Research on Parameter Design and Control Method for Current Source Inverter–Fed IM Drive Systems

: In the current source inverter (CSI) − fed inductor motor (IM) drive system, DC − bus induct ‐ ance and AC − filter capacitance have a direct impact on the dynamic response speed, power quality and power density. In addition, due to the addition of filter capacitors, the mathematical model formed by the IM and capacitor is a third − order system, which increases the difficulty of parameter tuning for the control loop. To solve the above problems in the CSI, a DC − bus inductance design method based on current response speed and the ripple of the DC − link current and an AC − filter capacitance design method based on current utilization and filtering characteristics are presented. Then, the analytical expressions between the open − loop cut − off frequency, phase margin and the PI controller parameters of the current loop and speed loop are derived. Finally, an experimental plat ‐ form is established to validate the proposed method.


Introduction
Inverters can be divided into voltage source inverter (VSI) and current source inverter (CSI) according to the characteristics of the power supply in the DC side [1,2]. For a long time, VSI has been a research hotspot and has been widely used in various AC drive and grid−connected applications [3,4]. However, the current ripple on the DC side of VSI is large, and it can only operate under the buck mode; deadtime is needed to prevent DC power supply from passing through [5]. Compared with VSI, CSI can boost the voltage and operate without short circuit protection [6]. Due to the presence of filter capacitors on the AC side, the voltage of the stator side of the motor has good waveform quality [7], so in many work situations, CSI is more suitable than VSI [8,9].
Although the CSI has the above advantages, due to some inherent defects, the application scope of CSI in motor drive systems is restricted. Instead of a constant current source, DC−link current is generated from a DC−bus inductor with a DC voltage source, and the DC−bus inductor is charged or discharged under different switching states, so the DC−link current of traditional CSI cannot be controlled [10,11]. Meanwhile, the CSI is unable to operate under the buck mode; otherwise, the DC−link current will increase continuously [12]. To achieve the control of DC−link current and a wide range of output voltage of CSI, many novel CSI topologies are proposed. In Refs. [13,14], a PWM current source rectifier is used as the front stage to control the DC−link current, but this topology is only suitable for the occasion of AC−DC−AC. In Ref. [15], a quasi−Z−source current source converter is added into the DC side, but the difficulty of control increases greatly with the addition of too many passive components. In Ref. [11], the bypass IGBT was paralleled on the DC−link inductor through the control of the switch device. Under the buck operation, the continuous increase of the DC-link current is effectively restrained. Compared to the above topologies, in Refs. [16,17], a bidirectional DC chopper is added in the front stage of the CSI; this topology is simple and consists of only a pair of switching tubes and diodes, and the DC−link current control can be achieved under the operations of boost, buck, and deceleration by regulating the switching state of the bidirectional DC chopper. Therefore, this paper takes the topology of CSI with bidirectional DC chopper as the research object.
In the AC side of CSI, due to the presence of filter capacitors, the mathematical model formed by the motor and capacitor is a third−order system, which increases the design difficulty of the control system of the inductor motor (IM). In recent years, the vector control technologies for the CSI−fed drive system have been studied by some scholars. In Ref. [13], the references of stator current in the d−q axis are obtained by the controllers of rotor flux and speed, respectively; then, the references of the output current of the inverter bridge in the d−q axis are calculated according to the steady state equation of the IM. However, this control strategy is more sensitive to the accuracy of the IM parameters. In Ref. [18], a small signal mathematical model of a CSI−fed permanent magnet synchronous motor is constructed through the state space averaging method; a speed control system based on duty cycle regulation is proposed, but the steady−state performance of this method is not ideal, and the stator current has obvious harmonic distortion. In Ref. [19], various decoupling methods and the active damping method are presented and compared, and a control structure of voltage inner loop and current outer loop are proposed. In addition to the method of decoupling and controlling, DC−bus inductance and AC−filter capacitance also have a direct impact on the dynamic response speed, power quality, and power density [20]; however, research on parameter design methods for DC−bus inductance and AC−filter capacitance is rarely reported.
In this paper, the design principles of DC−bus inductance and AC−filter capacitance are firstly studied. Then, to reduce the number of the control loops, a vector control technology without a voltage loop is proposed, and the analytical expressions of the control parameters are derived. Finally, under various conditions, experimental validation of the proposed method is carried out.

