# Steady-State Analysis and Control of Active Current Injection Based Three-Phase PFC Rectifier

Ramkrishan Maheshwari, Ankur Srivastava, Prashant Surana, Thomas Ebel, Lasse Chr Larsen, Egon Hansen

► To cite this version:

Ramkrishan Maheshwari, Ankur Srivastava, Prashant Surana, Thomas Ebel, Lasse Chr Larsen, et al.. Steady-State Analysis and Control of Active Current Injection Based Three-Phase PFC Rectifier. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0208 . hal-05100728

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# Steady-State Analysis and Control of Active Current Injection Based Three-Phase PFC Rectifier

Ramkrishan Maheshwari¹, Ankur Srivastava¹, Prashant Surana², Thomas Ebel¹, Lasse Chr. Larsen³, Egon Hansen³

¹Centre for Industrial Electronics, University of Southern Denmark, Soenderborg, Denmark

²Dep. of Hydro and Renewable Energy, Indian Institute of Technology Roorkee, Roorkee, India

³OJ Electronics A/S, Soenderborg, Denmark
E-Mail: ramkrishan@sdu.dk

## Keywords

Harmonic injection, Grid-connected converter control, Steady-State Analysis, Converter control.

## Abstract

The paper investigates a boost-integrated current injection-based unity-power-factor three-phase ac-dc converter. It presents a steady-state stress analysis and identifies various circuit modes which are required to design the converter. A simple feedforward controller is proposed and simulated.

## Introduction

Unity power factor (UPF) rectifiers are commonly used as ac-dc converters in applications with stringent grid power quality requirements [1]. Typically, pulse width modulated (PWM) rectifiers are used. However, they are expensive and less efficient [2]-[3]. To reduce the cost and losses, a diode-bridge rectifier incorporating a current-injected (CI) circuit may be employed. The CI circuits utilize either passive [4]-[6] or active components [7]-[9] for controlled current injection. Passive component-based circuits are bulky compared to the active component-based circuits [10].

A simple active CI-based circuit is proposed in [8] which is capable of injecting the current to make input current in-phase with the input voltage. However, it is not possible to control and regulate the output voltage. The output voltage is the same as the output voltage of the six-pulse diode bridge rectifier. To control the output

voltage, a boost converter is required at the output of the converter [7]. In addition to the voltage control, the boost converter needs to control the inductor current in a way that the CI-based rectifier circuit sees the boost converter as a constant power load. Furthermore, the boost converter stage also reduces the overall efficiency of the rectifier circuit.

It is also possible to integrate a boost converter with the CI circuit [11]. This boost integrated current injection (BICI) circuit requires output voltage as well as two currents to be controlled. Therefore, it has two control inputs to the converter. A detailed analysis of the controller design was presented in [11]. The controller is designed for a dc-operating point which was obtained by averaging over a fundamental cycle not in a switching period. However, the detailed analysis over a switching period is very crucial for designing the circuit and controller.

In this paper, a detailed analysis of the circuit in a switching period is presented. Different modes of the circuit are analyzed. Based on this analysis, the circuit model averaged over a switching period is presented, and this model is used to design a controller.

To simplify the derivation of the model, a three-phase symmetry of the input voltage is utilized. Based on the symmetry, six different modes are identified similar to [11]. However, it is shown that only three modes are needed for analysis in this paper. Based on the analysis, various components, such as inductors and capacitors can be selected. Furthermore, this analysis is used to find the current stress in various components such as inductors, MOSFETs and IGBTs. These current values are needed for component
selection, loss calculation, sizing of the heat sink etc. In addition, a controller is also presented which utilizes the steady-state values as feedforward terms.

This paper presents the detailed circuit operation in Section II. Based on this, a detailed stress analysis for various components is presented in Section III. Furthermore, equations to select inductance and capacitance are also presented in Section III. The control approach using a system model averaged over a switching period is presented in Section IV. Using the control approach, the simulation results with the controller are presented in Section IV. Conclusions are presented in Section V.

