A Cross Regulation Analysis for Single-Inductor Dual-Output CCM Buck Converters

Abstract

Cross regulation is a key technical issue of single-inductor multiple-output (SIMO) DC-DC converters. This paper investigates the cross regulation in single-inductor dual-output (SIDO) Buck converters with continuous conduction mode (CCM) operation. The expressions of the DC voltage gain, control to the output transfer function, cross regulation transfer function, cross coupled transfer function and impedance transfer function of the converter are presented by the time averaging equivalent circuit approach. A small signal model of a SIDO CCM Buck converter is built to analyze this cross regulation. The laws of cross regulation with respect to various load conditions are investigated. Simulation and experiment results verify the theoretical analysis. This study will be helpful for converter design to reduce the cross regulation. In addition, a control strategy to reduce cross regulation is performed.

I. INTRODUCTION

These days, portable devices such as digital assistants and mobile phones require multiple power supplies with different output voltages [1]-[3]. A single-inductor multiple-output (SIMO) DC-DC converter with only one inductor is a good candidate for providing multiple outputs. It has attracted a lot of attention by the researchers due to its advantages of low cost, high efficiency and small volume size [4]-[6].

According to difference in the inductor current operation mode, a single-inductor dual-output (SIDO) DC-DC converter has three operation modes including the inductor current discontinuous conduction mode (DCM), the pseudo-continuous conduction mode (PCCM) and the continuous conduction mode (CCM). Time multiplexing control techniques in the DCM are used in [6] and [7]. Although these approaches can eliminate cross regulation under light loads, a large inductor current and output voltage ripple exist under heavy loads. To overcome this large ripple, a SIDO PCCM DC-DC converter was proposed in [8]. However, an additional freewheeling switch results in power loss during the freewheeling stage [9]-[10].

Compared with the SIDO DCM and PCCM DC-DC converters, the SIDO CCM DC-DC converter has the advantages of a low ripple and less power loss due to it not having a zero inductor current stage or a freewheeling stage. However, when one SIDO CCM DC-DC converter load changes, the other output voltage changes. This is refered to as cross regulation and it plays a major role in the performance of SIDO converters [11]-[13]. Cross regulation influences the normal work of a converter and can even cause system instability. In spite of this, the SIDO CCM DC-DC converter has been researched due to its numerous advantages. Therefore, it is important to research the cross regulation of the SIDO CCM DC-DC converter. Control techniques for suppressing the cross regulation of SIDO CCM Buck converters was researched in [14]-[17]. However, a theoretical analysis of cross regulation is lacking.

In this paper, the cross regulation of SIDO CCM Buck converters is studied. According to the time averaging equivalent circuit approach [18], a small signal model to analysis the cross regulation of a SIDO CCM Buck converter is proposed. Based on this, a small signal model of a voltage controlled SIDO CCM Buck converter is established to analyze the law of cross regulation with the load changed. The analysis results can be used for the designing of the converter load condition to reduce cross regulation.

This paper is organized as follows: Section II illustrates the SIDO CCM Buck converter including the operation principle and small signal model. The small signal model of a voltage controlled SIDO CCM Buck converter is established in Section III. Cross regulation analyses with different load parameters are performed in Section IV. A control strategy to reduce cross regulation is provided in Section V. Simulation and experiment verification is performed in Section VI. A conclusion is provided in Section VII.

 

II. SIDO CCM BUCK CONVERTER

A. Operation Principle

Fig. 1 shows a circuit diagram of a SIDO CCM Buck converter with an output A and an output B. It consists of one inductor L, two output capacitors Ca and Cb, one input voltage source Vin, and two outputs with voltages Voa and Vob. The power switch Si and freewheeling diode D control the input power, and the power switches Sa and Sb are used to control the power distribution from the input to the two outputs. The output voltages Voa and Vob are regulated by adjusting the duty cycles Di, Da and Db. When a SIDO Buck converter operates in CCM, there is Da+Db=1, i.e. control pulses of output A and output B are complementary.

Fig. 1.Circuit diagram of SIDO CCM Buck converter.

There are three operation timings due to the duty cycles Di>Db, Di=Db and Di

Fig. 2.Steady-state timing and operation mode of SIDO CCM Buck converter.

Mode I: The power switches Si and Sb are turned on, Sa and the freewheeling diode D are turned off, the inductor current iL increases to the peak current IL1 with a slope of (Vin-Vob)/L.

Mode II: Si is turned off, D is turned on, Sa is kept off and Sb is kept on, while iL decrease to IL2 with a slope of -Vob/L.

Mode III: Si is kept off and D is kept on, Sa is turned on and Sb is turned off, and iL decreases with a slope of -Voa/L until the beginning of next switching cycle.

