9.3 Con.finjcdoii of ihe Important Quantities I/I I + T) aitd 7/(1 ¥ T} and the Closed-Loop Transfer Functions 337

Equation (9.4) also predicis, thai ihe convener ouiput impedance is reduced, from 2 ,л) to

So the feedbaclt loop also reduces ihe convener oulpul impedance by a facior of 1/(1 + 1\s)), and the influence of load current variations on the ouiput voltage is reduced.

9.2.2 Feedback Causes the Transfer Function from the Reference Input to the Output to be Insensitive to Variations in the tiains in the Forward Path of the Loop

According to Eq. (9.4), the closeddoop transfer function from v lo v is

v(j)

1 Г()

H(s) i + T(s) i9J)

If the loop gain is large in magnitude, thai is, ! T э> 1, then (I + Г) = T and Г/( 1 + 7 Г/7= 1. The transfer function then becomes

f /) H()

which is independent of V, nd OJ,). So provided that the loop gain is large in magnitude, then variations in Gj.(j), Уц, and OJs) have negligible effect on the output voltage. Of course, in the dc regulator application, is constant and t = 0. But Eq. (9.Й) applies equally well to the dc values. For example, if the system is linear, then we can write

V ; И(0) 1 + 7-(0) H(0)

So to make the dc output voltage Vaccurately follow the dc reference V we need only ensure lhat the dc sensor gain ЩУ) and dc reference V,j:are well-known and accuraie, and lhai 1(0) is large. Precision resisiotM are normally used lo realize H, but componenis wilh tighlly-controlled values need not be u.sed tn G., the pulse-width modulator, or the power stage. The sensitivity of the output voltage to the gains in Ihe forward path is reduced, while the sensitivity of v to the feedback gain H and the reference input vy is increased.

9.3 CONSTRUCTION OF THE IMPORTANT QUANTITIES 1/(1 + T) AND r/(l + T) AND THE CLOSED-LOOP TRANSFER FUNCTIONS

The transfer functions in Eqs. (9.4) to (9.9) can be easily consiructed using the algebra-on-the-graph method [4j, Let us assume that we have analyzed the blocks in our feedback system, and have plotted the Bode diagram of Ц T[s) Ц.То use a concrete example, suppose that the result is given in Fig. 9.5, for which Д.9) is

80 dB

60 dB

40 dB

20 dB

-20 dB

-40 dB

1 Hz 10 Hz [00 Bz I kHz

Fig. 9.5 Magnitude nf the loop gain example, E15, (9.10).

lO kHz

100 kHz

(9.10)

This example appears sotnewhat complicated. But the loop gains of practical voltage regulators are often even more complex, and may contain four, five, or more poles. Evaluation of Eqs. (9.5) to (9.7), to determine the closed-loop transfer functions, requires quite a bit of worlt, The loop gain Tmust be added to 1, and the resulting numerator and denominator must be refactored. Using this approach, it is difficult to obtain physical insight into the relationship between the closed-loop transfer functions and the loop gain. In consequence, design of the feedback кюр to meet specifications is difficult.

Using the algebra-on-the-graph method, the closed-loop transfer functions can be constructed by inspection, and hence the relation between these transfer functions and the loop gain becomes obvious. Let us first investigate how to pk)t l 77(1 +7) . It can be seen from Fig. 9.5 that there is a frequency called the crossover frequency, where Г = 1. At frequencies less than Д., Г > 1; indeed, II ГЦ 1 for/- ;/. Hence, at low frequency, (1 -b 7) = Г, and + T) T/T ]. At frequencies greater than /j, II ГЦ < 1, and T I for /:*/,.. So at high frequency, (1 + Г) 1 and T/{1 + T) = T/l = T. So we have

I Гоггэ.1 T ffrrj- I

(9.11)

The asymptotes corresponding to Eq. (9.11) are relatively easy to construct. The low-frequency asymptote, for /</r> is 1 or 0 dB. The high-frequency asymptotes, for f>f., follow Г. The result is shown in Fig. 9,6.

So at low frequency, where 71 is large, the reference-to-output transfer function is

I m I

v jis)-lm \ + ns)~H{s)

(9,12)

9j Consnucrioji of tk€ imporiant Qmmnties l/(I +Tl and ЩI + T) and the Closed-Loop Transfer Funaions

SOdB 60 dB 40 dB 20 dB OdB -20 dB

Cmssover /j frequency

20 dB/d=cade\/c

-40dB/deeade

1 Hz

lOHz

I kHz

10 kHz

100 kHz

Fig. 9.6 Uraj)liieal construction of the asymptotes of 7/(1 +7) [. Exact curves are ornitted.

This is the desired behavior, and the feedback loop works well at frequencies where II T\\ is large. At high frequency (J~ where { ГЦ is small, the reference-to-output transfer function is

v fis) Я(.0 1 + Т(л)

T(.0 TU) G,.(a)C,/f}

(9.13)

This is not the desired behavior; in fact, this is the gain with the feedback connection removed 0). At high frequencies, the feedback loop is unable to reject the disturbance because the bandwidth of Г is limited. The reference-to-output transfer function can be constructed on the graph by multiplying the r/(l + D asymptotes of Fig. 9.6 by 1/Я.

We can plot the asymptotes of 1/(1 -K 7) using similar arguments. At low frequencies where II TH l.then {i+T) = 7. and hence 1/(1 -l- T) = 1/Г. At high frequencies where T\\ l,then (1 +7:i = 1 and ЩХ-Т) ]. So we have

1 + T{s)

for7 ]:*- 1 for i Гi i

(9.14)

The asymptotes for the Tl.s-) example of Fig. 9.5 are plotted in Fig. 9.7.

At low frequencies where T is large, the disturbance transfer function from v, to v is

C (i) G (.f)

Ш.

v(.f) l4-r(..) T(i)

(9.15)

magnitude reduced by the factor 1/[ ГЦ. So if, for example, we want to reduce this transfer function by a factor of 20 at 120 FIz, then we need a кюр gain T\\ of at least 20 26 dB at 120 FIz. The disturbance transfer function froin v to v can be constructed on the graph, by multiplying the asymptotes of Fig. 9.7 by the asymptotes for GJs).

Similar arguments apply to the output impedance. The closed-loop output impedance at low fre-