Small-signal analysis of a single-stage bridgeless boost half- bridge AC/DC converter with bidirectional switch

Received Mar 22, 2021 Revised Sep 28, 2021 Accepted Oct 5, 2021 A small-signal analysis of a single-stage bridgeless boost half-bridge alternating current/direct current (AC/DC) converter with bidirectional switches is performed using circuit averaging method. The comprehensive approach to develop the small signal model from the steady state analysis is discussed. The small-signal model is then simulated with MATLAB/Simulink. The small-signal model is verified through the comparison of the bode-plot obtained from MATLAB/Simulink and the simulated large signal model in piecewise linear electrical circuit simulation (PLECS). The mathematical model obtain from the small-signal analysis is then used to determine the proportional gain Kp and integral gain Ki. In addition, the switch large-signal model is developed by considering the current and voltage waveforms during load transients and steady-state conditions.


INTRODUCTION
Nowadays, the usage of electronic equipment's has increased drastically. The requirement set up by a regulatory body such as 80Plus [1] and "Energy Star" [2], [3] has gain consumer attention which demand less harmonic, high power factor and higher efficiency system. Significantly, these trend makes the power factor correction (PFC) becoming important aspect even for low power electronic equipment's. Thus, researchers have proposed a new PFC technique which comply the setup standard such as IEC555-2 and IEC6100-4-7. In addition, the galvanic isolation is also one of the state requirements in international electrotechnical commission (IEC) standard [4] for the safety of the end user. However, the total cost is also one of the key factors to meet the setup requirements. Thus, as the comparison to two-stage scheme, single stage scheme is cost effective [5]. Single stage power converter integrates the PFC and direct current-direct current (DC-DC) converter circuit which significantly reduce the number of components [6]. However, the slow input bridge diodes and bulky DC link capacitor significantly leads to power losses. The proposed single stage semi-bridgeless converter without bulky DC link capacitor [7], [8] significantly reduce the converter power losses. Thus, improved the conversion efficiency with less components count.
Any new proposed circuit topologies should consider the main analysis of both large-signal model and small-signal model [9]. The large-signal analysis is for components design such as transformer, inductance and capacitance. The small-signal model is to investigate the behavior of a power convertor.
Several methods have been presented to derive the small signal model of a switched mode converter namely circuit averaging [10]- [17], state-space averaging [18]- [23] and current injected equivalent circuit approach [21], [22]. In circuit averaging technique, the averaging and linearization operations are performed directly from the converter circuit. The small-signal equation is derived from the steady state operation of the converter circuit. The steady-state equation is derived from the circuit operation and waveform analysis in one switching period. It is assumed that the components are lossless and ideal. This is to simplify the equation which derive from steady-state and small-signal analysis. In this paper, the small-signal of a bridgeless AC/DC converter with bidirectional switches as depicted in Figure 1 is presented. The full bridgeless of the proposed circuit topology eliminated the bridge rectifier of a conventional circuit topology and the semi-bridgeless diodes [7], [8] in a conventional bridgeless circuit. The idea of the proposed converter is based on the primary side [18] but without diodes presented which further eliminate the line frequency rectifier losses. In addition, the bidirectional switch concept is derived from the topologies in [24]- [31]. The bidirectional switches simplify the mode of circuit operation and control signal. The small-signal model is develop using the similar procedure in [19], [23]. The small-signal model is then used to find the control-to-output transfer function of the converter. Then, the derived transfer function of the proposed circuit is simulated in MATLAB/Simulink. The bode plot obtain from the MATLAB/Simulink is then verified with the bode-plot obtained from the large circuit model which simulated with PLECS. In addition, a switch large-signal model is proposed which is derive from the steadystate equation. The model is designed with the circuit parameters as stated in Table 1.
The circuit level simulation and experimental verification results will be presented in another report. This report will only be discussed on the verification of the derived small signal equation with MATLAB and PLECS software. In a nutshell, this works.
-Eliminates the drawbacks of a bridge rectifier in conventional AC/DC converter circuit and a semibridgeless AC/DC conventional circuit -Eliminates the crossover distortion of input current at high frequency of input supply -Input current harmonic which comply to the IEC 6000-3-2 standard -Improve input power factor -Reduce the components count -Simplify the modes of circuit operation and control signal In section 2, the details steady state analysis of the proposed circuit topology will be discussed. The modes of circuit operation with switching waveforms are illustrated to support the derivation of releated equations. Section 3 presented the small signal analysis and derivation steps to obtained the transfer function; control-to-ouput, output-to-input, and impedance. Next, the converter small signal block diagram in Simulink and in PECS is proposed. Section 4 presented the results from both MATLAB/Simulink and PLECS for verification. In addition, the large signal switch model is obtained. Finally, section 5 gives the conclusion.

