DE102019210266B4 - Method and device for heating dielectric objects as uniformly as possible by means of high-frequency radiation - Google Patents

Abstract

Method for heating dielectric objects with the most uniform possible heat distribution by means of high-frequency radiation, in which an object (3) or a group of objects is introduced into an irradiation zone (1) and via an HF transmission unit (4) with N ≥ 2 separate HF transmission channels with coherent High-frequency radiation is irradiated, - before the start of the heating process, either reflection measurements or at least one measurement of scattering parameters are or will be carried out, in which the high-frequency radiation backscattered from the irradiation zone (1) is detected at irradiation locations of the RF transmission channels, - from the measurement or measurements, heating matrices for N different sub-areas of the object (3) or the object group are estimated, of which each sub-area is assigned to one of the N HF transmission channels, 3) or de r lead object group, and the HF transmission channels are then operated to heat the object (3) or the object group with the 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 as uniform a heat distribution as possible 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, particularly 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, hereinafter also referred to as load or material to be heated, are located in a closed, electromagnetically isolated (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.

State of the art

For uniform heating of one or more objects in an irradiation zone, in particular a metallic, electromagnetically insulated cavity, different techniques have been known so far, some of which are briefly listed below.

Thus, for a better uniformity of the heating, a movement of the load within the cavity, for example by placing it on a rotating plate, the use of mechanical mode mixers or also of movable antennas is proposed. One of the most important techniques for optimizing uniformity 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.

Furthermore, the deterministic calculation of the modes or realizable fields in the loaded cavity is known, through the superposition of which the uniformity of the resulting field pattern can be optimized. 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.

Although the above techniques can improve the heating uniformity in a microwave cavity, they each have some disadvantages or limitations. These include, for example, the need for mechanically moving parts, which are often expensive to manufacture and complex to maintain or may have to be replaced due to mechanical wear, the need for complex measurement technology for a deterministic calculation, high demands on the control hardware and the power amplifier when generating a stochastic mixture or a non-optimal uniformity of heating.

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.

From the EP 3 154 314 A1 a device for generating desired electromagnetic field distributions for heating an object in a cavity is known. For example, a matrix is used / determined which reflects the spatial absorption behavior in the cavity. The field distribution for every possible excitation vector in the empty cavity is assumed to be known.

The object of the present invention is to provide a method and a device for heating dielectric objects by means of high-frequency radiation, with which the uniformity of the heating can be optimized without additional hardware expenditure.

Presentation of the invention

The object is achieved with the method and the device according to claims 1 and 8. 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, an object or a group of objects is introduced into an irradiation zone, e.g. in a metallic, electromagnetically insulated cavity, and irradiated with coherent high-frequency radiation via an HF transmission unit with N ≥ 2 separate HF transmission channels at one or more frequencies. The RF transmission channels each include 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. Before the start of the heating process, in the proposed method, either reflection measurements or at least one scattering parameter measurement is carried out, in which or the high-frequency radiation backscattered from the irradiation zone is detected with a suitable measuring device at the irradiation locations of the RF transmission channels. In the case of a reflection measurement, the reflected powers are measured, which are reflected back from the irradiation zone or cavity at the irradiation gates. With a scattering parameter measurement, not only the reflected power but also the phase of the backscattered high-frequency radiation and thus the complex scattering parameters (S-parameters) of the loaded irradiation zone are measured. These measurements can be carried out at low RF transmission power. From the at least one scatter parameter measurement or the reflection measurements, heating matrices are then estimated for N different partial areas of the object or the object group, each partial area of which is assigned to one of the N RF transmission channels. The method is based on the assumption that each antenna or antenna group fed by an RF transmission channel mainly heats a different sub-area of the material to be heated, ie the object or the object group, than all other antennas fed by the remaining RF transmission channels and that the power consumed in this sub-range is correlated with the power reflected on the respective RF channel. Using these assumptions, heating matrices, also referred to as Q-matrices, can be determined for the individual partial areas of the material to be heated from the test measurements carried out, ie the reflection measurements or the measurement of scattering parameters. The determination of these heating matrices requires a larger number of reflection measurements with different excitation of the individual RF transmission channels, as is the case, for example, from the DE 10 2016 202 234 B3 is known. When carrying out a scattering parameter measurement, this one measurement is sufficient to be able to calculate the heating matrices from the complex scattering parameters determined therefrom or the complex scattering matrix obtained therewith. With the help of these heating matrices, operating or excitation parameters (amplitude and phase) for the RF transmission channels of the RF transmission unit used for heating can then be determined for each frequency selected for the irradiation, which lead to the most uniform possible heat distribution in the object (s) . A suitable optimization process is used for this. The RF transmission channels are then operated to heat the objects or group of objects with the operating or excitation parameters determined from the heating matrices.

A plurality of sets of excitation parameters or excitation vectors are preferably determined for each of the selected frequencies on the basis of the heating matrices, which are used one after the other during the heating process in order to carry out mode mixing. The excitation of the RF transmission channels with these different excitation vectors then preferably takes place in the form of a periodic time slot method in which different excitation parameters are used for each time slot. This is explained in more detail in the context of the exemplary embodiments.

The proposed method enables an improvement or optimization of the uniformity of the heating of a lossy dielectric material to be heated by means of radiation of high-frequency radiation. In order to protect the environment from undesired electromagnetic radiation and to concentrate the electromagnetic power efficiently on the heating load, the heating process often takes place in a closed, electromagnetically isolated cavity. However, this can cause standing waves to form within the cavity and the load, which disrupt the uniformity of heating. With the proposed method, however, a high degree of uniformity in the heating can be achieved. The proposed method and the associated device do not require any mechanically moving parts, which are usually expensive to manufacture, are complex to maintain or also have to be replaced frequently. The method does not require any stochastic mode noise, so that the high-power amplifier to be used in the RF transmission unit does not have to switch between different excitation parameters as often. The process does not require any measurements of the physical state of the load or the interior of the cavity or the irradiation zone, since the physical quantities required for the process (reflected powers or S-parameters) are measured at the inlets of the irradiation zone at which the high-frequency power is fed can be. The use of expensive, time-consuming and complicated sensors in the interior of the irradiation zone is thus unnecessary.

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 irradiation zone 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. 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 irradiation zone 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. One or more antennas, which radiate into the irradiation zone, are arranged at each inlet as part of the respective RF transmission channel. Furthermore, there is either a measuring device that can measure the reflected power of the irradiation zone loaded with the material to be heated at the N ≥ 2 irradiation locations within the frequency bandwidth of the amplifier, or a measuring device that displays the scatter parameters (S matrix) of the irradiation zone loaded with the material to be heated can measure the N ≥ 2 irradiation locations within the frequency bandwidth of the amplifier.

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. The device also has a measuring device with which either reflection measurements of high-frequency radiation backscattered from the loaded irradiation zone or a measurement of the complex scatter parameters of the loaded irradiation zone can be carried out at the irradiation locations of the RF transmission channels in the irradiation zone. The RF transmission channels can have one or more transmission antennas. The RF transmission unit and the measuring device are controlled via a control and evaluation device which is designed to carry out the method in accordance with one or more of the method configurations described in the present patent application.

