CN109842307B - Direct power boundary control method based on three-phase three-opening two-level rectifier - Google Patents
Direct power boundary control method based on three-phase three-opening two-level rectifier Download PDFInfo
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Abstract
A direct power boundary control method based on a three-phase three-opening two-level rectifier is characterized by firstly establishing a mathematical model of the three-phase three-opening two-level rectifier under a two-phase synchronous rotating coordinate system; establishing a power model of the rectifier under a rotating coordinate system by combining an instantaneous power theory and adopting a voltage orientation principle; establishing a standard phase plane by taking the voltage at the direct current side as the horizontal axis of the phase plane and the current at the alternating current side as the vertical axis of the phase plane, and analyzing the change of the natural track of the rectifier when the current at the alternating current side of the rectifier in the standard phase plane is reduced and increased; the method is characterized in that a novel direct power control is created by utilizing boundary control conditions on the basis of power quantity, firstly, a natural switch surface of the boundary control is selected on the basis of a natural track of a rectifying system when the alternating current side current is reduced and increased, then, the rule of the output of a power hysteresis comparator in the direct power control is updated by using the natural switch surface, and finally, the direct power boundary control method is obtained. The invention effectively improves the dynamic performance of the DC side of the three-phase three-switch two-level rectifier aiming at the problem of load sudden change in practical application.
Description
Technical Field
The invention relates to the technical field of three-phase three-opening two-level rectifier control, in particular to a direct power boundary control method based on a three-phase three-opening two-level rectifier.
Background
In recent years, due to rapid development of economy, the demand for energy is increased, low-carbon energy conservation becomes a common requirement of all countries in the world, and green and low-carbon development of electric energy has great significance for energy conservation and environmental protection. Therefore, how to obtain high-quality electric energy becomes the focus of current research, and in order to solve the problem of harmonic pollution, a PWM technology is introduced into a rectifier to generate a PWM rectifier, which has the advantages of sinusoidal network side current, theoretically realizable unit power factor, low network side current distortion rate and the like. Currently, the grid-side current control strategy of the three-phase PWM rectifier is divided into an indirect current control strategy and a direct current control strategy, wherein the latter is dominant. Hysteresis current control, feedforward decoupling PI control, predictive current control, direct power control and the like are common direct current control strategies at present.
Direct Power Control (DPC) is one of the most efficient control strategies for three-phase PWM rectifiers, and has the following advantages over other PWM rectifier control strategies: operating with unit power factor; secondly, the dynamic response is very fast; and thirdly, the structure is simple and clear. The switching state of the Direct Power Control (DPC) strategy is selected by a switching table according to the instantaneous power error and the input voltage vector position, and the direct power control is improved by using an updated switching table, adaptive control, fuzzy logic selection, sliding mode control, duty cycle optimization or prediction method, which can cope with the problem of uncertain parameters, improve the anti-interference capability, etc., but these methods have limited improvement on the dynamic performance of the dc output voltage.
Boundary control is a geometry-based control method suitable for power electronic converters with switching action, which finds application in many power electronic converters. Various studies have been made on different switching planes, such as first, second and higher order switching planes, in which the natural switching plane has good dynamic properties. However, most of the articles only study the boundary control scheme of the single-phase power electronic converter, and the boundary control of the three-phase PWM rectifier, especially the natural switch surface is rarely studied.
Disclosure of Invention
In order to improve the dynamic performance of the direct-current output voltage of the three-phase PWM rectifier, the invention provides a direct power boundary control method based on a three-phase three-throw two-level rectifier, which can effectively improve the dynamic performance of the direct-current output voltage of the three-phase PWM rectifier. The control method is based on a mathematical model under a dq coordinate system, and deduces a natural track of a rectifying system when the current on the d-axis alternating current side is lowered and raised; combining with instantaneous power theory to obtain P, Q-variable direct power control model, and outputting S to direct power hysteresis comparatorqAnd a new rule is adopted, and the analyzed natural track is used as a switch surface, so that the effect of boundary control is achieved.
