen Anregungsvektoren a₁, a₂, ..., a_N sind die entsprechenden Zeilen von A, d.h. [a_1j], [a_2j], ..., [a_Nj].

Finally, the method for optimizing uniformity can be summarized as follows:

• • Input: A set of L heating matrices Q ₁ , Q ₂ , ..., Q _L , at a certain frequency f _n .
• • Output: A set of N excitation vectors a ₁ , a ₂ , ..., a _N , which are used one after the other for τ _n1 , τ _n2 , ..., τ _nN seconds as excitation parameters of the amplifier while it is at frequency f _n should be (cf. 2 ).
• • Steps:
  1. 1. Compute the basic matrices ${\mathbf{C.}}_m=\left[c_{ij}^m\right].$
     
  2. 2. Using the normalized Q-matrices ${\mathbf{Q}}_k/V_k=\left[{\overline{q}}_{ij}^k\right]$
     
     compute the N ² basic pattern $$p_k^m={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N{c}_{ij}^m{\overline{q}}_{ij}^k}},m=\mathrm{1.2,}\mathrm{...},{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">N</figure-callout>}}^2$$
     
  3. 3. Normalize the basic pattern to a sum of 1 watt: ${\widehat{p}}_k^m=p_k^m/{\displaystyle {\sum }_{k=1}^{\mathrm{L.}}{p}_k^m}.$
     
  4. 4. Compute the matrix ${\mathbf{P.}}_t=\left[{\pi }_{mn}^t\right],\text{Where}{\pi }_{mn}^t={\displaystyle {\sum }_{k=1}^{\mathrm{L.}}{p}_k^m/{\displaystyle {\sum }_{k=1}^{\mathrm{L.}}{\widehat{p}}_k^m{\widehat{p}}_k^n.}}$
     
  5. 5. Minimize (e.g. by convex optimization) the function f (γ̂) = γ̂ ^T P _t γ̂ under the following conditions:
     • ■ ${\displaystyle {\sum }_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\widehat{\gamma }}_m}=1$
       
       ■ The matrix $\mathbf{C.}={\displaystyle {\sum }_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\gamma }_m{\mathbf{C.}}_m.}$
       
       is positive semi-definite.
  6. 6. Compute non-normalized coefficients by ${\gamma }_m={\widehat{\gamma }}_m/{\displaystyle {\sum }_{k=1}^{\mathrm{L.}}{p}_k^m}.$
     
  7. 7. Compute C-matrix $\mathbf{C.}={\displaystyle {\sum }_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\gamma }_m{\mathbf{C.}}_m.}$
     
  8. 8. Perform the eigenvalue decomposition of C: C = VDV ^H.
  9. 9. Calculate the excitation matrix by A = VD ^1/2 V ^H. The optimal excitation vectors a ₁ , a ₂ , ..., a _N are the corresponding rows of A, ie [a _1j] , [a _2j] , ..., [a _Nj] .

The optimization method described above is only an exemplary embodiment and can also be replaced by an equivalent method, for example by a method as shown in FIG DE 10 2016 202 234 B3 (Paragraphs [0052] to [0055]). In the latter case, only one excitation vector (instead of N) per frequency would then be determined and applied.

In the previous procedure, the heating uniformity was optimized at a single frequency. If this method is used at several frequencies, one obtains a series of optimized power distributions p _k (f _n ) as a function of the frequency. You can excite these frequencies one after the other with the corresponding excitation parameters in order to get an average performance pattern: $${\overline{p}}_k=\frac{1}{N_{ƒ}}{\displaystyle \sum _{n=1}^{N_{ƒ}}{p}_k\left({ƒ}_n\right)}$$

wobei w_n positive Zahlen sind. Im Falle der Gleichgewichtung w₁ = w₂ = ... = w_{N f} = 1/N_f, erhält man den üblich definierten Durchschnitt aller Leistungsmuster, d.h. p̃_k = p̅_k.However, this superimposition of the corresponding monofrequency patterns can also be weighted differently, either by scaling the excitations at the individual frequencies accordingly, or by lengthening or shortening the dwell time at each frequency accordingly. The end effect is the same, namely that the equivalent power pattern that ultimately results is a weighted superposition of the monofrequency patterns: $${\overline{p}}_k={\displaystyle \sum _{n=1}^{N_{ƒ}}{w}_n{p}_k\left({ƒ}_n\right)}$$

where w _{n are} positive numbers. In the case of equal weighting w ₁ = w ₂ = ... = w _{N f} = 1 / N _f , one obtains the commonly defined average of all performance _{patterns, ie p̃ k} = p̅ _k .

The degree of non-uniformity of the resulting pattern is, as in method B, the normalized standard deviation of the power distribution among the sub-areas G _k (see formula (9)). One can express this measure again as an analytical function of the weights w _n. To simplify the formula, it makes sense to normalize the power densities at each individual frequency so that their sum 1 Watts / m ³ . The variable weights must also be scaled accordingly: $${\tilde{p}}_k={\displaystyle \sum _{n=1}^{N_{ƒ}}{\widehat{w}}_n{\widehat{p}}_k\left({ƒ}_n\right),{\widehat{p}}_k\left({ƒ}_n\right)=\frac{p_k\left({ƒ}_n\right)}{{\displaystyle {\sum }_{k=1}^{\mathrm{L.}}{p}_k\left({ƒ}_n\right)}}},{\widehat{w}}_n=w_n{\displaystyle \sum _{k=1}^{\mathrm{L.}}{p}_k\left({ƒ}_n\right)}$$

wobei $\widehat{w}={\left[{\widehat{w}}_1\cdots {\widehat{w}}_{N_{ƒ}}\right]}^T$

der Vektor der Gewichte und ${\mathbf{P}}_{ƒ}=\left[{\pi }_{mn}^{ƒ}\right]$

eine N_f × N_f symmetrische Matrix mit Elementen ${\pi }_{mn}^{ƒ}={\displaystyle {\sum }_{k=1}^L{\widehat{p}}_k}\left({ƒ}_m\right){\widehat{p}}_k\left({ƒ}_n\right)$

ist. Da eine Skalierung der Gewichte ŵ_n um den gleichen Faktor keinen Unterschied in der Gleichmäßigkeit verursacht, kann man annehmen, dass die Summe der Gewichte auf 1 normiert ist, d.h. Σ_n Ŵ_n = 1. Dementsprechend nimmt Formel (14) die Form ${\widehat{\sigma }}_p\left(\widehat{w}\right)=\sqrt{N_{ƒ}{\widehat{w}}^T{\mathbf{P}}_{ƒ}\widehat{w}-1}$

an. Um die Ungleichmäßigkeit σ_p(Ŵ) zu minimieren, reicht es also, die quadratische Form f(w)̂ = ŵ^TP_fŵ zu minimieren, wobei man beachten muss, dass die Summe der Gewichte auf 1 normiert ist und dass alle Gewichte nichtnegativ sind (es ist physikalisch unmöglich durch inkohärente Überlagerung von Leistungsmustern eine negative Gewichtung hervorzurufen, da es wegen der Inkohärenz keine destruktive Interferenz geben kann). Daraus folgt, dass folgendes Optimierungsproblem gelöst werden muss:

• • Finde einen reellen N_f -dimensionalen Vektor ŵ der folgenden Bedingungen genügt:
  • [i] ${\displaystyle {\sum }_{n=1}^{N_{ƒ}}{\widehat{w}}_n=1}$
    
  • [ii] ŵ_n ≥ 0, n = 1,2, ..., N_f.
  • [iii] Er minimiert die Funktion f(ŵ) = ŵ^TP_fŵ.

The normalized standard deviation of the final performance _pattern is therefore p̃ k $${\widehat{\sigma }}_p\left(\widehat{w}\right)=\sqrt{\mathrm{L.}\frac{{\widehat{w}}^T{\mathbf{P.}}_t\widehat{w}}{{\left({\displaystyle {\sum }_m{\widehat{w}}_m}\right)}^2}-1}$$

whereby $\widehat{w}={\left[{\widehat{w}}_1\cdots {\widehat{w}}_{N_{ƒ}}\right]}^T$

the vector of weights and ${\mathbf{P.}}_{ƒ}=\left[{\pi }_{mn}^{ƒ}\right]$

an N _f × N _f symmetric matrix with elements ${\pi }_{mn}^{ƒ}={\displaystyle {\sum }_{k=1}^{\mathrm{L.}}{\widehat{p}}_k}\left({ƒ}_m\right){\widehat{p}}_k\left({ƒ}_n\right)$

is. Since scaling the weights ŵ _n by the same factor does not cause a difference in uniformity, one can assume that the sum of the weights is normalized to 1, ie Σ _n Ŵ _n = 1. Formula (14) takes the form accordingly ${\widehat{\sigma }}_p\left(\widehat{w}\right)=\sqrt{N_{ƒ}{\widehat{w}}^T{\mathbf{P.}}_{ƒ}\widehat{w}-1}$

on. In order to minimize the unevenness σ _p (Ŵ), it is sufficient to minimize the quadratic form f (w) ̂ = ŵ ^T P _f ŵ, whereby one must note that the sum of the weights is normalized to 1 and that all weights are not negative (it is physically impossible to produce a negative weighting through incoherent superimposition of performance patterns, since there can be no destructive interference due to the incoherence). It follows that the following optimization problem has to be solved:

• • Find a real N _f -dimensional vector ŵ that satisfies the following conditions:
  • [i] ${\displaystyle {\sum }_{n=1}^{N_{ƒ}}{\widehat{w}}_n=1}$
    
  • [ii] ŵ _n ≥ 0, n = 1,2, ..., N _f .
  • [iii] It minimizes the function f (ŵ) = ŵ ^T P _f ŵ.