Topology of CSI−Fed IM Drive System with Bidirectional DC Chopper
To realize the decoupling control of the DC−link current and IM, a bidirectional DC chopper is added between the DC voltage source and DC−link inductor; the topology is shown in Figure 1. In the figure, udc is the DC input voltage; idc is the current of DC−link inductor Ldc; S1-S6 are six switching tubes of the inverter bridge; D1-D6 are the diodes connected in series with S1-S6; Ca, Cb and Cc are the three−phase filter capacitors in the AC side; iat, ibt and ict are the three−phase output currents of the inverter bridge; ias, ibs and ics are the stator currents of the IM; uas, ubs and ucs are the three−phase voltages of the filter capacitors. The bidirectional DC chopper is composed of the diodes Da, Db and the switching tubes Sa, Sb. When the control method proposed in Ref. [17] is adopted, the circuit of DC input side can be regarded as a controlled current source. In the process of mathematical modeling and control system design for the AC side of CSI, the influence on the DC−link current can be ignored, and the topology of the CSI−fed IM drive system with bidirectional DC chopper can be simplified as seen in the circuit shown in Figure 2. Figure 2. The equivalent circuit of CSI−fed IM drive system with bidirectional DC chopper.

Mathematical Model of Filter Capacitors and IM
where ids and iqs are the stator currents in the d−q axis; uds and uqs are the voltages of filter capacitor in the d−q axis; ψr is the rotor flux; σ, Rs, Lm, Lr and ws are the flux leakage coefficient, stator resistor, mutual inductance, rotor inductance and synchronous angular frequency, respectively.
According to Figure 2, the current equation in three−phase stationary coordinate frame can be expressed as as at as bs bt bs cs ct cs where J is the moment of inertia, ωr is the mechanical speed, and Te and TL are the electromagnetic torque and load torque, respectively.
In summary, the mathematical model of filter capacitors and IM in the AC side of CSI can be represented by Equation (1) to Equation (5).

Selection Range of DC−Link Inductance
A strong output impedance and constant current source characteristics of the DC side in CSI can be obtained by a larger DC−link inductance. However, the current regulation becomes slower as the inductance increases, which leads to overmodulation and generates low−order current harmonics. Therefore, it is necessary to determine a selection range of DC−link inductance.
In the start−up stage, the DC−link current is too low to provide the required current for the IM, so the zero switching state is adopted, and Ldc is charged by udc; the equivalent circuit is shown in Figure 3a. The state equation of idc is expressed as where Tup_max is the maximum allowable charging time, making sure that the charging time for idc to increase from zero state to maximum DC−link current idc_max is less than Tup_max. Ldc should satisfy the following condition: According to the conclusion in Ref. [21], the current fluctuation ∆idc can be calculated when the IM is powered independently by Ldc, which contains two active switching states in one carried period. For example, in sector I, the equivalent discharge circuits are shown in Figure 3b,c, and ∆idc is expressed as where Ts is the carried period, equal to 100 μs; U is the fundamental amplitude of the stator voltage; I is the fundamental amplitude of the stator current; θ is the phase deviation between stator voltage and stator current. In other sectors, the voltage of Ldc and the switching state are shown in Table 1, and the expression of ∆idc is consistent with that of sector I, as shown in Equation (8). Making sure that ∆idc is less than the maximum ripple ∆idc_max under θ = 1, Ldc should satisfy the follow condition: where mimax and mumax are the maximum modulation ratio and boost ratio of CSI, respectively. Combining Equations (7) and (9), the selection range of Ldc is expressed as (b) equivalent discharging circuit when S1 and S6 are turned on; (c) equivalent discharging circuit when S1 and S2 are turned on.