STEADY STATE ANALYSIS
In this analysis, all switches and components are assumed to be ideal and the analysis only carried out at the positive half line cycle. Figure 2 shows the four mode of circuit operation at positive half cycle of input voltage . At Mode 1, the Kirchhoff voltage law (KVL) equation at primary side is: As depicted in Figure 2 (a), capacitor 1 is connected in parallel with the transformer primary winding. Capacitor 1 discharged to primary winding. The value of capacitor 1 and 2 are also equal, and capacitor 2 is in charging mode. As depicted from the key waveform, the capacitor voltage 1 and 2 is equal to the difference between the input voltage and the boost inductor voltage. The capacitor voltage is at minimum or maximum at starting of each switching mode. This is due to the capacitor which is almost fully charged or fully discharged at the end of switching mode. Thus, assume the capacitor 2 voltage; 2 ≈ 0 and the capacitor 1 voltage; 1 equal to transformer primary voltage . The value of the inductor 1 and 2 are equal. Therefore, (1) can be simplified as.
The transformer turn ratio is given by: where transformer winding number of turns at primary side and secondary side . Thus, the voltage equation at secondary side + in Mode 1 is.
At Mode 2 in Figure 2 (b), the KVL equation at primary side is.
Thus, the inductor voltage can be written as: Mode 3 in Figure 2 (c) is similar to Mode 1. Thus, the voltage equation at secondary side − ; where is voltage drop at ouput inductor and is the load voltage. The boost mode operation at primary side in Mode 4 is the same as Mode 2. Thus, the KVL equation for mode 4 in Figure 2 (d) is the same as (5). The voltage drop equation at output inductor is. (1), thus the boost inductor voltage is; The average voltage across inductor within each switching period is zero. Thus, it can be concluded in (8) that for each half line period.
Hence, (7) can be simplified as. The boost inductor is in charging in Mode 1 and 3 and discharge at Mode 2 and 4 of the switching periods. It is assumed that the magnitude of input voltage is constant. Consequently, the input current also assumes to be constant. The boost inductor current works in continuous current mode. Therefore, based on the balanced volt-second; Hence, by (12), the relationship of duty cycle at Mode 1 1 and Mode 2 2 , The capacitor voltage in one switching period is zero volt. Therefore, (13) can be simplified as.
The total period of Mode 1 and Mode 2 is half of the switching period. Therefore; Substitute (15) in (14), thus; The input current directly flows to the boost inductor 1 and 2 . Thus, the boost inductor current is equal to input current . Assume that all input power transferred to output. Thus, the input current is.
where is power factor angle and is load resistance. Hence, the boost inductor is choose such that.
where is the switching frequency As depicted from switching waveform, the input current flow to the half bridge capacitor 1 and 2 at a period of (1 − 1 ) . Thus; the current flow trough capacitor 1 ; 1 and 2 ; 2 .
Therefore, the expression of half bridge capacitor is.
where ∆ is the minimum to maximum charging voltage of half bridge capacitor. As depicted from Figure 3, each half bridge capacitor is charged approximately from zero to peak supply voltage. There is no connection between the input boost circuit to output circuit at Mode 2 and Mode 4. Therefore, no energy transferred from primary side to secondary side of the transformer in this mode. Thus, the output inductor and output capacitor are discharged to the load . Therefore; By assuming the output capacitor is fully charge, the output inductor current is.

= =
Thus, the maximum and minimum output inductor current , and , are.
In order to ensure output inductor working in continuous conduction mode; Since the output capacitor current wave-shape same as output inductor current, the charge in output capacitor is equal to the area under the wave-shape. Thus, the ∆ is; By examining the above waveform, the peak inductor current can be represented as.
Based on the Kirchhoff current law, the current at capacitor 1 is.
By substituting (2) and (30) in (31). Thus; At 1 , the output inductor is connected to the transformer secondary winding through the rectifier diode. Therefore, output inductor current same as secondary current of the transformer. Thus, In addition, the output capacitor current ripple is.
At 2 , since all switches are in OFF state, the boost circuit has no connection to the secondary circuit loop. Therefore, the half bridge capacitor current same as boost inductor current. Thus, As depicted from the capacitor current waveform, capacitor current is constant across the period of (1 − 1 ). Thus.
At 2 , because has no energy transfer from primary circuit, hence.
This derived equation will further apply in development of small signal and steady state modelling.

