Improved control-to-output characteristics of a PWM buck-boost converter

Authors


Abstract

An indirect control variable for improving the control-to-output characteristics of a Pulse Width Modulation (PWM) buck-boost converter is introduced in this letter. The voltage gain and the small-signal model of the buck-boost converter are reviewed. The actual voltage command at one input of the PWM comparator is from the proposed indirect control variable and the peak value of the high-frequency PWM carrier. The resulted voltage gain function appears proportional to this indirect control command. Also the dependence of the DC gain of the control-to-output transfer function on the duty cycle is eliminated. Experimental results conform well to the theoretical analysis. Copyright © 2010 John Wiley & Sons, Ltd.

1. INTRODUCTION

DC-to-DC switching converters are widely used in the power electronics industry to serve as various AC-to-DC preregulators and DC power sources 1–4. Among many nonisolated topologies, the buck-boost converter is one capable of supplying an output voltage that can be stepped down or up from the input voltage 5–11. However, the voltage gain of the buck-boost converter is not a linear function of the duty cycle, and thus the control signal. Moreover, the small-signal control-to-output transfer function varies with the steady-state duty cycle. This is an undesirable feature when designing the feedback loop, especially if an adjustable output voltage is required. Computer simulations are performed to evaluate an adaptive feedback controller presented in 11. A nonlinear PWM control that requires the generation of an exponential carrier is proposed in 12 to improve the closed-loop performance. The control circuitry is somewhat complicated and the nonlinear modulating approach is an approximated solution rather than an exact one.

In this letter, the DC voltage gain and small-signal characteristics of a PWM buck-boost converter under continuous conduction mode (CCM) are reviewed. An indirect voltage command is introduced to exhibit a linearized DC voltage gain, and to eliminate the dependence of the DC gain of the control-to-output transfer function on the duty cycle 13. Experimental results on a prototype converter are given to verify the effectiveness of the presented control scheme.

2. PROPOSED INDIRECT CONTROL VARIABLE

Figure 1 shows a simplified schematic of the buck-boost power stage. RC is the equivalent series resistance (ESR) of the capacitor. R represents the load seen by the power stage output. When the switch is turned on, the inductor is charged by the input voltage. When the switch is turned off, the diode provides a conducting path for the inductor current, and the inductor discharges to the load. Not considering the parasitics, the voltage gain under CCM is expressed as

equation image(1)

where d is the duty ratio of the switch. Typically, the duty ratio is proportional to a control voltage, vd, which is compared with a triangular or a sawtooth-like high-frequency carrier, vt, to produce the gating signals. From (1) it is evident that the voltage gain is a nonlinear function of d and thus vd. Moreover, at a higher output voltage, a small perturbation in vd will result in a much larger swing in v0. The nonlinear transfer characteristic between vd and the voltage gain can be revealed in the following expression:

equation image(2)

where VT is the peak value of the carrier signal vt.

Figure 1.

Buck-boost power stage schematic.

The small-signal control-to-output transfer function in the s-domain is 14

equation image(3)

where equation image and equation image are the small-signal expressions for v0 and vd. D is the steady-state value of the duty cycle. Q and ω0 are the quality factor and the natural frequency. ωz1 is a zero due to the ESR of the capacitor. ωz2 is a right-half-plane (RHP) zero which depends on the load level and the duty cycle.

equation image(4)
equation image(5)
equation image(6)
equation image(7)

It is clearly seen that a term relating to the duty ratio is included in the denominator in (3). Therefore, the DC gain of T(s) is varied as the duty ratio changes, which is unfavorable for the feedback loop design with an adjustable output voltage.

From (2), to achieve the linearity between a new control signal vd′ and the voltage gain, it is required that

equation image(8)

From (8), vd can be related to vd′ as

equation image(9)

That is, the new control voltage or the indirect control voltage, vd′, is first multiplied by the peak of the carrier signal, which is usually a constant, then divided by the sum of VT and vd′ itself. The result is fed into the PWM comparator. The block diagram illustrating the proposed control mechanism is shown in Figure 2. From (8), the small-signal ratio between these two control variables can be derived. It is assumed that

equation image(10)
equation image(11)

where variables in upper-case represent the DC quantities and the italic lower cases with a cap are the small signals. Substituting (10) and (11) into (8), and neglecting the product of two small perturbations yields

equation image(12)
Figure 2.

