CN109507482B - Three-phase alternating current circuit impedance test system and method - Google Patents
Three-phase alternating current circuit impedance test system and method Download PDFInfo
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Abstract
The invention discloses a three-phase alternating current circuit impedance test system and a method, wherein the system comprises a disturbing signal injection module, a signal processing module, a signal amplification module, a signal detection module, a frequency response measurement module and an impedance calculation module; the method comprises the following steps: firstly, generating a disturbance signal; secondly, converting the disturbance signal in the dq coordinate system into an abc three-phase alternating current signal; thirdly, amplifying the disturbance signal and injecting the amplified disturbance signal into a three-phase alternating current circuit; fourthly, the signal detection module measures a power supply voltage signal and a port response signal and carries out filtering conditioning on the signals; fifthly, converting the port response signal from the abc coordinate system to the dq coordinate system; sixthly, obtaining frequency response; and seventhly, calculating to obtain the impedance of the three-phase alternating current circuit. The invention has novel and reasonable design, can conveniently and accurately test the three-phase alternating-current impedance, has important significance for researching the stability analysis of the three-phase alternating-current circuit, and has strong practicability and high popularization and application value.
Description
Technical Field
The invention belongs to the technical field of impedance testing, and particularly relates to a three-phase alternating-current circuit impedance testing system and method.
Background
The distributed power supply system has the advantages of high reliability, easy modularization realization, convenient maintenance and the like, is more and more popular in the industry, and is widely applied to power supply systems of spacecrafts, ships, communication base stations and large computers. In a distributed power supply system, each device is independently designed by different suppliers, the stable operation of the device is only ensured during the design, and when the devices are cascaded into a system, the performance of the system is reduced and even the system is unstable due to the negative input impedance characteristic of a constant power load.
The impedance stability criterion is widely applied to direct current systems. When a space station power supply system is researched, professor Fred c.lee does a great deal of research work aiming at the impedance stability analysis of a direct current system, and provides a method for measuring the impedance of the direct current system, wherein the direct current system has a determined direct current working point, the port impedance can be directly measured, and the port characteristics of the direct current system can be obtained by using commercial test equipment, such as a network analyzer of agilent corporation. For an alternating current system, voltage and current signals of an alternating current port are time-varying, so that a direct current working point does not exist, coordinate transformation is needed, a three-phase static coordinate system is transformed into a two-phase rotating coordinate system, and input and output impedance of the alternating current system needs to be measured under a dq coordinate system. The coordinate transformation needs to use the rotation angle of the alternating current system, the rotation angle is generally obtained through a phase-locked loop, but the disturbance signal influences the precision of the phase-locked loop and further influences the measurement result. On the other hand, since the dq coordinate system is a virtual coordinate system, the three-phase ac circuit cannot directly perform signal analysis and subsequent testing under the coordinate system.
The method is used for researching the stability of the three-phase alternating current circuit and the measurement of the impedance, so that the research of the testing principle of the impedance of the three-phase alternating current circuit is of great significance for researching the stability analysis of the three-phase alternating current circuit.
Disclosure of Invention
The technical problem to be solved by the present invention is to provide a three-phase ac circuit impedance testing system, which has a simple structure, a novel and reasonable design, and a convenient implementation, and can conveniently and accurately test three-phase ac impedance, in order to overcome the defects in the prior art.
In order to solve the technical problems, the invention adopts the technical scheme that: a three-phase alternating current circuit impedance test system comprises a disturbing signal injection module, a signal processing module and a signal amplification module which are sequentially connected, a signal detection module connected with the input end of the signal processing module, a frequency response measurement module connected with the output end of the signal processing module and an impedance calculation module connected with the frequency response measurement module; the disturbance signal injection module is used for generating a disturbance signal under a dq coordinate system for injecting a three-phase alternating current circuit, and transmitting the disturbance signal to the frequency response measurement module and the signal processing module; the signal processing module is used for converting a disturbance signal output by the disturbance signal injection module under a dq coordinate system into an abc three-phase alternating current signal and outputting the abc three-phase alternating current signal to the signal amplification module, processing a power supply voltage signal measured by the signal detection module through a phase-locked loop to obtain a value of a phase angle theta used for dq conversion, and converting a port response signal measured by the signal detection module from the abc coordinate system to the dq coordinate system by adopting dq conversion; the signal amplification module is used for amplifying the disturbance signal output by the signal processing module and injecting the amplified disturbance signal into the three-phase alternating current circuit; the signal detection module is used for measuring a power supply voltage signal and a port response signal in an abc coordinate system and filtering and conditioning the signal; the frequency response measurement module is used for obtaining corresponding frequency response through the disturbance signal and the port response signal in the dq coordinate system and outputting the corresponding frequency response to the impedance calculation module; the impedance calculation module is used for calculating and obtaining the impedance of the three-phase alternating current circuit.
