Analysis and Calculation of Inductor Parameter for PWM rectifier

In the PWM rectifier, the rationality of the AC side filter inductance design directly affects the performance of the whole system. It not only limits the power output, but also affects the dynamic and static response of the current loop. It has an important influence on the four quadrant stable operation of the motor, the isolation of the grid electromotive force and the harmonic suppression of the current. According to the waveform characteristics of the sinusoidal current, the maximum change rate of the current occurs at the zero-crossing point. In this paper, the inductor is designed to meet the requirement of the change rate of the current near the zero-crossing point, so as to meet the tracking requirement of the current at any time in the whole cycle. In addition, for the design of the AC side inductance, this paper comprehensively designs from the perspective of meeting the requirements of power index and meeting the requirements of suppressing harmonic current.


Introduction
With the advent of the ' electrification ' era, large-scale power electronic devices affect people 's basic necessities of life, such as frequency converters, inverters, high-frequency switching power supplies and other converter devices [1].The essence of power electronic equipment is a power switching device.In most cases, it works in a switching state except when operating in a linear power amplification situation.If effective control measures are not taken for this working state, it will cause 'harmonic pollution 'to the power system, and even lead to safety accidents.Usually, these devices rely on rectifiers to obtain the required DC voltage level.However, due to the defects of low power factor and high harmonic content of AC current in the operation process of phase-controlled rectifier and uncontrolled rectifier, they cannot meet the control requirements [2].
Pulse width modulation technology is born in this era.PWM modulation technology greatly improves the performance of the converter, expands its application range, and improves energy efficiency [3].PWM rectifier is a four-quadrant operation converter device with controllable AC and DC sides, and can control the grid side current and power factor.When the PWM rectifier works normally, the energy flows between the AC side and the DC side.The inductor is responsible for storing energy in the process, which is a necessary condition to ensure the normal operation of the rectifier.Whether the design of AC side filter inductance is reasonable directly affects the performance of the whole system.
It not only limits the power output, but also affects the dynamic and static response of the current loop [4].It has an important influence on the four-quadrant stable operation of the motor, the isolation of the grid electromotive force and the harmonic suppression of the current [5].Therefore, it is necessary to select the inductance parameters within a reasonable range.

PWM rectifier circuit model
The main circuit of PWM rectifier is shown in figure 1.The main circuit mainly includes AC voltage source, filter inductance L, IGBT switch tube and diode in parallel with its direction, bus capacitor.The main function of the filter inductor is to filter out the PWM harmonic current on the AC side.The value not only affects the response speed of the current, but also restricts the output power of the whole system.The function of bus capacitor C is to resist load disturbance and stabilize DC bus voltage.The capacity of capacitor will directly affect the antidisturbance and following performance of the whole system.To simplify the calculation, the following assumptions are made : (1)The AC voltage source is an ideal three-phase symmetrical voltage source.
(2)The incoming reactor is completely linear and does not consider its saturation effect.
(3)A loss resistor is used to equivalent the loss of the switch tube, and the IGBT in the switch tube is equivalent to a series of ideal switches and a loss resistor.
(4)In order to describe the two-way flow process of system energy, the load of the system is represented by a resistor and a DC power supply in series.
According to the above assumptions, we can get the equivalent circuit diagram of the PWM rectifier under ideal conditions, shown in figure 2.

Electric operation analysis
There are six switches in the back-end inverter of the motor as shown in figure 3.These six switches have 8 combinations of switching states.Taking the a-phase bridge arm as an example, the definition of V1 conduction V2 turn-off state is 1, V2 conduction V1 turn-off state is 0, and the definition of b-phase and c-phase bridge arm state is the same as a.The switch combination state of the inverter link is shown in table 1.
Randomly select a set of switch combinations to analyze the current direction and the operating state of the motor.Assuming that the motor is in the 010 switching state, the current flow is shown in figure 3.In the diagram, V2, V3 and V6 are in the conduction state, and V1, V4 and V5 are in the shutdown state.The electric energy of the motor is converted into mechanical energy through the medium of the motor.

