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IEEE TRANSACTIONS ON EDUCATION, VOL. 48, NO. 1, FEBRUARY 2005 53

Simplified Analysis of Feedback AmplifiersJos Luis Rodrguez Marrero

AbstractA very simple and general method for the analysis offeedback amplifiers with large-loop gain is presented in this paper.The general properties of feedback amplifiers, such as gain andinput and output resistances, are obtained using an open-loop cir-cuit where the loading effect of the feedback network is easily takeninto account. Emphasis is placed on quick, intuitive, and reliablecalculations, useful for both the analysis and design of feedbackamplifiers.

Index TermsNegative feedback, practical feedback amplifiers.

I. INTRODUCTION

THE CONTENTS of most analog electronics textbooks arewell suited for a sequence of two semester courses. A firstcourse is usually devoted to devices and basic circuits, usingoperational amplifiers, diodes, and transistors. More advancedtopics, such as differential amplifiers, multistage design, fre-quency response, and feedback are covered in a second course.

The study of negative feedback is useful because it producesmany important benefits for amplifiers, such as gain insensi-tivity against parameter changes and precise control of input andoutput impedances and bandwidth. A typical chapter on feed-back in most textbooks starts with the analysis of the proper-ties of the ideal feedback configuration [1][5]. The feedbackamplifier signal gain is obtained; it is independent of the basicamplifier characteristics when the loop gain is large. Since theinput and output signals can be voltages or currents, one needsto introduce four basic topologies in the study of amplifiers withnegative feedback. Up to this point, the analysis of feedbackamplifiers is straightforward. However, the analysis of practicalfeedback amplifiers becomes complicated because the feedbacknetwork is not unidirectional and because it loads the basic am-plifier stage. The simplest approaches are often seen by the stu-dents as a collection of recipes [3], [4], whereas more formalapproaches, such as the two-port representations of the basicamplifier and the feedback network, lack insight into the advan-tages and properties of these amplifiers and make their analysisa very complicated task [5]. Some books offer an alternativemethod, the return-ratio method [4][6], for the calculation ofthe loop gain. Although this method is not very intuitive, it lookssimpler.

Although the two-port analysis and the return-ratio methodmay be used to study feedback amplifiers, important differencesmay be present in the closed-loop formulas because the loopgain obtained by means of two-port analysis does not always

Manuscript received July 15, 2003; revised December 15, 2003. This workwas supported in part by the Ministerio de Ciencia y Tecnologa of Spain underGrant MAT2002-04246-C05-02.

The author is with the Universidad Pontificia Comillas, 28015 Madrid, Spain(e-mail: [email protected]).

Digital Object Identifier 10.1109/TE.2004.832878

agree with the return ratio [7], [8]. The reason for the disagree-ment is that some forward signal transfers through the feedbacknetwork. Since this network is usually a passive one, this for-ward signal transfer can be ignored at low frequencies, but itmay become important at high frequencies where the gain ofthe basic amplifier falls [9], [10].

All feedback amplifiers can be analyzed as circuits. However,such treatment becomes very tedious and difficult in most prac-tical cases, and, most important, the key aspects of circuit per-formance are not transparent. A different approach to analyzefeedback amplifiers, where the feedback loop plays an impor-tant role, is presented in this paper. This approach is appropriatefor the analysis of feedback amplifiers where it is easy to recog-nize the kind of connection at both the input and output of theamplifier. It is based on the calculation of the return ratio, andit gives accurate results when the loop gain is very large, whichshould be the case for a well-designed feedback amplifier [5].Emphasis is placed on general properties common to all feed-back topologies, using a straightforward and informal approachsuitable for quick-hand analysis. This method has proven verysuccessful in introductory courses where motivation and insightto circuit design is very important and where detailed analysisis not needed.

There are, however, feedback circuits where one cannot easilyidentify the kind of connection at both the input and output of theamplifier or where the circuits do not fit into the ideal feedbackamplifier configuration. In some current generators, such as thebipolar Wilson current mirror, the return-ratio method seems theappropriate choice for the analysis [5].

