AC and DC Impedance Extraction for 3-Phase and 9-Phase Diode Rectiﬁers Utilizing Improved Average Mathematical Models

: Switching models possess discontinuous and nonlinear behavior, rendering difﬁculties in simulations in terms of time consumption and computational complexity, leading to mathematical instability and an increase in its vulnerability to errors. This issue can be countered by averaging detailed models over the entire switching period. An attempt is made for deriving improved dynamic average models of three phase (six-pulse) and nine phase (18-pulse) diode rectiﬁers by approximating load current through ﬁrst order Taylor series. Small signal AC/DC impedances transfer functions of the average models are obtained using a small signal current injection technique in Simulink, while transfer functions are obtained through identiﬁcation of the frequency response into the second order system. For the switch models in Simulink and the experimental setup, a small signal line to line shunt current injection technique is used and the obtained frequency response is then identiﬁed into second order systems. Sufﬁcient matching among these results proves the validity of the modelling procedure. Exact impedances of the integral parts, in interconnected AC/DC/AC systems, are required for determining the stability through input-output impedances.


Introduction
The demand for power electronic converters has increased with the dependence on electronic appliances, digital products and computer systems in both industrial and household applications [1]. Power converters are required almost in every field that deals with electronics such as aircrafts, sea ships, communication systems, renewable energy generations, such as wind generation systems, photovoltaic systems and fuel cells etc. [2,3]. Since past few decades, engineers and scientists have focused on developing new control techniques as well as efficient models for various power converters so that to meet the IEEE prescribed standards, reliability and performance. Multi-pulse AC/DC converters are found to be one of the best solutions for providing loads with output voltages and currents having less ripples, reliable as well as rarely polluting the ac supply sources. These converters help in keeping lower total harmonic distortion (THD) at the ac side and hence less power consumption for the rated devices. The most common type of multi-pulse converter is the conventional six-pulse (three-phase) line-commutated rectifier, typically used as the input stage in low-to-medium-power variable frequency drives and motor loads common in industrial and commercial applications [4][5][6]. However, an increased number of pulses is employed for improved performance, comprising of impedances of the rectifiers through identification process and tested under steady and dynamic states. The dc side of the rectifier (output) has only one impedance parameter "Z out " while ac side has four parameters which are Z dd , Z dq , Z qd , Z qq . Extracting exact impedances transfer functions is the main goal of all research focused in this area since these are required for stability analysis of interconnected systems. Comparison of these transfer functions with those obtained from the detailed models as well as hardware prototypes reveals the validity and efficiency of the proposed average value models.
Energies 2018, 11, x FOR PEER REVIEW 3 of 20 steady and dynamic states. The dc side of the rectifier (output) has only one impedance parameter " " while ac side has four parameters which are , , , . Extracting exact impedances transfer functions is the main goal of all research focused in this area since these are required for stability analysis of interconnected systems. Comparison of these transfer functions with those obtained from the detailed models as well as hardware prototypes reveals the validity and efficiency of the proposed average value models.

Mathematical Average Value Models
In static models, load current is assumed to be constant dc but in actual, it varies according to load variation. It is therefore required to develop such a model which brings the load variation into consideration so that the model is much in accordance to that of real circuit under steady state as well as dynamic state.

Dynamic Average Value Model of 3-Phase Diode Rectifiers
For a three-phase rectifier with parameters given in Table 1, µ representing commutation angle, representing the average value of and K representing the rate of load current variation during this period of time ( ⁄ ), assume that load current varies linearly and is represented by evaluating first order Taylor series as: Commutation and conduction periods of a three-phase rectifier are shown in Figure 2a,b, along with the current waveforms in Figure 3. Current waveforms are trapezoidal shaped since inductors used at dc side, filters out peak lobes hence feeding load with constant DC current. Although, commutation period is affected by the higher filtering inductor value which could cause spikes in load voltage but proper selection of dc filter values could be the ultimate solution. Considering as the peak value of supply voltages given by Equation (2),

Mathematical Average Value Models
In static models, load current is assumed to be constant dc but in actual, it varies according to load variation. It is therefore required to develop such a model which brings the load variation into consideration so that the model is much in accordance to that of real circuit under steady state as well as dynamic state.