The technical scheme adopted by the invention is as follows:
the direct power boundary control method based on the three-phase three-opening two-level rectifier comprises the following steps of:
step 1: analyzing the working process of the three-phase three-switch two-level rectifier, and establishing a mathematical model of the rectifier under a synchronous rotation dq coordinate system by using coordinate transformation;
step 2: combining an instantaneous power theory, converting a mathematical model of the rectifier under the synchronous rotation dq coordinate system into a power model with P and Q as variables under the dq coordinate system;
and step 3: analyzing the boundary control conditions of the three-phase three-switch two-level rectifier, namely establishing a standard phase plane by taking the voltage at the direct current side as the horizontal axis of the phase plane and the current at the alternating current side as the vertical axis of the phase plane; in a standard phase plane, the rectifier has different natural tracks in different states, and the natural tracks of the rectifier when the current on the alternating current side is reduced and increased are analyzed;
and 4, step 4: based on the power quantity, firstly, a natural switch surface of boundary control is selected according to a natural track of the rectifier when the alternating current side current is reduced and increased, then the rule of the output of a power hysteresis comparator in direct power control is updated by using the natural switch surface, and finally the direct power boundary control method based on the three-phase three-switch two-level rectifier is obtained.
In step 2, obtaining a calculation formula of instantaneous active power and instantaneous reactive power based on a dq coordinate system according to an instantaneous power theory; and (3) adopting the voltage orientation of the power grid, selecting the initial phase angle of the d axis to be equal to the initial phase angle of the a phase, and substituting the calculation formula into the mathematical model under the synchronous rotation dq coordinate system established in the step 1 to obtain a power model with P and Q as variables under the dq coordinate system.
And 3, establishing a phase plane by taking the voltage at the direct current side as the horizontal axis of the phase plane and the current at the alternating current side as the vertical axis of the phase plane, and obtaining the natural tracks of the rectifier when the current at the alternating current side is reduced and when the current is increased in the standardized phase plane through simple standardized calculation.
In step 4, in the direct power boundary control method, the reactive power hysteresis comparator outputs SQNumerical valueThe rule and the switching vector table are the same as the traditional direct power control method, and only the output S of the direct active power hysteresis comparatorPThe rule of the values is different from the conventional direct power control method.
The invention discloses a direct power boundary control method based on a three-phase three-opening two-level rectifier, which has the following technical effects:
1: direct power control is created by using boundary control conditions based on power quantities, using a new rule to control the power hysteresis comparator output SpThe numerical value of (c).
2: aiming at the problem of load sudden change in practical application, the dynamic performance of the direct-current side voltage of the three-phase three-throw two-level rectifier can be effectively improved.
Drawings
Fig. 1 is a topology structure diagram of a three-phase three-switch two-level rectifier.
Fig. 2 is a natural track diagram of a rectification system when the current on the alternating current side of the three-phase three-switch two-level rectifier is reduced.
Fig. 3 is a natural track diagram of a rectifying system when the current on the alternating current side of the three-phase three-switch two-level rectifier rises.
Fig. 4 is a boundary condition diagram for direct power control of a three-phase three-switch two-level rectifier.
Fig. 5 is a waveform diagram of the a-phase voltage and current on the ac side under rated load of the rectifier.
Fig. 6 is a waveform diagram of the a-phase current on the ac side when the rectifier suddenly loads 1 time from the rated load.
Fig. 7 is a diagram showing waveforms of voltage and current on the dc side when the rectifier is loaded 1 times from the rated load.
Detailed Description
Step 1: analyzing the working process of the three-phase three-switch two-level rectifier, and establishing a mathematical model of the rectifier under a synchronous rotation dq coordinate system by using coordinate transformation;
step 2: combining an instantaneous power theory, converting a mathematical model of the rectifier under the synchronous rotation dq coordinate system into a power model with P and Q as variables under the dq coordinate system;
and step 3: analyzing the boundary control conditions of the three-phase three-switch two-level rectifier, namely establishing a standard phase plane by taking the voltage at the direct current side as the horizontal axis of the phase plane and the current at the alternating current side as the vertical axis of the phase plane; in a standard phase plane, the rectifier has different natural tracks in different states, and the natural tracks of the rectifier when the current on the alternating current side is reduced and increased are analyzed;
and 4, step 4: based on the power quantity, firstly, a natural switch surface of boundary control is selected according to a natural track of the rectifier when the alternating current side current is reduced and increased, then the rule of the output of a power hysteresis comparator in direct power control is updated by using the natural switch surface, and finally the direct power boundary control method based on the three-phase three-switch two-level rectifier is obtained.
