Controller Design

9.1 INTRODUCTION

In all switching converters, the output voltage v(/) is a function of the input line voltage vj,i), the duty cycle d(t), and the load current /щ). as well as the converter circuit element values. In a dc-dc converter application, it is desired to obtain a constant output voltage v(t) = V, in spite of disturbances in v(t) and jr a/). und in spite of variations in the convertercircuit element values. The sources of these disturbances and variations are many, and a typical situation is illustrated in Fig. 9.1. The input voltage v.(()of an off-line power supply may typically contain periodic variations at the second harmonic of the ac power system frequency (100 Hz or 120 Hz), produced by a rectifier circuit. The magnitude of v(t) may also vaty when neighboring power system loads are switched on or off. The load current ij nt)may contain variations of significant amplitude, and a typical power supply specification is that the output voltage must remain within a specified range (for example, 3.3 V ± 0,05 V) when the load current takes a step change from, for example, full rated load current to 50% of the rated current, and vice versa. The values ofthe circuit elements are constructed to a certain tolerance, and so in high-volume manufacturing of a converter, converters are constructed whose output voltages lie in some distribution. It is desired that essentially all of this distribution fall within the specified range; however, this is not practical to achieve without the use of negative feedback. Similar considerations apply to inverter applications, except that the output voltage is ac.

So we cannot expect to simply set the dc-dc converter duty cycle to a single value, and obtain a given constant output voltage under all conditions. The idea behind the use of negative feedback is to build a circuit that automatically adjusts the duty cycle as necessary, to obtain the destred output voltage with high accuracy, regardless of disturbances in v({) or ij, ,0 or variations in component values. This is

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Fig. 9.1 llie output voltage of a typical switchiug couvertetr is a function of tlie line input voltage u, the duty cycle d, and die load current i/. (a) open-loop bucit converter; (b) functional diagram illustrating dependence of v on the independent quantities v., d, and j,

a useful thing to do whenever there are variations and imknowns that otherwise prevent the system from attaining the desired performance.

A block diagram of a feedback system is shown in Fig. 9.2. The output voltage v{t) is measured, using a sensor with gain M(s). In a dc voltage regulator or dc-ac inverter, the sensor circuit is usually a voltage divider, comprised of precision resistors. The sensor output signal M(.s)v(.<} is compared with a reference input voltage Vj), The objective is to make H{.t)v{.t) equal to i.*). so that v(*) accurately follows VjyC.f) regardless of disturbances or ctmipment variations in the compensator, pulse-width modulator, gate driver, or converter power stage.

The difference between the reference input v,s) and the sensor output His)v{s) is called the error signal v(s). If the feedback system works perfectly, then vJis) = H(s)vU), and hence the error signal is zero. In practice, the error signal is usually nonzero but nonedieless small. Obtaining a small enor is one of the objectives in adding a compensator network G.(,s) as shown in Fig. 9.2. Note that the output voltage v(s) is equal to the error signal vjs}, multiplied by the gains of the compensator, pulse-width modulator, and converter power stage. If the compensator gain G(s) is large enough in magnitude, then a small error signal can produce the required output voltage v{t) = Г for a dc regulator {Q: how should Я and i-then be chosen?). So a large compensator gain leads to a small error, and therefore the otitput follows the reference input with g(k>d accuracy. This is the key idea behind feedback systems.

The averaged small-signal converter models derived in Chapter 7 are used in the following sections to find the effects of feedback on the small-signal transfer functions of the regulator. The loop gain T(.s) is defined as the product of the small-signal gains in the forward and feedback paths of the feedback

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Fig. S.2 Feedback loop for regulatian ofthe output voltage: (a) buck oonvertei, with feedback loop btock diagram; (b) functional block diagram of the feedback system.

1сюр. It is found that the transfer function frotn a disturbance to the output is multiplied by the factor 1/(1 + TTs)). Hence, when the loop gain Tis large in magnitude, then the influence of disturbances on the output voltage is small. A large loop gain also causes the output voltage v{s) to be nearly equal to v s)IH{s), with very little dependence on the gains in the forward path of the feedback 1<кгр. So the loop giiin magnitude [[T[ is a measure of how well the feedback system works. All of these gains can be easily constructed using the algebra-on-the-graph method; this allows easy evaluation of important closed-loop performance measures, such as the output voltage ripple resulting frcmi 121) Hz rectification ripple in v,(f) or the closed-loop output impedance.

Stability is another important issue in feedback systems. Adding a feedback Iwp can cause an otherwise well-behaved circuit to exhibit oscillations, ringing and ovetxhoot, and other undesirable behavior. An in-depth treatment of stability is beyond the scope of this book; however, the simple phase margin criterion for assessing stability is used here. When the phase inargin ofthe loop gain Tis positive, then the feedback system is stable. Moreover, increasing the phase margin causes the system transient response to be better behaved, with less overshoot and ringing. The relation between phase margin and closed-loop response is quantified in Section 9.4.

.An exaraple is given in Section 9.5, in which a compensator network is designed for a dc regu-