The above optimization problem is convex and can be numerically solved using known mathematical methods for convex optimization, among other things. After determining the normalized weights ŵ _n , the non-normalized weights can be determined using the following formula $$w_n=\frac{{\widehat{w}}_n}{{\displaystyle {\sum }_{k=1}^{\mathrm{L.}}{p}_k\left({ƒ}_n\right)}}$$

These can turn to grand total 1 be normalized, ie w ' _n = w _n / ∑ _n w _n and the final weighting can be implemented either by scaling the dwell time of the signal source in the respective frequency, or by scaling the corresponding power level of the amplifier channels.

• • Input: Ein Satz von Leistungsmustern {p_k{f_n), k = 1,2, ...,L} gemessen oder berechnet bei einem vorbestimmten Stichprobensatz von Frequenzen {f_n,n = 1,2, ...,N_f}.
• • Output: Ein Satz von Gewichten {w_n, n = 1,2, ..., N_f}, die entweder als Zeit- oder Amplitudenmodulation des Verweilens des Verstärkers bei der jeweiligen Frequenz f_n angewandt werden können, um die Gleichmäßigkeit des am Ende resultierenden Musters zu optimieren.
• • Schritte:
  1. 1. Normiere Leistungsmuster mithilfe der Formel ${\widehat{p}}_k\left({ƒ}_n\right)=\frac{p_k\left({ƒ}_n\right)}{{\displaystyle {\sum }_{k=1}^L{p}_k\left({ƒ}_n\right)}}.$
     
  2. 2. Berechne Matrix ${\mathbf{P}}_{ƒ}=\left[{\pi }_{mn}^{ƒ}\right]\text{ durch }{\pi }_{mn}^{ƒ}={\displaystyle {\sum }_{k=1}^L\left({ƒ}_m\right){\widehat{p}}_k\left({ƒ}_n\right)}$
     
  3. 3. Minimiere Funktion f(ŵ) = Ŵ^TP_fŴ unter folgenden Bedingungen: $$\begin{array}{l}·{\displaystyle {\sum }_{n=1}^{N_{ƒ}}{\widehat{w}}_n}=1\\ ·{\widehat{w}}_n\ge \mathrm{0,}n=\mathrm{1,2,}\mathrm{...},N_{ƒ}\end{array}$$
     
     Die Minimierung kann u.a. mit Verfahren der konvexen Optimierung durchgeführt werden.
  4. 4. Berechne die nichtskalierten Gewichte durch $w_n={\widehat{w}}_n/{\displaystyle {\sum }_{k=1}^{N_{ƒ}}{p}_k\left({ƒ}_n\right)}.$
     
     (Optional) Normiere die Gewichte auf 1 durch $w_n^{\text{'}}=w_n/{\displaystyle {\sum }_{n=1}^{N_{ƒ}}{w}_n}.$
     

Finally, the latter process can be summarized as follows:

• • Input: A set of performance patterns {p _k {f _n ), k = 1,2, ..., L} measured or calculated at a predetermined sample set of frequencies {f _n, n = 1,2, ..., N _f }.
• • Output: A set of weights {w _n , n = 1,2, ..., N _f }, which can be applied either as time or amplitude modulation of the dwell of the amplifier at the respective frequency f _n in order to ensure the evenness of the optimize the resulting pattern at the end.
• • Steps:
  1. 1. Normalize performance patterns using the formula ${\widehat{p}}_k\left({ƒ}_n\right)=\frac{p_k\left({ƒ}_n\right)}{{\displaystyle {\sum }_{k=1}^{\mathrm{L.}}{p}_k\left({ƒ}_n\right)}}.$
     
  2. 2. Compute matrix ${\mathbf{P.}}_{ƒ}=\left[{\pi }_{mn}^{ƒ}\right]\text{through}{\pi }_{mn}^{ƒ}={\displaystyle {\sum }_{k=1}^{\mathrm{L.}}\left({ƒ}_m\right){\widehat{p}}_k\left({ƒ}_n\right)}$
     
  3. 3. Minimize function f (ŵ) = Ŵ ^T P _f Ŵ under the following conditions: $$\begin{array}{l}·{\displaystyle {\sum }_{n=1}^{N_{ƒ}}{\widehat{w}}_n}=1\\ ·{\widehat{w}}_n\ge \mathrm{0,}n=\mathrm{1.2,}\mathrm{...},N_{ƒ}\end{array}$$
     
     The minimization can be carried out using methods of convex optimization, among other things.
  4. 4. Calculate the unscaled weights $w_n={\widehat{w}}_n/{\displaystyle {\sum }_{k=1}^{N_{ƒ}}{p}_k\left({ƒ}_n\right)}.$
     
     (Optional) Normalize the weights to 1 $w_n^{\text{'}}=w_n/{\displaystyle {\sum }_{n=1}^{N_{ƒ}}{w}_n}.$
     

3) Any distribution of the performance values:

wobei $p_l^0$

der erwünschte Wert und ∈_l die entsprechende erlaubte Toleranz sind. Ein Algorithmus, der dieses Ziel erreichen kann, ist beispielsweise in der DE 10 2016 202 234 B3 in den Absätzen [0052]-[0055] beschrieben. Dieser Algorithmus fällt in die Kategorie der sogenannten Algorithmen der alternierenden Projektionen.Between the two extreme cases mentioned above (focusing or even distribution) there is the intermediate case in which the user strives for a certain distribution of the EM power among the sub-areas of the material to be heated, which is neither absolutely focused on a sub-area, nor is it completely uniform. In some applications it might be desirable, for example, that the power that is consumed in each sub-area has a certain value within certain tolerances: $$\left|p_l-{\mathrm{<figure-callout\; id="0"\; label="p\; l"\; filenames="DE102019210264B4\_0153.png"\; state="\{\{state\}\}">p</figure-callout>}}_l^{}\right|<{\in }_l$$

whereby ${\mathrm{<figure-callout\; id="0"\; label="p\; l"\; filenames="DE102019210264B4\_0153.png"\; state="\{\{state\}\}">p</figure-callout>}}_l^{}$

is the desired value and ∈ _{l is} the corresponding allowable tolerance. One algorithm that can achieve this goal is, for example, in the DE 10 2016 202 234 B3 described in paragraphs [0052] - [0055]. This algorithm falls into the category of the so-called algorithms of alternating projections.

The basic principle behind this algorithm is that a compromise is found between “realizable” and “desired” performance patterns. All performance patterns that can be caused by any excitation vector a in the material to be heated are called “realizable”, i.e. all patterns of the form p _l = a ^H Q _l a, l = 1,2, ..., L. As “desirable “One denotes all the patterns that are desired by Performance patterns deviate less than predefined tolerance limits, so all patterns p _l meet condition (15). Ideally, there are patterns that are both feasible and desirable. If not, a feasible pattern is sought which deviates the least from any desired pattern.

wobei a der N-dimensionale komplexe Vektor der Kanalanregungen und ${\text{Q}}_l^{1/2}$

die Quadratwurzel der Q-Matrix von G_l darstellen. Die quadrierte euklidische Norm ||F_l∥² = a^HQ_la dieses Vektors ist gleich der verbrauchten Leistung p_l in diesem Teilbereich, wenn die Kanäle des Leistungsverstärkers 4 mit Phasen und Amplituden, die im Vektor a enthalten sind, angeregt werden.In the context of this algorithm, an N-dimensional complex power vector is defined for each sub-area G _l ${\text{F.}}_l={\text{Q}}_l^{1/2}\text{a},$

where a is the N-dimensional complex vector of the channel excitations and ${\text{<figure-callout id="1" label="Q l" filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png" state="{{state}}">Q</figure-callout>}}_l^{}$

represent the square root of the Q matrix of G _l . The squared Euclidean norm || F _l ∥ ² = a ^H Q _l a of this vector is equal to the consumed power p _l in this sub-range if the channels of the power amplifier 4th are excited with phases and amplitudes contained in vector a.

• - Menge A enthält alle „realisierbaren“ Leistungsvektorensätze, d.h. alle Leistungsvektorensätze {F₁, F₂,..., F_L}, für die ein beliebiger Anregungsvektor existiert a, sodass $F_l={\mathrm{<figure-callout\; id="1"\; label="Q\; l"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">Q</figure-callout>}}_l^{}a\text{.}$
  
• - Menge B enthält alle „erwünschten“ Leistungsvektorensätze, d.h. alle Leistungsvektorensäzte {F̃₁, F₂,..., F̃_L}, für die Bedingung (15) erfüllt ist $\left(\left|{\Vert {\tilde{F}}_l\Vert }^2-{\mathrm{<figure-callout\; id="0"\; label="p\; l"\; filenames="DE102019210264B4\_0153.png"\; state="\{\{state\}\}">p</figure-callout>}}_l^{}\right|<{\epsilon }_l,l=\mathrm{1,}\mathrm{2,}\mathrm{...,}L\right)$
  

Using the above definition, one can assign a corresponding power pattern _{{p 1} , p ₂ , ..., p _{L } to each power vector set {F 1} , F ₂ , ..., F _L }, where p _l = || F _l ^{∥ 2} (each power value of the sample is the squared norm of the corresponding power vector). Based on this, the definition of the “desired” and “realizable” performance patterns can be transferred to performance vector sets. A performance vector set {F ₁ , F ₂ , ..., F _L } can be implemented or desired if the corresponding performance pattern is also realizable or desired. Accordingly, one defines two sets:

• - Set A contains all “realizable” power vector sets, ie all power vector sets {F ₁ , F ₂ , ..., F _L } for which an arbitrary excitation vector exists a, so that ${\mathrm{F.}}_l=Q_l^{1/2}a\text{.}$
  
• - Set B contains all “desired” service vector sets, ie all service vector sets {F̃ ₁ , F ₂ , ..., F̃ _L }, for which condition (15) is fulfilled $\left(\left|{\Vert {\widetilde{\mathrm{F.}}}_l\Vert }^2-p_l^0\right|<{\epsilon }_l,l=\mathrm{1,}\mathrm{2,}\mathrm{...,}\mathrm{L.}\right)$
  

Instead of looking for performance patterns that are both desirable and feasible, one can look for corresponding performance vectors, which is easier from a mathematical point of view. The aim of the procedure is to find an element in the intersection A ∩ B, i.e. a power vector set that is both desirable and realizable. If the intersection is empty (A ∩ B = Ø), the procedure searches for a feasible set in A that is as little as possible from the desired set B.