In step 2, obtaining a calculation formula of instantaneous active power and instantaneous reactive power based on a dq coordinate system according to an instantaneous power theory; and (3) adopting the voltage orientation of the power grid, selecting the initial phase angle of the d axis to be equal to the initial phase angle of the a phase, and substituting the calculation formula into the mathematical model under the synchronous rotation dq coordinate system established in the step 1 to obtain a power model with P and Q as variables under the dq coordinate system.
And 3, establishing a phase plane by taking the voltage at the direct current side as the horizontal axis of the phase plane and the current at the alternating current side as the vertical axis of the phase plane, and obtaining the natural tracks of the rectifier when the current at the alternating current side is reduced and when the current is increased in the standardized phase plane through simple standardized calculation.
In step 4, in the direct power boundary control method, the reactive power hysteresis comparator outputs SQThe rule of the numerical value and the switching vector table are the same as the traditional direct power control method, and only the direct active power hysteresis comparator outputsGoes out of SPThe rule of the values is different from the conventional direct power control method.
The method comprises the following specific steps:
step 1: as can be seen from the analysis of the topology structure of the three-phase three-switch two-level rectifier shown in fig. 1, the three-phase three-switch two-level rectifier is a PWM rectifier, the rectifying circuit includes three switching tubes, and the output voltage at the dc side is only vdcAnd-vdcTwo types of three-phase three-switch two-level rectifiers have the characteristics of simplicity, strong robustness, easy acquisition of power modules and auxiliary devices and the like.
ea、eb、ecThe three-phase static coordinate system is the power grid electromotive force; i.e. ia、ib、icThree-phase current at the alternating current side under a three-phase static coordinate system; l isA、LB、LCIs an AC side inductor, LA＝LB＝LCL; r is an equivalent resistance of an alternating current side circuit; c is a direct current side capacitor; rLIs a direct current side load; v. ofdcIs a direct current side voltage; e.g. of the typed、eq、id、iqThe voltage and current of the AC side under the two-phase synchronous rotating coordinate system.
In order to establish a general mathematical model of a three-phase three-opening two-level rectifier, the following assumptions are made: 1) the electromotive force of the power grid is an ideal three-phase sine wave; 2) the alternating current filter inductor is linear and is not saturated; 3) the power switch tube ignores dead time and is an ideal switch.
Defining a switching function:
a. b and c represent a three-phase stationary coordinate system.
Establishing a general mathematical model of a three-phase three-switch two-level rectifier:
Sa、Sb、Screpresenting abc three-phase switching functions, respectively.
In addition, kirchhoff's current law is applied to the positive node of the direct current measurement capacitor, and the following results are obtained:
c represents a DC capacitor, RLRepresenting a dc load.
And (3) adopting coordinate transformation to the equations (2) and (3), transforming the three-phase stationary coordinate system (a, b, c) to the two-phase synchronous rotating coordinate system (d, q), and obtaining a mathematical model of the three-phase three-switch two-level rectifier as follows:
in the formula: w is angular velocity, Sd、SqIs a switching function transformed into dq coordinate system.
Step 2: and (3) balancing the three-phase power grid, and obtaining the calculation formula of instantaneous active power and instantaneous reactive power under the dq coordinate system by combining an instantaneous power theory:
P＝edid+eqiq，Q＝eqid-ediq (5)
by using grid voltage orientation, e is obtainedqWhen the above formula is substituted by formula (5), 0 can be obtained:
P＝edid，Q＝-ediq (6)
multiplying both sides of the formula (4) by edAnd obtaining a power model with P and Q as variables under the dq coordinate system as follows:
and step 3: firstly, two axes of a phase plane are selected, and according to the existence and uniqueness theory of differential equation solutions, under the condition that any initial data condition is selected, a phase track is required to correspond to the phase plane on the phase plane. For a rectifying system, any of its states corresponds to a point in the phase plane, so that system motion can be represented in the phase plane. The movement of points on the phase plane corresponds to the change of the system state with time, and these movement trajectories are the phase trajectories. The switch surface used by the boundary control of the invention is a natural switch surface, so that the alternating current side current is selected as the longitudinal axis of the phase plane, the direct current side voltage is selected as the transverse axis of the phase plane, the phase plane is established, and the natural track is selected as the phase track.
To obtain the specific power factor, i is controlledqWhen the inductance equivalent resistance R is negligibly small, i.e., R is 0, equation (4) can be simplified as:
v in formula (8)Sd＝vdcSd，
According to the operating principle of three-phase three-switch two-level rectifiers vSdThe maximum value of (d) is:
when the AC side current decreases, vSdShould be positive, i.e.