In the impedance testing system for the three-phase alternating-current circuit, the disturbing signal injection module and the frequency response measuring module are composed of a network analyzer.
According to the three-phase alternating current circuit impedance testing system, the signal processing module comprises the FPGA board card.
In the three-phase alternating current circuit impedance testing system, the FPGA board card is an NI USB-7845R multifunctional data acquisition card.
In the aforementioned three-phase ac circuit impedance testing system, the signal amplification module includes a power amplifier.
In the above three-phase ac circuit impedance testing system, the impedance calculating module includes a computer.
The invention also provides a three-phase alternating current circuit impedance test method which aims at the condition that signals under dq axes are difficult to process and analyze in the three-phase alternating current circuit impedance test, provides an alternating current system impedance test scheme based on the FPGA, can effectively process test signals and analyze final test results, and can conveniently and accurately test three-phase alternating current impedance, and the method comprises the following steps:
step one, a disturbance signal is generated by a disturbance signal injection module and transmitted to a frequency response measurement module and a signal processing module;
secondly, the signal processing module calls a coordinate transformation module from a dq coordinate system to an abc coordinate system, converts a disturbance signal in the dq coordinate system generated by the disturbance signal injection module into an abc three-phase alternating current signal and outputs the three-phase alternating current signal to the signal amplification module;
thirdly, amplifying the disturbance signal output by the signal processing module by the signal amplification module and injecting the amplified disturbance signal into the three-phase alternating current circuit;
measuring a power supply voltage signal and a port response signal by the signal detection module, filtering and conditioning the signal, and outputting the signal to the signal processing module;
processing the power supply voltage signal by the signal processing module through a phase-locked loop to obtain a phase angle theta value used for dq conversion, and converting the port response signal from an abc coordinate system to a dq coordinate system by adopting the dq conversion;
step six, the frequency response measurement module obtains corresponding frequency response through the disturbance signal and the port response signal under the dq axis and outputs the corresponding frequency response to the impedance calculation module;
and seventhly, generating two different disturbance signals by the disturbance signal injection module, repeatedly executing the step one to the step six for eight times, thereby obtaining port current and voltage data groups corresponding to the two different disturbance signals, and calculating the impedance of the three-phase alternating current circuit by the impedance calculation module according to the port current and voltage data groups corresponding to the two different disturbance signals.
In the method, the disturbance signal injection module and the frequency response measurement module are formed by a network analyzer, the signal processing module comprises an FPGA board, the signal amplification module comprises a power amplifier, and the impedance calculation module comprises a computer.
In the method, in the sixth step, the network analyzer obtains corresponding frequency response through the disturbance signal and the port response signal under the dq axis and stores the frequency response as an Excel file, and in the seventh step, when the computer calculates the impedance of the three-phase alternating current circuit according to the port current and voltage data groups corresponding to two different disturbance signals, the Excel file is firstly imported into MATLAB software, then the frd function is used for creating the frequency response data into a frequency response data model, then the frequency response data model is fitted into a state space model through the fitfrd function, and then the impedance of the three-phase alternating current circuit is calculated according to the state space model.
In the above method, the expression of the state space model is:
wherein, Ud1Representing the component on the d-axis, U, of the first port voltage response to the first perturbation signalq1Representing the component of the first port voltage response on the q-axis, U, corresponding to the first perturbation signald2A second port voltage response corresponding to the second disturbance signal is representedComponent on d-axis, Uq2Representing a component of the second port voltage response on the q-axis, I, corresponding to the second disturbance signald1Representing the component on the d-axis of the first port current response to the first perturbation signal, Iq1Representing the component of the first port current response in the q-axis, I, to the first disturbance signald2Representing the component of the second port current response to the second disturbance signal on the d-axis, Iq2Representing a component of a second port current response corresponding to the second perturbation signal on the q-axis; zddRepresenting the impedance of the d-axis, ZdqRepresenting the coupling impedance of the d-axis and the q-axis, ZqdRepresenting the coupling impedance of the q-axis with the d-axis, ZqqRepresenting the impedance of the q-axis;
obtaining a solution for the impedance of the three-phase alternating current circuit (11) according to equation (F1) as:
compared with the prior art, the invention has the following advantages:
1. the three-phase alternating current circuit impedance testing system adopts a modular design, and has the advantages of simple structure, novel and reasonable design and convenient realization.