Selection Range of AC−Filter Capacitance
The IM is represented by the T−type equivalent circuit; the phase−a circuit model of IM and the capacitor is shown in Figure 4, where Lsσ and Lrσ are the stator leakage inductor and rotor leakage inductor, respectively; Rr is the rotor resistor; Iac is the current vector of Ca; and sr is the slip ratio. The vector diagram composed of Iat, Ias, Iac and Uas is shown in Figure 5, and the following relationship can be obtained: at ac at s as as In order to maximize the AC current utilization rate, the following equation should be satisfied: at as According to the geometric relationship in Figure 5, Iac should be satisfied as follows: sin ac as Substituting Equation (11) into Equation (13), the expression of maximum capacitance Cmax can be obtained as The mathematical relationship of Uas, Ias and sinθ can be obtained according to the T−type equivalent circuit of IM shown in Figure 4, and Equation (14) can be rewritten as From Equations (15) and (16), Cmax is a function of sr, and ωs is set as the rated angular frequency of 100π rad/s. Substituting the IM parameters listed in Table 2 into Equations (14) and (15), the relationship curve between Cmax and sr can be obtained, which is shown in Figure 6. It is obvious that a minimum value occurs in Cmax, so the AC−filter capacitance should be less than 2.37 mF. sr Figure 6. The relationship curve between Cmax and sr.
Next, the influence mechanism on harmonic characteristics of the stator current caused by AC−filter capacitance will be discussed. The T−type equivalent circuit of IM is a third−order system; it is difficult to derive the transfer function and mathematical relationship between the capacitance and bandwidth. Since the high−order harmonic contents near the switching frequency of ids (iqs) are same as the stator currents in three−phase stationary coordinate, according to Equations (1) and (2), the equivalent circuits of the IM and AC−filter capacitor in the d−q axis based on rotor flux orientation are established, as shown in Figure 7. idt and iqt are composed of a DC component and several high−order harmonics components near the switching frequency. The equivalent controlled current sources, voltage sources, and back EMFs in Figure 7 cannot amplify or attenuate harmonics, so the suppression effect of high−order harmonics is determined by Rs, σLs and C, and the transfer function Git2is(s) from idt (iqt) to ids (iqs) can be expressed as and the oscillation angular frequency ωn can be expressed as 1 Substituting the value of Rs and Ls into Equation (16), the Bode plot of Git2is(s) is drawn under different AC−filter capacitance, which is shown in Figure 8. It can be seen that Git2is(s) has a resonance spike, and the cutoff frequency decreases with the increase of C. In order to obtain the high quality stator current, Git2is(s) should effectively suppress the harmonic components in iat near the switching frequency fs, so the oscillation frequency should be far below fs. This paper stipulates that the oscillation frequency is less than fs/2; according to Equation (17), AC−filter capacitance should be satisfied as Amplitude /dB Substituting the IM parameters listed in Table 2 into Equation (19), the minimum value of C is 2.5 μF.

Design Principle for Vector Control Strategy of CSI−Fed IM Drive System
According to Section 1, the mathematical model formed by the IM and capacitor is a third−order system; when the three closed−loop control strategy is adopted, more voltage sensors are needed, and the dynamic response to torque is slower than the VSI−fed IM drive system. In this section, a novel vector control strategy without a voltage inner loop for the CSI−fed IM drive system will be proposed, which includes current loop, flux loop and speed loop.

Analysis and Parameter Design for Current Loop
The switching delay and digital control delay are equivalent to a first−order transfer function, which is expressed as Due to the presence of high−frequency noise in the stator current sampling signal, a filter circuit is adopted in this paper, and its equivalent transfer function is where Tcf is the filter time constant, equal to 6 × 10 −5 in this paper.
Ignoring the coupling terms in Equations (1) and (3), such as ωsσLsiqs, ωsσLsids, ωsCuqs and ωsCuds, in the d−q axis, the control diagram of the current loop without stator voltage sampling and control is shown in Figure 9, where kcp and ci  In this paper, the target cut−off frequency and phase margin of current loop are represented as ωcc and φcc. According to Figure 9, Qc_amp, the amplitude of the current open−loop at ωcc, can be derived as follows: To ensure that the phase angle margin of the current loop with the PI controller is greater than φcc, ci  should be satisfied as follows: To ensure that the amplitude of the current loop with the PI controller is equal to 1, kcp should be satisfied as follows: In order to achieve fast dynamic response and the effective suppression of high−order harmonics in the output current of the inverter bridge, ωcc is set to 2000π rad/s, which is one−tenth of the switching frequency. Meanwhile, to obtain a better stability, φcc is set to π/4 rad. Substituting the parameters listed in Table 2 Figure 10; the bandwidth of Gi(s) is 1kHz, and the amplitude remains 0 dB when the frequency is less than 1kHz. When the frequency is greater than 1 kHz, the amplitude decays with a slope close to −60 dB/dec, which can effectively suppress the high−order harmonics generated in the switching process. This explains that a good steady−state and dynamic performance can be achieved by the PI parameter optimization proposed in this paper. In order to facilitate the subsequent parameter design of the external loop control system, Gi(s) is simplified to a second−order transfer function and expressed as