Order to
Substituting the compound in the formula (8) to obtain:
by using the following trigonometric identity:
equation (10) can be transformed into the following form:
wherein k is and ide、vdcInitial value-dependent constants.
Let vn＝vdc，
edn＝ed，
And substituting the natural path lambda of the rectifying system when the alternating current side current is reduced in an expression (11)downComprises the following steps:
in the normal phase plane, λ is shown in FIG. 2downIs one to
A circle with a radius of l as the center of the circle.
When the AC side current rises, vSdShould be negative, i.e.
The natural track lambda of the rectifying system is obtained when the current of the alternating current side rises by adopting the same derivation method when the current of the alternating current side fallsupComprises the following steps:
in the normal phase plane, λ, as shown in FIG. 3upIs one to
A circle with m as the radius as the center of the circle.
And 4, step 4: since only the active power has an effect on the DC output voltage, and inCan be represented by a change in the instantaneous active power P, and a change in the state of the switch will cause a change in P. Therefore, the system operation track can be controlled by the method.
In order to obtain the direct power boundary control rule of the three-phase three-switch two-level rectifier, the following definitions are firstly made:
i in formulae (14) and (15)n_T、vn_TAre the current and voltage at the operating target point on the phase plane.
By natural trajectory λ when the alternating side current is reduceddownAnd a natural locus lambda of when the alternating-current side current risesupThe selected boundary condition natural switch surfaces are:
RT_downindicating a falling boundary resistance
ednRepresenting the voltage to ground on the DC side, RT_upRepresenting the rising boundary resistance.
The α β plane is divided into 12 vector sectors, each of which is 30 degrees in the α β plane, and the phase angle range thereof can be represented by equation (18).
When the space where the grid voltage vector is located needs to be determined, the phase angle of the grid voltage vector E is calculated firstly
eα、eβThe interval of the grid voltage vector E is determined by using an equation (18) for the alternating-current side voltage in the alpha beta coordinate system.
Reactive power hysteresis comparator output S in direct power boundary control algorithm of three-phase three-throw two-level rectifierQThe rules for the values are the same as the conventional direct power control rules:
in the formula (19), q is an instantaneous reactive power estimated value, qrFor instantaneous reactive power reference value, HqThe hysteresis width of the reactive power hysteresis comparator. Generally, the hysteresis width of the hysteresis comparator is affected by parameters in the main circuit, such as the inductance L on the ac side and the voltage v on the dc sidedc. If the hysteresis width is too small, the switching frequency is too high, and as a result, the loss of the system switch is increased, and the aging of the switch is accelerated; if the hysteresis width is too large, the power tracking is too slow to satisfy real-time control. Therefore, when designing a system, the size of the hysteresis loop width should be selected to meet the system requirement and meet the practical requirement.
Active power hysteresis in direct power boundary control algorithm of three-phase three-switch two-level rectifierOutput S of loop comparatorPThe value adopts a new rule:
i: if v isdc<vn_TOnly when σdown<At 0, S P1, otherwise SP＝0；
II: if v isdc>vn_TOnly when σup>At 0, S P0, otherwise SP＝1。
Firstly, a sector where a power grid voltage vector is located is judged by adopting a formula (18), and then S is output according to a power hysteresis comparatorP、SQThe sector of the grid voltage vector and the S outputP、SQAnd sending the switching vector into a switching table to select the switching vector.
The switch table used in the direct power boundary control algorithm of the three-phase three-switch two-level rectifier is the same as the switch table used in the traditional direct power control:
TABLE 1 direct power boundary control algorithm switching table for three-phase three-switch two-level rectifier
Setting line parameters of a three-phase three-switch two-level rectifier: the effective value of the three-phase circuit is 220V/50Hz, and the input inductance L at the alternating current sideA＝LB＝LCThe line equivalent resistance R is 0.1 Ω, the dc-side filter capacitance C is 3300 μ F, the load R is 30 Ω, the switching frequency is 20kHz, and at 0.15s, the load is doubled on the dc side.
When the load is rated, the waveform diagram of the A-phase voltage current at the AC side of the three-phase three-switch two-level rectifier is shown in figure 5, and the current at the network side of the three-phase three-switch two-level rectifier adopting a direct power boundary control algorithm can be obtained from the waveform diagram, so that the sine is realized, the voltage current is basically in the same phase, and the power factor is close to 1.