# Circuit Operation

## Topology description

A block schematic of the topology under consideration is given in Fig. 1. The topology can be divided into two parts: passive rectifier and BICI circuit. The BICI circuit consists of two inductors and one capacitor in addition to 6 low-frequency switches ($S_1-S_6$), 2 high-frequency switches ($S_7-S_8$), and a high-frequency diode ($D$). This circuit is connected to the output of the passive diode bridge rectifier. The diode bridge rectifier connects the phase with the maximum and minimum voltage values to terminal $P$ and $N$, respectively. In addition, the switches $S_1-S_6$ are turned on such that they connect the phase with the middle voltage value to terminal $M$. This simplifies the circuit to be analyzed which is as

shown in Fig. 2. The circuit can be considered to have two inputs $V_{PM}$ and $V_{MN}$. The circuit takes the advantage of symmetry of the balanced three-phase voltages which are given by

$$ 
\begin{aligned}
v_a &= V_m \sin(\omega t) \\
v_b &= V_m \sin\left(\omega t - \frac{2\pi}{3}\right) \\
v_c &= V_m \sin\left(\omega t - \frac{4\pi}{3}\right)
\end{aligned} 
\qquad (1) 
$$

Therefore, the circuit needs to be analyzed for $-\pi/6 < \omega t < \pi/6$, where $\omega$ is the supply frequency in rad/s. The circuit operation is described in the following subsection.

## Modes of operation

There are two time-varying inputs in the simplified circuit and one output voltage which should be a dc value in the steady state. To get dc output voltage from two time-varying input voltages, the circuit needs two independent control duty cycles, $d_{S7}$ and $d_{S8}$. In addition, the circuit should control the inductor current in a way that the three-phase input voltage and current are in-phase. If this constraint is included, the positive and negative inductor currents, for $-\pi/6 < \omega t < \pi/6$, are given by

$$ i_P = I_m \sin\left(\omega t - \frac{4\pi}{3}\right) \qquad (2) $$

and

$$ i_N = I_m \sin\left(\omega t - \frac{2\pi}{3}\right) \qquad (3) $$

<figure><img src="image_2.png" /><figcaption>Fig. 1: Block schematic of the boost integrated current injection circuit-based rectifier.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Fig. 2: Reduced simplified circuit</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3: Different modes during condition $d_{S7} > d_{S8}$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4: Different modes during condition $d_{S7} < d_{S8}$.</figcaption></figure>

respectively. To achieve the desired current and voltage, the circuit has three modes in a switching period. In addition, there are two conditions for the duty cycles: $d_{S7} > d_{S8}$ and $d_{S7} < d_{S8}$. Due to these facts, the circuit needs to be analyzed in 6 modes as shown in Fig. 3 and Fig. 4. For both conditions, the volt-second balance can be applied to find the steady state value of $d_{S7}$ and $d_{S8}$. For

condition 1, the volt-second balance equation for inductors $L_p$ and $L_n$ are given by

$$ (v_{\max} - v_{\text{mid}})d_{S7} + (v_{\max} - v_{\text{mid}} - v_o)(1 - d_{S7}) = 0 \quad (4) $$

$$ (v_{\text{mid}} - v_{\text{min}} - v_o)(d_{S7} - d_{S8}) + (v_{\text{mid}} - v_{\text{min}})(1 - d_{S7} + d_{S8}) = 0 \quad (5), $$

respectively. Therefore,
$$d_{S7} = 1 - \frac{v_{\max} - v_{\min}}{v_o} \qquad (6)$$

$$d_{S7} - d_{S8} = \frac{v_{\max} - v_{\min}}{v_o} \qquad (7)$$

$$d_{S8} = 1 - \frac{v_{\max} - v_{\min}}{v_o} \qquad (8)$$

Similarly, for condition 2, $d_{S7} < d_{S8}$,

$$d_{S7} = 1 - \frac{v_{\max} - v_{\min}}{v_o} \qquad (9)$$

$$d_{S8} - d_{S7} = \frac{v_{\max} - v_{\min}}{v_o} \qquad (10)$$

$$d_{S8} = 1 - \frac{v_{\min} - v_{\max}}{v_o} \qquad (11)$$

where $v_{\max}$, $v_{\min}$, and $v_{\min}$ are the maximum, middle, and minimum voltage values among three-phase voltages. $v_o$ is the output voltage. These equations of the duty cycles, the input voltages, and the inductor currents are shown in Fig. 3 and Fig. 4 for $-\pi/6 < \omega t < \pi/6$ for different modes of two conditions. The equation for the duty cycles, the input voltages, and the inductor currents for mode 1 and mode 5 are identical. Similarly, the equations for mode 2 and mode 6 and mode 3 and mode 5 are identical. Therefore, only three modes are needed to analyze the peak, average, and rms currents of various components. These analyses are presented in the following section as they are needed to select the components for the converters.