Analysis and Parameter Design for Flux Loop
A PI controller is adopted to adjust rotor flux ψr and is combined with Equation (4); the diagram of the flux loop with a PI controller is shown in Figure 11. kψp and i   are the proportional coefficient and integral coefficient of the PI controller. To simplify the design method of the control parameters, zero−pole cancellation is adopted, and i   is set to Tr. Then, the root locus plot is utilized to determine kψp, which is shown in Figure 12. It is obvious that the condition for the stability of the flux loop is as follows: (0,6.21/ )   Figure 13. The cut−off frequency is 600 Hz, and the phase margin is 60°.

Analysis and Parameter Design for Speed Loop
The speed of IM is sampled by an orthogonal encoder, and the sampling frequency is half of fs; the transfer function of speed sampling and delay is expressed as In accordance with Equations (5) and (6), the block diagram of the speed loop under the PI controller is shown in Figure 14, where ksp and si  are the proportional coefficient and integral coefficient of the PI controller for adjusting the stator current, respectively.
where Qs_amp and φs_phase are the amplitude and phase of the speed open loop at ωss before the PI controller, respectively. ksp and si  are obtained by substituting the equation ωss = 200π rad/s and φss = π/6 rad into Equation (31). The specific expression of Gs_close(s) is as follows: The Bode plots of open−loop and closed−loop transfer functions are shown in Figure  15. The cut−off frequency of the speed open−loop transfer function is 100 Hz, and the phase margin is 30°. The closed−loop transfer function has a good following characteristic in the bandwidth of 40 Hz. The unit step response of the speed closed−loop transfer function is shown in Figure 16; the rise time is only 2 ms, the overshoot is little, and no steady−state error occurs.

Simulation Results
The simulation model of the CSI−fed IM drive system with bidirectional DC chopper is built on MATLAB Simulink, including the power module, control module and modulation module. The parameters of IM adopted are consistent with Table 2, and other simulation parameters are shown in Table 3.

Load Acceleration Condition
The value of load torque is set to 3 N•m, the initial reference speed is 1500 rpm, and the reference speed is 3000 rpm when the time is 0.8 s. The simulation waveforms of stator phase a current, stator currents in the d−q axis, electromagnetic torque, motor speed and rotor flux in the whole simulation process are shown in Figure 17.
To improve DC−link current utilization and reduce losses and harmonic distortion, according to the MTPA method proposed in Ref. [17], the references of rotor flux and DC−link current are adjusted with the change of operating point; the specific principle is as follows: A. In the dynamic process, in order to provide greater torque, the reference DC−link current is set to 50 A, which is the maximum current of the DC−link inductor. B. In the steady state, the references of rotor flux ψ r * and DC−link current i dc * are adjusted according to the load torque and speed, which are shown in Table 4. The whole simulation process is divided into three stages. Start−up stage: in order to make the motor speed reach the reference speed as soon as possible, the hysteresis control strategy of DC−link current is adopted to regulate idc to 50 A, which is the max reference, so as to meet the demand of current. The optimal reference value of rotor flux ψ r * is 0.068 Wb. After 0.05 s, ψ r * reaches the reference value, ids is around 15 A, iqs is around 43 A, and Te is 8 N•m. After 0.4 s, the speed is adjusted to 1500 rpm, and Te is reduced to 3 N•m. Steady−state operation stage at 1500 rpm: idc is regulated to 23.5 A, as shown in Figure  17a. It can be seen from Figure 17e that when idc is reduced to 23.5 A, the ripple of torque is significantly reduced.
Second acceleration stage: at the time of 0.8 s, the value of step speed is 3000 rpm. idc reach to 50 A again, which is the same as the start−up stage. When ψ r * is adjusted to 0.064 Wb, ids is reduced immediately, ψ r follows the given value quickly, iqs and Te are the same as the start-up. Finally, at 1.08 s, the speed reaches to 3000 rpm.