When the load is suddenly changed to 2 times of load from the rated load, the A-phase current waveform diagram of the AC side of the three-phase three-switch two-level rectifier is shown in fig. 6, and the current of the network side of the three-phase three-switch two-level rectifier adopting the direct power boundary control algorithm can quickly reach a stable state when the load fluctuates.
When the load suddenly changes from the rated load to 2 times of the load, the voltage and current waveform diagram of the direct current side of the three-phase three-switch two-level rectifier is shown in fig. 7, and the direct current side voltage and current of the three-phase three-switch two-level rectifier adopting the direct power boundary control algorithm can be obtained from the diagram and reach the stable state again in a short time when the load fluctuates, so that the three-phase three-switch two-level rectifier has good dynamic performance.
From the analysis of the oscillogram, the direct power boundary control algorithm based on the three-phase three-switch two-level rectifier can effectively improve the dynamic performance of the direct current output current and the direct current output voltage of the three-phase three-switch two-level rectifier, achieves the expected experimental result and has certain practical value.
Claims (7)
1. The direct power boundary control method based on the three-phase three-opening two-level rectifier is characterized by comprising the following steps of:
step 1: analyzing the working process of the three-phase three-switch two-level rectifier, and establishing a mathematical model of the rectifier under a synchronous rotation dq coordinate system by using coordinate transformation;
step 2: combining an instantaneous power theory, converting a mathematical model of the rectifier under the synchronous rotation dq coordinate system into a power model with P and Q as variables under the dq coordinate system;
and step 3: analyzing the boundary control conditions of the three-phase three-switch two-level rectifier, namely establishing a standard phase plane by taking the voltage at the direct current side as the horizontal axis of the phase plane and the current at the alternating current side as the vertical axis of the phase plane; in a standard phase plane, the rectifier has different natural tracks in different states, and the natural tracks of the rectifier when the current on the alternating current side is reduced and increased are analyzed;
and 4, step 4: on the basis of power quantity, firstly, selecting a natural switch surface of boundary control according to a natural track of a rectifier when the alternating current side current is reduced and increased, then updating a rule output by a power hysteresis comparator in direct power control by using the natural switch surface, and finally obtaining a direct power boundary control method based on a three-phase three-switch two-level rectifier;
output S of active power hysteresis comparatorpThe values are then scaled as follows:
i: if v isdc＜vn_TOnly when σdownWhen < 0, Sp1, otherwise Sp＝0；
II: if v isdc>vn_TOnly when σup>At 0, Sp0, otherwise Sp＝1；
The direct power boundary control method based on the three-phase three-switch two-level rectifier comprises the following steps:
the following definitions are firstly made:
i in formulae (14) and (15)n_T、vn_TIs the current and voltage of the operating target point on the phase plane;
by natural trajectory λ when the alternating side current is reduceddownAnd a natural locus lambda of when the alternating-current side current risesupThe selected boundary condition natural switch surfaces are:
dividing an α β plane into 12 vector sectors, each sector being 30 degrees in the α β plane, and a phase angle range thereof can be represented by equation (18);
when the space where the grid voltage vector is located needs to be determined, the phase angle of the grid voltage vector E is calculated firstly
eα、eβDetermining the interval of a power grid voltage vector E by using an equation (18) for the alternating-current side voltage under an alpha beta coordinate system;
reactive power hysteresis comparator output S in direct power boundary control algorithm of three-phase three-throw two-level rectifierqThe rules for the values are:
in the formula (19), q is an instantaneous reactive power estimated value, qrFor instantaneous reactive power reference value, HqThe hysteresis width of the reactive power hysteresis comparator.
2. The direct power boundary control method based on the three-phase three-open two-level rectifier according to claim 1, wherein: step 1, analyzing the working process of the three-phase three-switch two-level rectifier, defining a switch function, and establishing a mathematical model of the three-phase three-switch two-level rectifier under a three-phase static coordinate system; and introducing coordinate transformation into a modeling process of the system, and obtaining a mathematical model of the three-phase three-switch two-level rectifier under a synchronous rotation dq coordinate system by utilizing the coordinate transformation.