Dynamic Average Value Model of 3-Phase Diode Rectifiers
For a three-phase rectifier with parameters given in Table 1, µ representing commutation angle, I dc0 representing the average value of I dc and K representing the rate of load current variation during this period of time (dI dc /dωt), assume that load current varies linearly and is represented by evaluating first order Taylor series as: Commutation and conduction periods of a three-phase rectifier are shown in Figure 2a,b, along with the current waveforms in Figure 3. Current waveforms are trapezoidal shaped since inductors used at dc side, filters out peak lobes hence feeding load with constant DC current. Although, commutation period is affected by the higher filtering inductor value which could cause spikes in load voltage but proper selection of dc filter values could be the ultimate solution. Considering V m as the peak value of supply voltages given by Equation (2),   From the circuit in Figure 2 during commutation period, considering , to be the resistance and inductance on the ac side while , be the resistance and inductance on the dc side, and representing the positive and negative terminal potentials with respect to neutral point of the rectifier ac supply, system dynamic equations are:    From the circuit in Figure 2 during commutation period, considering R ac , L ac to be the resistance and inductance on the ac side while R dc , L dc be the resistance and inductance on the dc side, U p and U n representing the positive and negative terminal potentials with respect to neutral point of the rectifier ac supply, system dynamic equations are: Now during conduction period, system dynamic equations are: V a = R ac i a + L ac di a dt + U p and i a = I dc V c = R ac i c + L ac di c dt + U n and i b = 0, i c = −I dc Energies 2018, 11, 550 5 of 19 Solving the above equations for U p , U n and rearranging, Integrating (7) over 0 to µ and (8) over µ to π where R 1 = R dc + 3 2 R ac , L 1 = L dc + 3 2 L ac , R 2 = R dc + 2R ac and L 2 = L dc + 2L ac Averaging (9) and (10) over 0 → π AC Current Equation During commutation period, it can be seen from Figure 2a that During conduction period, it is evident from Figure 2b that For commutation angle derivation, evaluating i c (θ) (Equation (12)) at θ = µ To find i d and i q components from AC currents During commutation period: During conduction period: Now integrating (16) for commutation ( θ = 0 → µ ) and (17) for conduction ( θ = µ → π 3 ) periods and averaging i d , The complete average value model for three phase rectifiers is represented by Equations (11), (18) and (19).

Dynamic Average Value Model for 9-Phase Diode Rectifier
For a nine-phase diode rectifier with specifications given in Table 2, ac currents are derived by approximating load current using Equation (1). Commutation and conduction periods of the rectifiers are given in Figure 4a,b respectively where current flow during commutation and conduction period is shown by dotted lines. Current waveforms are shown separately in Figure 5.