Braking operation analysis
The motor is in the state of energy regeneration when braking.Before analyzing the process of energy regeneration, it is necessary to clarify the definition of energy regeneration.Reference [6] pointed out that the process of energy regeneration means that no matter how large the voltage output by the driver is, the energy in the load is transmitted to the DC side of the driver through the motor.
The motor uses an AC speed control system.From the above definition, there are two ways of energy regeneration in the system [6].One is that the internal back EMF of the motor exceeds its terminal voltage and charges the bus capacitor.Another case is that the back EMF of the motor is lower than the terminal voltage of the motor.Although the back EMF is lower than the terminal voltage, due to the existence of the freewheeling diode, the current in the motor winding will flow through the freewheeling diode to the DC side to charge the bus capacitor.Both cases can produce regenerative energy.The above two cases are introduced below.

The back EMF in the motor does not exceed its terminal voltage but also charges the DC bus
capacitor.This situation usually occurs during the switch state transition of the motor driver.figure 3 shows the situation when the motor is electrically operated in the 010 switching state.When the driver is converted from the 010 switching state to the 011 switching state, in order to avoid the simultaneous conduction of the upper and lower arms, a dead time needs to be added.In the dead time, although the six switch tubes are turned off at the same time, because the current in the motor cannot mutate, the bus capacitor must be charged by the diode freewheeling in reverse parallel with V1, V4 and V5, which leads to the increase of the bus voltage and the regeneration energy.The current flow in the regeneration process is shown in figure 4 Motor operation in handoff process of switch mode When the dead time is over, the switch state is converted to 011 state.At this time, V2, V3 and V5 turn on VD1, VD4 and VD5 reverse cut-off motors to start electric operation.The current direction is shown in figure 5.
. Electric operation of motor in 011 on-off state

The case that the back EMF. in the motor exceeds the terminal voltage of the motor and charges the DC bus capacitor.
Affected by external factors, the sum of the voltage drop of the motor inductance and the back EMF may exceed the output voltage of the inverter.At this time, the motor is equivalent to a generator.In this case, the current will change the direction energy to the DC bus side.At this time, the driver provides the motor with braking torque.This kind of energy regeneration usually occurs when the motor runs from electric to stop, the speed of the motor changes suddenly during operation, and the motor runs in the power generation mode.
During the braking process of the motor, the terminal voltage of the driver added to the motor is zero.However, due to the existence of mechanical inertia, the motor cannot immediately stop and it will continue to rotate to generate back EMF.The current in the motor winding flows to the DC side through the freewheeling diode to form energy feedback.During the operation of the motor, it is sometimes necessary to slow down the motor or even change the speed direction of the motor.In this case, there will also be energy feedback phenomenon, in which the amount of feedback energy depends on the speed change rate and moment of inertia of the motor.The power generation operation of the motor usually occurs when the load drives the motor.In this case, the back EMF of the motor exceeds its terminal voltage.In this case, the role of the actuator itself is the braking effect.

PWM rectifier control strategy
The control of PWM rectifier usually adopts double closed-loop structure.The voltage loop is used as the outer loop to stabilize and control the voltage of the DC bus and generate active current to quantify.The current loop is used as the inner loop to make the current of the system track the given current to achieve unity power factor rectification or feedback.The system generates the active current reference of the current inner loop through the PI adjustment of the voltage outer loop.In order to make the system work under the unit power factor, the reactive current is given to the quantitative 0. In this way, the ( d, q ) axis component of the required reference voltage vector is generated by the PI adjustment combined with the voltage control equation, and the reference voltage vector is then transformed by the coordinate transformation and the SVPWM link to generate the PWM wave to drive the switch tube.Figure 6 is the PWM rectifier schematic diagram used in this paper.The design process of each part will be introduced in detail below.