2. The three-phase alternating current circuit impedance test system provided by the invention adopts the network analyzer to realize the disturbing signal injection module, can inject the sweep sine wave, has variable frequency range and amplitude and has a signal isolation effect.
3. The three-phase alternating current circuit impedance test system adopts the network analyzer to realize the disturbing signal injection module and the frequency response measurement module, the network analyzer has complete functions, and the system structure is simplified.
4. According to the three-phase alternating current circuit impedance testing system, the FPGA board card is adopted to realize the signal processing module, the FPGA board card is provided with the visual interface, the gain control item of the signal is arranged on the visual interface, the amplitude of the disturbance signal can be adjusted according to the requirement, the mutual conversion of the dq coordinate system signal and the abc three-phase alternating current signal can be realized, the parameter of the phase-locked loop can be adjusted on line, and the functions of monitoring the change of the signal in real time, alarming when the signal is out of limit and the like can be realized.
5. The three-phase alternating current circuit impedance test method provided by the invention provides an alternating current system impedance test scheme based on the FPGA aiming at the condition that signals under dq axes are difficult to process and analyze in the three-phase alternating current circuit impedance test, can effectively process test signals and analyze final test results, and can conveniently and accurately test the three-phase alternating current impedance.
6. The method has important significance for researching the stability analysis of the three-phase alternating current circuit, and is high in practicability and popularization and application value.
In conclusion, the three-phase alternating-current impedance testing device is novel and reasonable in design, can conveniently and accurately test the three-phase alternating-current impedance, has important significance for researching the stability analysis of the three-phase alternating-current circuit, and is high in practicability and popularization and application value.
The technical solution of the present invention is further described in detail by the accompanying drawings and embodiments.
Drawings
Fig. 1 is a schematic circuit block diagram of a three-phase ac circuit impedance testing system according to the present invention.
Fig. 2 is a schematic diagram of a circuit connection relationship of a three-phase ac circuit impedance testing system according to an embodiment of the present invention.
Fig. 3 is a schematic diagram of a visual interface of the FPGA board of the present invention.
FIG. 4 is a block diagram of a method for testing impedance of a three-phase AC circuit according to the present invention.
FIG. 5A shows the impedance Z measured by the method of the present inventionddIs compared with the waveform of the theoretical value.
FIG. 5B shows the impedance Z measured by the method of the present inventiondqIs compared with the waveform of the theoretical value.
FIG. 5C shows the impedance Z obtained by the method of the present inventionqdIs compared with the waveform of the theoretical value.
FIG. 5D is a graph showing the impedance Z obtained by the method of the present inventionqqIs compared with the waveform of the theoretical value.
Description of reference numerals:
1-a disturbing signal injection module; 2-a signal processing module; 3-a signal amplification module;
4-a signal detection module; 5-a frequency response measurement module; 6-an impedance calculation module;
7-a network analyzer; 8-FPGA board card; 9-a power amplifier;
10-a computer; 11-three-phase alternating current circuit.