## Component Stress Analysis and Selection

### Inductor selection

To select the inductor for the topology, it is required to find the ripple current as given by

$$\Delta i_{Lp} = \frac{v_o(1-d_{S7})d_{S7}T_S}{L_p}, \quad \text{for } d_{S7} > d_{S8}, \qquad (12)$$

and

$$\Delta i_{Ln} = \frac{v_o(1-d_{S8})d_{S8}T_S}{L_n} \quad \text{for } d_{S7} > d_{S8}. \qquad (13)$$

From (12) or (13), the maximum value of the current ripple will occur at 0.5 duty cycle if such conditions occur. Therefore,

$$L_p = L_n = \frac{v_o T_S}{4 \Delta i_{\max}} \qquad (14)$$

where $T_S$ is the switching period, $V_o$ denotes the dc output voltage, and $\Delta i_{\max}$ is the maximum peak current ripple.

The rms value of the inductor current can be obtained by

$$I_{\text{rms},L} = \sqrt{\frac{3}{\pi} \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} I_m^2 \sin^2\left(\omega t - \frac{4\pi}{3}\right) d\omega t} = \\ I_m \sqrt{\frac{4\pi + 3\sqrt{3}}{8\pi}} \qquad (15)$$

and the peak value of the inductor current is $I_m$.

## MOSFET(S7-S8) Stress Analysis

The switches S7-S8 are realized by MOSFETs. To select a MOSFET, it is necessary to find the rms current flowing through the MOSFETs. For this, the current through MOSFETs should be calculated in each mode in addition to the duration of each mode. The current through the MOSFETs and duration for each mode are shown in Fig. 3 and Fig. 4. Using them, the rms current through MOSFET S7 is given by

$$I_{\text{rms},S7} = \left( \frac{3}{\pi} \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \left( I_m^2 \sin^2\left(\omega t - \frac{4\pi}{3}\right) \left(1 - \frac{\sqrt{3}v_m \cos(\omega t)}{v_o}\right) + I_m^2 \sin^2(\omega t) \frac{\sqrt{3}v_m \sin(\omega t + \frac{\pi}{6})}{v_o} \right) d\omega t \right)^{\frac{1}{2}} \qquad (16)$$

This can be simplified to

$$I_{\text{rms},S7} = I_m \sqrt{\frac{3}{\pi} \left( \frac{4\pi + 3\sqrt{3}}{24} - \frac{2\sqrt{3}v_m}{3v_o} \right)} \qquad (17)$$

Similarly, it can be shown that

$$I_{\text{rms},S8} = I_{\text{rms},S7} \qquad (18)$$

The maximum blocking voltage across a MOSFET is equal to output voltage $V_o$.

## IGBTs (S1-S6) Stress Analysis

These devices are realized using IGBTs and switch at frequency of 50 Hz. The maximum blocking voltage for the device is $\sqrt{3}V_m$. The rms and average current stress for the device are given by [9]

$$I_{\text{rms},S IGBT} = I_m \sqrt{\left(\frac{1}{12} - \frac{\sqrt{3}}{8\pi}\right)} \qquad (19)$$

$$I_{av,S IGBT} = \frac{1}{\pi} \int_{0}^{\frac{\pi}{6}} I_m \sin(\omega t) d\omega t = \frac{l_m}{\pi} \left(1 - \frac{\sqrt{3}}{2}\right) \qquad (20)$$

## High Frequency Diode (D)

The rms and average current through the diode are needed to select a diode. In the converter under discussion, a high switching frequency diode 'D' is connected before the output capacitor, and the current stress analysis can be done using the
expressions of the current and duration of each mode from Fig. 3 and Fig. 4. The rms and average currents are given by

$$I_{\text{rms,D}} = \left( \frac{3}{\pi} \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \left( I_m^2 \sin^2 \left( \omega t - \frac{2\pi}{3} \right) \frac{\sqrt{3}V_m \sin\left(\omega t + \frac{\pi}{6}\right)}{V_O} + I_m^2 \sin^2 \left( \omega t - \frac{4\pi}{3} \right) \frac{\sqrt{3}V_m \sin\left(\omega t + \frac{5\pi}{6}\right)}{V_O} \right) d\omega t \right)^{\frac{1}{2}} \quad (21)$$

$$I_{\text{rms,D}} = I_m \sqrt{\frac{5 \sqrt{3} V_m}{2\pi V_O}} \quad (22)$$

$$I_{\text{av,D}} = \frac{3}{\pi} \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \left( -I_m \sin \left( \omega t - \frac{2\pi}{3} \right) \frac{\sqrt{3}V_m \sin \left( \omega t + \frac{\pi}{6} \right)}{V_O} + I_m \sin \left( \omega t - \frac{4\pi}{3} \right) \frac{\sqrt{3}V_m \sin \left( \omega t + \frac{5\pi}{6} \right)}{V_O} \right) d\omega t \quad (23)$$

$$I_{\text{av,D}} = \frac{3V_m I_m}{2V_O} = \frac{P_{\text{in}}}{V_O} \quad (24)$$