Dynamic Average Value Model for 9-Phase Diode Rectifier
For a nine-phase diode rectifier with specifications given in Table 2, ac currents are derived by approximating load current using Equation (1). Commutation and conduction periods of the rectifiers are given in Figure 4a,b respectively where current flow during commutation and conduction period is shown by dotted lines. Current waveforms are shown separately in Figure 5.      Considering V m as the peak value of supply voltages by auto transformer given by Equation (20), average model for this rectifier is derived as mentioned below: (20) From the circuit in Figure 4 during commutation and conduction period separately, considering R ac , L ac to be the resistance and inductance on the ac side of each rectifier while R dc , L dc be the resistance and inductance on the dc side as shown in Figure 4, U p and U n representing the positive and negative terminal potentials with respect to neutral point of the rectifier ac supply, during commutation period system dynamic equations are: While during conduction period, system dynamic equations are: Solving the above circuit in Figure 4 during commutation and conduction period separately Let Integrating (27) over 0 to µ and (28) over µ to π 9 and averaging both over 0 to π AC Current Equations Now finding the DQ components of AC currents in Figure 5 i Consider current commutates from a 1 to a 0 as shown in above figure. Phase current equations is derived as: Integrating Equation (31) from 0 to θ range which includes commutation and conduction period Similarly For i a 1 (θ), put i a 0 + i a 1 = I dc in Equation (31) and Integrate both side from 0 to θ For commutation angle derivation, evaluate Equation (32) Vm ωL sin 2 π 9 (cos(2θ) − 2 cos(θ) + 1) + Kθ 2 2 sin(θ) 1 + cos π 9 + I dc0 sin(θ) 1 + cos π 9 + cos(θ) sin π 9 − Kµ 2 sin(θ) 1 + cos π 9 + cos(θ) sin π 9 (37) While during conduction period, cos(θ) cos π 9 + sin(θ) sin π Integrating (36) and (37) over 0 to µ and (39) and (40) over µ to π 9 range and averaging both over 0 to π 9 , 2ωL sin 2 π 9 (cos(2µ) − 4 cos(µ) + 3) + 2I dc0 sin π 9 cos(µ) + K sin π 9 π 9 − sin(µ) Vm 2ωL sin 2 π 9 (sin(2µ) − 4 sin(µ) + 2µ) + 2I dc0 sin(µ) sin π 9 + K sin π 9 (1 + cos(µ)) − π 9 − π 9 cos π 9 The complete average value model for nine phase rectifiers is represented by Equations (29), (41) and (42).

Simulations and Experimental Results
This section discusses the simulation and hardware results for three-phase and nine-phase rectifiers. Detailed models (switch models) possess a high frequency electromagnetic field and electromagnetic compatibility behavior, leading to thermal and mechanical stressing. These high order models suffer from slower execution process, computational complexity and mathematical instability due to bandwidth in megahertz region. Dynamic AVMs for these switching models proved to be faster, more efficient and effective tools for analyzing these systems, which is evident from the results shown here in this section. Using MATLAB 2015a environment, switch and average models both were simulated for 1.1 s with ode23tb solver and relative tolerance set to 1 × 10 −4 since it performs best in switching simulations. With the Solver Jacobian method set to full analytical, the switching model for the three phase rectifier switching model was completely executed in 3530 s while that of the nine phase rectifier switching model was completely executed in 3706 s. However, on the other hand, average model for three phase rectifier took 33 s and that of nine phase rectifier took 36 s for complete execution. These average models proved to be 107 and 104 times faster than switch model respectively.  Figure 12. In frequency domain, the dc output impedance Z out and ac input impedances, Z dd and Z qq of the three-phase rectifier are shown in Figures 13, 19 and 20 respectively. In frequency domain, the dc output impedance Z out and ac input impedances Z dd and Z qq of the nine-phase rectifier are shown in Figures 14, 21 and 22 respectively. The results of the derived model, switching model and prototype waveforms closely resembles hence validating the derived AVMs. For stability analysis, it is imperative to obtain the ac output impedance of the power supply connected to three phase loads. Using the technique discussed in Section 4, ac output impedance of the three-phase power supply is measured and compared with that of switch model as shown in Figure 18. Load voltage, current and input currents from the experimental setup were extracted using an HBM Gen7t data acquisition module (7T, HBM, Darmstadt, Germany) while an Agilent Technologies E5061B series (E5061B, Santa Clara, CA, USA) along with a Hushan PA 300 amplifier (PA-300, Guangzhou, China) were used for measuring the output impedance. For the sake of clarity in figures, waveforms for switching versus average model and those of average model versus prototype are plotted separately. Output impedance measuring setup is shown in Figure 12. In frequency domain, the dc output impedance and ac input impedances, and of the three-phase rectifier are shown in Figures 13, 19 and 20 respectively. In frequency domain, the dc output impedance and ac input impedances and of the nine-phase rectifier are shown in Figures 14, 21 and 22 respectively. The results of the derived model, switching model and prototype waveforms closely resembles hence validating the derived AVMs. For stability analysis, it is imperative to obtain the ac output impedance of the power supply connected to three phase loads. Using the technique discussed in Section 4, ac output impedance of the three-phase power supply is measured and compared with that of switch model as shown in Figure 18. Load voltage, current and input currents from the experimental setup were extracted using an HBM Gen7t data acquisition module (7T, HBM, Darmstadt, Germany) while an Agilent Technologies E5061B series (E5061B, Santa Clara, CA, USA) along with a Hushan PA 300 amplifier (PA-300, Guangzhou, China) were used for measuring the output impedance.