Detailed Description
As shown in fig. 1, the three-phase ac circuit impedance testing system of the present invention includes a disturbing signal injection module 1, a signal processing module 2, and a signal amplification module 3, which are connected in sequence, and a signal detection module 4 connected to an input end of the signal processing module 2, a frequency response measurement module 5 connected to an output end of the signal processing module 2, and an impedance calculation module 6 connected to the frequency response measurement module 5; the disturbance signal injection module 1 is used for generating a disturbance signal under a dq coordinate system for injecting the three-phase alternating current circuit 11, and transmitting the disturbance signal to the frequency response measurement module 5 and the signal processing module 2; the signal processing module 2 is configured to convert a disturbance signal in a dq coordinate system output by the disturbance signal injection module 1 into an abc three-phase alternating-current signal and output the abc three-phase alternating-current signal to the signal amplification module 3, process a power supply voltage signal measured by the signal detection module 4 through a phase-locked loop to obtain a value of a phase angle θ used for dq conversion, and convert a port response signal measured by the signal detection module 4 from the abc coordinate system to the dq coordinate system by using dq conversion; the signal amplification module 3 is used for amplifying the disturbance signal output by the signal processing module 2 and injecting the amplified disturbance signal into the three-phase alternating current circuit 11; the signal detection module 4 is used for measuring a power supply voltage signal and a port response signal in an abc coordinate system, and filtering and conditioning the signal; the frequency response measuring module 5 is used for obtaining a corresponding frequency response through the disturbance signal and the port response signal in the dq coordinate system and outputting the corresponding frequency response to the impedance calculating module 6; the impedance calculation module 6 is used for calculating the impedance of the three-phase alternating current circuit 11.
In this embodiment, as shown in fig. 2, the disturbing signal injecting module 1 and the frequency response measuring module 5 are composed of a network analyzer 7.
In specific implementation, the disturbance signal generated by the disturbance signal injection module 1 and used for injecting into the dq coordinate system of the three-phase ac circuit 11 is a voltage signal, and the port response signal measured by the signal detection module 4 is a current response signal.
The adoption of the network analyzer 7 can inject the sweep sine wave, the frequency range and the amplitude of the sweep sine wave are variable, and the sweep sine wave has a signal isolation effect.
In this embodiment, as shown in fig. 2, the signal processing module 2 includes an FPGA board card 8. Preferably, the FPGA board 8 is an NI USB-7845R multifunctional data acquisition card. The FPGA board card is provided with a visual interface, as shown in FIG. 3, a gain control item of a signal is arranged on the visual interface, the amplitude of a disturbance signal can be adjusted according to requirements, the mutual conversion of a dq coordinate system signal and an abc three-phase alternating current signal can be realized, the parameter of a phase-locked loop can be adjusted on line, and the functions of monitoring the change of the signal in real time, alarming when the signal is out of limit and the like can be realized.
In this embodiment, as shown in fig. 2, the signal amplification module 3 includes a power amplifier 9.
In this embodiment, as shown in fig. 2, the impedance calculating module 6 includes a computer 10.
As shown in fig. 4, the impedance testing method for the three-phase ac circuit 11 of the present invention includes the following steps:
step one, a disturbance signal is generated by a disturbance signal injection module 1 and transmitted to a frequency response measurement module 5 and a signal processing module 2;
step two, the signal processing module 2 calls a coordinate transformation module from a dq coordinate system to an abc coordinate system, converts a disturbance signal in the dq coordinate system generated by the disturbance signal injection module 1 into an abc three-phase alternating current signal, and outputs the three-phase alternating current signal to the signal amplification module 3;
thirdly, the signal amplification module 3 amplifies the disturbance signal output by the signal processing module 2 and injects the amplified disturbance signal into the three-phase alternating current circuit 11;
step four, the signal detection module 4 measures a power supply voltage signal and a port response signal, performs filtering conditioning on the signal and outputs the signal to the signal processing module 2;
step five, the signal processing module 2 processes the power supply voltage signal through a phase-locked loop to obtain a phase angle theta value used for dq transformation, and then port response signals are transformed from an abc coordinate system to a dq coordinate system through the dq transformation;
step six, the frequency response measuring module 5 obtains corresponding frequency response through the disturbance signal and the port response signal under the dq axis and outputs the corresponding frequency response to the impedance calculating module 6;
and seventhly, generating two different disturbance signals by the disturbance signal injection module 1, and repeatedly executing the step from one to six for eight times to obtain port current and voltage data groups corresponding to the two different disturbance signals, wherein the impedance calculation module 6 calculates the impedance of the three-phase alternating current circuit 11 according to the port current and voltage data groups corresponding to the two different disturbance signals.
In this embodiment, the disturbing signal injection module 1 and the frequency response measurement module 5 are composed of a network analyzer 7, the signal processing module 2 includes an FPGA board 8, the signal amplification module 3 includes a power amplifier 9, and the impedance calculation module 6 includes a computer 10.