Current loop PI regulator design
It can be seen from ( 1) that the d and q axis components of the PWM rectifier are coupled to each other, which increases the difficulty of the controller design.In order to solve this problem, a feedforward decoupling method can be used.In order to solve this problem, feedforward decoupling can be used.When the feedforward decoupling of the equation and the PI adjustment algorithm is applied to the d, q axis current, the control equations of ud and uq can be obtained as follows : where, ud = SdVdc, uq = SqVdc, id*、iq* is the given value of the current, iq is the feedback value of the current, Kip、KiI is the proportional coefficient and integral coefficient of the current inner loop regulator.
Figure 7 is the current decoupling block diagram of PWM rectifier.Due to the symmetry of the two current inner loops, only the d-axis current is taken as an example to design the regulator of the current inner loop.Since the current inner loop signal of the system has sampling delay, and the PWM control has small inertia characteristics, the structure of the decoupled id current loop can be expressed as figure 8.
The signal period of SVPWM triangular carrier is also the sampling period of current.KPWM is the equivalent gain of controllable rectifier bridge.Usually Ts is a small time constant, in order to simplify the analysis of ignoring the disturbance of ed and merge the two inertial links in the block diagram into an inertial link.In order to facilitate the tuning of the parameters, the PI regulator is usually written as a zero-pole form as shown in (2), and a simplified control block diagram can be obtained as shown in figure 9. ( In (4), θ is the initial phase angle and m is the modulation ratio.Unit power factor control is required whether the system works in active inverter or controllable rectifier state.So the grid side current of the whole system can be expressed as: cos( ) cos( 120) cos( 120) The current Idc on the DC side can be expressed by three-phase current and switching function:

I i S S S i S S S i S S S i i S S S i i S S S i i S S S i i i S S S i S i S i S
The ( 4) and ( 5) can be derived: Based on the above analysis, the control block diagram of the voltage outer loop is constructed , as shown in figure 10.
. Control structure of voltage loop in the PWM rectifier In the block diagram, Tvs is the voltage outer ring sampling period and has τv = Kvp/KvI.In the previous analysis process, the current loop has been approximated as a first-order inertial link.In order to simplify the design process, the small time constant of the current loop and the small time constant of the voltage loop are combined, namely Tiv = Tvs + 3Ts.In the voltage loop structure diagram, 0.75 mcosθ is a link that changes with time, which is not conducive to the design of the control system.Obviously, the maximum proportional gain of this link is 0.75, and the maximum proportional gain has the greatest influence on the stability of the system.Therefore, it can be replaced by its maximum proportional gain of 0.75.In order to simplify the analysis, the influence of the load current is ignored, and the structure of the voltage loop is shown in figure 11.

The realization process of SVPWM algorithm 4.3.1
The determination of the sector where the vector is located.In order to modulate the required vector Uout, we must first determine the sector where the vector is located so that it can be modulated with its adjacent vectors.In the process of sector judgment, it is necessary to first convert the voltage value from the three-phase stationary coordinate system to the two-phase stationary coordinate system.It is assumed that the voltage values in the two-phase stationary coordinate system are Uα and Uβ, respectively.Define the following three variables.
Let N = sign(a1) + 2×sign(a2)+4×sign(a3), then the sector where the voltage vector is located corresponds to the value.Now the corresponding relationship between the sector number and the N value is beneficial to table 2.
The action time of two adjacent vectors in the two-phase stationary coordinate： The peak phase voltage of ( 11) is normalized to obtain: The calculation of the vector action time in the other five sectors can refer to the derivation process of the action time of two adjacent vectors in the first sector.

Calculation of switching point of action time.
The following three variables are defined according to the conduction law.

T T T T T T T T T T
After calculating the three variables in (13), the switching point of the vector action time can be obtained from table 3.

Design of AC side inductance
In the PWM rectifier, the rationality of the AC side filter inductance design directly affects the performance of the whole system.It not only limits the power output, but also affects the dynamic and static response of the current loop.It has an important influence on the four quadrant stable operation of the motor, the isolation of the grid electromotive force and the harmonic suppression of the current.For the design of the AC side inductance, it should be comprehensively designed from the perspective of meeting the power index, fast current tracking requirements, and harmonic current suppression requirements.

Inductance design to meet the power index
The PWM rectifier in the motor usually works in the controllable rectification mode.The active inverter process of energy feedback is only applied when braking, so the selection of inductance value is based on the rectification mode.Ignoring the input resistance of the AC side, the vector diagram of the system during steady-state operation is shown in figure 12[7].In the figure 12, U is the AC side voltage vector, E is the grid voltage vector, UL is the grid side inductance voltage vector, and φ is the power factor Angle.According to the law of cosines sin sin When the DC side voltage Vdc of PWM rectifier is determined, the AC side phase voltage peak range can also be determined by (16).
In this paper, 3 3 m = is used, and ( 16) is brought into (15) to obtain: Considering that the power factor is 1 when the system works under the controlled rectifier state, sin =0  .Then (17) can be written as In order to obtain the range of the peak current of the AC side, the active and reactive power equations of the PWM rectifier are listed as follows: In order to ensure the normal operation of the system, the input power Pp of the network side must be greater than or equal to the given power P * of the system.Combined with (20), it can be derived that * 2 3 The peak range of the AC side phase current is obtained by using (20).The value range of the inductance when the power index is satisfied can be obtained by combining (18).