In an advantageous embodiment of the method, the heating is carried out with coherent high-frequency radiation in a time sequence at several different frequencies of the high-frequency radiation, this cycle of the sequential frequency excitation also being able to be repeated as often as desired. In this case, the reflection measurements or the scattering parameter measurement are also used for each of these carried out at different frequencies. As a result, the heating matrices are also determined for the different frequencies within the operating spectrum of the signal source or the power amplifier of the device. The uniformity can thus be optimized at several frequencies. The excitation of the RF transmission channels can then take place at all frequencies used with the respectively optimized excitation parameters, with several sets of excitation parameters in each case one after the other. It is also possible to weight the excitation (s) separately for each individual frequency, either through different power inputs or through different dwell times at the respective frequency. This weighting can further improve the uniformity of the heating. The weighting is preferably also determined by a suitable optimization method.

After all frequencies have been processed, the cycle of excitation can be repeated as often as desired at different frequencies. In one embodiment of the method, the determination of the heating matrices by renewed reflection measurements or scatter parameter measurements can be repeated one or more times during the heating process in order to adapt the excitation parameters to the time-variable irradiation zone and load parameters - and potentially also to the needs of the time-based heating profile.

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 uniformly 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. 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 uniform heating of tissues, for thawing frozen blood supplies or transplant organs.

Figure list

• 1 eine schematische Darstellung eines Beispiels der vorgeschlagenen Vorrichtung für die Erwärmung eines oder mehrerer Objekte;
• 2 ein Beispiel für die periodische Anwendung der Betriebsparameter bzw. Anregungsvektoren bei unterschiedlichen Frequenzen gemäß dem vorgeschlagenen Verfahren; und
• 3 eine beispielhafte Unterteilung eines dielektrischen Objekts in drei fiktive Bereiche, je nach Einfluss der Antennen der in diesem Fall drei HF-Sendekanäle.

The proposed method and the associated device are described again in more detail below on the basis of 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 example of the periodic application of the operating parameters or excitation vectors at different frequencies according to the proposed method; and
• 3 an exemplary subdivision of a dielectric object into three fictitious areas, depending on the influence of the antennas of the three RF transmission channels in this case.

Ways of Carrying Out the Invention

• • Eine Bestrahlungszone in Form einer elektromagnetisch isolierten Kavität 1.
• • 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 Kavität 1 einspeisen können und durch die HF-Sendekanäle der Vorrichtung gespeist werden. 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.
• • Einer Messeinrichtung 5, die entweder
  • - die komplexen Streuparameter bzw. Streumatrizen mit vordefinierter Frequenzauflösung innerhalb der Betriebsbandbreite zu willkürlichen Zeitpunkten vor und/oder während dem Erwärmungsprozess messen kann, oder
  • - die reflektierte Leistung, die an jedem Verstärkerkanal ankommt, mit vordefinierter Frequenzauflösung innerhalb der Betriebsbandbreite zu willkürlichen Zeitpunkten vor oder während dem Erwärmungsprozess messen kann. Diese Messeinrichtung kann entweder in die oben genannte Signalquelle eingebaut oder separat davon angeordnet sein.
• • Einer elektronischen Steuer- und Auswerteeinrichtung 9 die an vier Datenströmen angebunden ist: einen benutzerdefinierten Eingangsdatenstrom 7, einen Befehlsdatenstrom 8, einen Rückmeldungsstrom 6 aus dem Hochleistungsverstärker 4 und (optional) einen Rückmeldungsstrom 10 aus der Kavität 1. Datenstrom 6 kann unter anderem Daten entweder der S-Parametermessungen oder der Leistungsreflexionsmessungen aus Messeinrichtung 5 enthalten. Datenstrom 7 enthält Anweisungen vom Benutzer (beispielsweise, aber nicht ausschließlich, Leistungspegel, Dauer der Erwärmung, Frequenzauflösung, Anzahl der Zyklen, Erwärmungsprofil usw.). Datenstrom 8 der

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

• • An irradiation zone in the form of an electromagnetically isolated cavity 1 .
• • A coherent high frequency power amplifier 4th with HF source, which 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 , the high-frequency electromagnetic radiation into the cavity 1 can feed and are fed through the RF transmission channels of the device. 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 either
  • - can measure the complex scatter parameters or scatter matrices with predefined frequency resolution within the operating bandwidth at arbitrary times before and / or during the heating process, or
  • - can measure the reflected power arriving at each amplifier channel with a predefined frequency resolution within the operating bandwidth at arbitrary times before or during the heating process. This measuring device can either be built into the above-mentioned signal source or arranged separately from it.
• • An electronic control and evaluation device 9 which is connected to four data streams: a user-defined input data stream 7th , a command stream 8th , a feedback stream 6th from the high-performance amplifier 4th and (optionally) a feedback stream 10 out of the cavity 1 . Data stream 6th can include data from either the S-parameter measurements or the power reflection measurements from the measuring device 5 contain. 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.). Data stream 8th the

Control and evaluation unit contains the excitation parameters for the different channels of the amplifier (including frequency, amplitude, phase difference, duration of the signal). Data stream 10 can take sensor data from the cavity 1 and / or the material to be heated (e.g. temperature, humidity level, etc.), if such sensors are present. The control and evaluation device 9 contains software that carries out different mathematical processes A and B for determining the heating matrices and determining the operating and excitation parameters. Method A uses the S parameter measurement or the measurement of the reflected powers to determine the heating matrices, also referred to below as Q matrices, for a number of partial areas of the material to be heated, which is equal to the number of channels in the system. Method B uses the Q matrices that were calculated with method A to determine a set of antenna excitation parameters or operating parameters for each frequency within the operating bandwidth of the RF source, which compensates for the consumed power of the named sub-areas of the material to be heated. Optionally, the software can also execute a further mathematical method C, which can use the results of method B to determine the appropriate weights for each frequency in order to further improve the uniformity of the heating of 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 see that in the cavity 1 was introduced and can consist of one or more dielectric objects.

1. 1. Bestimmung der Erwärmungsmatrizen für N Teilbereiche des Erwärmungsguts mithilfe eines semi-empirischen Verfahrens A, wobei N der Anzahl der Sendekanäle entspricht.
2. 2. Bestimmung der Anregungsparameter der Kanäle (Amplituden und Phasen), auch als Anregungsvektoren bezeichnet, bei jeder einzelnen der Frequenzen ƒ_n, n = 1,2,...,N_ƒ, die vom Benutzer oder vom System innerhalb der Betriebsbandbreite ausgewählt wurden, mithilfe eines mathematischen Verfahrens B.
3. 3. (Optional) Bestimmung von Gewichten, mit denen die Anregungsparameter und/oder Bestrahlungszeitdauern der oben ausgewählten Frequenzen gewichtet werden müssen, mithilfe eines mathematischen Verfahrens C.
4. 4. Anwendung der Anregungsparameter, die in Schritten 2 und 3 bestimmt wurden, und Wiederholung der Anwendung für eine Anzahl von Erwärmungszyklen, die vom Benutzer oder automatisch vom System bestimmt werden kann. Das System (die Steuer- und Auswerteeinheit 9) kann diese Anzahl entweder einmal am Anfang des Erwärmungsprozesses bestimmen, oder adaptiv, je nach Zustand des Erwärmungsgutes, anpassen.
5. 5. (Optional) Wiederholung der Schritte 1 bis 4, um die Anregungsparameter an potentiell veränderte Bedingungen der Last 3 anzupassen.