In this embodiment, in the sixth step, the network analyzer 7 obtains a corresponding frequency response through the disturbance signal and the port response signal under the dq axis and stores the frequency response as an Excel file, and in the seventh step, when the computer 10 calculates the impedance of the three-phase ac circuit 11 according to the port current and voltage data sets corresponding to two different disturbance signals, the Excel file is first imported into MATLAB software, then the frd function is used to create the frequency response data into a frequency response data model, then the frequency response data model is fitted into a state space model through the fitfrd function, and then the impedance of the three-phase ac circuit 11 is calculated according to the state space model.
In this embodiment, the expression of the state space model is as follows:
wherein, Ud1Representing the component on the d-axis, U, of the first port voltage response to the first perturbation signalq1Representing the component of the first port voltage response on the q-axis, U, corresponding to the first perturbation signald2Representing the component on the d-axis, U, of the second port voltage response corresponding to the second perturbation signalq2Representing a component of the second port voltage response on the q-axis, I, corresponding to the second disturbance signald1Representing the component on the d-axis of the first port current response to the first perturbation signal, Iq1Representing the component of the first port current response in the q-axis, I, to the first disturbance signald2Representing the component of the second port current response to the second disturbance signal on the d-axis, Iq2Representing a component of a second port current response corresponding to the second perturbation signal on the q-axis; zddRepresenting the impedance of the d-axis, ZdqRepresenting the coupling impedance of the d-axis and the q-axis, ZqdRepresenting the coupling impedance of the q-axis with the d-axis, ZqqRepresenting the impedance of the q-axis;
the solution to the impedance of the three-phase ac circuit 11 is obtained according to the formula F1:
to verify the technical effect of the present invention, a three-phase RL network circuit is taken as an example, a comparison graph of the test value and the theoretical value of the impedance obtained by the test of the method of the present invention is drawn as shown in fig. 5A to 5D, and the impedance Z obtained by the test is shown in fig. 5A to 5Ddd、Zdq、Zqd、ZqqThe impedance value inosculated with the theory is very high, only in the coupling term Zdq、ZqdSome differences exist between the test result and the theoretical result, and the analysis error sources are as follows: on the one hand due to Zdq、ZqdThe amplitude of the method is small and sensitive to errors, and on the other hand, the errors are introduced in a simulation calculation process and a parameter identification process.
The above description is only a preferred embodiment of the present invention, and is not intended to limit the present invention, and all simple modifications, changes and equivalent structural changes made to the above embodiment according to the technical spirit of the present invention still fall within the protection scope of the technical solution of the present invention.
Claims (6)
1. A three-phase alternating current circuit impedance test system is characterized in that: the device comprises a disturbing signal injection module (1), a signal processing module (2) and a signal amplification module (3) which are sequentially connected, a signal detection module (4) connected with the input end of the signal processing module (2), a frequency response measurement module (5) connected with the output end of the signal processing module (2) and an impedance calculation module (6) connected with the frequency response measurement module (5); the disturbance signal injection module (1) is used for generating a disturbance signal under a dq coordinate system for injecting a three-phase alternating current circuit (11), and transmitting the disturbance signal to the frequency response measurement module (5) and the signal processing module (2); the signal processing module (2) is used for converting a disturbance signal output by the disturbance signal injection module (1) in a dq coordinate system into an abc three-phase alternating current signal and outputting the abc three-phase alternating current signal to the signal amplification module (3), is used for processing a power supply voltage signal measured by the signal detection module (4) through a phase-locked loop to obtain a value of a phase angle theta used for dq conversion, and is used for converting a port response signal measured by the signal detection module (4) from the abc coordinate system to the dq coordinate system by adopting the dq conversion; the signal amplification module (3) is used for amplifying the disturbance signal output by the signal processing module (2) and injecting the amplified disturbance signal into the three-phase alternating current circuit (11); the signal detection module (4) is used for measuring a power supply voltage signal and a port response signal in an abc coordinate system, and filtering and conditioning the signal; the frequency response measuring module (5) is used for obtaining corresponding frequency response through the disturbance signal and the port response signal in the dq coordinate system and outputting the corresponding frequency response to the impedance calculating module (6); the impedance calculation module (6) is used for calculating and obtaining the impedance of the three-phase alternating current circuit (11); the disturbance signal injection module (1) and the frequency response measurement module (5) are composed of a network analyzer (7); the signal processing module (2) comprises an FPGA board card (8); the disturbance signal generated by the disturbance signal injection module (1) and used for being injected into a dq coordinate system of the three-phase alternating current circuit (11) is a voltage signal, and the port response signal measured by the signal detection module (4) is a current response signal.