3. The direct power boundary control method based on the three-phase three-open two-level rectifier according to claim 1, wherein: in step 2, obtaining a calculation formula of instantaneous active power and instantaneous reactive power based on a dq coordinate system according to an instantaneous power theory; and (3) adopting the voltage orientation of the power grid, selecting the initial phase angle of the d axis to be equal to the initial phase angle of the a phase, and substituting the calculation formula into the mathematical model under the synchronous rotation dq coordinate system established in the step 1 to obtain a power model with P and Q as variables under the dq coordinate system.
4. The direct power boundary control method based on the three-phase three-open two-level rectifier according to claim 1, wherein: and 3, establishing a phase plane by taking the voltage at the direct current side as the horizontal axis of the phase plane and the current at the alternating current side as the vertical axis of the phase plane, and obtaining the natural tracks of the rectifier when the current at the alternating current side is reduced and when the current is increased in the standardized phase plane through standardized calculation.
5. The direct power boundary control method based on the three-phase three-open two-level rectifier according to claim 2, wherein: the step 1 comprises the following steps:
ea、eb、ecthe three-phase static coordinate system is the power grid electromotive force; i.e. ia、ib、icThree-phase current at the alternating current side under a three-phase static coordinate system; l isA、LB、LCIs an AC side inductor, LA＝LB＝LCL; r is an equivalent resistance of an alternating current side circuit; c is a direct current side capacitor; rLIs a direct current side load; v. ofdcIs a direct current side voltage; e.g. of the typed、eq、id、iqThe voltage and the current of the alternating current side under the two-phase synchronous rotating coordinate system;
in order to establish a general mathematical model of a three-phase three-opening two-level rectifier, the following assumptions are made: 1) the electromotive force of the power grid is an ideal three-phase sine wave; 2) the alternating current filter inductor is linear and is not saturated; 3) the power switch tubes ignore dead time and are ideal switches;
defining a switching function:
a. b and c represent a three-phase static coordinate system;
establishing a general mathematical model of a three-phase three-switch two-level rectifier:
in addition, kirchhoff's current law is applied to the positive node of the direct current measurement capacitor, and the following results are obtained:
and (3) adopting coordinate transformation to the equations (2) and (3), transforming the three-phase stationary coordinate system (a, b, c) to the two-phase synchronous rotating coordinate system (d, q), and obtaining a mathematical model of the three-phase three-switch two-level rectifier as follows:
in the formula: w is angular velocity, Sd、SqIs a switching function transformed into dq coordinate system.
6. The direct power boundary control method based on the three-phase three-open two-level rectifier according to claim 2, wherein: the step 2 comprises the following steps:
by combining the instantaneous power theory, the calculation formula of the instantaneous active power and the instantaneous reactive power under the dq coordinate system is obtained as follows:
by using grid voltage orientation, e is obtainedqWhen the above formula is substituted by formula (5), 0 can be obtained:
P＝edid，Q＝-ediq (6)
multiplying both sides of the formula (4) by edAnd obtaining a power model with P and Q as variables under the dq coordinate system as follows:
7. the direct power boundary control method based on the three-phase three-open two-level rectifier according to claim 3, wherein: the step 3 comprises the following steps: the boundary control uses a natural switch surface, so that alternating current side current is selected as a longitudinal axis of a phase plane, direct current side voltage is selected as a transverse axis of the phase plane, the phase plane is established, and a natural track is selected as a phase track;
to obtain the specific power factor, i is controlledqWhen the inductance equivalent resistance R is negligibly small, i.e., R is 0, equation (4) can be simplified as:
in the formula (8), vSd＝vdcSd，
According to the operating principle of three-phase three-switch two-level rectifiers vSdThe maximum value of (d) is:
when the AC side current decreases, vSdIs positive, i.e.
Order to
Substituting the compound in the formula (8) to obtain:
by using the following trigonometric identity:
equation (10) can be transformed into the following form:
wherein k is and ide、vdcAn initial value-dependent constant;
let vn＝vdc，
edn＝ed，
And substituting the natural path lambda of the rectifying system when the alternating current side current is reduced in an expression (11)downComprises the following steps:
in the normal phase plane, λdownIs one to
A circle with a radius of l as the center of the circle;
when the current rises, vSdShould be negative, i.e.
The natural track lambda of the rectifying system is obtained when the current on the alternating current side rises by adopting the same derivation method when the current is reducedupComprises the following steps:
in the normal phase plane, λupIs one to
A circle with m as the radius as the center of the circle.
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