A preferred embodiment of the method for heating one or more objects within the cavity as uniformly as possible is described in more detail below. The procedure comprises the following steps:

1. 1. Determination of the heating matrices for N partial areas of the material to be heated using a semi-empirical method A, where N corresponds to the number of transmission channels.
2. 2. Determination of the excitation parameters of the channels (amplitudes and phases), also known as excitation vectors, at each of the frequencies ƒ _n , n = 1,2, ..., N _ƒ selected by the user or the system within the operating bandwidth , using a mathematical procedure B.
3. 3. (Optional) Determination of weights with which the excitation parameters and / or irradiation times of the frequencies selected above must be weighted using a mathematical method C.
4. 4. Apply the excitation parameters determined in steps 2 and 3 and repeat the application for a number of heating cycles that can be determined by the user or automatically by the system. The system (the control and evaluation unit 9 ) this number can either be once Determine at the beginning of the heating process or adaptively, depending on the condition of the material to be heated.
5. 5. (Optional) Repeat steps 1 to 4 to adjust the excitation parameters to potentially changed conditions of the load 3 adapt.

1. 1. Für n = 1 bis N_f
   1. a. Verweile bei Frequenz ƒ_n für t_n Sekunden.
   2. b. Für j = 1 bis N
      1. i. Betreibe Leistungsverstärker 4 kohärent mit Anregungsvektor ${\mathbf{a}}_j^n$
         
         für τ_nj Sekunden.
      2. ii. Gehe zu Schritt b.
   3. c. Gehe zu Schritt a.
2. 2. Wiederhole Schritt 1 bis der Erwärmungsprozess beendet ist.

Steps 1 to 3 represent a test process (detection phase) in which the system uses targeted measurements and numerical calculations to explore the coupling of the EM power with the material being heated and optimally adapts the excitation parameters to this coupling. Therefore, these steps can take place at a low input power. Step 4 represents the actual heating process, in which the level of the HF power is adapted to the needs of the requirements made by the user for the heating of the material to be heated. This step is preferably carried out in the form of a periodic time slot method in which different excitation parameters are used for each time slot. The time slot method is shown schematically and by way of example in 2 for the case of three frequencies and four channels. Accordingly, it consists of the following three nested time loops:

1. 1. For n = 1 to N _f
   1. a. Dwell at frequency ƒ _n for t _n seconds.
   2. b. For j = 1 to N
      1. i. Operate power amplifiers 4th coherent with excitation vector ${\mathbf{a}}_j^n$
         
         for τ _nj seconds.
      2. ii. Go to step b.
   3. c. Go to step a.
2. 2. Repeat step 1 until the heating process is finished.

The following describes the three methods (A, B and C) with which the excitation parameters of the RF transmission channels are determined.

Method A is a semi-empirical method that is based on the same principle as the method of DE 10 2016 202 234 B3 based.

wobei r den Ortsvektor innerhalb des Erwärmungsguts darstellt. Man definiere den „Hauptbereich“ B_k des Kanals k als die Ortsmenge innerhalb des Erwärmungsguts G, wobei der Leistungspegel von $p_k^0\left(\mathbf{r}\right)$

höher als der Leistungspegel aller übrigen Leistungsmuster ist: $$B{\text{}}_k=\left\{\mathbf{r}\in G:p_k^0\left(\mathbf{r}\right)\ge p_l^0\left(\mathbf{r}\right),\forall l\ne k\right\}$$

When each channel is excited individually, the radiation from the antenna or antennas connected to it creates a power pattern within the material to be heated. The power pattern excited by channel k alone is $p_k^0\left(\mathbf{r}\right),$

where r represents the position vector within the material to be heated. Define the “main area” B _{k of} the channel k as the local amount within the material to be heated G, where the power level of $p_k^0\left(\mathbf{r}\right)$

is higher than the performance level of all other performance patterns: $$\mathrm{B.}{}_k=\left\{\mathbf{r}\in G:p_k^0\left(\mathbf{r}\right)\ge p_l^0\left(\mathbf{r}\right),\forall l\ne k\right\}$$

These areas B _k form a natural subdivision of the material to be heated into N separate, non-overlapping local quantities, where N corresponds to the number of channels. Such a subdivision is exemplary for the case N = 3 in 3 shown.

wobei a = [a₁ a₂ ... a_N] der komplexe Anregungsvektor der Kanäle ist. Damit ist die Leistung, mit der Kanal k angeregt wird, |a_k|² und die Phase arg a_k = arctan(Im{a_k}/Re{a_k}). Q_k ist eine quadratische N × N Matrix, auch Erwärmungsmatrix oder Q-Matrix des Bereichs B_k genannt, und ihre Elemente sind durch folgende Formel gegeben: $$q_{ij}^k=\frac{1}{2}{\displaystyle {\iiint }_{B_k}\sigma \left(\mathbf{r}\right){\mathbf{E}}_i^{\ast }\left(\mathbf{r}\right)\cdot {\mathbf{E}}_j\left(\mathbf{r}\right)dV}$$

wobei E_k(r) das elektrische Feld, das von Kanal k erzeugt wird, und σ(r) die äquivalente elektrische Leitfähigkeit im Erwärmungsgut ist.The total power in the range B _k when all antennas (or a part of them) are excited is given by the following formula: $$p_k={\mathbf{a}}^H{\mathbf{Q}}_k\mathbf{a}$$

where a = [a ₁ a ₂ ... a _N ] is the complex excitation vector of the channels. This means that the power with which channel k is excited is | a _k | ² and the phase arg a _k = arctan (Im {a _k } / Re {a _k }). Q _k is a square N × N matrix, also called the heating matrix or Q matrix of the area B _k , and its elements are given by the following formula: $$q_{ij}^k=\frac{1}{2}{\displaystyle {\iiint }_{{\mathrm{B.}}_k}\sigma \left(\mathbf{r}\right){\mathbf{E.}}_i^{\ast }\left(\mathbf{r}\right)\cdot {\mathbf{E.}}_j\left(\mathbf{r}\right)dV}$$

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

The aim of method A is to find the Q-matrices of each sub-area B _k. This can be done by “questioning” the system using targeted suggestions. Since Q _k is an N × N Hermitian matrix, there are N ² real unknowns that need to be determined. Therefore the system has to be ^{“questioned” N 2} times, ie N ² different stimuli have to be applied to the channels and the corresponding powers within the sub-areas B _{k have to be} measured or estimated. Then, using the equation p _k = a ^H Q _k a, an N ² × N ² linear system of equations can be set up, the solution of which is the elements of the matrix Q _k . The necessary suggestions are detailed, for example, in the DE 10 2016 202 234 B3 (Paragraphs [0039] to [0041]).