• - Ein Punkt in der Schnittmenge A n B wurde gefunden.
• - Der Abstand zwischen aufeinanderfolgenden Elementen der Folge x₁, x₂, ..., x_n, ... wird kleiner als eine vordefinierte Schwelle ∈, d.h. ∥x_n - x_n+1∥ < ∈.
• - Eine maximale Anzahl an Iterationen Nmax wurde überschritten, d.h. n > N_max

On the basis of the above description, the search for a performance pattern that corresponds to an arbitrary performance distribution has been reformulated into a search for an intersection. A well-known mathematical method that can find an element in the intersection of two sets A and B is the iterative method of alternating projections. Accordingly, one begins with an arbitrary point x ₁ ∈ A and projects it onto set B. This results in a point x̃ ₁ ∈ B. This is projected back onto set A and gets a point x̃ ₂ ∈ A. This becomes again on B projected to get a point x̃ ₂ , etc. The process is repeated until one of the following termination criteria is met:

• - A point in the intersection A n B was found.
• - The distance between successive elements of the sequence x ₁ , x ₂ , ..., x _n , ... becomes smaller than a predefined threshold ∈, ie ∥x _n - x _{n + 1} ∥ <∈.
• - A maximum number of iterations Nmax was exceeded, ie n> N _max

How the above method can be implemented in the specific case of the quantities A and B of the desired or realizable power vectors will be described shortly.

berechnet. Falls Bedingung (15) nicht erfüllt wird, d.h. falls $$\Vert {\mathbf{F}}_l\Vert >\sqrt{p_l^0+{\epsilon }_l}oder\Vert {\mathbf{F}}_l\Vert <\sqrt{p_l^0-{\epsilon }_l}$$

dann wird der Vektor F_l entsprechend folgender Formel skaliert: $${\mathbf{F}}_l^{\text{'}}=\frac{{\mathbf{F}}_l}{\Vert {\mathbf{F}}_l\Vert }\sqrt{p_l^0+{\epsilon }_l}{F}_l^{\text{'}}=\frac{{\mathbf{F}}_l}{\Vert {\mathbf{F}}_l\Vert }\sqrt{p_l^0-{\epsilon }_l}$$

One starts with an arbitrary start vector a ₀ (eg a ₀ = [1 1 ... 1] ^T ). The resulting power in sub-area G _l is given by the formula ${\Vert {\mathrm{F.}}_l\Vert }^2=a_0^H{Q}_l{a}_0$

calculated. If condition (15) is not met, ie if $$\Vert {\mathbf{F.}}_l\Vert >\sqrt{{\mathrm{<figure-callout\; id="0"\; label="p\; l"\; filenames="DE102019210264B4\_0153.png"\; state="\{\{state\}\}">p</figure-callout>}}_l^{}+{\epsilon }_l}Oder\Vert {\mathbf{F.}}_l\Vert <\sqrt{p_l^0-{\epsilon }_l}$$

then the vector F _l is scaled according to the following formula: $${\mathbf{F.}}_l^{\text{'}}=\frac{{\mathbf{F.}}_l}{\Vert {\mathbf{F.}}_l\Vert }\sqrt{{\mathrm{<figure-callout\; id="0"\; label="p\; l"\; filenames="DE102019210264B4\_0153.png"\; state="\{\{state\}\}">p</figure-callout>}}_l^{}+{\epsilon }_l}{\mathrm{F.}}_l^{\text{'}}=\frac{{\mathbf{F.}}_l}{\Vert {\mathbf{F.}}_l\Vert }\sqrt{p_l^0-{\epsilon }_l}$$

In partial areas in which the boundary conditions (15) are met by the original vector F _l , the vector is left the same, ie ${\mathbf{F.}}_l^{\text{'}}={\mathbf{F.}}_l.$

erzeugt, der die Randbedingungen (15) des Problems erfüllt. Als Nächstes wird nach einem neuen Anregungsvektor a₁ gesucht, der einen Leistungsvektor erzeugen kann, der den Vektor ${\mathbf{F}}_l^{\text{'}}$

am besten approximieren kann. Das heißt, dass man den Vektor a₁ sucht, der die Fehlernorm ${\displaystyle {\sum }_{l=1}^L{\Vert {\mathbf{Q}}_l^{\frac{1}{2}}{a}_1-{\mathbf{F}}_l^{\text{'}}\Vert }^2}$

minimiert. Dieses Problem kann beispielsweise durch die Methode der kleinsten Quadrate gelöst werden. Der so gewonnene Anregungsvektor a₁ ist näher dem optimalen Anregungsvektor als der Startvektor a₀. Die gleiche Prozedur wird mit a₁ als Startvektor wiederholt und ein neuer Vektor a₂ erzeugt. Durch iterative Wiederholung der oben beschriebenen Schritte wird eine Folge von Anregungsvektoren a₁, a₂, a₃, ..., an, ... erzeugt. Das Verfahren wird abgebrochen, wenn ein Konvergenz- oder Abbruchkriterium erreicht wird, oder wenn eine maximale Anzahl von Iterationen erreicht wird. Ein Abbruchkriterium könnte bspw. die vollständige Erfüllung von Kriterien (15) und ein Konvergenzkriterium könnte die Konvergenz obiger Folge sein (d.h. wenn ∥a_n - a_n-1∥ < δ, wobei δ ein vorbestimmter Grenzwert ist).The previous procedure creates a new vector ${\mathbf{F.}}_l^{\text{'}}$

generated that fulfills the boundary conditions (15) of the problem. Next, a search is made for a new excitation _{vector a 1} that can generate a power vector that contains the vector ${\mathbf{F.}}_l^{\text{'}}$

can best approximate. That means that one _{looks for the vector a 1} , which the error norm ${\displaystyle {\sum }_{l=1}^{\mathrm{L.}}{<figure-callout\; id="1"\; label="\Vert \; Q\; l"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">\Vert </figure-callout>{\mathbf{Q}}_l^{}{}_{\frac{1}{2}}^{}{a}_1-{\mathbf{F.}}_l^{\text{'}}\Vert }^2}$

minimized. For example, this problem can be solved by the least squares method. _{The excitation vector a 1} obtained in this way is closer to the optimal excitation vector than the start vector a ₀ . The same procedure is _{repeated with a 1} as the start vector and a new vector a _{2 is} generated. By iteratively repeating the steps described above, a sequence of excitation vectors a ₁ , a ₂ , a ₃ , ..., an, ... is generated. The process is terminated if a convergence or termination criterion is reached, or if a maximum number of iterations is reached. A termination criterion could, for example, be the complete fulfillment of criteria (15) and a convergence criterion could be the convergence of the above sequence (ie if ∥a _n - a _n-1 ∥ <δ, where δ is a predetermined limit value).

In the proposed method, the above algorithms can be used to calculate or estimate heating matrices for all subregions of the material to be heated. The knowledge of these matrices enables the user or system to solve a very broad spectrum of optimization problems with regard to the power distribution. With the help of these matrices, not only can the power be focused, but also distributed evenly, or, in the general case, distributed arbitrarily. Some algorithms that can be used for this have been described in the previous units as examples.

Phase 3 (heating of the load):

After the excitation vectors that lead to the desired power distribution within the material to be heated have been determined in phase 2 through the corresponding optimization process, they are sent to the amplifier 4th via data stream 8th transferred and the phase of the actual heating of the material to be heated 3 begins. During this phase, the corresponding channels of the amplifier 4th stimulated by the corresponding parameter sets. The duration and total power of the excitation can either be specified by the user through data stream 7th can be specified, or automatically by the system (control and evaluation unit 9 ) and, if necessary, dynamically adjusted. Both user information (data stream 7th , e.g. recipe, load quantity or material information) as well as feedback from the amplifier (reflected power via data stream 6th ) and (optional) from sensors within the EMAZ (e.g. temperature, humidity, etc. via data stream 10 ) be used.

It is possible that for some specialized heating processes the optimization goals have to be adjusted or changed in terms of time. For example, it could happen that a heating process should run uniformly at the beginning and should only start focusing after a certain temperature. Accordingly, phase 2 would have to be repeated in order to adapt the excitation parameters to the changed heating targets.

