contains two high-freqtiency zeroes that are not present in 1\.<y Depending on the Q-factor of these zeroes, the phase of at the crossover frequency could be influenced. To ensure lhat Ihe phase margin is correctly measured, it is important that 2[/Zj be sufficiently small in magnilude.

9.6J Current injection

The results of the preceding paragraphs can also be obtained in dual fornt, where the loop gain is measured by ctirrent injection [3]. As illuslrated in Fig. 9.3S, we can model bloclt 1 and the analyzer injection source by Iheir Norton equivalents, and tise current probes to measure and ly The gain measured by current injection is

(9.77)

It can be shown that

Us) = 7f J)

Z,(.v)

(9,78)

Hence,

Tis) = 7(j) provided

(i) i 2з(.0 i ii Z,(.!) , and

( ) In*-)i

Zj(,v) Z,(.v)

(9.79)

St) to obtain an accurate measurement of the loop gain by current injecti[)n, we must find a point in the network where block 2 has sufficiently small input impedance. Again, note that the injection sotirce impedance Z. does not affect the measurement. In fact, we can realize (, by use of aThevenin-equivalent source, as illustrated in Fig, 9.39. The network analyzer injection source is represented by voltage source

Block I

G,(i)v,W

I- Zi(j)

я(1)

Block 2

Zils)

Giis)C,{s)= Us)

Гщ. 9.3B MeasurenKiiLofloop gain by CLirrcul iiijeetion.

v) lis)

Fig. %39 Curretit injectinci using ThCTcnin-equivaicnt source.

and Dutpiit resistance A series capacitor, C, is inserted to avoid disrupting tlte dc bias at the injection point.

9.6.3 Meiisurement of Unstable Systems

When the prototype feedback system is unstable, we are even mote eager to measure the loop gain-to find out what went wrong. But measurements cannot be made while the system oscillates. We need lo stabilize the system, yet measure the original unstable loop gain. It is possible to do this by recognizing that the injection source impedance Z. does not influence the measured loop gain [3]. As illustrated in Fig. 9.40, we can even add additional resistance Л , effectively increasing the source impedance Z,. The measured loop gain T.(s) is unaffected.

Adding series impedance generally lowers the кюр gain nf a system, leading to a lower crossover frequency and a more positive phase margin. Hence, it is usually possible to add a resistor that is sufficiently large to stabilize the system. The gain T(x), Eq. (9.67), continues to be approximately equal to the original unstable loop gain, according to Eq. (9.75). To avoid disturbing the dc bias conditions, it may be necessary to bypass Й then it will not influence the stability of the modified system.

with itiductor L . If the inductance value is sufficiently large,

Block 1

L...

Block 2

Pj,(i)

Mis)

Ci(i)v,(i) = m

Fig. 9.40 MensurcMieiit of nn un.sUible loop gain by voltage injeciion.

9.7 SUMMARY OF KEY POINTS

\. Negiilive feedback causes ihe sysiem ouiput to closely follow ihe reference input, according to ihe gain \/Hls). The influence on the output of disturbances and variation of gains in the forward path is reduced.

2. Tlie loop gain T{s) is equal lo the products of the iiins in ihe forward and feedback paths. The кюр ain is a measure of how well the feedback system works; a large кюр gain leads lo belter regulation of the outpul. The crossover frequency s the frequency al which the loop gain 7 has unity magnitude, and is a measure ofthe bandwidth ofthe control system.

3. The introduction of feedback causes the iransfer funcdons from disturbances lo the oulpul lo be multiplied by the factor + 7f!)). Al frequencies where T is large in magnitude (i.e.. below the crossover frequency), this facior is approximately equal to l/T(!). Hence, the influence of low-frequency dislurbances on the oulpul is reduced by a factor of l/Tis). At frequencies where T is small in magnitude (i.e., above the crossover frequency), the factor is approximately equal lo 1. The feedback кюр then has no effect. Closed-loop dislurbiince-to-outpul iriinsfer functions, such as ihe line-lo-ouipul transfer function or the outpul impedance, can easily be constructed using ihe algebra-on-the-graph method.

4. Stability can be a.4Sessed using the pha.4e margin test. The phase of Tis evaluated at the crossover frequency, and the stability of the imporlani closed-loop quaniities 77(1 + T) and 1/(1 + 7 is then deduced. Inadequate phase margin leads lo ringing and overshoot in the system transient response, and peaking in the closed-loop transfer functions.

5. Compensators are added in ihe forward paths of feedback ltюps lo shape the loop gain, such lhat desired performance is obtained. Lead compensators, or PD controllers, are added U) improve the phase margin and extend the contral system bandwidlh. PI conlrallers are used lo increase ihe low-frequency кюр gain, 10 improve the rejection of low-frequency dislurbances and reduce the sleiidy-stale error.

6. Loop gains can be experimenlally measured by use of voltage or current injection. This approach avoids the problem of establishing the correct quiescen! operaiing conditions in the system, a common difficulty in systems having a large dc loop gain. An injection point must be found where interstage loading is not significant. Unstable кюр gains can also be ineasured.

References

[1] B. Kvo. AiitoHiatif Confml Sy.stems, New York: Prenlice-Hall. Inc.

[2] J. DAzzo imd C. Houpis. Linear Control System Analysis and Design: Conventional and Modern. New

York: McGraw-Hill. 1995.

[3] R. D. MlDDLEBMXJK, Measurement of Loop Gain in Feedback Systems, International Journal of Elec-

tronic!. Vol. 3S, No. 4, pp. 4S5-512, 1975.

[4] R. D. Middlebrook, Design-Oriented Analysis of Feedback Amplifiers, Proceedings National Elec-

tronic.4 Conference, Vol. XX, October 1964, pp. 234-238.

Proeleiws

9.1 Derive both forms ofEq. (9.25).

; flyback converter system of Fig, 9.41 coniains a feedback loop for regulation of the main outpul lage Vj, An auxiliary outpul produces voltage fj. The dc input voliage lies in the range 280 V < v < 380 V. The compensator network has transfer function

9.2 The

vohage