bestimmt werden, wobei ${\mathbf{S}}_n=\left[S_{kj}^n\right]$

die komplexe Streumatrix des Garraums bei der angeregten Frequenz ƒ_n und a_j die komplexe Anregungsamplitude von Antenne j darstellen. Letztere indirekte Bestimmung erfordert, dass die Messeinrichtung 5 in der Lage ist, die komplexe Streumatrix des Garraums bei den Frequenzen zu messen, die von der Signalquelle 4 erzeugt werden können.A method is now described by means of which one can estimate the power p _k in the sub-range B _k for any excitation of the channels. The core of the concept is to correlate the power consumed in the sub-range B _k with the power r _k that is reflected back to the k-th channel. This reflected power can either be done directly by the measuring device 5 measured or indirectly by formula ${\mathbf{r}}_k={\left|{\sum }_j{\mathrm{S.}}_{kj}^n{a}_j\right|}^2$

be determined, where ${\mathbf{S.}}_n=\left[{\mathrm{S.}}_{kj}^n\right]$

the complex scatter matrix of the cooking chamber at the excited frequency ƒ _n and a _{j represent} the complex excitation amplitude of antenna j. The latter indirect determination requires that the measuring device 5 is able to measure the complex scattering matrix of the cooking chamber at the frequencies used by the signal source 4th can be generated.

Die reflektierte Leistung, die vom Bereich B_k an einen Kanal zurückkehrt, ist proportional zur einfallenden Leistung auf diesen Bereich. Daher ist r_k = ce_k, wobei c ein positiver Reflexionskoeffizient ist. Diese Gleichung beinhaltet die zwei einzigen Vereinfachungen bzw. Annahmen des Verfahrens, dass

1. (a) der Reflexionskoeffizient für alle Teilbereiche gleich ist
2. (b) die reflektierte Leistung am k-ten Kanal nur aus Teilbereich B_k hervorgeht.

Annahme (a) ignoriert die möglichen materiellen Unterschiede zwischen den Teilbereichen und den Einfallsrichtungen der Strahlung auf die jeweiligen Teilbereiche (beides beeinflusst in der Praxis den Reflexionskoeffizienten). Annahme (b) ignoriert die Tatsache, dass die reflektierte Leistung am k-ten Kanal die Überlagerung der Reflexionen aus allen Teilbereichen ist. Stattdessen wird angenommen, dass die Reflexion aus dem Teilbereich B_k den Hauptanteil der reflektierten Leistung am k -ten Kanal ausmacht und dass die Reflexionsbeiträge aus den restlichen Teilbereichen vernachlässigbar sind.Let the incident power on a sub-area B _{k be e k} . If the total power that is fed into all channels is normalized to 1 watt, then follows ${\sum }_k{e}_k=1$

The reflected power _{returning to a channel from area B k} is proportional to the power incident on that area. Hence r _k = ce _k , where c is a positive reflection coefficient. This equation contains the only two simplifications or assumptions of the method that

1. (a) the reflection coefficient is the same for all sub-areas
2. (b) the reflected power on the kth channel can only be derived from sub-area B _k .

Assumption (a) ignores the possible material differences between the sub-areas and the directions of incidence of the radiation on the respective sub-areas (both influence the reflection coefficient in practice). Assumption (b) ignores the fact that the reflected power on the kth channel is the superposition of the reflections from all partial areas. Instead, it is assumed that the reflection from sub-area B _k makes up the main portion of the reflected power on the k th channel and that the reflection contributions from the remaining sub-areas are negligible.

Wenn man also die gesamtreflektierte Leistung ${\sum }_k{r}_k$

als r_tot bezeichnet, hat man c = r_tot.If one adds up all equations of the form r _k = cek, one obtains a formula for the reflection coefficient c, because ${\sum }_k{r}_k=c{\sum }_k{e}_k=c.$

So if you consider the overall performance ${\sum }_k{r}_k$

referred to as r _dead , one has c = r _dead .

The power consumed in the k -th sub-range is the difference between the incident and the reflected power, ie $$p_k=e_k-r_k=\left(\frac{1}{c}-1\right)r_k=\frac{1-r_{tOt}}{r_{tOt}}{r}_k$$

The above formula can be used to estimate the consumed power for each sub-area B _k and any stimuli by measuring (directly or indirectly) the reflected power on all channels.

Method A represents a test phase or learning or recognition phase, during which the system recognizes or learns the coupling between the excitation of the channels of the amplifier and the heating of the different partial areas of the material to be heated. Either N ² test measurements per Excitation frequency can be carried out in real time, during which the incident and reflected powers are measured on each channel, or the complex scattering matrix of the cavity must be determined at each excitation frequency, and with the help of which the reflected powers on each channel ^{are calculated for N 2} virtual excitations. The latter stimuli are referred to as “virtual” because they do not take place in the real system, but rather their effect is emulated by the control software with the help of the measured scatter matrices.

• • Erste Ausführungsvariante
  1. 1. Wähle N_ƒ Frequenzen innerhalb der Betriebsbandbreite aus: ƒ₁,ƒ₂, ...,ƒ_Nƒ.
  2. 2. Für n = 1 bis N_ƒ
     1. a. Für i = 1 bis N
        1. i. Rege Kanal i an.
        2. ii. Miss reflektierte Leistung r_k (k = 1,2, ... N) an allen Kanälen und berechne Gesamtreflektion $r_{tot}={\sum }_k{r}_k.$
           
           r_tot
        3. iii. Berechne konsumierte Leistung in B_k aus $p_k=\left(r_{tot}^{-1}-1\right)r_k$
           
        4. iv. Für j = i + 1 bis N
           1. (a) Rege Kanäle i und j mit Phasendifferenz φ₁, an.
           2. (b) Miss reflektierte Leistung r_k (k = 1,2, ... N) an allen Kanälen und berechne Gesamtreflektion $r_{tot}={\sum }_k{r}_k.$
              
              r_tot
           3. (c) Berechne konsumierte Leistung in B_k aus p_k = $\left(r_{tot}^{-1}-1\right)r_k$
              
           4. (d) Wiederhole (a) bis (c) für Phasendifferenz φ2.
     2. b. Aus den vorangegangenen Messungen berechne Q_k bei ƒ_n.
• • Zweite Ausführungsvariante
  1. 1. Wähle N_ƒ Frequenzen innerhalb der Betriebsbandbreite aus: ƒ₁,ƒ₂, ...,ƒ_Nƒ.
  2. 2. Für n = 1 bis N_ƒ
     1. a. Miss S-Parametermatrix S(ƒ_n)
     2. b. Für i = 1 bis N
        1. i. Berechne reflektierte Leistung an allen Kanälen, wenn nur Kanal i aktiv ist, durch r_k = |S_ki|².
        2. ii. Berechne konsumierte Leistung in B_k aus $p_k=\left(r_{tot}^{-1}-1\right)r_k$
           
        3. iii. Für = i + 1 bis N
           1. (a) Berechne reflektierte Leistung an allen Kanälen k, wenn Kanäle i und j gleiche Phase haben, aus r_k = |S_kj + S_ki|².
           2. (b) Berechne konsumierte Leistung in B_k aus p_k = $\left(r_{tot}^{-1}-1\right)r_k$
              
           3. (c) Berechne reflektierte Leistung an allen Kanälen k, wenn Kanäle i und j eine Phasendifferenz φ,bspw. von 90°, haben, aus r_k = |S_kje^iφ + S_ki|².
           4. (d) Berechne konsumierte Leistung in B_k aus p_k = $\left(r_{tot}^{-1}-1\right)r_k.$
              
     3. c. Aus den vorangegangenen Berechnungen berechne Q_k bei ƒ_n.