Deceleration Condition with No Load
This section simulates the deceleration condition of the motor with no load. When the IM speed reaches the rated speed of 1500 rpm, the reference speed is changed to 900 rpm at 0.6 s. The simulation waveforms of electromagnetic torque, speed, DC−link current and output power of DC power supply are shown in Figure 18, respectively. When the speed decelerates, idc is around 30 A. The electromagnetic torque plays a braking role and reaches to −2.3 N•m, and the speed reaches stability again after 0.28 s. The DC power supply intermittently absorbs the energy feedback from the motor decelerating process, and the output power is less than zero, so the energy feedback is realized.  Figure 19 are the simulation waveforms of load disturbance, when the speed remains at 1500 rpm, the load torque is reduced from 3 N•m to 1 N•m at 0.1 s, and the speed controller makes the electromagnetic torque drop rapidly to 1 N•m, where the speed remains unchanged. At the same time, the ψr and idc are reduced to the optimal value of 0.039 Wb and the minimum value of 14.8 A, respectively. When the load torque is increased to 2 N•m at 0.3 s, the speed is stabilized at 1500 rpm after 0.16 s, and the maximum error of speed is only 14 rpm. The ψr and idc are adjusted to 0.055 Wb and 20.9 A, respectively.

Experimental Results
In order to verify the feasibility of parameter design for DC−link inductance and AC−filter capacitance and the effectiveness of the vector control strategy proposed in this paper, the experimental platform was established, as shown in Figure 20; the experimental parameters are shown in Table 5. The value of DC−link inductance and AC−filter capacitance are satisfied with the selection range specified in Section 3. Because the motor speed, rotor flux, and stator d−q axis current cannot be measured by the oscilloscope, in order to facilitate debugging as well as the observation and storage of these signals, this paper develops a host computer program on the GUI platform of MATLAB to realize the data communication with DSP.  Next, the experimental results will be carried out under the three operation conditions: acceleration with load, deceleration with no−load and change of load.

Performance under Acceleration with Load of 3 N•m
Experimental waveforms of accelerated conditions are shown in Figure 21, under the load of 3 N•m. The experimental process consists of two stages. The reference speed changes from 1500 rpm to 3000 rpm, in order to reduce the harmonics and losses, and to improve current utilization, the references of idc and ψr are also adjusted according to the calculation method for the minimum DC−link current and the optimal flux proposed in Ref. [17]. In the dynamic process, to obtain a larger torque and reduce the dynamic response time, the reference idc is set to 50 A. In the first stage, the reference speed is set to 1500 rpm, and the reference idc and ψr are set to 23.5 A and 0.068 Wb, respectively. In the second stage, the reference speed is set to 3000 rpm, and the reference idc and ψr are set to 22.5 A and 0.065 Wb, respectively. From Figure 21, the rise time of idc increasing from 0 A to 50 A is only 15 ms, and the ripple of idc is less than 0.5 A, which demonstrates that the parameter design method of the DC−link inductor is feasible. In two stages, the speed and rotor flux can quickly track the references, and the THD of the stator current is less than 2%.

Performance under Deceleration with No Load
The experimental waveforms of the reference speed decreasing from 1500 rpm to 900 rpm with no load are shown in Figure 22. Unlike with the operational condition where the load is 3 N•m, the amplitude of the stator voltage is 18.5 V, which is less than udc. idc remains at 12 A with little ripple in the steady state, and during the dynamic process of speed decreasing from 1500 rpm to 900 rpm, the reference of idc is set to 30 A, and the mechanical energy from IM is recovered by the DC voltage source. Due to a fast dynamic and little ripple of Te, dynamic and steady performance of speed can be obtained, the rotor flux can quickly track the references, and the stator current has very little harmonic distortion.

Performance under Different Loads
The reference speed is set to 1500 rpm. The IM starts with the load of 3 N•m, then the load is suddenly reduced to 1 N•m, and finally the load is increased to 2 N•m. The experimental waveforms are shown for the reference speed decreasing from 1500 rpm to 900 rpm with no load are shown in Figure 23. Since the torque can be quickly adjusted when the sudden change occurs on the load, the fluctuation of the speed is less than 3%. Meanwhile, idc and ψr can quickly track the references.

Conclusions
In order to improve the performance of the current source inverter-fed inductor motor drive system, this paper presents a novel design principle of DC−bus inductance and AC−filter capacitance, and a vector control technology without voltage loop. The feasibility and correctness are verified by the simulation and experimental results under various operating conditions. The main contributions can be summarized as follows: (1) Under consideration of the dynamic response speed and current ripple, the range expression of the DC−link inductance is derived.