Based on time averaging equivalent circuit approach, the DC voltage gains of output A and output B are as:

As shown in (1) and (2), the DC voltage gains of a SIDO CCM Buck converter are associated with Di and Db. The output voltages Voa and Vob can be regulated by adjusting Di and Db. In addition, the DC voltage gains of a SIDO CCM Buck converter are associated with Ra and Rb. The variation of one output will affect the other output, i.e. cross regulation exists in this converter.

B. Small Signal Modeling for a SIDO CCM Buck Converter

According to the time averaging equivalent circuit approach, Si and Sa can be replaced by the controlled voltage sources si , while D and Sb can be replaced by the controlled current source . Then the time averaging AC small singal equivalent circuit can be given in Fig. 3, where are given by:

Where are the small signal disturbance variables of the inductor current iL, Di and Db, while in are the small signal disturbance variables of Vin, Voa and Vob, respectively.

Fig. 3.AC small signal equivalent circuit of SIDO CCM Buck converter.

From the AC small signal equivalent circuit of a SIDO CCM Buck converter, as shown in Fig. 3, it can have:

where

From (3) to (5), the control to output transfer functions G11(s) and G22(s) of output A and output B are:

The cross coupled transfer functions G21(s) and G12(s) of output A and output B are:

The output impedances Z11(s) and Z22(s) of output A and output B are:

The cross regulation transfer functions, Z21(s) of output B to output A and Z12(s) of output A to output B, are:

From (12), it can be seen that Z21(s) = Z12(s).

Select the currents of the two outputs in a SIDO CCM Buck converter as system inputs. Similarly, select the voltages of the two outputs in a SIDO CCM Buck converter as system outputs. Thus, a SIDO CCM Buck converter can be equivalent to a dual-input and dual-output system. According to the transfer functions from (6) to (12), the small signal model of a SIDO CCM Buck converter is established as shown in Fig. 4. This is used to analyze the cross regulation between the two outputs of the converter in this paper.

Fig. 4.The small signal model of SIDO CCM Buck converter.

As shown in Fig. 4, the output current of output A affects the output voltage of output B through the cross regulation transfer function Z12(s), and the output current of output B affects the output voltage of output A through the cross regulation transfer function Z21(s).

 

III. MODELING OF A VOLTAGE CONTROLLED SIDO CCM BUCK CONVERTER

There are two outputs in a SIDO CCM Buck converter. Therefore, at least two control loops are required to realize the control of a SIDO CCM Buck converter. For the small signal model of the SIDO CCM Buck converter shown in Fig. 4, two PI compensation functions F11(s), F22(s) and two PWM transfer functions Gm1(s), Gm2(s) are introduced from the outputs to keep the output voltage at their order values. Thus, the small signal model of a voltage controlled SIDO CCM Buck converter can be obtained, as shown in Fig. 5(a). The circuit diagram of the voltage controlled SIDO CCM Buck converter is shown in Fig. 5(b). As shown in Fig. 5(b), the feedback voltage Voa determines the duty cycle Di of switch Si, and feedback voltage Vob determines the duty cycle Db of switch Sb. Since Da+Db=1, the duty cycle Da of switch Sa is also determined by Vob.

Fig. 5.Voltage controlled SIDO CCM Buck converter.

F11(s), F22(s) and Gm1(s), Gm2(s) are:

To simplify the analysis, assume that Vm1 = Vm2 = 1 in (14). Thus, the transfer function Gm1(s) = Gm2(s) = 1. According to Fig. 5(a), the output impedances of output A and of output B are:

The cross regulation transfer functions of output B to output A and of output A to output B are:

Based on (17) and (18), the cross regulation of the voltage controlled SIDO CCM Buck converter can be analyzed.

 

IV. CROSS REGULATION ANALYSIS OF THE CONVERTER

A. Frequency Domain Analysis

1) Cross Regulation of Output B to Output A with Different Rb or of Output A to Output B with Different Ra.: Fig. 6(a) gives the transfer function with different Rb. This shows that the low frequency gain of the transfer function decreases with a decrease of Rb, i.e. the cross regulation of output B to output A decreases gradually with a decrease of Rb.

Fig. 6.The cross regulation of one output with different parameters to another output.

Fig. 6(b) gives the transfer function with different Ra. This shows that the low frequency gain of the transfer function decreases with a decrease of Ra, i.e. the cross regulation of output A to output B decreases gradually with a decrease of Ra.

From the frequency domain analysis shown in Fig. 6, it can be concluded that a heavier cross regulation from output A to output B occurs when the load of output A is lighter. It is same for the cross regulation from output B to output A.

2) Cross Regulation of Output A to Output B with Different Rb or of Output B to Output A with Different Ra.: The cross regulation of output A to output B with different Rb and of output B to output A with different Ra are investigated in this section.