Design of inductor to meet the requirements of fast current tracking
According to the waveform characteristics of the sinusoidal current, the maximum change rate of the current occurs at the zero-crossing point.The design of the inductor must meet the change rate requirement of the current near the zero-crossing point, so that the current tracking requirement can be satisfied at any time of the entire cycle.The following will take phase a as an example to analyze the value of inductance when the change rate near the current zero crossing point is satisfied.In order to facilitate the analysis of the action time of ignoring the zero vector.
The a-phase voltage equation has been derived previously.For the convenience of description, it is re-written as follows: For (23), the waveform at the zero-crossing point of the current in one switching cycle is discussed, as shown in figure 13: The grid voltage near the zero-crossing point is very small to facilitate analysis and ignore it.( 2 ) 3 In order to achieve the standard of fast current tracking, (26) should be established.
When Sb and Sc are 1, the upper limit value of inductance can be obtained by ( 24),( 25)and (26).
Finally, the inductance satisfying harmonic suppression can be obtained : In the above formula, max I  is the maximum current fluctuation allowed near the peak.

Simulation result
In order to verify the inductance design method in this paper, the PWM rectifier is simulated.The simulation model of PWM rectifier built in MATLAB / Simulink is shown in figure 15.The simulation parameters used in the operation of the simulation model are as follows : the peak input voltage of the AC side of the system is 933V, the power frequency is 50Hz, the input inductance of the AC side is 6mH, the resistance value is 0.5Ω, the filter capacitance of the DC side is 2200uF, the given value of the DC bus voltage is 1750V, the output power of the system is 80kW, and the pumping voltage during state transition is 1900V.Figure 16 shows the voltage response curve when the system works in the controllable rectification state.From the DC voltage response curve of the system, it can be seen that the system has a faster reaction speed from the start of the system to the DC bus voltage reaching the given 1750 V in 0.06 s.In the simulation process, the maximum value of the bus voltage is 1800V overshoot within the  Figure 17 shows the waveforms of grid voltage and current on the AC side of the system under controllable rectification.By observing the waveform, it can be seen that the input current of the PWM rectifier tends to be stable after 0.02 s adjustment, and remains in phase with the grid voltage after stability, realizing the unit power factor rectification operation.When the motor is in electric operation, the system works in a controllable rectifier state, and when the motor needs to be braked, the system is required to quickly switch from the controllable rectifier state to the energy feedback state.Therefore, we simulate the conversion process of the system from the controllable rectifier state to the energy feedback state.
Figure 18 is the voltage response curve of the system from the controllable rectification state to the energy feedback state.The system runs in a controllable rectifier state before 0.3 s.At 0.3 s, it began to change to the energy feedback state.After 0.06 s adjustment, the bus voltage was re-adjusted to 1750 V at 0.36 s and operated stably.Figure 19 is the AC side grid voltage and AC side current curve when the system changes from the controllable rectification state to the energy feedback state.It can be seen from the diagram that the current curve changes at 0.3s.After 0.02s adjustment, the current is stabilized again, and the phase difference with the grid voltage changes from 00 to 1800, that is to say, the system completes the transition from the unit power factor controllable rectification state to the unit power factor energy feedback state.

Conclusion
This paper analyzes the PWM rectifier circuit diagram under ideal conditions.For the design of the AC side inductance, a comprehensive design is carried out to meet the power index, fast current tracking requirements, and harmonic current suppression requirements.The correctness of the design is proved by MATLAB / Simulink simulation.

