7-3 BENEFITS OF FEEDBACK

Once a system design is chosen, any departure from the ideal represents a penalty in performance. Departures in system gain result in increases in thermal noise if the gain is less than the design value or in intermodulation noise if the gain is greater than desired.

Furthermore, in the latter situation the system may become over-loaded. In addition to the performance penalties, such gain departures carry a cost penalty because they must be compensated by some form of equalization to correct the gain/frequency or delay/frequency characteristic, or both, to within tolerable limits over the transmission band.

Equation (7-2) shows that the gain of a feedback amplifier is nearly independent of the JL circuit. Thus, departures from the ideal gain/frequency characteristic (Le., departures from design values) that are caused by changes in the JL circuit are effectively reduced

, I

Chap. 7 Negative Feedback Amplifiers 171 by feedback. These changes may be caused by manufacturing, aging, and temperature-induced variations in p.-circuit components, which include the active devices. Gain variations caused by power supply fluctuations are also reduced.

The nonlinear input/output characteristics of all active devices are another source of impairment in broadband electronic circuits. This type of impairment, often referred to as harmonic distortion or inter-modulation noise, is also reduced by the use of negative feedback. If no other benefits accrued from using feedback, this alone would justify application in analog cable transmission systems and in FM terminal equipment of microwave radio systems.

Additional feedback benefits accrue in the resolution of problems involving amplifier input and output impedances. Usually it is re-quired that these impedances, or at least their absolute values, match the impedances of the circuits to which they connect. In nonfeedback amplifiers it is nearly always difficult to meet this requirement be-cause the desired impedances are incompatible with the impedances of the devices used in the amplifiers. Circuit compromises often must be made to achieve an acceptable impedance match. In feedback ampli-fiers, however, the provision of feedback increases the flexibility of the design choices that can be made, and it is usually possible to achieve a better impedance match over a wide bandwidth by using a feedback amplifier than by using a nonfeedback amplifier.

Example 7-1: Feedback Effects

This simple example illustrates how a ft-gain change of about 0.8 dB may be suppressed by feedback to an amplifier gain change of approximately 0.1 dB.

Let the overall gain of an amplifier be 10 dB; that is,

e2 e2

20 log - = 10; - ~ 3.16.

el el

From Equation (7-2),

1

!

^p.f3

~

^3.16.

Assume the /L gain (without feedback) is 20 log /L

==

30 dB; then /L

==

31.6

and, by substitution,

/3 ==

^0.284.

Now, let the /L gain increase from 30 dB to 30.8 dB; that is, /L increases by about 10 percent from 31.6 to 34.8.

Then the overall amplifier gain is

e2 34.8 32

el 1 - 34.8 (-0.284) - . and

20 log ~

==

20 log 3.2

==

^{10.09 dB.}

el

Thus a 10 percent change in JL-circuit gain is held to about a 1.3 percent change in overall gain (0.09 dB).

The fact that the amplifier gain increased as the /L gain in-creased is due to the phase relationships implied by the simple substitutions made. In complex feedback structures, the amplifier gain might increase or decrease over limited portions of the band and within a limited range of the /L-gain change.

7-4 CIRCUIT CONFIGURATIONS

The principal circuit configurations useful in feedback circuits can be classified most easily in terms of the way in which the JL and

f3

circuits are connected to each other and to the external interconnec-tions at amplifier input and output. The variety of connecinterconnec-tions that can be made cannot be clearly demonstrated by a simple drawing such as that of Figure 7-1. The actual situation is that shown broadly by Figure 7-2 in which the JL,

/3,

input, and output circuits are interconnected by means of six-terminal networks. The classi-fication of feedback circuits then depends on the forms which these six-terminal networks assume.

,I II

Chap. 7 Negative Feedback Amplifiers 173

at ao

bl bo

Input Output

Input coupling coupling Output

network network

CI Co

II fo

tit do

Figure 7-2. Feedback amplifier representation.

Illustrations of some of the more common feedback amplifier structures are given in Figures 7-3 through 7-6. Where appropriate, the network terminals are identified in accordance with the notation used in Figure 7-2. The JL circuits commonly have one, two, or three stages of gain; an unlimited number of network configurations may be found in the passive networks shown in the figures. To avoid complexity here, the internal network configurations are generally omitted in the figures.