Second execution

The application scenario of the second exemplary embodiment of the proposed method relates to material to be heated that does not necessarily have to or cannot be divided into sub-areas by the user, but rather that consists of materials whose dielectric properties preferably change relatively strongly with temperature. An important and common example of such scenarios is the thawing of frozen objects such as food, dishes, blood samples, transplant organs, etc. Such objects contain a relatively high proportion of water, and the dielectric properties of water vary greatly with temperature, especially at the transition point from solid to liquid state of aggregation.

wobei E(r) die elektrische Feldstärke am Punkt r des Erwärmungsguts ist. Wenn also σ(r) an einem Punkt r₀, oder innerhalb eines kleinen Volumens r ∈ V_h sehr stark ansteigt, wächst die Leistungsdichte dort auch proportional damit. Dies führt wiederum zu weiterem Ansteigen der örtlichen Temperatur T(r) und des Verlustfaktors σ(r). Daraus lässt sich ableiten, dass ohne geeignete Gegenmaßnahmen die Temperatur eines Hotspots sich selbst hochschaukelt und viel schneller als die Temperatur seiner Umgebung steigen wird. Dieses Phänomen wird „thermal runaway“ Effekt genannt und hat sehr negative Auswirkungen auf die Qualität der aufgewärmten bzw. aufgetauten Objekte und auf den Prozess der Erwärmung bzw. des Auftauens überhaupt.As already mentioned, the challenge in such scenarios is to heat or thaw such objects as evenly as possible. Since the power distribution, which was initiated by the EMAZ 1 with any set of excitation parameters a of the amplifier 4th is caused in the food to be heated or thawed, which is mostly uneven, this also results in an uneven temperature distribution over time. As a result, so-called “hotspots” are formed, i.e. areas of the material to be heated in which temperatures are above average compared to the rest of the material to be heated. Many materials, such as water, have the property that their dielectric losses increase with increasing temperature. The electromagnetic power p (r) converted into heat depends directly on the equivalent conductivity σ (r), which can be derived from the dielectric losses: $$p\left(r\right)=\frac{1}{2}\sigma \left(\mathbf{r}\right){\left|\mathbf{E.}\left(\mathbf{r}\right)\right|}^2$$

where E (r) is the electric field strength at point r of the material to be heated. So if σ (r) _{increases very strongly at a point r 0} , or within a small volume r ∈ V _h , the power density there also increases proportionally. This in turn leads to a further increase in the local temperature T (r) and the loss factor σ (r). It can be deduced from this that, without suitable countermeasures, the temperature of a hotspot will rock itself up and will rise much faster than the temperature of its surroundings. This phenomenon is called the "thermal runaway" effect and has very negative effects on the quality of the heated or thawed objects and on the process of heating or thawing in general.

The purpose of the second embodiment of the method is to counteract this effect and, more generally, to uniformly heat materials with temperature-dependent dielectric properties.

In the second embodiment, the S-parameters of successive measurements are compared in order to determine the heating matrices of the areas that were heated particularly strongly during the time interval between the two measurements (hotspots). With the help of these matrices, new excitation parameters are then determined, which prevent or at least reduce the further heating of said hotspots.

To make this clear, assume that the S parameters S (f) of the EMAZ within the operating frequency spectrum f ₁ <f <f _{2 were} measured _{at times t n} and t _{n + 1} > t _n. The corresponding measurements are designated as S _n and S _{n + 1} . During the time interval [t _n , t _{n + 1} ], the excitation vector an is applied and a corresponding power distribution p _n (r) is generated in the material to be heated. In the left part of the 6th you can see the power distribution within an exemplary material to be heated 3 at time t = t _n and in the right part of the 6th at time t = t _{n + 1} . It can be seen that at the end of the affected time interval, new areas have been created in which the power density is particularly high, namely the hotspots that are in 6th are identified by the reference numeral 3a.

In addition, assume that the losses of the material to be heated increase with the temperature. Then the total consumption of the EM power has increased during the time interval [t _n , t _{n + 1]} and the total reflected power has decreased accordingly. The change in the reflected power is given by the following matrix $$\Delta \left({\mathbf{S.}}^H\mathbf{S.}\right)={\mathbf{S.}}_{n+1}^H{\mathbf{S.}}_{n+1}-{\mathbf{S.}}_n^H{\mathbf{S.}}_n.$$

According to the law of conservation of energy, the power consumed in the load and the parasitic losses must have increased by the negative amount above. If one assumes that the additional power was only consumed in the regions of the material to be heated where the temperature has increased significantly (see areas 3a in 6th ) then is the cumulative warming matrix of these regions $${\mathbf{Q}}_H={\mathbf{S.}}_n^H{\mathbf{S.}}_n-{\mathbf{S.}}_{n+1}^H{\mathbf{S.}}_{n+1}=-\Delta \left({\mathbf{S.}}^H\mathbf{S.}\right)$$

gegeben, der dem kleinsten Eigenwert λ_min entspricht. Um die weitere Erwärmung der bereits entstandenen Hotspots zu vermeiden, muss man also im nächsten Zeitintervall [t_n+1, t_n+2] die Kanäle des Verstärkers mit Anregungsvektor a_n+1 = v_min anregen.In order to avoid further heating of these areas, one must ^{minimize the entered power a H} Q _h a. The suitable excitation vector is given by the eigenvector v _{min of} the problem $${\mathbf{Q}}_H\mathbf{v}=\lambda \mathbf{v}\iff \left({\mathbf{S.}}_n^H{\mathbf{S.}}_n-{\mathbf{S.}}_{n+1}^H{\mathbf{S.}}_{n+1}\right)\mathbf{v}=\lambda \mathbf{v}$$

which corresponds to the smallest eigenvalue λ _min. In order to avoid further heating of the hotspots that have already arisen, one must excite the channels of the amplifier with excitation vector a _{n + 1} = v _{min in the next time interval [t n + 1} , t _{n + 2].}

Another optimization option is to minimize the relative power consumed in the hotspots in relation to the rest of the heating material. For this you have to minimize the following quotient: $$\frac{{\mathbf{a}}^H{\mathbf{Q}}_H\mathbf{a}}{{\mathbf{a}}^H\left(\mathbf{I.}-{\mathbf{S.}}^H\mathbf{S.}-{\mathbf{Q}}_H\right)\mathbf{a}}$$

der dem kleinsten Eigenwert λ_min entspricht.The numerator is the power that is consumed in the hotspots, and the denominator is the power that is consumed in the rest of the heating material (incident power minus reflected minus undesired power). The corresponding optimized vector is the eigenvector v _min following generalized eigenvalue problem $${\mathbf{Q}}_H\mathbf{v}=\lambda \left(\mathbf{I.}-{\mathbf{S.}}^H\mathbf{S.}-{\mathbf{Q}}_H\right)\mathbf{v}$$

which corresponds to the smallest eigenvalue λ _min.

The above algorithm is based on the assumption that the increase in power consumption during the time interval [t _n , t _{n + 1] is} only consumed in the hotspots. This is only correct as an approximation, because in general the temperature, and therefore the power consumed, will increase throughout the entire product being heated or thawed. Therefore the matrix -Δ (S ^H S) will in reality be a mixture of the Q matrix of the hotspots and the rest of the material to be heated. However, the greatest increase in power consumption will take place where the temperature has risen the most.

In order to quantify these qualitative statements, the Q matrix of the entire material to be heated for t = t _{n can be divided} into two parts, one part Q _h for the areas V _h in which in the future (t = t _{n + 1} ) the hotspots will form, and part Q _c for the rest of the load V _c : $${\left[\mathbf{Q}\right]}_{ij}={\left[{\mathbf{Q}}_H\right]}_{ij}+{\left[{\mathbf{Q}}_c\right]}_{ij}=\frac{1}{2}{\displaystyle {\iiint }_{V_c}\sigma \left(\mathbf{r}\right){\mathbf{E.}}_i^{\ast }\left(\mathbf{r}\right)}\cdot {\mathbf{E.}}_j\left(\mathbf{r}\right)dV+\frac{1}{2}{\displaystyle {\iiint }_{V_H}\sigma \left(\mathbf{r}\right){\mathbf{E.}}_i^{\ast }\left(r\right)\cdot {\mathrm{E.}}_{j}dV.}$$

wobei r₁, r₂ ∈ V_c und r₃, r₄ ∈ V_h. Die Durchschnittswerte σ̅_c, σ̅_h der Verluste sind wie folgt definiert: $\overline{\sigma }=\frac{1}{V}{\displaystyle {\iiint }_V\sigma \left(\mathbf{r}\right)dV.}$