In summary, method A can be broken down into the following steps:

• • First variant
  1. 1. Select N _ƒ frequencies within the operating bandwidth: ƒ ₁ , ƒ ₂ , ..., ƒ _Nƒ .
  2. 2. For n = 1 to N _ƒ
     1. a. For i = 1 to N
        1. i. Stimulate channel i.
        2. ii. Miss reflected power r _k (k = 1,2, ... N) on all channels and calculate total reflection $r_{tOt}={\sum }_k{r}_k.$
           
           r _dead
        3. iii. Calculate the power consumed in B _k $p_k=\left(r_{tOt}^{-1}-1\right)r_k$
           
        4. iv. For j = i + 1 to N
           1. (a) Active channels i and j with phase difference φ ₁ , on.
           2. (b) Miss reflected power r _k (k = 1,2, ... N) on all channels and calculate total reflection $r_{tOt}={\sum }_k{r}_k.$
              
              r _dead
           3. (c) Calculate the power consumed in B _k from p _k = $\left(r_{tOt}^{-1}-1\right)r_k$
              
           4. (d) Repeat (a) to (c) for phase difference φ2.
     2. b. From the previous measurements calculate Q _k at ƒ _n .
• • Second variant
  1. 1. Select N _ƒ frequencies within the operating bandwidth: ƒ ₁ , ƒ ₂ , ..., ƒ _Nƒ .
  2. 2. For n = 1 to N _ƒ
     1. a. Miss S parameter matrix S (ƒ _n )
     2. b. For i = 1 to N
        1. i. Compute reflected power on all channels, if only channel i is active, by r _k = | S _ki | ² .
        2. ii. Calculate the power consumed in B _k $p_k=\left(r_{tOt}^{-1}-1\right)r_k$
           
        3. iii. For = i + 1 to N
           1. (a) Compute reflected power on all channels k, if channels i and j have the same phase, from r _k = | S _kj + S _ki | ² .
           2. (b) Calculate the power consumed in B _k from p _k = $\left(r_{tOt}^{-1}-1\right)r_k$
              
           3. (c) Calculate reflected power on all channels k if channels i and j have a phase difference φ, e.g. of 90 °, from r _k = | S _kj e ^iφ + S _ki | ² .
           4. (d) Calculate the consumed power in B _k from p _k = $\left(r_{tOt}^{-1}-1\right)r_k.$
              
     3. c. From the previous calculations compute Q _k at ƒ _n .

After determining the heating matrices Q _{k of} each sub-area B _k , it is possible to determine the consumed EM power in all sub-areas for any antenna excitations a using the formula p _k = a ^H Q _k a. A very frequently asked requirement for the heating process is that the power _{values p 1} , p ₂ , ..., p _k , ..., p _N differ from one another as little as possible (uniform heating of the material to be heated). The aim of method B is to find the appropriate excitations a for the channels that lead to the most uniform possible heating, ie to values p ₁ , p ₂ ,..., P _N which have a small standard deviation.

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 method B is based on several sets or vectors of excitation _{while staying at a frequency ƒ n} ${\mathbf{a}}_1^n,{\mathbf{a}}_2^n,{\mathbf{a}}_3^n,...$

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 Leistung im k -ten Teilbereich ist $p_k^l={\mathbf{a}}_l^H{\mathbf{Q}}_k{\mathbf{a}}_l.$

Die zeitliche Überlagerung der unterschiedlichen Leistungen hat als Endeffekt, dass die äquivalente Leistung am Ende von t_n Sekunden wie folgt aussieht: $$p_k=\frac{1}{t_n}{\displaystyle \sum _{l=1}^{N_T}{\tau }_{nl}{\mathbf{a}}_l^H{\mathbf{Q}}_k{\mathbf{a}}_l}$$

Assume that the signal source dwells at a frequency ƒ = ƒ _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 _n3 , ..., τ _{nN T} Seconds (see 2 ). 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 generated in this way is in the k -th sub-range $p_k^l={\mathbf{a}}_l^H{\mathbf{Q}}_k{\mathbf{a}}_l.$

The final effect of the temporal superposition of the different powers is that the equivalent power looks like this _{at the end of t n seconds:} $$p_k=\frac{1}{t_n}{\displaystyle \sum _{l=1}^{N_T}{\tau }_{nl}{\mathbf{a}}_l^H{\mathbf{Q}}_k{\mathbf{a}}_l}$$

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

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

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: $$p_k={\displaystyle \sum _{l=1}^{N_T}{\mathbf{a}}_l^H{\mathbf{Q}}_k{\mathbf{a}}_l}$$

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

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 (1), 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 a _lj , then the above formula takes the form: $$p_k={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N\left({\displaystyle \sum _{l=1}^{N_T}{a}_{lj}{a}_{li}^{\ast }}\right)q_{ij}^k={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N{c}_{ij}{q}_{ij}^k}}}}$$

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

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 (1) 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. It is therefore possible to choose the number of time slots while staying at a frequency equal to the number of channels, ie N _T = N, without loss of generality. More degrees of freedom for A would not increase the number of degrees of freedom for C.

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

der durchschnittliche Wert aller Teilleistungen p_k ist.In the following, method B aims at the skilful choice of matrix C in order to achieve _{values p k that deviate as little as possible.} The measure of uniformity that is used is the normalized standard deviation of the values p _k , ie $${\widehat{\sigma }}_p=\frac{1}{\sqrt{N}}\cdot \frac{\sqrt{{\displaystyle {\sum }_{k=1}^N{\left(p_k-\overline{p}\right)}^2}}}{{}^{\overline{p}}}$$

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

is the average value of all partial services p _k .

wo C_m positiv-definite hermitesche Matrizen. Eine mögliche Basis des oben genannten Vektorraumes kann mithilfe folgender Matrizen aufgebaut werden: $$\begin{array}{l}{\mathbf{E}}_k=\left[e_{ij}^k\right],wo{e}_{ij}^k=\{\begin{array}{cc}\mathrm{1,}& wenni=j=k\\ \mathrm{0,}& sonst\end{array}\\ {\mathbf{F}}_{kl}=\left[f_{ij}^{kl}\right],wo{f}_{ij}^{kl}=\{\begin{array}{cc}\mathrm{1,}& wenni=k,j=l,oderi=l,j=k\\ \mathrm{0,}& sonst\end{array}\\ G_{kl}=\left[g_{ij}^{kl}\right],wo{g}_{ij}^{kl}=\{\begin{array}{ll}\sqrt{-1},\hfill & wenni=k,j=l\hfill \\ -\sqrt{-\mathrm{1,}}\hfill & wenni=l,j=k\hfill \\ \mathrm{0,}\hfill & sonst\hfill \end{array}\end{array}$$

In order _{to minimize σ̂ 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 $$\mathbf{C.}={\displaystyle {\sum }_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0080.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\gamma }_m{\mathbf{C.}}_m}$$