The topic will be presented as it appears in class notes, omit-ting important aspects of feedback, such as a detailed analysis ofgain sensitivity, bandwidth extension, or effect of feedback ondistortion. These topics and the frequency response of feedbackamplifiers can be studied after these basic ideas have been pre-sented. Section II is devoted to the review of the analysis of theideal feedback amplifier, where the loading effect of the feed-back network is ignored. This effect is studied in Section III,where a general and simple method for the analysis and designof practical feedback amplifiers is proposed.

II. IDEAL FEEDBACK AMPLIFIERS

Fig. 1 shows the basic configuration of a feedback amplifier.It is termed ideal feedback configuration because one assumesthat 1) the feedback network does not load the basic amplifier

and that 2) each block is unidirectional. In this figure andare the input and output signals. The feedback network

samples the output signal and produces a signalthat is fed back to the amplifier input, where an error signal

is generated. The basic amplifier amplifies

0018-9359/$20.00 2005 IEEE

54 IEEE TRANSACTIONS ON EDUCATION, VOL. 48, NO. 1, FEBRUARY 2005

Fig. 1. Ideal feedback amplifier.

this signal to produce the output signal . Hence, thefeedback amplifier signal gain is easily obtained as

(1)

where the loop gain has been introduced. For large-loop gains , the feedback amplifier signal gain dependsonly on the feedback network

(2)

For the feedback amplifier to have gain, , which withimplies that . Equation (2) shows that

feedback with large-loop gain makes the feedback gain in-sensitive to changes of the basic amplifier gain . These changesmay be a result of temperature dependence, aging, or bias con-ditions of the active devices of the basic amplifier. When theloop gain is higher, the feedback amplifier becomes more in-sensitive and has smaller gain. This result is the penalty paidfor a well-defined gain and is one disadvantage of feedback (theother one being potential instability).

Using (1), one sees that

(3)

For large-loop gains, the error signal is much smaller thanthe input signal , and since , then . Inthis case, inverting , one obtains the result of (2)

(4)

If , the error signal becomes zero, which means thatboth and are equal. This result is the main objective ofthe feedback amplifier, an important result that will be used toestablish very general procedures to determine bias conditionsand to compute the gain of feedback amplifiers.

The feedback network samples the output signal to producea signal that is fed back to the amplifier input. Since these sig-nals can be voltages or currents, four basic configurations arepossible. The properties of the feedback amplifier are closelyrelated to these configurations and, hence, to the type of feed-back network. As an example, the properties of the seriesshuntfeedback configuration are reviewed subsequently. The proper-ties of the other configurations are easily obtained from theseconsiderations.

Fig. 2 shows the ideal configuration of a seriesshunt feed-back amplifier. In this configuration, both the feedback and

Fig. 2. Ideal seriesshunt feedback amplifier.

Fig. 3. Diagram to calculate the feedback amplifier output resistance.

output signals are voltage signals with the feedback networkconnected in series with the basic amplifier input (voltagesare added at the input) and in parallel at the output (both thebasic amplifier and the feedback network share the same outputvoltage). The properties of the ideal feedback network areeasily obtained because it does not load the basic amplifierand has zero output resistance (because it is connected in serieswith the basic amplifier input) and infinite input resistance(because it is connected in parallel with the basic amplifieroutput).

Since the feedback and output signals are voltage signals, thegain of the feedback amplifier is obtained from (1) by replacing

and by and , respectively.

(5)

where the last step holds for large-loop gains.The feedback amplifier input current is common to both the

basic amplifier input and the feedback network output so that. Hence, the feedback amplifier input resistance is

(6)

where is the basic amplifier input resistance.In order to calculate the feedback amplifier output resistance,

one may refer to Fig. 3. The basic amplifier output resistancehas been added to the figure, where an ideal zero-output resis-tance basic amplifier is considered. To calculate the output re-sistance , an external output voltage is applied with theinput voltage turned off. Since the feedback network is idealand it is connected in parallel at the output of the basic ampli-fier, it draws no current at its input and hence

(7)

where has been used, since the inputvoltage .