With the help of the intermediate value theorem, the above integrals can be rewritten as follows: $$\begin{array}{l}{\left[\mathbf{Q}\right]}_{ij}=\frac{1}{2}\left(\text{re}\left\{{\mathbf{E.}}_i^{\ast }\left({\mathbf{r}}_1\right)\cdot {\mathbf{E.}}_j\left({\mathbf{r}}_1\right)\right\}+\text{in the}\left\{{\mathbf{E.}}_i^{\ast }\left({\mathbf{r}}_2\right)\cdot {\mathbf{E.}}_j\left({\mathbf{r}}_2\right)\right\}\right)V_c{\overline{\sigma }}_c\\ \text{+}\frac{1}{2}\left(\text{re}\left\{{\mathbf{E.}}_i^{\ast }\left({\mathbf{r}}_3\right)\cdot {\mathbf{E.}}_j\left({\mathbf{r}}_3\right)\right\}+\mathrm{I.}m\left\{{\mathbf{E.}}_i^{\ast }\left({\mathbf{r}}_{\mathrm{4th}}\right)\cdot {\mathbf{E.}}_j\left({\mathbf{r}}_{\mathrm{4th}}\right)\right\}\right)V_H{\overline{\sigma }}_H,\end{array}$$

where r ₁ , r ₂ ∈ V _c and r ₃ , r ₄ ∈ V _h . The average values σ̅ _c , σ̅ _{h of} the losses are defined as follows: $\overline{\sigma }=\frac{1}{V}{\displaystyle {\iiint }_V\sigma \left(\mathbf{r}\right)dV.}$

sind, dann ist Δσ̅h = σ̅'_h - σ̅_h >> σ̅'_c - σ̅_c, = Δσ̅_c. Wenn man zusätzlich annimmt, dass sich die Feldwerte nicht wesentlich verändert haben, ist die Veränderung in der Q-Matrix $$\begin{array}{l}{\left[\Delta \mathbf{Q}\right]}_{ij}={\left[{\mathbf{Q}}^{\text{'}}\right]}_{ij}-{\left[\mathbf{Q}\right]}_{ij}\\ \simeq \frac{1}{2}\left(\text{Re}\left\{{\mathbf{E}}_i^{\ast }\left({\mathbf{r}}_1\right)\cdot {\mathbf{E}}_j\left({\mathbf{r}}_1\right)\right\}+\text{Im}\left\{{\mathbf{E}}_i^{\ast }\left({\mathbf{r}}_2\right)\cdot {\mathbf{E}}_j\left({\mathbf{r}}_2\right)\right\}\right)V_c\Delta {\overline{\sigma }}_c\\ +\frac{1}{2}\left(\text{Re}\left\{{\mathbf{E}}_i^{\ast }\left({\mathbf{r}}_3\right)\cdot {\mathbf{E}}_j\left({\mathbf{r}}_3\right)\right\}+\text{Im}\left\{{\mathbf{E}}_i^{\ast }\left({\mathbf{r}}_4\right)\cdot {\mathbf{E}}_j\left({\mathbf{r}}_4\right)\right\}\right)V_h\Delta {\overline{\sigma }}_h\\ =\frac{\Delta {\overline{\sigma }}_c}{{\overline{\sigma }}_c}{\left[{\mathbf{Q}}_c\right]}_{ij}+\frac{\Delta {\overline{\sigma }}_h}{{\overline{\sigma }}_h}{\left[{\mathbf{Q}}_h\right]}_{ij}.\end{array}$$

At the time t = t _{n + 1} , the average values of the losses σ̅ _c , σ̅ _{h have} increased. By definition, the losses in the hotspot area V _h have grown much more rapidly than in the remaining area V _c . When the new values ${\overline{\sigma }}_c^{\text{'}},{\overline{\sigma }}_H^{\text{'}}$

are, then Δσ̅h = σ̅ ' _h - σ̅ _h >>σ̅' _c - σ̅ _c , = Δσ̅ _c . If one also assumes that the field values have not changed significantly, the change is in the Q matrix $$\begin{array}{l}{\left[\Delta \mathbf{Q}\right]}_{ij}={\left[{\mathbf{Q}}^{\text{'}}\right]}_{ij}-{\left[\mathbf{Q}\right]}_{ij}\\ \simeq \frac{1}{2}\left(\text{re}\left\{{\mathbf{E.}}_i^{\ast }\left({\mathbf{r}}_1\right)\cdot {\mathbf{E.}}_j\left({\mathbf{r}}_1\right)\right\}+\text{in the}\left\{{\mathbf{E.}}_i^{\ast }\left({\mathbf{r}}_2\right)\cdot {\mathbf{E.}}_j\left({\mathbf{r}}_2\right)\right\}\right)V_c\Delta {\overline{\sigma }}_c\\ +\frac{1}{2}\left(\text{re}\left\{{\mathbf{E.}}_i^{\ast }\left({\mathbf{r}}_3\right)\cdot {\mathbf{E.}}_j\left({\mathbf{r}}_3\right)\right\}+\text{in the}\left\{{\mathbf{E.}}_i^{\ast }\left({\mathbf{r}}_{\mathrm{4th}}\right)\cdot {\mathbf{E.}}_j\left({\mathbf{r}}_{\mathrm{4th}}\right)\right\}\right)V_H\Delta {\overline{\sigma }}_H\\ =\frac{\Delta {\overline{\sigma }}_c}{{\overline{\sigma }}_c}{\left[{\mathbf{Q}}_c\right]}_{ij}+\frac{\Delta {\overline{\sigma }}_H}{{\overline{\sigma }}_H}{\left[{\mathbf{Q}}_H\right]}_{ij}.\end{array}$$

Since the change in the Q matrix (consumed power) is the negative of the change in reflection, i.e. ΔQ = -Δ (S ^H S), the above equation can be summarized as follows: $$-\Delta \left({\mathbf{S.}}^H\mathbf{S.}\right)\simeq \frac{\Delta {\overline{\sigma }}_c}{{\overline{\sigma }}_c}{\mathbf{Q}}_c+\frac{\Delta {\overline{\sigma }}_H}{{\overline{\sigma }}_H}{\mathbf{Q}}_H.$$

minimieren. Aus dem oben beschriebenen Vorgang p_c a^HQ_ca hat man jedoch nur Zugriff auf folgende Mischung beider Werte: $$-{\mathbf{a}}^H\Delta \left({\mathbf{S}}^H\mathbf{S}\right)\mathbf{a}\simeq \frac{\Delta {\overline{\sigma }}_c}{{\overline{\sigma }}_c}{\mathbf{a}}^H{\mathbf{Q}}_c\mathbf{a}+\frac{\Delta {\overline{\sigma }}_h}{{\overline{\sigma }}_h}{\mathbf{a}}^H{\mathbf{Q}}_h\mathbf{a}=\kappa p_c+\lambda p_h,$$

wobei $\kappa \equiv \frac{\Delta {\overline{\sigma }}_c}{{\overline{\sigma }}_c},\lambda \frac{\Delta {\overline{\sigma }}_h}{{\overline{\sigma }}_h}.$

Wenn wir das Verhältnis obiger Mischung zu Gesamtleistung minimieren, haben wir folgende Minimierungsfunktion: $$\frac{\kappa p_c+\lambda p_h}{p_c+p_h}=\frac{\kappa +\lambda x}{1+x}\equiv g\left(x\right)$$

The above equation shows that - if the electromagnetic fields remain relatively undisturbed by the temperature change - the negative change in the matrix S ^H S approximates a linear combination (mixture) of the Q matrices of the “cool” and the overheated area. One would like to minimize the ratio of the power in the “hot” area to the power in the “cool” area, ie the quotient $x=\frac{p_H}{p_c}=\frac{{\mathbf{a}}^H{\mathbf{Q}}_H\mathbf{a}}{{\mathbf{a}}^H{\mathbf{Q}}_c\mathbf{a}}$

minimize. However, from the process p _c a ^H Q _c a described above, one only has access to the following mixture of both values: $$-{\mathbf{a}}^H\Delta \left({\mathbf{S.}}^H\mathbf{S.}\right)\mathbf{a}\simeq \frac{\Delta {\overline{\sigma }}_c}{{\overline{\sigma }}_c}{\mathbf{a}}^H{\mathbf{Q}}_c\mathbf{a}+\frac{\Delta {\overline{\sigma }}_H}{{\overline{\sigma }}_H}{\mathbf{a}}^H{\mathbf{Q}}_H\mathbf{a}=\kappa p_c+\lambda p_H,$$

whereby $\kappa \equiv \frac{\Delta {\overline{\sigma }}_c}{{\overline{\sigma }}_c},\lambda \frac{\Delta {\overline{\sigma }}_H}{{\overline{\sigma }}_H}.$

If we minimize the ratio of the above mixture to the total output, we have the following minimization function: $$\frac{\kappa p_c+\lambda p_H}{p_c+p_H}=\frac{\kappa +\lambda x}{1+x}\equiv G\left(x\right)$$

D.h. wenn $\lambda >\kappa \iff \frac{\Delta {\overline{\sigma }}_h}{{\overline{\sigma }}_h}>\frac{\Delta {\overline{\sigma }}_c}{{\overline{\sigma }}_c},$

dann ist g(x) monoton steigend. Dies ist tatsächlich der Fall für unsere Anwendung, denn per Definition ist die relative Erhöhung der Verluste im heißen Bereich größer als im kalten Bereich. Deswegen können wir statt der direkt erwünschten Größe x = p_h/p_c die abgeleitete und messbare Größe g(x) minimieren. Wegen der Monotonie von g erreichen beide Größen ihr Minimum für denselben Anregungsvektor a.The first derivative of g is $G\text{'}\left(x\right)=\frac{\lambda -\kappa }{{\left(1+x\right)}^2}.$

Ie if $\lambda >\kappa \iff \frac{\Delta {\overline{\sigma }}_H}{{\overline{\sigma }}_H}>\frac{\Delta {\overline{\sigma }}_c}{{\overline{\sigma }}_c},$

then g (x) is monotonically increasing. This is actually the case for our application, because by definition the relative increase in losses is greater in the hot area than in the cold area. Therefore, instead of the directly desired quantity x = p _{h /} p _c , we can minimize the derived and measurable quantity g (x). Because of the monotony of g, both quantities reach their minimum for the same excitation vector a.

The above presentation of the physical-mathematical principles of the method has shown that the algorithm is also effective when the losses increase not only in the hotspots, but also generally in the entire heating material (which is to be assumed physically more realistically). The only assumption or simplification is that the electromagnetic fields do not change significantly between two successive measurements of the S-parameters. In order to maintain this assumption, it is advisable to measure the S-parameters often enough, i.e. before large changes in the loss and field distribution in the material to be heated have taken place.