SMALL-SIGNAL ANALYSIS
For the proposed circuit topology, the circuit averaging and average switch model are used to develop the small signal model. The similar procedure [10] applied to develop the small signal model. The small-signal model is then used to find the control-to-output transfer function of the converter. In the case of the proposed bridgeless converter, the input current is the same current flow to the inductor and also to the half bridge capacitors. Thus, only inductor loop and output capacitor loop are presented.
In the steady state, the average inductor voltage is zero but there is no net change in inductor current over one switching period. During transients or ac variations, the average inductor voltage is not zero and this leads to net variations in inductor current. The net change in inductor current over one switching period is The small ripple approximation equation at Mode 2 interval is In the previous steady state analysis, the input voltage is considered to be constant. However, in the next derivation of small signal model, the sinusoidal input voltage will be used. Thus, the small signal equation describes on how the low frequency circuit components waveform evolve in time. The input sinusoidal can be represented as. DC terms only containing DC quantities. Therefore, DC term will be removed in order to get small signal linearize equation. The second order ac terms are nonlinear because it contains the product of ac quantities. On the other hand, the second order ac terms are much smaller than the first order form. Therefore, the second order terms will be neglected. Thus, only first order equation left. The first order term contains a single ac quantity which mostly multiplied by a constant coefficient such as DC term. These first order equation is the linearize equation which describes small signal ac variations.
The proposed circuit average model is depicted in Figure 4 (a) which derived from the of (47) and (48). The average switch network in Figure 4   (53) The model in Figure 5 is simulated with control design tools in the MATLAB/Simulink. The Simulink control design is used to tune the value of controller proportional and integral coefficient. The control circuit parameters are determined based on the obtained steady state equation. The voltage follower control method is used to verify the operation of the proposed converter in circuit simulator and experimentally. This control method is used to regulate the output voltage to the desired value. Thus, the controller is expected to provide the appropriate duty cycle for any load changes. This closed loop feedback controller is designed using the control-to-output transfer function expressed by (49).
It is then compared with the transfer function that obtained from the switch model in PLECS/Simulink. This is to prove that the derived transfer function (49) is correct. The sinusoidal waveform with frequency of 10-1000 Hz and amplitude range of (0.1-4) V is used to perturb the converter duty cycle in the PLECS model. Thus, the duty cycle that drive the MOSFET's in the PLECS circuit is a summation of 1 and the perturbed signal. Figure 6 shows the block diagram of the switch Model with perturbation of duty cycle in the PLECS/Simulink.

RESULTS
The circuit model in PLECS is used to verify the developed transfer function in Simulink. The bodeplot diagram obtained from the mathematical expression is compared with the bode-plot diagram obtained in the PLECS/Simulink switch model as depicted in Figure 7. The plot shows the same magnitude and Q-factor obtained from the mathematical expression and PLECS/Simulink switch model. However, a small difference of corner frequency around 10 Hz is observed. This due to ideal consideration in developing the mathematical expression. Thus, the results show good agreement between the mathematical expression control to output transfer function and the PLECS/Simulink switch model. Therefore, the mathematical expression (49) which developed using circuit averaging techniques is validated. In this work, PI controller is used such that the output voltage is regulated at 20 regardless the output power condition. As depicted in Figure 8, the crossover frequency is at 2.92 Hz with overall system phase margin of 90°. Table 2 shows the system performance and robustness which is obtained from the MATLAB/Simulink automated proportional-integrated-derivate (PID) tuning apps. It shows that the system meets the Nyquist stability criterion whereby the system is stable if the phase lag at the crossover frequency is less than 180°. However, an overshoot of 0.474% occurred which have to be accepted.
The small-signal model can be used to determine the stability of the propose system. However, there still have some loop hole even though small signal model can ensure the stability of the system. A small signal model only considered a small variation of circuit signal. Therefore, the large signal model is important to observe the overall time response analysis either during steady-state or transient. The large signal affects the operating point and non-linear components. In this work, the large-signal model as depicted in Figure 9 is developed by considering the current and voltage waveforms during load transients and steadystate conditions.

CONCLUSION
The small-signal and steady-state analysis has been derived and discussed in detail. The results show good agreement between the mathematical expression control to output transfer function and the

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PLECS/Simulink switch model. The mathematical expression of a small-signal model is used to design the PI controller. It shows that the system is stable and meet the Nyquist stability criterion. In addition, the large signal model based on switch model is developed. The details large signal circuit and experimental results will be discuss in another report.

FUTURE WORKS
This study is to verify the mathematical expression of a small signal model with Simulink/PLECS. The details design, circuit simulation, and experimental verification will be reported in another paper. The proposed circuit topology performances tested at input supply 115 Vac 50 Hz with output 20 Vdc 5 A. Hence, for future works, it is suggested to study the circuit performances at another parameters. In addition, the secondary side synchronous rectifier can be applied to furthermore reduce the overall power lossess.