2. A three-phase ac circuit impedance testing system according to claim 1, wherein: the FPGA board card (8) is an NI USB-7845R multifunctional data acquisition card.
3. A three-phase ac circuit impedance testing system according to claim 1, wherein: the signal amplification module (3) comprises a power amplifier (9).
4. A three-phase ac circuit impedance testing system according to claim 1, wherein: the impedance calculation module (6) comprises a computer (10).
5. A method of performing three-phase ac circuit impedance testing using the system of claim 1, the method comprising the steps of:
step one, a disturbance signal is generated by a disturbance signal injection module (1) and transmitted to a frequency response measurement module (5) and a signal processing module (2);
step two, the signal processing module (2) calls a coordinate transformation module from a dq coordinate system to an abc coordinate system, converts a disturbance signal in the dq coordinate system generated by the disturbance signal injection module (1) into an abc three-phase alternating current signal, and outputs the three-phase alternating current signal to the signal amplification module (3);
thirdly, the signal amplification module (3) amplifies the disturbance signal output by the signal processing module (2) and injects the amplified disturbance signal into the three-phase alternating current circuit (11);
step four, the signal detection module (4) measures a power supply voltage signal and a port response signal, filters and conditions the signal and outputs the signal to the signal processing module (2);
step five, the signal processing module (2) processes the power supply voltage signal through a phase-locked loop to obtain a value of a phase angle theta used for dq transformation, and then port response signals are transformed from an abc coordinate system to a dq coordinate system through the dq transformation;
step six, the frequency response measuring module (5) obtains corresponding frequency response through the disturbance signal and the port response signal under the dq axis and outputs the corresponding frequency response to the impedance calculating module (6);
seventhly, generating two different disturbance signals by the disturbance signal injection module (1), and repeatedly executing the step one to the step six for eight times to obtain port current and voltage data groups corresponding to the two different disturbance signals, wherein the impedance calculation module (6) calculates the impedance of the three-phase alternating current circuit (11) according to the port current and voltage data groups corresponding to the two different disturbance signals;
the disturbance signal injection module (1) and the frequency response measurement module (5) are composed of a network analyzer (7), the signal processing module (2) comprises an FPGA board card (8), the signal amplification module (3) comprises a power amplifier (9), and the impedance calculation module (6) comprises a computer (10);
the disturbance signal generated by the disturbance signal injection module (1) and used for being injected into a dq coordinate system of a three-phase alternating current circuit (11) is a voltage signal, and a port response signal measured by the signal detection module (4) is a current response signal;
and in the sixth step, the network analyzer (7) obtains corresponding frequency response through the disturbance signals and the port response signals under the dq axis and stores the frequency response as an Excel file, and in the seventh step, when the computer (10) calculates the impedance of the three-phase alternating current circuit (11) according to the port current and voltage data groups corresponding to the two different disturbance signals, the Excel file is firstly led into MATLAB software, then frequency response data is established into a frequency response data model by utilizing a frd function, then the frequency response data model is fitted into a state space model by utilizing a fitfrd function, and then the impedance of the three-phase alternating current circuit (11) is calculated according to the state space model.
6. The method of claim 5, wherein: the expression of the state space model is as follows:
wherein, Ud1Representing the component on the d-axis, U, of the first port voltage response to the first perturbation signalq1Representing the component of the first port voltage response on the q-axis, U, corresponding to the first perturbation signald2Representing the component on the d-axis, U, of the second port voltage response corresponding to the second perturbation signalq2Representing a component of the second port voltage response on the q-axis, I, corresponding to the second disturbance signald1Representing the component on the d-axis of the first port current response to the first perturbation signal, Iq1Representing the component of the first port current response in the q-axis, I, to the first disturbance signald2Representing the component of the second port current response to the second disturbance signal on the d-axis, Iq2Representing a component of a second port current response corresponding to the second perturbation signal on the q-axis; zddRepresenting the impedance of the d-axis, ZdqRepresenting the coupling impedance of the d-axis and the q-axis, ZqdRepresenting the coupling impedance of the q-axis with the d-axis, ZqqRepresenting the impedance of the q-axis;
obtaining a solution for the impedance of the three-phase alternating current circuit (11) according to equation (F1) as:
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