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: $$\begin{array}{l}{\mathbf{E.}}_k=\left[e_{ij}^k\right],wO{e}_{ij}^k=\{\begin{array}{cc}\mathrm{1,}& wenni=j=\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE102019210266B4\_0079.png"\; state="\{\{state\}\}">k</figure-callout>}\\ \mathrm{0,}& sOnst\end{array}\\ {\mathbf{F.}}_{kl}=\left[f_{ij}^{kl}\right],wO{f}_{ij}^{kl}=\{\begin{array}{cc}\mathrm{1,}& wenni=k,j=l,Oderi=l,j=\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE102019210266B4\_0079.png"\; state="\{\{state\}\}">k</figure-callout>}\\ \mathrm{0,}& sOnst\end{array}\\ G_{kl}=\left[G_{ij}^{kl}\right],wO{G}_{ij}^{kl}=\{\begin{array}{ll}\sqrt{-1},\hfill & wenni=k,j=l\hfill \\ -\sqrt{-\mathrm{1,}}\hfill & wenni=l,j=\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE102019210266B4\_0079.png"\; state="\{\{state\}\}">k</figure-callout>}\hfill \\ \mathrm{0,}\hfill & sOnst\hfill \end{array}\end{array}$$

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

If you number the above matrices continuously from m = 1 to m = N ² , you get the base (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 $\left(\mathbf{C.}={\displaystyle {\sum }_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0080.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\gamma }_m{\mathbf{C.}}_m}\right)$

one must take care that the coefficients γ _m cannot be arbitrary, but must have such values that C ≽ 0, that is, C must be positive semi-definite (it must only have nonnegative eigenvalues). This follows from the definition of C = A ^H A.

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

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

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

According to the above decomposition, the individual elements of C can be written as follows: $c_{ij}={\displaystyle {\sum }_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0080.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\gamma }_m{c}_{ij}^m,wO}{c}_{ij}^m$

the elements of the basic matrix C _m . Therefore equation (1) takes the form: $$p_k={\displaystyle {\sum }_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0080.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\gamma }_m}{\displaystyle {\sum }_{i,j}{c}_{ij}^m{q}_{ij}^k={\displaystyle {\sum }_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0080.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\gamma }_m{p}_k^m}}$$

whereby ${\mathrm{<figure-callout\; id="1"\; label="p\; k"\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0079.png"\; state="\{\{state\}\}">p</figure-callout>}}_k^{},{\mathrm{<figure-callout\; id="2"\; label="p\; k"\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0080.png"\; state="\{\{state\}\}">p</figure-callout>}}_k^{},...,p_k^m,...,{\mathrm{<figure-callout\; id="2"\; label="p\; k\; N"\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0080.png"\; state="\{\{state\}\}">p</figure-callout>}}_{k{N}^{}}^{{}^2}$

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

Decomposition (4) 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.

zusammenfassen kann. Die analytische Abhängigkeit der Optimierungsfunktion σ̂_p von γ kann anhand der Zerlegung $p_k={\displaystyle \sum {}_{m=1}^{N^2}}{\mathrm{\gamma }}_m{p}_k^m$

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

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

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

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

wobei ${\text{P}}_t=\left[{\mathrm{\pi }}_{mn}^t\right]$

eine N² × N² symmetrische reelle Matrix mit Elementen ${\mathrm{\pi }}_{mn}^t=$

${\displaystyle \sum {}_{m=1}^N}{\widehat{p}}_k^m{\widehat{p}}_k^n\text{,}$

Wenn man die Summe der unbekannten Koeffizienten auf 1 normiert, d.h. ${\displaystyle \sum {}_m{\widehat{\gamma }}_m}=\mathrm{1,}$

dann ist die Minimierung der Ungleichmäßigkeit laut Formel (5) äquivalent zur Minimierung der quadratischen Form ƒ(γ) = γ̂^TP_tγ̂. Daher läuft Verfahren B auf folgendes Optimierungsproblem hinaus:

• • Finde einen reellen N² -dimensionalen Vektor γ̂, der folgenden Bedingungen genügt:
  • [i] ${\displaystyle \sum {}_{m=1}^{N^2}}{\widehat{\gamma }}_m=1.$
    
  • [ii] Die Matrix $\text{c}={\displaystyle \sum {}_{m=1}^{N^2}}{\mathrm{\gamma }}_m{\text{c}}_m$
    
    ist positiv semi-definit.
  • [iii] Er minimiert die Funktion ƒ(γ) = 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 can be converted into a real one vector $\gamma ={\left[{\mathrm{<figure-callout\; id="1"\; label="\gamma "\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0079.png"\; state="\{\{state\}\}">\gamma </figure-callout>}}_1{\mathrm{<figure-callout\; id="2"\; label="\gamma "\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0080.png"\; state="\{\{state\}\}">\gamma </figure-callout>}}_{2...}{\gamma }_{N^2}\right]}^T$

can summarize. The analytical dependence of the optimization function σ̂ _p on γ can be determined using the decomposition $p_k={\displaystyle \sum {}_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0080.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}}{\mathrm{\gamma }}_m{p}_k^m$

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

normalized in such a way that their sum is 1 watt, and new coefficients are defined: $$p_k={\displaystyle \sum _{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0080.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\mathrm{\gamma }}_m{p}_k^m}={\displaystyle \sum _{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0080.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}{\widehat{\gamma }}_m{\widehat{p}}_k^m}$$

whereby ${\widehat{p}}_k^m=p_k^m/{\displaystyle \sum {}_{m=1}^N}{p}_k^m\text{and}{\widehat{\gamma }}_m={\mathrm{\gamma }}_m{\displaystyle \sum {}_{m=1}^N}{p}_k^m\text{.}$

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

whereby ${\text{P.}}_t=\left[{\mathrm{\pi }}_{mn}^t\right]$

an N ² × N ² symmetric real matrix with elements ${\mathrm{\pi }}_{mn}^t=$

${\displaystyle \sum {}_{m=1}^N}{\widehat{p}}_k^m{\widehat{p}}_k^n\text{,}$

If one normalizes the sum of the unknown coefficients to 1, ie ${\displaystyle \sum {}_m{\widehat{\gamma }}_m}=\mathrm{1,}$

then the minimization of the unevenness according to formula (5) is equivalent to the minimization of the quadratic form ƒ (γ) = γ̂ ^T P _t γ̂. Method B therefore results in the following optimization problem:

• • Find a real N ² -dimensional vector γ̂ that satisfies the following conditions:
  • [i] ${\displaystyle \sum {}_{m=1}^{N^2}}{\widehat{\gamma }}_m=1.$
    
  • [ii] The matrix $\text{c}={\displaystyle \sum {}_{m=1}^{N^2}}{\mathrm{\gamma }}_m{\text{c}}_m$
    
    is positive semi-definite.
  • [iii] It minimizes the function ƒ (γ) = 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 (3), the matrix of the optimal channel excitations A can be found by solving the 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: $$\text{A.}={\text{<figure-callout id="1" label="VD" filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0079.png" state="{{state}}">VD</figure-callout>}}^{1\text{/}2}{\text{V}}^H$$