The block diagram of the proposed control scheme.

Therefore, (12) can be rearranged as

equation image(13)

From (8), thus

equation image(14)

By substituting (14) into (3), the modified small-signal control-to-output transfer characteristic with the new control voltage vd′ becomes

equation image(15)

From (15), it is noticed that for a constant input voltage, as VT is constant, the incremental DC gain remains constant for the CCM buck-boost converter under the proposed control method. Also, the locations of the dominant open-loop zeros and poles are theoretically unchanged. There is no need to reconsider the issues of the stability and dynamic response.

3. EXPERIMENTAL RESULTS

A prototype converter is built to perform the experiments with the following parameters: Vi = 50V, L = 2.85mH, C = 300µF/450V, and R = 100Ω. The switching frequency is 60 kHz. One possible implementation of the block diagram in Figure 2 is shown in Figure 3. TL494 is adopted as the PWM comparator. VT is about 3.35 V in a TL494. A low-cost analog multiplier, such as AD633, is placed in the feedback loop of a noninverting amplifier to perform the division operation in (9) 15. The output of the multiplier is obtained as

equation image(16)
Figure 3.

One possible implementation of the proposed control.

As the multiplier is in the feedback path of the noninverting amplifier, W can also be derived as

equation image(17)

By equating (16) and (17), it can be proven that

equation image(18)

By setting R1 = R2 = Rf = 10kΩ and R3 = 49.7kΩ, M is calculated to be 3.35, which is equal to VT. Therefore, the conversion between vd and vd′ as described in (9) can be fulfilled.

Figures 4(a) and (b) show the recorded nonlinear and linear transfer characteristics between V0 and the control voltages. A fairly constant gain is observed for the whole operating range in Figure 4(b).

Figure 4.

Transfer characteristics between (a) V0 and vd and (b) V0 and vd′.

Figures 5(a) and (b) depict, respectively, the open-loop frequency responses (the gains are scaled down by 42, or equivalently 32 dB, for measurements) under the conventional and the proposed control schemes for Vi = 120V, D = 0.4, and V0 = 80V. The dominant pole frequencies (at the phase of − 90) for both cases are around 100 Hz. It is thus concluded that the transients of the compensated system are stable as well. The DC gain for the proposed system reduces slightly from 8 to − 0.9dB. The difference of the DC gain, − 8.9dB, is equivalent to (1 − D)2.

Figure 5.

Frequency responses under (a) the conventional and (b) proposed control schemes.

Figure 6 shows the output voltage regulation under a step load current change with a proportional (P) feedback loop. In Figure 6(a), when I0 drops suddenly from 2 to 0 A with the conventional control method, v0 rises and then stabilizes within 50 ms. The same dynamic performance can be observed in Figure 6(b) for the proposed system. However, the proposed system has a larger steady-state error due to a smaller DC gain. Nevertheless, this nonzero steady-state error can be eliminated by applying a proportional-integral (PI) feedback technique.

Figure 6.

Transients and steady-state behaviors under (a) the conventional and (b) proposed control schemes (V0: 1V/div., I0: 1A/div., Time: 10ms/div.)

4. CONCLUSION

A novel method for improving the control-to-output transfer function of a PWM buck-boost converter is realized in this letter. An indirect control signal is introduced. From the DC and the small-signal characteristics of the discussed buck-boost converter, the linearization of the steady-state voltage gain is achieved. Also the dependence of the DC gain of the control-to-output transfer function on the duty cycle is eliminated. The implementation of the presented controller requires additional analog IC's. Nevertheless, the proposed control method can be implemented via digital approach as well, and applied to any buck-boost derived converter topologies.

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