DC Impedance Measurement
The Agilent Network analyzer (E5061B, Santa Clara, CA, USA), having a frequency range from 5 Hz to 3 GHz, was used for measuring the small signal output impedance. Circuit setup is shown in Figure 12 where 1 Ω resistor is used for sensing the current in the return path and a capacitor of a higher rating than that of the output voltage is used for blocking high DC voltage so as to protect the network analyzer from being damaged. Sweep frequency signal of 34 mV ranging from 10 Hz to

DC Impedance Measurement
The Agilent Network analyzer (E5061B, Santa Clara, CA, USA), having a frequency range from 5 Hz to 3 GHz, was used for measuring the small signal output impedance. Circuit setup is shown in Figure 12 where 1 Ω resistor is used for sensing the current in the return path and a capacitor of a higher rating than that of the output voltage is used for blocking high DC voltage so as to protect the

DC Impedance Measurement
The Agilent Network analyzer (E5061B, Santa Clara, CA, USA), having a frequency range from 5 Hz to 3 GHz, was used for measuring the small signal output impedance. Circuit setup is shown in Figure 12 where 1 Ω resistor is used for sensing the current in the return path and a capacitor of a higher rating than that of the output voltage is used for blocking high DC voltage so as to protect the network analyzer from being damaged. Sweep frequency signal of 34 mV ranging from 10 Hz to 5 kHz was injected into the circuit through an amplifier and a small signal voltage and the current values were fed-back to the network analyzer through T and R terminals. Capacitor rated double that of operating load voltage was used for dc blocking. The current was measured using 1 Ω sense resistor as shown in Figure 12. Signal collected at T is divided by signal collected at R to obtain output impedance. This output impedance data, measured through impedance analyzer, was exported to MATLAB, identified into second order systems and redrawn for comparison with the output impedance obtained from derived average model as shown in Figures 13 and 14.

AC Impedance Measurement
The four ac impedance terms , , , and described in Section 1, can be measured using Equations (43) and (44). For determining and at , perturbation frequency is injected in while current is set to zero. However, for determining and , perturbation frequency is injected in while is to be set to zero. This frequency response in the form of magnitude and phase is further identified into second order transfer functions which could be utilized for stability analysis using Nyquist criteria discussed in Section 1.