## Capacitor

To find the value of the output capacitance, it is required to find the current through the capacitor. For this, a circuit as shown in Fig. 5. The system of Fig. 5 has diode current as an input current source, and the averaged diode current over a switching period is given by

$$I_{\text{av,D}}(\omega t) = -I_m \sin \left( \omega t - \frac{2\pi}{3} \right) \frac{\sqrt{3}V_m \sin \left( \omega t + \frac{\pi}{6} \right)}{V_O} + I_m \sin \left( \omega t - \frac{4\pi}{3} \right) \frac{\sqrt{3}V_m \sin \left( \omega t + \frac{5\pi}{6} \right)}{V_O} \quad (25)$$

$$I_{\text{av,D}}(\omega t) = \frac{3V_m I_m}{2V_O} = \frac{P_{\text{in}}}{V_O} \quad (26)$$

<figure><img src="image_6.png" /><figcaption>Fig. 5: Equivalent circuit for capacitor stress analysis.</figcaption></figure>

As the average current of the diode over a switching period is constant and equal to the load current, the capacitor current will have only a switching frequency component, and the capacitance decides the voltage ripple caused by the product of diode conduction time and the difference of the diode current and the load

current. The maximum value of the product will cause maximum charge in the capacitor, and the capacitance should be selected in a way that the voltage ripple requirement is met. Therefore,

$$\frac{c_{\min \Delta v_{o,max}}}{T_s} = \max \left( \left( -I_m \sin \left( \omega t - \frac{2\pi}{3} \right) - I_O \right) \frac{\sqrt{3}V_m \sin \left( \omega t + \frac{\pi}{6} \right)}{V_O} + \left( I_m \sin \left( \omega t - \frac{4\pi}{3} \right) - I_O \right) \frac{\sqrt{3}V_m \sin \left( \omega t + \frac{5\pi}{6} \right)}{V_O} \right) \quad (27)$$

$$\frac{c_{\min \Delta v_{o,max}}}{T_s} = \max \left( \frac{3V_m I_m}{2V_O} - \frac{\sqrt{3}V_m I_O}{V_O} \left( \sin \left( \omega t + \frac{\pi}{6} \right) + \sin \left( \omega t + \frac{5\pi}{6} \right) \right) \right) \quad (28)$$

$$\frac{c_{\min \Delta v_{o,max}}}{T_s} = \frac{3V_m I_m}{2V_O} - \frac{\sqrt{3}V_m I_O}{V_O} \left( \min \left( \sin \left( \omega t + \frac{\pi}{6} \right) + \sin \left( \omega t + \frac{5\pi}{6} \right) \right) \right) \quad (29)$$

The function $\left(\sin\left(\omega t + \frac{\pi}{6}\right) + \sin\left(\omega t + \frac{5\pi}{6}\right)\right)$ has minimum value at $\omega t = \pm \frac{\pi}{6}$ for $-\pi/6 < \omega t < \pi/6$. Therefore,

$$\frac{c_{\min \Delta v_{o,max}}}{T_s} = \frac{3V_m I_m}{2V_O} - \frac{\sqrt{3}V_m I_O}{V_O} \left( \frac{\sqrt{3}}{2} \right) \quad (30)$$

$$C_{\min} = \frac{3V_m (I_m - I_O)T_s}{2V_O \Delta v_{o,max}} \quad (31)$$

## Control approach

The converter output voltage is required to be controlled. Therefore, an outer loop with a voltage controller is used. The output of the voltage controller is used as the current reference which is multiplied with normalized maximum and minimum voltage values among three-phase ac voltages. This will yield two current references for the top and bottom inductors. The current references are compared with the measured inductor currents and the current error is fed to the current controllers. The outputs of the current controllers are the duty cycles of the top ($S_7$) and bottom ($S_8$) devices. One important point to be noted is that the duty cycles of $S_7$ and $S_8$ are not complementary as there are modes when both devices are turned on. Therefore, the converter needs two independent signals for duty cycles of $S_7$ and $S_8$. However, the duty cycles are not constant in the steady state. They are functions of voltage signals as given by (6), (8), (9) and (11). These duty cycles are pre-calculated and provided as a feedforward term in the controller. They are added to the output of the current controller as
shown in the overall controller block diagram of Fig. 6. Furthermore, these variables are calculated by the circuit model averaged over a switching period as compared to the model averaged over a fundamental cycle given in [11].