1. 1. Am Anfang (t_n = 0) wähle einen beliebigen Anregungsvektor a aus (bspw. a = [11...1]^T, d.h. alle Amplituden gleich und null Phasendifferenz).
2. 2. Für t = t_n miss die S-Matrix der EMAZ S_n(f).
3. 3. Wende Anregungsvektor a an.
4. 4. Nach dem Ablauf eines vordefinierten Zeitintervalls Δt, d.h. für t = t_n+1 = t_n + Δt, miss erneut die S-Matrix S_n+1(f).
5. 5. Berechne die Veränderung in der Matrix $\Delta \left({\mathbf{S}}^H\mathbf{S}\right)={\mathbf{S}}_{n+1}^H{\mathbf{S}}_{n+1}-{\mathbf{S}}_n^H{\mathbf{S}}_n.$
   
   • • (Optional) Falls die absolute oder relative Veränderung zu groß ist, d.h. falls ||Δ(S^HS)|| oder $\Vert \Delta \left({\mathbf{S}}^H\mathbf{S}\right)\Vert /\Vert {\mathbf{S}}_n^H{\mathbf{S}}_n\Vert$
     
     eine vordefinierte Grenze überschreitet, verringere den Zeitraum Δt zwischen zwei S-Parametermessungen.
6. 6. Finde den Vektor a, der den $-\frac{{\mathbf{a}}^H\Delta \left({\mathbf{S}}^H\mathbf{S}\right)\mathbf{a}}{{\mathbf{a}}^H\left(\mathbf{I}-{\mathbf{S}}_{n+1}^H{\mathbf{S}}_{n+1}\right)\mathbf{a}}$
   
   minimiert.
   • • Dieser Vektor ist der Eigenvektor der Gleichung $-\Delta {\left(S^{H}S\right)}_a=\lambda \left(\mathbf{I}-{\mathbf{S}}_{n+1}^H{\mathbf{S}}_{n+1}\right)\mathbf{a},$
     
     der dem kleinsten Eigenwert λ entspricht.
7. 7. Falls der Erwärmungs- bzw. Auftauprozess noch nicht zu Ende ist, gehe zu Schritt 3. Sonst beende den Prozess.

In summary, the second execution of the method can be divided into steps as follows:

1. 1. At the beginning (t _n = 0) select any excitation vector a (for example a = [11 ... 1] ^T , ie all amplitudes equal and zero phase difference).
2. 2. For t = t _n measure the S-matrix of the EMAZ S _n (f).
3. 3. Apply excitation vector a.
4. 4. After a predefined time interval Δt has elapsed, ie for t = t _{n + 1} = t _n + Δt, measure the S matrix S _{n + 1} (f) again.
5. 5. Calculate the change in the matrix $\Delta \left({\mathbf{S.}}^H\mathbf{S.}\right)={\mathbf{S.}}_{n+1}^H{\mathbf{S.}}_{n+1}-{\mathbf{S.}}_n^H{\mathbf{S.}}_n.$
   
   • • (Optional) If the absolute or relative change is too large, ie if || Δ (S ^H S) || or $\Vert \Delta \left({\mathbf{S.}}^H\mathbf{S.}\right)\Vert /\Vert {\mathbf{S.}}_n^H{\mathbf{S.}}_n\Vert$
     
     exceeds a predefined limit, reduce the time period Δt between two S-parameter measurements.
6. 6. Find the vector a that corresponds to the $-\frac{{\mathbf{a}}^H\Delta \left({\mathbf{S.}}^H\mathbf{S.}\right)\mathbf{a}}{{\mathbf{a}}^H\left(\mathbf{I.}-{\mathbf{S.}}_{n+1}^H{\mathbf{S.}}_{n+1}\right)\mathbf{a}}$
   
   minimized.
   • • This vector is the eigenvector of the equation $-\Delta {\left({\mathrm{S.}}^H\mathrm{S.}\right)}_a=\lambda \left(\mathbf{I.}-{\mathbf{S.}}_{n+1}^H{\mathbf{S.}}_{n+1}\right)\mathbf{a},$
     
     which corresponds to the smallest eigenvalue λ.
7. 7. If the warming or thawing process has not ended, go to step 3 . Otherwise end the process.

The termination of the above process can be decided based on different criteria. For example, the temperature of the material to be heated can be monitored at one or more points by temperature sensors (see data stream 10 ) and the process will be canceled as soon as the temperature has reached a certain limit value. Another option is to define a specific time or energy consumption before starting the process. When this time has passed or when this amount of energy has been absorbed by the material to be heated (the latter can easily be determined by means of the S-parameter measurements), the heating process can be ended.

A termination criterion for thawing processes, or in general processes during which a phase change of the material to be heated takes place, can be obtained by monitoring the change in the S-parameters or the power consumption capacity over time. For example, the power consumption capacity of frozen food does not grow as much after defrosting as it does during defrosting. Accordingly, one could monitor the time curve of the power consumption capacity over time and then stop the process if it stops increasing or if the first derivative of the same curve drops sharply. These quantitative changes would indicate a qualitative phase transition.

Similar approaches are already known. It was proposed to monitor the absorption rate (i.e. the ratio of absorbed to fed-in power) for as many excitation vectors a and frequencies f as possible. Accordingly, some frequencies and excitation vectors a could be better suited to detect the phase transition process. It has also been suggested to take the mean value of the absorption rate over all applied vectors and frequencies as an indicator.

wobei σ_min der kleinste Singulärwert von S ist. Dieser Wert ist repräsentativer als der Absorptionswert bei einem beliebigen Anregungsvektor. Wenn sich nämlich die Absorption bei einem konstanten Anregungsvektor reduziert, bedeutet dies nicht zwangsläufig, dass die Absorptionskapazität der Last gesunken ist. Es könnte sein, dass sich das Feld, das durch diesen Vektor angeregt wird, wegen der Veränderung der Lasttemperatur verändert hat und ineffizienter geworden ist. Es könnte jedoch andere realisierbare Felder geben, die effizienter sind.Two new methods of detecting this phase transition are described below. The first method is to use the S-parameters to calculate the maximum absorption rate for each frequency. This value is $1-{\sigma }_{min}^2$

where σ _{min is} the smallest singular value of S. This value is more representative than the absorption value for any excitation vector. If the absorption is reduced for a constant excitation vector, this does not necessarily mean that the absorption capacity of the load has decreased. It could be that the field excited by this vector has changed due to the change in load temperature and has become more inefficient. However, there could be other viable fields that are more efficient.

The average value of the absorption rate is also a good indicator of the state of the load. The procedure previously used to determine this average value is, however, imprecise, very complex and inefficient, since the reflected power has to be measured with as many excitation parameters as possible. An alternative way of calculating this average in the second method at a frequency is presented below.

wobei Q die Erwärmungsmatrix des gesamten Erwärmungsguts darstellt. Wenn man die durchschnittliche Absorptionsrate η(a) über alle möglichen Anregungsvektoren a berechnen möchte, muss man über alle normierten Vektoren integrieren, d.h. auf der Oberfläche der Kugel ||a||² = 1. Um über reelle Variablen zu integrieren, kann man a^HQa wie folgt umschreiben: $${\mathbf{a}}^H\mathbf{Q}\mathbf{a}=\left[{\mathbf{x}}^T{\mathbf{y}}^T\right]\cdot \left[\begin{array}{cc}\mathbf{A}& -B\\ \mathbf{B}& \mathbf{A}\end{array}\right]\cdot \left[\begin{array}{c}\mathbf{x}\\ \mathbf{y}\end{array}\right]\equiv {\mathbf{r}}^T\tilde{Q}\mathbf{r},$$

wobei $\mathbf{x}=\text{Re}\left(\mathbf{a}\right),\mathbf{y}=\text{Im}\left(\mathbf{a}\right),\mathbf{A}=\text{Re}\left(\mathbf{Q}\right),\mathbf{B}=\text{Im}\left(\mathbf{Q}\right),\mathbf{r}=\left[\begin{array}{c}\mathbf{x}\\ \mathbf{y}\end{array}\right],\tilde{Q}=\left[\begin{array}{cc}\mathbf{A}& -B\\ \mathbf{B}& \mathbf{A}\end{array}\right]$

. Somit wurde eine Funktion von N komplexen Variablen η(a) in eine Funktion von 2N-reellen Variablen η(r) umgewandelt. Daher ist der Durchschnitt der Absorptionsrate in den neuen Variablen gleich $$\overline{\eta }=\frac{1}{S_{2N-1}}{\displaystyle \oint r^T\tilde{Q}r}dS\text{,}$$

wobei S_2N-1 der Flächeninhalt der Einheitskugel in 2N Dimensionen ist. Laut Divergenzsatz und der Symmetrie Q̃^T = Q̃ kann obiges Integral wie folgt berechnet werden: $${{\displaystyle \oint \left(\tilde{Q}r\right)}}^T\text{.}\widehat{r}dS={\displaystyle \int {}_{V_{2N}}}\nabla \cdot \left(\tilde{Q}r\right)dV={\displaystyle \int {}_{V_{2N}}}\text{tr}\left(\tilde{Q}\right)dV=\text{tr}\left(\tilde{Q}\right)V_{2N}\text{,}$$

wobei $\text{tr}\left(\tilde{Q}\right)=2\text{tr}\left(Q\right)=2{\displaystyle \sum {}_{i=1}^N}{q}_{ii}$

die Spur der Matrix Q und V_2N das Volumen einer 2N-dimensionalen Einheitskugel darstellen. Für N-Dimensionale Kugeln gilt $\frac{V_N}{S_{N-1}}=\frac{1}{N}\text{.}$