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 (6) 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 N Erwärmungsmatrizen Q₁, Q₂, ...,Q_N, bei einer bestimmten Frequenz ƒ_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 ƒ_n angewandt werden sollen (vgl. 2).
• • Schritte:
  1. 1. Berechne die Basismatrizen ${\text{C}}_m=\left[c_{ij}^m\right]$
     
  2. 2. Mithilfe der Q-Matrizen $Q_k=\left[q_{ij}^k\right]$
     
     berechne die N² Basismuster durch $$\begin{array}{cc}{p}_k^m={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N{c}_{ij}^m}}{q}_{ij}^k\text{,}& m=\mathrm{1,2,}\dots \text{,}{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0080.png"\; state="\{\{state\}\}">N</figure-callout>}}^2\end{array}$$
     
  3. 3. Normiere die Basismuster auf Summe von 1 Watt: ${\widehat{p}}_k^m=p_k^m/{\displaystyle \sum {}_{m=1}^N}{p}_k^m\text{.}$
     
  4. 4. Berechne die Matrix ${\text{P}}_t=\left[{\mathrm{\pi }}_{mn}^t\right]\text{, won }{\mathrm{\pi }}_{mn}^t={\displaystyle \sum {}_{m=1}^N}{\widehat{p}}_k^m{\widehat{p}}_k^n\text{.}$
     
  5. 5. Minimiere (bspw. durch konvexe Optimierung) die Funktion ƒ(γ) = γ̂^TP_tγ̂ unter folgenden Bedingungen:
     • ■ ${\displaystyle \sum {}_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0080.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}}{\widehat{\gamma }}_m=1$
       
     • ■ Die Matrix $\mathbf{C}={\displaystyle \sum {}_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0080.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}}{\mathrm{\gamma }}_m{\mathbf{C}}_m$
       
       ist positiv semi-definit.
  6. 6. Berechne nichtnormierte Koeffizienten durch ${\mathrm{\gamma }}_m={\widehat{\gamma }}_m/{\displaystyle \sum {}_{m=1}^N}{p}_k^m\text{.}$
     
  7. 7. Berechne C-Matrix durch $C={\displaystyle \sum {}_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0080.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}}{\mathrm{\gamma }}_m{C}_m\text{.}$
     
  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_lj], [a_2j], ..., [a_Nj].

Finally, method B can be summarized as follows:

• • Input: A set of N heating matrices Q ₁ , Q ₂ , ..., Q _N , at a certain frequency ƒ _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 the amplifier's excitation parameters while it is at frequency ƒ _n should be (cf. 2 ).
• • Steps:
  1. 1. Compute the basic matrices ${\text{C.}}_m=\left[c_{ij}^m\right]$
     
  2. 2. Using the Q-matrices $Q_k=\left[q_{ij}^k\right]$
     
     compute the N ² basic pattern $$\begin{array}{cc}{p}_k^m={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N{c}_{ij}^m}}{q}_{ij}^k\text{,}& m=\mathrm{1.2,}...\text{,}{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0080.png"\; state="\{\{state\}\}">N</figure-callout>}}^2\end{array}$$
     
  3. 3. Normalize the basic pattern to a sum of 1 watt: ${\widehat{p}}_k^m=p_k^m/{\displaystyle \sum {}_{m=1}^N}{p}_k^m\text{.}$
     
  4. 4. Compute the matrix ${\text{P.}}_t=\left[{\mathrm{\pi }}_{mn}^t\right]\text{, won}{\mathrm{\pi }}_{mn}^t={\displaystyle \sum {}_{m=1}^N}{\widehat{p}}_k^m{\widehat{p}}_k^n\text{.}$
     
  5. 5. Minimize (e.g. by convex optimization) the function ƒ (γ) = γ̂ ^T P _t γ̂ under the following conditions:
     • ■ ${\displaystyle \sum {}_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0080.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}}{\widehat{\gamma }}_m=1$
       
     • ■ The matrix $\mathbf{C.}={\displaystyle \sum {}_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0080.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}}{\mathrm{\gamma }}_m{\mathbf{C.}}_m$
       
       is positive semi-definite.
  6. 6. Compute non-normalized coefficients by ${\mathrm{\gamma }}_m={\widehat{\gamma }}_m/{\displaystyle \sum {}_{m=1}^N}{p}_k^m\text{.}$
     
  7. 7. Compute C-matrix $\mathrm{C.}={\displaystyle \sum {}_{m=1}^{{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="DE102019210266B4\_0078.png,DE102019210266B4\_0080.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}}{\mathrm{\gamma }}_m{\mathrm{C.}}_m\text{.}$
     
  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 _lj ], [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 method B, 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 (ƒ _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 ƒ} = 1/N_ƒ, 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: $${\widehat{p}}_k={\displaystyle \sum _{n=1}^{N_{ƒ}}{\text{w}}_n{p}_k\left({ƒ}_n\right)}$$

where w _{n are} positive numbers. In the case of equal weighting w ₁ = w ₂ = ... = w _{N ƒ} = 1 / N _ƒ , 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 B _k (see formula (2)). 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 pattern for each individual frequency so that their total is 1 watt. The variable weights must also be scaled accordingly: $${\widehat{p}}_k={\displaystyle \sum _{n=1}^{N_{ƒ}}{\widehat{\text{w}}}_k{\widehat{p}}_k\left({ƒ}_n\right)}\text{,}{\widehat{p}}_k\left({ƒ}_n\right)=\frac{p_k\left({ƒ}_n\right)}{{\displaystyle \sum {}_{k=1}^N}{p}_k\left({ƒ}_n\right)}{\widehat{\text{w}}}_n=w_n{\displaystyle \sum _{k=1}^N{p}_k\left({ƒ}_n\right)}$$

wobei ŵ = [ŵ₁ ... ŵ_{N ƒ} ]^T der Vektor der Gewichte und ${\text{P}}_{ƒ}=\left[{\mathrm{\pi }}_{mn}^{ƒ}\right]{\text{eine N}}_{ƒ}\times {\text{N}}_{ƒ}$

symmetrische Matrix mit Elementen ${\mathrm{\pi }}_{mn}^{ƒ}={\displaystyle \sum {}_{m=1}^N}{\widehat{p}}_k\left({ƒ}_m\right){\widehat{p}}_k\left({ƒ}_n\right)$

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. ${\displaystyle \sum {}_n{\widehat{\text{w}}}_n}=\mathrm{1,}$

Dementsprechend nimmt Formel (7) die Form ${\widehat{\sigma }}_p\left(\widehat{\text{w}}\right)=\sqrt{N_{ƒ}{\widehat{\text{w}}}^T{P}_{ƒ}\widehat{\text{w}}-1}$

an. Um die Ungleichmäßigkeit σ̂_p(ŵ) zu minimieren, reicht es also, die quadratische Form ƒ(ŵ) = ŵ^TP_ƒŵ 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] ${\displaystyle \sum {}_{n=1}^{N_{ƒ}}}{\widehat{\text{w}}}_n=1$
    
  • [ii] ŵ_n ≥ 0, n = 1,2, ..., N_ƒ.
  • [iii] Er minimiert die Funktion ƒ(ŵ) = ŵ^TP_fŵ.