Figure 1 .
Figure 1.PWM rectifier topology diagram.The main circuit mainly includes AC voltage source, filter inductance L, IGBT switch tube and diode in parallel with its direction, bus capacitor.The main function of the filter inductor is to filter out the PWM harmonic current on the AC side.The value not only affects the response speed of the current, but also restricts the output power of the whole system.The function of bus capacitor C is to resist load disturbance and stabilize DC bus voltage.The capacity of capacitor will directly affect the antidisturbance and following performance of the whole system.To simplify the calculation, the following assumptions are made :(1)The AC voltage source is an ideal three-phase symmetrical voltage source.(2)Theincoming reactor is completely linear and does not consider its saturation effect.(3)Aloss resistor is used to equivalent the loss of the switch tube, and the IGBT in the switch tube is equivalent to a series of ideal switches and a loss resistor.(4)Inorder to describe the two-way flow process of system energy, the load of the system is represented by a resistor and a DC power supply in series.

Figure 3 .
Figure 3. Electric operation of motor in 010 on-off state

Figure 6 .
Figure 6.Control schematics of the PWM rectifier

Figure 7 .
Figure 7.Control structure of current decoupling in the PWM rectifier Due to the symmetry of the two current inner loops, only the d-axis current is taken as an example to design the regulator of the current inner loop.Since the current inner loop signal of the system has sampling delay, and the PWM control has small inertia characteristics, the structure of the decoupled id current loop can be expressed as figure8.

Figure 9 .
Figure 9. Structure of id current loop in the case of ignoring disturbance

Figure 11 .
Figure 11.Control structure of voltage loop in the PWM rectifier after simplification.

Figure 12 .
Figure 12.PWM rectifier steady-state operation vector diagram.In the figure12, U is the AC side voltage vector, E is the grid voltage vector, UL is the grid side inductance voltage vector, and φ is the power factor Angle.According to the law of cosines .1088/1742-6596/2589/1/012036 10 Among them, Pp and Pq are active component and reactive component respectively, and Em and Im are the peak value of phase voltage and phase current respectively.Similarly, considering the case of unit power factor rectification, (19) can be written as

Figure 13 .
Figure 13.The current tracking waveform at zero current in a PWM switching cycle.

5 . 3 .Figure 14 .
Figure 14.The current tracking waveform at the peak current in a PWM switching cycle.When 1 0 tT  , there is 0 a S =

Figure 15 .
Figure 15.PWM rectifier simulation model diagram.Figure16shows the voltage response curve when the system works in the controllable rectification state.From the DC voltage response curve of the system, it can be seen that the system has a faster reaction speed from the start of the system to the DC bus voltage reaching the given 1750 V in 0.06 s.In the simulation process, the maximum value of the bus voltage is 1800V overshoot within the of Physics: Conference Series 2589 (2023) 012036 allowable range.It can be seen from the local amplification diagram of the corresponding voltage curve that the steady-state error is within 0.6V.

Figure 16 .
Figure16.The voltage response curve of PWM rectifier working in controllable rectifier state.Figure17shows the waveforms of grid voltage and current on the AC side of the system under controllable rectification.By observing the waveform, it can be seen that the input current of the PWM rectifier tends to be stable after 0.02 s adjustment, and remains in phase with the grid voltage after stability, realizing the unit power factor rectification operation.

Figure 17 .
Figure17.The a-phase voltage and current curve of the AC side when the PWM rectifier works in a controllable rectifier state.When the motor is in electric operation, the system works in a controllable rectifier state, and when the motor needs to be braked, the system is required to quickly switch from the controllable rectifier state to the energy feedback state.Therefore, we simulate the conversion process of the system from the controllable rectifier state to the energy feedback state.Figure18is the voltage response curve of the system from the controllable rectification state to the energy feedback state.The system runs in a controllable rectifier state before 0.3 s.At 0.3 s, it began to change to the energy feedback state.After 0.06 s adjustment, the bus voltage was re-adjusted to 1750 V at 0.36 s and operated stably.

Figure 18 .
Figure 18.Voltage response curve of PWM rectifier during state transition process.

Figure 19 .
Figure 19.The a-phase voltage and current curve of the AC side during the state transition of the PWM rectifier.

Table 1 .
Switch state table of motor driver

Table of
Calculation of action time of two adjacent space voltage vectors.The sine function value and cosine function value of angle θ are expressed by the components Uα and Uβ, of vector Uout in the twophase stationary coordinate system.

Table 3 .
Switching point of vector action time