60 dB

- OdB

1 Hi

10 Шх 100 kHz

10 Hi 100 Hz 1 kHz

FiB. 9.31 ConstiwiionofllriUndll 1/(1 + 7> Ц wirii the WD-compeiisator of Fig, 9.30,

unchanged by the inverted zero. The lo<)p oontinues to exhibit a crossover frequency of 5 kHz.

So that the inverted zero does not significantly degrade the phase tnargin, let tis (somewhat arbitrarily) choose/l to be tme-tenth ofthe crossover freqtiency, or 5Ш Hz. The inverted zero will then increase the loop gain at frequencies below 500 Hz, improving the low-frequency regulation of the output voltage. The loop gain of Fig. 9.31 is obtained. The magnitude of the quantity 1/(1 +7) is also con-slnicted. h can be seen thai the inverted zero at 500 Hz causes the magnitude of 1/(1 + T) at 100 Hz to be reduced by a factor of approximately (100 Hz)/(500 Hz) = 1/5. The total attenuation of 1/(1 + T) al 100 Hz is -32.7dB. A 1 V, 1(X) Hz variation in v(r) would now induce a 12 mV variatitm in v(0. Further improvements could be obtained by increasing/l; however, this would require redesign ofthe PD portion of the compensator to maintain an adequate phase margin.

The line-to-outptit transfer ftmction is constructed iit Fig. 9.32. Both the open-loop tiansfer function G j,(j), Eq. (9.51), and the closed-loop transfer function G,,(.i)/(1 + ,v)), are constructed using the algebra-on-the-graph method. The two transfer functions coincide at frequencies greater than the crossover frequency. At frequencies less than Ihe CTOssover frequency / , the closed-ltwp transfer function is reduced by a factor of T{s). It can be seen that the poles of G,Js) are cancelled by zeroes of l/( 1 + 7). Hence Ihe closed-loop line-lo-output Iransfer function is approximately

(9,61)

So the algebra-on-the-graph method allows simple approximate disturbance-to-output closed-loop transfer functions to be writlen. Armed with such an analytical expression, the system designer can easily compute the output disturbances, and can gain the insight required to shape the loop gain Г(.?) such that system specifications are met. Computer simulations can then be used tojudge whetherthe specifications are met under all operating conditions, and over expected ranges of component parameter values. Results of computer simulations ofthe design example described in this section can be found in Appendix B, Section B.2.2.

20 dB OdB -20 dB dB -60dB -KUdB -100 dB

20 dB/dccadc

Open-loop \\G.\\ p /p

IHz lOHz lOOH: 1кНг

10 кШ 100 кНг

Fig, 9.32 Comparison of open-loop liiie-lo-outpLit iran.sfei function G and ctnsed-limp line-to-outpui transfer function of Ec. (9,61).

MEASUREMENT OF LOOP GAINS

It is gtxxi engineering practice to measure the loop gains of prototype feedback systems. The objective of such an exercise is to verify that the system has been correctly modeled. If so, then provided that a good controller design has been implemented, then the system behavior will meet expectations regarding transient overshoot (and рЬа.че margin), rejection of disturbances, dc output voltage regulation, etc. Unfortunately, there are reasons why practical system prototype.s are likely to differ from theoretical models. Phenotnena may [)ccur that were шЛ iiccouiited for in the tiriginal mtxJel, and that significantly influence the system behavior. Noise and electromagnetic interference (EMI) can be present, which cause the system transfer functions lo deviate in unexpected ways.

So let us consider the measurement of the loop gain r(.s) of the feedback system of Fig. 9.33.

Block 1 Block 2 гi a I................................I

-*--

I- Z,(i)

His)

TXs)

Fig. 9.33 It is desired to dBtcrniinc the loop gain lis) experimentally, hy nriiking measurements at point A.

Blocl i

dc bias

1 f-)

His)

Cv.is) = Hs)

Fig. 9.34 Measurement of loop gain by breaking tKe loop.

We will make measurements at some point .4, where two blocks of Ihe network are connecleii electrically. In Fig. 9.33, the tititptit p[>rt [>fhl[K.k 1 is represented by a Theveitin-eqtiivalent network, cotnposed ofthe tlependent voltage sotirce Gif and output impedance Z,. Bltxrk 1 is loaded by the inpnt impedance Zj of block 2. The remainder of the feedback system is represented by a Ь1[к;к diagratn as shown. The loop gain ofthe system is

ns) = G,M

(9.62)

Measurement of this loop gain presents several challenges not present in other frequency tespotise measurements.

In principle, one could break the loop at point A, and attempt to measure T{s) using die transfer function measurement method of the previous chapter. As illustrated in Fig. 9.34, a dc supply voltage Vand potentiometer would be used, to establish a dc bias in the voltage v, such that all of the elements of the network tiperate at the correct quiescent point. Ac voltage variatitms in viO are coupled into the injection point via a dc bl[K;king capacitor. Any other independent ac inputs to the system are disabled. A network analyzer is used to measure the relative magnitudes and phases ofthe ac components ofthe voltages vp) and Vjit):

(9.63)

The measured gain 7, (i) differs from the aclualgain Т{.ч) Ijecause, by breaking the connection between blocks 1 and 2 at the measurement point, we have removed the loading of block 2 on block 1. Solutitm of Fig. 9.34 for the measured gain TJ.s) leads to

T {s) = G,(s)G4.,)H(s) Equations (9.62) and (9.64) can be combined lo express T,s{s) in terms of T{s):

(9.64)

TJs) = T(s)

Zi(£)l

2,{s)

(9.65)