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

where ŵ = [ŵ ₁ ... ŵ _{N ƒ} ] ^{T is} the vector of the weights and ${\text{P.}}_{ƒ}=\left[{\mathrm{\pi }}_{mn}^{ƒ}\right]{\text{a}}_{ƒ}\times {\text{N}}_{ƒ}$

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

is. Since scaling the weights ŵ _n by the same factor does not cause any difference in uniformity, one can assume that the sum of the weights is normalized to 1, ie ${\displaystyle \sum {}_n{\widehat{\text{w}}}_n}=\mathrm{1,}$

Accordingly, Formula (7) takes the form ${\widehat{\sigma }}_p\left(\widehat{\text{w}}\right)=\sqrt{N_{ƒ}{\widehat{\text{w}}}^T{\mathrm{P.}}_{ƒ}\widehat{\text{w}}-1}$

on. In order to minimize the unevenness σ̂ _p (ŵ), it is sufficient to minimize the quadratic form ƒ (ŵ) = ŵ ^T P _ƒ ŵ, whereby one must note that the sum of the weights is normalized to 1 and that all weights are nonnegative (it is physically impossible to produce a negative weighting by incoherently superimposing performance patterns, since there cannot be any 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] ${\displaystyle \sum {}_{n=1}^{N_{ƒ}}}{\widehat{\text{w}}}_n=1$
    
  • [ii] ŵ _n ≥ 0, n = 1,2, ..., N _ƒ .
  • [iii] It minimizes the function ƒ (ŵ) = ŵ ^T P _f ŵ.

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

${\text{w}}_n/{\displaystyle \sum {}_n{w}_n}$

und die endgültige Gewichtung kann entweder durch Skalierung der Verweildauer der Signalquelle in der jeweiligen Frequenz, oder durch Skalierung des entsprechenden Leistungspegels der Verstärkerkanäle umgesetzt werden.These can in turn be normalized to a total of 1, ie ${\text{w}}_n^{\text{'}}$

${\text{w}}_n/{\displaystyle \sum {}_n{w}_n}$

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

Finally, method C can be summarized as follows:

• • Input: A set of performance patterns {p _k (ƒ _n ), k = 1,2, ..., N) measured or calculated at a predetermined sample set of frequencies {ƒ _n , n = 1,2, ..., N _ƒ }.
• • Output: A set of weights {w _n , n = 1,2, ..., N _ƒ }, which can be applied either as time or amplitude modulation of the dwell time of the amplifier at the respective frequency ƒ _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 ${\widehat{p}}_k\left({ƒ}_n\right)=\frac{p_k\left({ƒ}_n\right)}{{\displaystyle \sum {}_{k=1}^N{p}_k\left({ƒ}_n\right)}}\cdot$
     
  2. 2. Compute matrix ${\text{P.}}_{ƒ}=\left[{\mathrm{\pi }}_{mn}^{ƒ}\right]\text{through}{\mathrm{\pi }}_{mn}^{ƒ}={\displaystyle \sum {}_{k=1}^N{\widehat{p}}_k\left({ƒ}_m\right)}{\widehat{p}}_k\left({ƒ}_n\right)$
     
  3. 3. Minimize function ƒ (ŵ) = w ^T P _ƒ ŵ under the following conditions:
     • ■ ${\displaystyle \sum {}_{k=1}^{N_{ƒ}}{\widehat{\text{w}}}_n=1}$
       
     • ■ ŵ _n ≥ 0, n = 1,2, ..., N _ƒ
     The minimization can be carried out using methods of convex optimization, among other things.
  4. 4. Calculate the unscaled weights ${\text{w}}_n={\widehat{\text{w}}}_n/{\displaystyle \sum {}_{k=1}^N{p}_k\left({ƒ}_n\right)}\text{.}$
     
  5. 5. (Optional) Normalize the weights to 1 ${\text{w}}_n^{\text{'}}={\text{w}}_n/{\displaystyle \sum {}_{n=1}^{N_{ƒ}}{w}_n}\text{.}$
     

List of reference symbols

1

Cavity / irradiation zone

2

Antennas

3

Goods / load or objects to be heated

4th

Power amplifier with RF source / RF transmitter unit

5

Measuring device

6th

Feedback data stream

7th

Input data stream

8th

Command stream

9

Control and evaluation device

10

Feedback data stream

B1-B3

fictional areas

Claims (8)

Method for heating dielectric objects with the most uniform possible heat distribution by means of high-frequency radiation, in which - an object (3) or a group of objects is brought into an irradiation zone (1) and via an HF transmission unit (4) with N ≥ 2 separate HF transmission channels with coherent High-frequency radiation is irradiated, - before the start of the heating process, either reflection measurements or at least one measurement of scattering parameters are or will be carried out in which the high-frequency radiation backscattered from the irradiation zone (1) is detected at irradiation locations of the RF transmission channels, - Heating matrices for N different sub-areas of the object (3) or the object group are estimated from the measurement or measurements, each sub-area of which is assigned to one of the N RF transmission channels Heat distribution as evenly as possible in the object (3) or the object group, and - the HF transmission channels are then operated to heat the object (3) or the object group with the operating parameters determined from the heating matrices. Procedure according to Claim 1 , characterized in that a number of preferably N excitation vectors are determined from the heating matrices as operating parameters for the RF transmission channels, which are used to heat the object (3) or the group of objects one after the other. Procedure according to Claim 1 or 2 , characterized in that the reflection measurements or the at least one scatter parameter measurement are or will be 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 (3) or the object group once or are operated repeatedly one after the other at the different frequencies with the operating parameters determined from the heating matrices for these frequencies. Procedure according to Claim 3 , characterized in that the RF transmission channels are operated at the different frequencies for the same irradiation duration. Method according to one of the Claims 1 until 4th , characterized in that an optimization method is used to determine the operating parameters from the heating matrices in order to obtain the most uniform possible heating of the object (3) or the group of objects. Procedure according to Claim 3 , characterized in that an optimization method is used to determine the operating parameters from the heating matrices in order to obtain as uniform a heating of the object (3) or the object group as possible, with a further optimization method also including an irradiation duration and / or an irradiation power for each frequency is determined with which the RF transmission channels are operated at the respective frequency. Method according to one of the Claims 1 until 6th , characterized in that the reflection measurements or the at least one scattering parameter measurement are or will be repeated one or more times during the heating process in order to obtain updated heating matrices for the N different partial areas of the object (3) or the object group, and updated operating parameters from the updated heating matrices determined and used for further heating. Device for heating dielectric objects by means of high-frequency radiation, with - an HF transmission unit (4) with at least two HF transmission channels, via which objects (3) introduced into an irradiation zone (1) can be irradiated with coherent high-frequency radiation, - a measuring device (5), with which reflection measurements or scatter parameter measurements of the irradiation zone (1) with introduced objects (3) can be carried out at irradiation locations of the HF transmission channels, and - a control and evaluation device (9) for activating the HF transmission unit (4) and measuring device (5 ) and for the evaluation of the measurements carried out, which are necessary for the implementation of the method according to one or more of the Claims 1 until 7th is trained.

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