Daher ist $$\overline{\eta }=2\frac{V_{2N}}{S_{2N-1}}\text{tr}\left(Q\right)=2\frac{1}{2N}{\displaystyle \sum \text{eig}}\left(\text{I}-S^{H}S\right)=\frac{1}{N}\left(N-{\displaystyle \sum \text{eig}\left(S^{H}S\right)}\right)\text{.}$$

The absorption rate for any normalized excitation vector a (|| a || = 1) is $$\eta \left(\mathbf{a}\right)={\mathbf{a}}^H\left(\mathbf{I.}-{\mathbf{S.}}^H\mathbf{S.}\right)\mathbf{a}={\mathbf{a}}^H\mathbf{Q}\mathbf{a},$$

where Q represents the heating matrix of the entire material to be heated. If one wants to calculate the average absorption rate η (a) over all possible excitation vectors a, one has to integrate over all normalized vectors, ie on the surface of the sphere || a || ² = 1. To integrate over real variables, a ^H Qa can be rewritten as follows: $${\mathbf{a}}^H\mathbf{Q}\mathbf{a}=\left[{\mathbf{x}}^T{\mathbf{y}}^T\right]\cdot \left[\begin{array}{cc}\mathbf{A.}& -\mathrm{B.}\\ \mathbf{B.}& \mathbf{A.}\end{array}\right]\cdot \left[\begin{array}{c}\mathbf{x}\\ \mathbf{y}\end{array}\right]\equiv {\mathbf{r}}^T\tilde{Q}\mathbf{r},$$

whereby $\mathbf{x}=\text{re}\left(\mathbf{a}\right),\mathbf{y}=\text{in the}\left(\mathbf{a}\right),\mathbf{A.}=\text{re}\left(\mathbf{Q}\right),\mathbf{B.}=\text{in the}\left(\mathbf{Q}\right),\mathbf{r}=\left[\begin{array}{c}\mathbf{x}\\ \mathbf{y}\end{array}\right],\tilde{Q}=\left[\begin{array}{cc}\mathbf{A.}& -\mathrm{B.}\\ \mathbf{B.}& \mathbf{A.}\end{array}\right]$

. Thus, a function of N complex variables η (a) was converted into a function of 2N real variables η (r). Therefore the average of the absorption rate is the same in the new variables $$\overline{\eta }=\frac{1}{{\mathrm{S.}}_{2N-1}}{\displaystyle \oint r^T\tilde{Q}r}d\mathrm{S.}\text{,}$$

where S _{2N-1 is} the area of the unit sphere in 2N dimensions. According to the divergence ^{theorem and the symmetry Q̃ T} = Q̃ the above integral can be calculated as follows: $${{\displaystyle \oint \left(\tilde{Q}r\right)}}^T\text{.}\widehat{r}d\mathrm{S.}={\displaystyle \int {}_{V_{2N}}}\nabla \cdot \left(\tilde{Q}r\right)dV={\displaystyle \int {}_{V_{2N}}}\text{tr}\left(\tilde{Q}\right)dV=\text{tr}\left(\tilde{Q}\right)V_{2N}\text{,}$$

whereby $\text{tr}\left(\tilde{Q}\right)=2\text{tr}\left(Q\right)=2{\displaystyle \sum {}_{i=1}^N}{q}_{ii}$

the trace of the matrix Q and V _{2N represent} the volume of a 2N-dimensional unit sphere. The following applies to N-dimensional spheres $\frac{V_N}{{\mathrm{S.}}_{N-1}}=\frac{1}{N}\text{.}$

thats why $$\overline{\eta }=2\frac{V_{2N}}{{\mathrm{S.}}_{2N-1}}\text{tr}\left(Q\right)=2\frac{1}{2N}{\displaystyle \sum \text{own}}\left(\text{I.}-{\mathrm{S.}}^H\mathrm{S.}\right)=\frac{1}{N}\left(N-{\displaystyle \sum \text{own}\left({\mathrm{S.}}^H\mathrm{S.}\right)}\right)\text{.}$$

wobei σ₁,σ₂, ..., σ_N die Singulärwerte von S sind. D.h. die durchschnittliche Absorptionsrate über alle möglichen Anregungsvektoren bei einer Frequenz ist eins minus den Mittelwert der quadrierten Singulärwerte der Streumatrix bei derselben Frequenz.Since the eigenvalues of SHS are equal to the squared singular values of S, one finally has: $$\overline{\eta }=1-\frac{1}{N}{\displaystyle \sum _{n=1}^{\mathrm{<figure-callout\; id="2"\; label="N\; \sigma \; n"\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">N</figure-callout>}}{\mathrm{\sigma }}_n^{}{}_2^{}=1}-\overline{{\mathrm{<figure-callout\; id="2"\; label="\sigma "\; filenames="DE102019210264B4\_0150.png,DE102019210264B4\_0151.png"\; state="\{\{state\}\}">\sigma </figure-callout>}}^2}\text{,}$$

where σ ₁ , σ ₂ , ..., σ _{N are} the singular values of S. That is to say, the average absorption rate over all possible excitation vectors at a frequency is one minus the mean value of the squared singular values of the scatter matrix at the same frequency.

1. a. Führe die SVD-Zerlegung der in Schritt 4 gemessenen Streumatrix S_n+1(f) durch.
2. b. Aus den obigen Singulärwerten berechne die Absorptionsrate bei jeder Frequenz durch Formell $\overline{\eta }\left(ƒ\right)=1-\frac{1}{N}{\displaystyle \sum {}_{n=1}^N}{\sigma }_n^2\left(ƒ\right).$
   
3. c. Berechne den Mittelwert, den Maximalwert oder den Minimalwert der Absorptionsrate über die Frequenz: $\overline{\overline{\eta }}=\frac{1}{{ƒ}_{max}-{ƒ}_{min}}{\displaystyle \int {}_{{ƒ}_{min}}^{{ƒ}_{max}}}\overline{\eta }\left(ƒ\right)df$
   
   oder $\underset{ƒ}{\overline{\overline{\eta }}=\text{max}}\overline{\eta }\left(ƒ\right)\text{ oder }\overline{\overline{\eta }}=\underset{ƒ}{\text{min}}\overline{\eta }\left(ƒ\right)\text{.}$
   
4. d. Zeichne die dadurch entstandene Kurve der zeitlichen Entwicklung dieses Werts als $\overline{\overline{\eta }}\left(\text{t}\right)$
   
   oder als Zeitreihe ${\overline{\overline{\eta }}}_n$
   
   auf.
5. e. Berechne die zeitliche Ableitung der obigen Kurve $d\overline{\overline{\eta }}/dt$
   
   oder die Differenz ${\overline{\overline{\eta }}}_{n+1}-{\overline{\overline{\eta }}}_n\text{.}$
   
   Falls dieser Wert von positiv auf negativ umschlägt, oder sehr stark sinkt, brich den Erwärmungsprozess ab.

On the basis of the above calculation, a thawing process or similar heating process, during which a phase transition of the load or the material to be heated takes place, could be monitored using the following algorithm and possibly canceled (see step 7th the second execution of the procedure):

1. a. Perform the SVD decomposition in step 4th measured scattering matrix S _{n + 1} (f).
2. b. From the singular values above, calculate the absorption rate at each frequency by Formula $\overline{\eta }\left(ƒ\right)=1-\frac{1}{N}{\displaystyle \sum {}_{n=1}^N}{\sigma }_n^2\left(ƒ\right).$
   
3. c. Calculate the mean value, the maximum value or the minimum value of the absorption rate over the frequency: $\overline{\overline{\eta }}=\frac{1}{{ƒ}_{max}-{ƒ}_{min}}{\displaystyle \int {}_{{ƒ}_{min}}^{{ƒ}_{max}}}\overline{\eta }\left(ƒ\right)df$
   
   or $\underset{ƒ}{\overline{\overline{\eta }}=\text{Max}}\overline{\eta }\left(ƒ\right)\text{or}\overline{\overline{\eta }}=\underset{ƒ}{\text{min}}\overline{\eta }\left(ƒ\right)\text{.}$
   
4. d. Draw the resulting curve of the development of this value over time as $\overline{\overline{\eta }}\left(\text{t}\right)$
   
   or as a time series ${\overline{\overline{\eta }}}_n$
   
   on.
5. e. Calculate the time derivative of the curve above $d\overline{\overline{\eta }}/dt$
   
   or the difference ${\overline{\overline{\eta }}}_{n+1}-{\overline{\overline{\eta }}}_n\text{.}$
   
   If this value changes from positive to negative, or drops very sharply, stop the heating process.