AC Impedance Measurement
The four ac impedance terms Z dd , Z dq , Z qd , and Z qq described in Section 1, can be measured using Equations (43) and (44). For determining Z qq and Z dq at ω p , perturbation frequency is injected in i q while current i d is set to zero. However, for determining Z dd and Z qd , perturbation frequency is injected in i d while i q is to be set to zero. This frequency response in the form of magnitude and phase is further identified into second order transfer functions which could be utilized for stability analysis using Nyquist criteria discussed in Section 1.
For switch model and experimental setup, an easy approach of using single line to line current injection technique is used for three phase ac impedance measurement. General diagram of ac impedance measurement technique is shown in Figure 15 while line to line current injection technique for three phase diode rectifier impedance measurement is shown in Figure 16. This technique is also applied to nine phase diode rectification system by measuring the ac impedance of each rectifier individually. Specifications for current injection setup with three phase diode rectifier system, are given in Table 3. The chopper circuit, used for line to line shunt current injection, consists of a resistor and inductor in series with a bidirectional switch. Practically, line to line current is injected in any two of the three lines a, b and c by switching power MOSFETs (Motorola IRF 540, 150W, 27 Ampere, 100V) A and B alternatively, with 50% duty ratio, to introduce impedance variation at the interface junction. This causes current injection into system at the switching frequency. For calculating frequency response measurement at point ω p , two linearly independent frequency signals are injected at ω = ω p ± ω g . Matlab script, based on flow chart as given in Figure 17, is used for transformation of ABC (three-axes coordinate system of ac voltages and currents) time domain values into DQ0 (two-axes coordinate system of ac voltages and currents) synchronous reference frame and then performing fast Fourier transform (FFT) to extract injected frequency dq values of voltage and current. Output impedance of system 1 and input admittance of system 2 can be calculated by utilizing Equations (45) and (46) but, only Z dd and Z qq impedances of these systems are extracted as shown in Figures 18-22 respectively, for efficient utilization of space. These impedances are extracted from switch model as well as prototype using shunt current line to line injector circuit as shown in Figure 16. To avoid frequency overlapping, which may affect the results, multiples of fundamental frequency were not injected. This technique has a drawback of injecting considerable harmonics but they are mathematically removed by the FFT process.

Conclusions
With the addition of a linear variation term in the DC load current, it is evident from the results shown in the time domain and the frequency domain that there is a close resemblance, i.e., lower mean square error (MSE) with detailed model results. Similarly, increasing the order of equations further (adding quadratic term and higher) will fetch more improved results but the derivation will become more and more complex because of higher order derivative terms. Since the derived models are in close agreement with those of detailed models, the addition of higher terms to the dc load equation is exempted. To prove the validity and efficiency of derived average model, it is imperative to compare the results in time and frequency domain, under steady and dynamic states, with those of switching/detailed models/prototypes. The proposed average model is, therefore, compared in time domain and frequency domain, for three phase rectifier system as well as nine phase rectifier

Conclusions
With the addition of a linear variation term in the DC load current, it is evident from the results shown in the time domain and the frequency domain that there is a close resemblance, i.e., lower mean square error (MSE) with detailed model results. Similarly, increasing the order of equations further (adding quadratic term and higher) will fetch more improved results but the derivation will become more and more complex because of higher order derivative terms. Since the derived models are in close agreement with those of detailed models, the addition of higher terms to the dc load equation is exempted. To prove the validity and efficiency of derived average model, it is imperative to compare the results in time and frequency domain, under steady and dynamic states, with those of switching/detailed models/prototypes. The proposed average model is, therefore, compared in time domain and frequency domain, for three phase rectifier system as well as nine phase rectifier

Conclusions
With the addition of a linear variation term in the DC load current, it is evident from the results shown in the time domain and the frequency domain that there is a close resemblance, i.e., lower mean square error (MSE) with detailed model results. Similarly, increasing the order of equations further (adding quadratic term and higher) will fetch more improved results but the derivation will become more and more complex because of higher order derivative terms. Since the derived models are in close agreement with those of detailed models, the addition of higher terms to the dc load equation is exempted. To prove the validity and efficiency of derived average model, it is imperative to compare the results in time and frequency domain, under steady and dynamic states, with those of switching/detailed models/prototypes. The proposed average model is, therefore, compared in time domain and frequency domain, for three phase rectifier system as well as nine phase rectifier system shown in Sections 3 and 4, which shows sufficiently close agreement with those of the detailed/switch model results hence proving the validity of the proposed AVMs. Steady and dynamic states of the systems can be observed from a single figure since it will reduce the space usage. Appropriate extraction of AC/DC impedances of integral parts of the interconnected systems is important since it has to deal with the stability analysis of the system. The proposed AVMs are much faster than switch model/detailed models.