Fig. 7(a) gives the cross transfer function with different Ra. This shows that the low frequency gain of the transfer function decreases with an increase of Ra, i.e. the affection of output B to output A decreases gradually with an increase of Ra. Fig. 7(b) shows the cross transfer function with different Rb. This shows that the low frequency gain of the transfer function decreases with an increase of Rb, i.e. the affection of output A to output B decreases with an increase of Rb.

Fig. 7.The cross regulation of one output to the another output with different parameters.

The frequency domain analysis results shown in Fig. 6 and Fig. 7 show that a heavier cross regulation from output B to output A occurs when output B has a lighter load condition and when output A has a heavier load condition. Likewise, a weaker cross regulation from output B to output A occurs when output B has a heavier load condition and when output A has a lighter load condition. Thus, when the load change is in output B, to reduce the cross regulation to output A, output B should be designed with a heavier load condition and output A should be designed with a lighter load condition.

 

V. CROSS REGULATION REDUCTION CONTROL STRATEGY

In this section, a control strategy to reduce cross regulation is proposed based on the small signal model shown in Fig. 4(a). The voltages of output A and output B can be expressed as:

It can be seen that cross regulation can be reduced by adding output current compensation functions.

Based on a small signal equivalent model of a voltage control SIDO CCM Buck converter, the output current compensation functions Z1(s) and Z2(s), as shown in Fig. 8, are introduced to reduce cross regulation. With these output current compensation functions, the output voltage can be represented by:

with:

Fig. 8.The circuit schematic and small signal model of output current control SIDO Buck converter.

Theoretically, cross regulation can be eliminated by introducing Z1(s) and Z2(s). However, the derivation process involves some approximations. Therefore, cross regulation is not perfectly eliminated but it is greatly suppressed.

Fig. 9 depicts the frequency response for the cross regulation of output B to output A. The dashed line and solid line represent a voltage controlled and an output current controlled SIDO CCM Buck converter, respectively. It can be seen that the low frequency gain of the output current controlled SIDO CCM Buck converter is lower than that of the voltage control. This means that the cross regulation of the output current controlled SIDO CCM Buck converter is smaller.

Fig. 9.Bode plot of voltage control and output current control SIDO CCM Buck converter for Voa with Iob changed.

 

VI. SIMULATION AND EXPERIMENT RESULTS

A. Frequency Domain Analysis

MATLAB/Simulink based simulations of a SIDO CCM Buck converter with the circuit parameters listed in Table I are presented in this section to investigate the analysis results.

TABLE ICIRCUIT PARAMETERS OF SIDO CCM BUCK CONVERTER

1) Cross Regulation of Output B to Output A with Different Rb or of Output A to Output B with Different Ra.: Rb=66Ω and 33Ω correspond to the output currents Iob = 50mA and 100mA, respectively. Ra=36Ω and 18Ω correspond to the output currents Ioa=50mA and 100mA, respectively.

Fig. 10(a) shows the transient waveform when Iob is step changed from 50mA to 100mA. The cross regulation of output B to output A is 30mV. Fig. 10(b) shows the transient waveform when Iob is step changed from 100mA to 200mA. The cross regulation of output B to output A is 20mV. From Fig. 10(a) and Fig. 10(b), it can be seen that a heavier cross regulation of output B to output A occurs when Iob is step changed from 50mA to 100mA than when it is step changed from 100mA to 200mA. i.e. heavier cross regulation from output B to output A occurs at a lighter load of output B.

Fig. 10.The transient-state simulation waveforms of cross regulation of output B to output A with different Rb or output A to output B with different Ra.

Fig. 10(c) shows the transient waveform when Ioa is step changed from 50mA to 100mA. The cross regulation of output A to output B is 220mV. Fig. 10(d) shows the transient waveform when Ioa is step changed from 100mA to 200mA. The cross regulation of output A to output B is 190mV. From Fig. 10(c) and Fig. 10(d) it can be seen that a heavier cross regulation of output A to output B occurs when Ioa is step changed from 50mA to 100mA than when it is step changed from 100mA to 200mA, i.e. heavier cross regulation from output A to output B occur at a lighter load of output A.

2) Cross regulation of output A to output B with different Rb or of output B to output A with different Ra.: Rb=33Ω and 16.5Ω correspond to the output currents Iob=100mA and 200mA, respectively. Ra=18Ω and 9Ω correspond to the output currents Ioa=100mA and 200mA, respectively.

A transient state simulation waveform when Ioa is equal to 100mA and Iob is step changed from 100mA to 200mA is shown in Fig. 11(a). It is shown that the cross regulation of output B to output A is 20mV under this condition. The transient state simulation waveform when Ioa is equal to 200mA and Iob is step changed from 100mA to 200mA is shown in Fig. 11(b). As shown in Fig. 11(b), the cross regulation of output B to output A is 60mV. Thus, it is verified that when output A operates at a lighter load, a weaker cross regulation of output B to output A is produced. In addition, when output A operates at a heavier load, a greater cross regulation of output B to output A is produced.