## Simulation Results

Based on the previous section, various components can be selected for the converter for the specifications given in Table 1.

For the given load resistance, the input power rating of the converter is assumed to be 5 kW. This corresponds to $I_m \approx 10$ A. Using these values, the values of various components are given in Table 2. The inductance is calculated for 1 A pk-pk current ripple. The capacitance is calculated for 14 V pk-pk voltage ripple.

The output voltage of the system is controlled using a cascaded closed-loop approach as shown

<table><caption>Table 1. Converter Specification</caption><tr><td>V<sub>m</sub></td><td>325 V</td></tr><tr><td>V<sub>O</sub></td><td>700 V</td></tr><tr><td>Load Resistance (R)</td><td>100 Ω</td></tr><tr><td>Switching frequency</td><td>100 kHz</td></tr></table>

<table><caption>Table 2. Component parameters</caption><tr><td>L<sub>P</sub> = L<sub>N</sub></td><td>1.75 mH</td></tr><tr><td>C<sub>min</sub></td><td>1.5 μF</td></tr></table>

<figure><img src="image_7.png" /><figcaption>Fig. 6: Controller block diagram.</figcaption></figure>

in Fig. 6. The duty cycles for the MOSFETs as given by (6)-(11) are decided by the input voltage. The proposed control structure with the converter is simulated in MATLAB Simulink®. The simulation results are presented in Fig. 7 - Fig. 9. Fig. 7 shows the input voltage and input currents. Unity power factor is maintained with total harmonic distortion (THD) = 4.4% of the input current. The MOSFET currents, inductor currents, diode current, and load current are shown in Fig. 8. An important point to note is that due to switching between different conditions, the problem is controlling a switched system. Due to this, the output capacitance is kept at 50 μF to reduce the effect of switching. Fig. 9 shows control duty cycles, $d_{S7}$ and $d_{S8}$.

<figure><img src="image_8.png" /><figcaption>Fig. 7: Input voltage and input current.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 8: The MOSFET current, inductor current, diode current, and load current.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 9: Duty cycle for switches $S_7$ and $S_8$.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 10: Input voltage and input current response against step change of the load current from 7 A to 14 A at $t = 0.05$ s.</figcaption></figure>

<table><caption>Table 3. Component current stress</caption><thead><tr><td>Variable</td><td>Calculated</td><td>Simulated</td></tr></thead><tbody><tr><td>I<sub>rms,L</sub></td><td>8.4 A</td><td>8.41 A</td></tr><tr><td>I<sub>pk,L</sub></td><td>10 A</td><td>10 A</td></tr><tr><td>I<sub>rms,S<sub>7</sub></sub></td><td>4.4 A</td><td>4.5 A</td></tr><tr><td>I<sub>rms,D</sub></td><td>8 A</td><td>8.05 A</td></tr><tr><td>I<sub>av,D</sub></td><td>7 A</td><td>7 A</td></tr><tr><td>I<sub>rms,S<sub>IGBT</sub></sub></td><td>1.2 A</td><td>1.35 A</td></tr><tr><td>I<sub>av,S<sub>IGBT</sub></sub></td><td>0.4 A</td><td>0.5 A</td></tr></tbody></table>

The response of the overall system at the input side due to load disturbance is shown in Fig. 10. At $t = 0.05$ s, the load current is increased from 7 A to 14 A. The grid current increases in the same proportion without any overshoot.

The current stresses in various components are verified in the simulation. The calculated and simulated values are listed in Table 3. The comparison verifies the formulas derived in the previous section which can be used to select the components during design or for comparison with other topologies.

## Conclusion

This paper discusses a boost-integrated current injection-based unity power factor rectifier circuit. Different modes of the circuit are analyzed in detail. Based on the analysis, design equations are derived which can be used to find the inductance and capacitance values used in the circuit. Furthermore, the analysis also helps to find different current stress formulas that are necessary to select different components. In addition, these formulas can be used to compare different topologies.

The paper also discusses a simple control structure that utilizes the analysis to feedforward two terms. The proposed control structure is

simulated in MATLAB Simulink®. The simulation results verify the results obtained from the analysis.

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