Series and Shunt Feedback

The configuration of Figure 7-3 is called series feedback because, as seen from the input and output terminals, the JL and

f3

circuits are in series. The

f3

circuit, shown here as a '1T arrangement of three impedances (A, B, and C) may be much simpler or much more com-plex than that illustrated. The effective line terminals (ei,

Ii,

eo, and fo) are shown at the high sides of the transformers since the transformer characteristics in this case may be added directly to those of the connecting circuits.

Figure 7-4 shows how feedback may be provided by means of shunt connections. The

f3

circuit, here represented as a T network of the three impedances, may again take on any of an unlimited

b;

r - - - --,

^bo

CI I I

I I ^Co

I I

I I

I _I

II

I

_I fo

I I

d; do

I

I I

L _ _ _ _ _ _

-.J

Figure 7-3. Series feedback amplifier.

number of configurations. Note that the connecting terminals (input and output), f3 network, and JL network are all in parallel.

Series and shunt feedback designs are simple and they are con-venient for many applications. The feedback phenomenon tends to change the effective input and output impedances of the amplifier to very high or very low values. As a result, it is possible to build out these impedances conveniently by the use of discrete components to achieve a good impedance match to the connecting network or trans-mission line. A disadvantage is that the line or connecting impedances form a part of the JLf3 loop. As a result, variations in the line im-pedance, sometimes large and impossible to control, affect the JLf3 characteristic; in some cases, the effect may be great enough to cause amplifier instability.

Bridge-Type Feedback

These difficulties may be mitigated by using bridge-type feedback circuits. Many variations of these circuits exist, but the configuration

1,1 I analog cable systems, is the high-side hybrid feedback arrangement illustrated in Figure 7-5. Several network branches must be added in this configuration to provide hybrid balance and input and output impedance control. These branches are designated Zn and Zl in Figure 7-5. The advantages of this circuit include the achievement of minimum noise and improved intermodulation performance while controlling both the input and output impedances.

Figures 7-3 through 7-5 show symmetrical arrangements at each end of the amplifier. This has been done only to simplify the illustra-tions. The number of configurations is increased greatly by com-bining different types of connections at input and output. Further-more, circuit advantages can sometimes be realized by providing multiple loop configurations. An example of such a configuration is given in Figure 7-6. Here, a feedback amplifier with a series feed-back network Z{31, similar to that of Figure 7-3, is shown with local shunt feedback ^Z{32 around the last stage of a three-stage configura-tion in theJL path. The impedances Zil and Zi2 are interstage networks in the fL path.

Input[J

Ij

r

-I

I

I I I

B

~ I I ~

L _________ --1

Figure 7-5. Amplifier with high-side hybrid feedback.

Figure 7-6. Three-stage series feedback amplifier with local shunt feedback on last stage.

I' some important relationships and design limitations are discussed in order to provide an improved understanding of how transmission systems operate and how system performance is related to the design

Df the individual amplifier.

Gain and Feedback

The shape and magnitude Df the gain/frequency characteristic are basic design considerations. The closeness of the gain/frequency characteristic

to.

the desired characteristic may be determined by the degree Df circuit complexity that can be tolerated; however, the better the match, the better will be the ultimate transmission charac-teristic of the system. Equipment size and power dissipation may also be important considerations in making this first set of compro-mises in amplifier design.

Characteristic Shaping. The characteristics Df feedback amplifiers are all complex functiDns Df frequency which are importantly related to the transmission characteristics of all of the netwDrks making up the complete amplifier and its external terminations.

In many applications, it is desirable

to.

design the amplifier to a flat gain, one that is equal over the entire transmitted band. In the case of line repeaters for analog cable systems, it is usually desirable to have the gain of the amplifier sections of the repeaters match the loss of the cable section over the band of interest. In either case, the desired flat or shaped gain/frequency characteristic is produced primarily by proper design of the ,8-circuit network since the gain de-sirable in many cases

to.

shape the feedback/frequency characteristic of an amplifier. FDr example, it is possible to increase low-frequency feedback at the expense of high-frequency feedback. This can be

ac-co.mplished by careful designs o.f all networks in the J1-f3 lo.o.P, using frequency-dependent reactive co.mpo.nents, since the feedback is, by definitio.n, pro.Po.rtio.nal to. 11 (1-J1-f3).

Gain and Phase Margins. The selectio.n Df a circuit cDnfiguratio.n and the amo.unt o.f feedback to. be provided depend Dn the magnitudes o.f the gain and bandwidth required and o.n the characteristics o.f avail-able active devices. These co.nsiderations include the linearity o.f the device inputlo.utput characteristics, the no.ise figure o.f the input device, and the need fo.r minimizing variatio.ns in circuit parameters due to. device aging and ambient temperature changes.