Fig. 11.Transient-state simulation waveforms of cross regulation of output A to output B with different Rb or output B to output A with different Ra.

The transient state simulation waveform when Iob is equal to 100mA and Ioa is step changed from 100mA to 200mA is shown in Fig. 11(c). It can be seen that the cross regulation of output A to output B is 170mV. The transient state simulation waveform when Iob is equal to 200mA and Ioa is stepped changed from 100mA to 200mA is shown in Fig. 11(d). As shown in Fig. 11(d), the cross regulation of output A to output B is 200mV. Thus, it is verified that when output B operates at a lighter load condition, a weaker cross regulation of output A to output B is produced. In addition, when output B operates at a heavier load condition, a greater cross regulation of output A to output B is produced.

B. Experiment Results

The experimental study of a SIDO CCM Buck converter is employed to verify the analysis and simulation results. The prototype of the experiment is shown in Fig. 12 and the circuit parameters are listed in Table I, which are the same as the ones used in the simulation study. The power stage has been designed based on the circuit diagram shown in Fig. 1. The power switches Si, Sa and Sb are IRF540N MOSFETs (n-channel), and their gates are connected to an IR2110 MOS driver. The diode is a Fast Recovery MUR560. The control signals Vgsi, Vgsa and Vgsb are obtained as the outputs of a microcontroller DSP TMS320F2812, via processing the state variables (Voa and Vob), which are sampled once per period Ts. The microcontroller determines duty ratios considering the constant output voltages within a period. The integral action is implemented digitally. Hence, the PI parameters are loaded into the microcontroller memory. Experimental diagrams corresponding to the simulation results are shown in Fig. 13 and 14, respectively.

Fig. 12.The prototype for the experiment.

1) Cross Regulation of Output B to Output A with Different Rb or of Output A to Output B with Different Ra.: Fig. 13(a) and Fig. 13(b) show transient waveforms when Iob is step changed from 50mA to 150mA and from 100mA to 200mA. Fig. 13(c) and Fig. 13(d) show transient waveforms when Ioa is step changed from 50mA to 150mA and from 100mA to 200mA. The experiment results of the cross regulation are given in Table II.

Fig. 13.Transient-state experiment waveforms of cross regulation of output B to output A with different Rb or output A to output B with different Ra.

TABLE IIEXPERIMENT RESULTS

2) Cross regulation of output A to output B with different Rb or of output B to output A with different Ra.: Fig. 14(a) and Fig. 14(b) show transient waveforms when Iob is step changed from 100mA to 200mA and when Ioa is step changed from 100mA to 200mA. Fig. 14(c) and Fig. 14(d) show transient waveforms when Ioa is step changed from 100mA to 200mA and when Iob is step changed from 100mA and 200mA. The experimental results of cross regulation are also given in Table II.

Fig. 14.The transient-state experiment waveforms of cross regulation of output A to output B with different Rb or output B to output A with different Ra.

The data in Table II shows that when Iob is step changed from 50mA to 150mA (Ioa=100mA) and when Ioa is step changed from 50mA to 150mA (Iob=100mA), greater cross regulation of output B to output A and of output A to output B is produced, than when there is a step change from 100mA to 200mA. When output A and output B operate at Ioa=100mA (Iob=100mA→200mA) and Iob=100mA (Ioa=100mA→200mA), a weaker cross regulation of output B to output A and output A to output B will be produced, when compared with Ioa=200mA (Iob=100mA→200mA) and Iob=200mA (Ioa=100mA→200mA). Therefore, the experimental results efficiently verify the analysis and simulation results of the SIDO CCM Buck converter. In addition, it is obvious that the transient performance of the converter is good.

 

VII. CONCLUSIONS

In this paper, a small signal model that is suitable for analyzing the cross regulation of a SIDO CCM Buck converter, is established based on the time averaging equivalent circuit approach. The operation principle of a SIDO CCM Buck converter is introduced and a detailed small signal analysis process is presented. Based on the proposed small signal model, a small signal model of a voltage controlled SIDO CCM Buck converter is established. The law of cross regulation with load changes is studied. It is shown that when the load of output B changes, the disturbance on output A caused by cross regulation is heavier since output A operates at a heavier load condition and output B operates at a lighter load condition. To reduce the cross regulation caused by various load changes in output B, a Buck converter should be designed with a lighter load in output A and a heavier load in output B. When the load of output A changes, the opposite conclusion is reached. Furthermore, to reduce cross regulation based on the small signal model, a current feedback control strategy is developed. The frequency domain analysis shows that the designed feedback controller can significantly reduce cross regulation. Simulation and experimental results obtained from a SIDO Buck converter verified that the law of cross regulation exist in the converter.