List of reference symbols

1

Irradiation Zone (EMAZ)

2

Antennas

3

Material to be heated or object (s)

4th

Power amplifier with RF source

5

Measuring device for scattering parameters

6th

Feedback data stream

7th

Input data stream

8th

Command stream

9

Control and evaluation unit

10

Feedback data stream

Claims (19)

Method for heating dielectric objects with a predeterminable heat distribution by means of high-frequency radiation, in which - an object group consisting of several objects is introduced into an irradiation zone and irradiated with coherent high-frequency radiation via an RF transmission unit with at least two separate RF transmission channels, - with at least one differential measurement of scattering parameters being carried out before the object group is heated, in which high-frequency radiation backscattered from the irradiation zone is detected with a different number of objects in the object group in the irradiation zone at irradiation locations of the RF transmission channels, - Warming matrices for the objects of the object group are estimated from the differential measurement of scattering parameters, - Operating parameters for the RF transmission channels are determined from the heating matrices, which lead to a predetermined heat distribution in the object group, and - The RF transmission channels are then operated to heat the object group with the operating parameters determined from the heating matrices. Method for heating dielectric objects with a uniform heat distribution by means of high-frequency radiation, in which - an object or a group of objects is introduced into an irradiation zone and irradiated with coherent high-frequency radiation via an HF transmission unit with at least two separate HF transmission channels, - with one or more differential parameter measurements are carried out in which backscattered high-frequency radiation from the irradiation zone is detected before the beginning and at a time during the heating period and / or at different times during the heating period at irradiation locations of the RF transmission channels, - from each differential measurement of scattering parameters, heating matrices for relative to other areas of the object or the object group more strongly heated areas are estimated, - operating parameters for the RF transmission channels are determined from the heating matrices, which ode to a uniform heat distribution in the object r of the object group, and - the RF transmission channels are then operated for further heating of the object or the group of objects with the operating parameters determined from the heating matrices. Procedure according to Claim 1 , characterized in that, to carry out the differential measurement of scattering parameters, scattering parameters are recorded when all objects of the object group are brought into the irradiation zone and when different subsets of the objects of the object group are brought into the irradiation zone. Procedure according to Claim 3 , characterized in that each of the different subsets in a total number of L objects of the object group comprises L-1 objects and scatter parameter measurements are carried out on L or on L-1 different subsets. Procedure according to Claim 4 , characterized in that a scatter matrix S is generated for each scattering parameter measurement and the heating matrix of each object from a change in the scatter matrix product Δ (S ^H S) between the scatter matrix when all objects of the object group are introduced into the irradiation zone and the scatter matrix when a subset is introduced without the respective object is estimated in the irradiation zone. Method according to one of the Claims 1 and 3 until 5 , characterized in that the at least one differential measurement of scattering parameters is carried out at different frequencies of the high-frequency radiation in order to obtain heating matrices and operating parameters for each of these frequencies, the RF transmission channels for heating the object group in chronological order with those determined from the heating matrices for the different frequencies Operating parameters are operated. Method according to one of the Claims 1 and 3 until 6th , characterized in that an eigenvector of the equation is used as the predetermined heat distribution to determine the operating parameters for focusing in a partial area G _{l of the material to be heated} $$Q_{l}a=\lambda \left({\displaystyle \sum _{k\ne l}{Q}_k}\right)a$$

is determined as the excitation vector which corresponds to the greatest eigenvalue λ, where Q _l corresponds to the heating matrix of the sub-area C _l, Q _k heating matrices of the remaining sub-areas G _k and a correspond to excitation vectors. Method according to one of the Claims 1 and 3 until 6th , characterized in that to determine the operating parameters for uniform heating of the material to be heated, the relative standard deviation of the power density values of the sub-areas of the material to be heated is the predetermined heat distribution $$\text{RSD}=\frac{\sqrt{\frac{1}{\mathrm{L.}}{{\displaystyle \sum {}_{l=1}^{\mathrm{L.}}\left(\frac{p_l}{V_l}-\overline{p}\right)}}^2}}{\overline{p}}$$

is minimized with the help of a double periodic repetition loop of excitation vectors, in which in an inner loop N _T excitation vectors at the same frequency f _n are applied successively in time, while an outer loop _{applies such N T} excitation vectors at N _f different frequencies one after the other, where N _T ≥ N and N the number of RF transmission channels, p _l the power that is consumed in the sub-area G _l , V _l the volume of the sub-area, $\overline{p}={\displaystyle \sum {}_{l=1}^{\mathrm{L.}}{p}_l}/{\displaystyle \sum {}_{l=1}^{\mathrm{L.}}}{V}_l$

represent the average power density and L the number of sub-areas. Method according to one of the Claims 1 and 3 until 6th , characterized in that for the determination of excitation vectors for the generation of a predetermined heat distribution $\left\{p_1^0\text{,}{p}_2^0\text{,}...\text{,}{p}_{\mathrm{L.}}^0\right\}$

of the material to be heated, where $p_l^0$

represents the desired performance in sub-area G _l , - the following two quantities are defined: (a) a set A of complex L -dimensional vector sets F ₁ , F ₂ , ..., F _L , represented by an arbitrary vector a by formula ${\mathrm{F.}}_l=Q_l^{1/2}a$

are generated, where a is the N-dimensional complex vector of the channel excitations and $Q_l^{1/2}$

represent the square root of the warming _{matrix of G l} , (b) a set B of complex L -dimensional vector sets F ₁ , F ₂ , ..., F̃ _L , whose norms || F̃ _l || ² = p _l of the given performance $p_l^0$

deviate within specified tolerances ∈ _l: $p_l-p_l^0|<{\in }_l\text{,}l=\mathrm{1.2,}...\text{,}\mathrm{L.}\text{,}$

- an intersection between set A and set B is found, and - in the case that the intersection of A and B is empty, a vector set F _l , l = 1,2, ..., L is found in set A, which is as little removed from set B as possible. Procedure according to Claim 9 , characterized in that an iterative method is used to determine a vector set F ₁ , F ₂ , ..., F _L in set A, which converts an element of set A to set B, then back to set A, then again projected onto set B etc. until a convergence or termination criterion or a predeterminable maximum number of iterations is reached. Method according to one of the Claims 1 and 3 until 6th , characterized in that for the determination of excitation vectors for the generation of a predetermined heat distribution of the material to be heated, in which _{a power is consumed in each sub-area G l} which has a certain value within certain tolerances: $p_l-p_l^0|<{\in }_l$

∈ _l where $p_l^0$

the desired value and ∈ _l the corresponding permitted tolerance, - an N-dimensional complex power _{vector for each sub-area G l} ${\mathrm{F.}}_l=Q_l^{1/2}a$

is defined, where a is the N-dimensional complex vector of the channel excitations and $Q_l^{1/2}$

represent the square root of the warming _{matrix of G l} , - in an iterative process starting with an arbitrary starting vector $a_0$

in each case the resulting power in sub-area G _l using the formula ${\Vert {\mathrm{F.}}_l\Vert }^2>a_0^H{Q}_l{a}_0$

calculated and with the value $p_l^0$

the performance to be achieved is compared, and if so $$\Vert {\mathrm{F.}}_l\Vert >\sqrt{p_l^0+{\in }_l}\text{or}\Vert {\mathrm{F.}}_l\Vert <\sqrt{p_l^0-{\in }_l}$$

the vector F _l is scaled according to the following formula $${\mathrm{F.}}_l^{\text{'}}=\frac{{\mathrm{F.}}_l}{\Vert {\mathrm{F.}}_l\Vert }\sqrt{p_l^0+{\in }_l}bzw{\mathrm{F.}}_l^{\text{'}}=\frac{{\mathrm{F.}}_l}{\Vert {\mathrm{F.}}_l\Vert }<\sqrt{p_l^0-{\in }_l}$$

and then a new excitation _{vector a 1 is} determined, which is the error norm ${\displaystyle \sum {}_{l=1}^{\mathrm{L.}}}{\Vert Q_l^{\frac{1}{2}}{a}_1-{\mathrm{F.}}_l^{\text{'}}\Vert }^2$

minimized, the resulting power is calculated again with this new excitation vector and compared with the value p ° of the power to be achieved, and - the method is continued in this way until a convergence or termination criterion or a predeterminable maximum number of iterations is reached will. Procedure according to Claim 2 , characterized in that the differential scatter parameter measurements are carried out at regular time intervals during the heating period and the operating parameters are determined and applied. Procedure according to Claim 2 , characterized in that a time interval between the measurements of the scattering parameters for the differential scattering parameter measurements and / or time intervals between the differential scattering parameter measurements are adapted as a function of a magnitude of a change in the scattering parameters in the differential scattering parameter measurements. Method according to one of the Claims 2 , 12th or 13th , characterized in that the differential scatter parameter measurements are carried out at different frequencies of the high-frequency radiation in order to obtain heating matrices for the more heated areas and operating parameters for each of these frequencies, the HF transmission channels for heating the object or the group of objects in chronological order with those from the Heating matrices for the different frequencies are operated with certain operating parameters. Procedure according to Claim 14 , characterized in that a phase change in the object or the object group, in particular when thawing frozen products, is detected by determining a maximum of the absorption rate for one or more of the frequencies using the measured scattering parameters and using it as the time of the phase change. Procedure according to Claim 14 , characterized in that a phase change in the object or the group of objects, in particular when thawing frozen products, is detected by an average absorption rate for one or more of the frequencies $\overline{\eta }=1-{\sigma }^2$

is determined at the respective frequency and a maximum or a strong change in the average absorption rate, possibly averaged over the frequencies, is used as the point in time of the phase change, with $\overline{{\sigma }^2}$

corresponds to the mean value of the squared singular values of a scatter matrix that is generated for each scatter parameter measurement. Procedure according to Claim 15 or 16 , characterized in that the heating is ended when a phase change is detected. Method according to one of the Claims 1 until 17th , characterized in that the operating parameters are determined from the heating matrices using an optimization process. Device for heating dielectric objects by means of high-frequency radiation, with - an HF transmission unit with at least two HF transmission channels, via which objects introduced into an irradiation zone can be irradiated with coherent high-frequency radiation, HF transmission channels can be measured, and - a control and evaluation device for controlling the HF transmission unit and measuring device and for evaluating the scattering parameters measured by the measuring device, which are used to carry out the method according to one or more of the Claims 1 until 18th is trained.

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