As shown in Equation (7-2), the insertion gain o.f a feedback amplifier is

e2 J1- 1

el 1-/Lf3 ~ -

73·

The to.tal gain aro.und the feedback Io.o.P is defined as J1-f3, where J1- is the to.tal gain pro.vided by the active devices (and their related /L-circuit netwo.rks) and f3 is the lo.SS o.f the netwo.rk that co.nnects the o.utput back to. the input. From these relationships, the Io.o.P gain in dB is

20 lo.g J1-f3

==

20 lo.g /L

+

^{20 lo.g f3 ~} 20 lo.g P. -gR (7-3) where gR is the insertio.n gain o.f the co.mplete clo.sed-lo.o.P amplifier in dB. It is appro.ximately equal to. -20 lo.g

f3.

Thus,

20 lo.g /L ~ 20 lo.g 1Lf3

+

^{gR dB. (7-4)}

That is, the sum of the lo.o.P gain and insertio.n gain cannDt exceed the to.tal gain available in the J1- circuit. It is therefore impo.ssible

to.

get Io.DP gain in excess Df the difference between the JL gain and the desired insertion gain. When the desired Io.o.P gain is greater, the design is said to. be gain limited.

Most bro.adband amplifier designs, ho.wever, are not gain limited;

the need fDr adequate stability margins is usually contro.lling. In the gain expressio.n

JLI

(1-1Lf3) , the deno.minato.r may beco.me zero., de-pending o.n phase relatio.nships, when J-tf3

==

1. If /Lf3 is equal to. unity at any frequency, in-band Dr o.ut-o.f-band, the amplifier may beco.me unstable and break into. sPo.ntaneous oscillatio.n at that frequency if the phase o.f p.f3 is unfavo.rable. If it were Po.ssible to. ho.ld

I

p.f3

I

> >1

I

II

Chap. 7 Negative Feedback Amplifiers 179

for all frequencies, this would not be a problem, but every active device has some frequency above which its gain decreases monotoni-cally. The rate of decrease may be enhanced by circuit stray margins, it is said to meet the Nyquist stability criteria. Such margins are illustrated in Figure 7-7 where the characteristics are plotted on an arbitrary, normalized frequency scale. A phase margin of about 30 degrees and gain margin of about 10 dB, as illustrated, allow for variations in device characteristics which result from manufacturing processes, aging, and temperature effects.

The achievement of adequate phase and gain margins sets an upper limit on the achievable in-band feedback. When this limit is lower

Figure 7-7. Typical feedback amplifier characteristics.

'Iii'

expressiQn eQuId be written as 1/ (l+i}L{3[). Within the transmissiQn band the phase is Qften cQntrQlled to' apprQach this cQnditiQn. HQw-ever, Qut-of-band phase changes due to' phase shifts inherently as-sQciated with any gain/frequency characteristic, such as the gain cutQff mentiQned earlier. FurthermQre, fQr very high frequencies the prQpagatiQn time arQund the feedback IQQP contributes additiQnal phase shift which can be minimized, but nQt eliminated, by careful design.

Nonlinear Distortion and Overload

In additiQn to' the related consideratiQns Qf gain and achievable feedback, the related combinatiQn of overload, gain, and nQnlinear distQrtion must be cQnsidered in feedback amplifier design. These can be studied by first examining the phenQmenon of nQnlinear dis-tortiQn and its reductiQn by feedback and then relating these to. the problems Qf gain and overlQad.

Nonlinear Distortion. The generatiQn Qf intermodulatiQn products caused by nonlinear input/output characteristics Qf transistors is a very cQmplex phenQmenon. The analysis here is Qversimplified in Qrder to illustrate how products are generated, how feedback tends to.

suppress them, and how gain and Qverload are affected.

The nonlinear input/output vQltage relatiQnships of an amplifier may be represented by the expression

(7-5) where eo and ei are the output and input signal vQltages, and the a coefficients provide magnitude values Qf variQus wanted and un-wanted components in the output signal. If the input signal has many frequency components, EquatiQn (7-5) may be used to. study the intermQdulatiQn phenQmenQn by assuming

ei

==

A CQS at

+

B cos {3t

+

^{C CQS} rt.

When this value Qf ei is substituted in EquatiQn (7-5), the expressiQn can be expanded by trigQnometric identities. The Qutput voltage then cQntains an infinite number Qf terms cQnsisting of variQus CQm-binatiQns Qf input signal cQmponents; the magnitudes are

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