MARRERO: SIMPLIFIED ANALYSIS OF FEEDBACK AMPLIFIERS 55

These results are very general. If is the basic amplifierresistance at the port being considered, a series connection ofthe feedback network increases the resistance at that port by afactor of

(8)while a shunt connection of the feedback network reduces theresistance by the same factor.

(9)

III. PRACTICAL FEEDBACK AMPLIFIERS

In this section, the small-signal analysis of electronic circuitswill be carried out using the bipolar transistor model [4],[11]. A similar model can be used for the field-effect transistors(FETs). The small-signal voltages and currents will be repre-sented by lower case letters with lower case subscripts. For ex-ample, represents the small-signal collector voltage of tran-sistor , while represents the small-signal base current oftransistor . In the model, the transistor collector currentis modeled by a voltage-controlled current source with the con-trol voltage being , the small-signal base-emitter voltage. Thesmall-signal collector current is given by ,where is the transconductance of the bipolar tran-sistor at its bias point ( 25 mV at room temperature). Onewill assume that is very large so that ( willbe used hereafter). The resistance represents the small-signalresistance seen looking into the emitter, for the base held at aconstant voltage. The resistance seen looking into the base, forthe emitter held at a constant voltage, is simply , because theemitter current is times larger than the base current. A very el-egant use of this model appears in [11] and [12].

For a well-designed feedback amplifier, the feedback ampli-fier gain depends only on the characteristics of the feedback net-work. Hence, two important steps should be followed to startthe analysis of a feedback amplifier: 1) check that the circuithas negative feedback and 2) find the feedback network. Nega-tive feedback is required to have a well-defined gain, althoughlarge-loop gains may lead to oscillations. In order to find thefeedback network, one must recognize the kind of connectionat both the input and output of the feedback amplifier. This in-formation is not always easy to obtain, particularly for output se-ries connections involving bipolar transistor circuits. Once thefeedback network has been identified, many properties of thefeedback amplifier are easily obtained, including its gain as fol-lows from (2).

A. The Loading EffectThe division between basic amplifier and feedback network

is not always obvious when analyzing practical feedback ampli-fiers. This analysis becomes more complicated because 1) thefeedback network loads the basic amplifier and 2) the feedbacknetwork is not unidirectional. The effect of loading is usuallythe most important effect and must be taken into account.

The parameters that are needed to compute the characteris-tics of feedback amplifiers are , , and the input and output

Fig. 4. Open-loop amplifier with loading effect.

resistances of the basic amplifier. They can be obtained byopening the feedback loop. In practical feedback amplifiers,the loading effect of the feedback network must be consideredwhen opening the loop. This task is accomplished by termi-nating the basic amplifier output with a resistance equal tothat seen looking to the left of the feedback network [4], asshown in Fig. 4, which shows the feedback amplifier of Fig. 1with the loop open. From Fig. 4, the loop gain is obtained as

(10)

In fact, this result coincides with the return ratio [10] and isindependent of the signals chosen when opening the loop, aswill be shown in some examples hereafter.

The feedback network gain is easily obtained by dis-connecting the feedback network from the basic amplifier,avoiding the loading effect of the basic amplifier. For example,for voltage error signals

(11)

while for current error signals

(12)

Finally, the basic amplifier input and output resistances, in-cluding the loading effect of the feedback network, can be com-puted from Fig. 4 by setting the sampled signal equal to zero.The basic amplifier gain can also be obtained from this figure,but this quantity is not needed in this analysis.

B. SeriesShunt FeedbackFig. 5 shows a circuit with seriesshunt feedback. The resis-

tive voltage divider (inside the dashed box) connecting the am-plifier output with the base of transistor forms the feedbacknetwork. It samples the output voltage and produces a feed-back voltage . The divider is the simplest circuit thataccomplishes this task with passive components.

Several points are worth discussing: bias points of the transis-tors, voltage gain and input, and output resistances of the feed-back amplifier. Transistors and form a differential pairbiased by the left current source, with the common emitteramplifying its output. These transistors and the biasing compo-nents make up the basic amplifier. Since both the differential


Fig. 5. Seriesshunt feedback amplifier.

pair and the common emitter have high gains, one can see thatthe open-loop gain is very large. Then, (5) shows that

(13)

The feedback network must be disconnected in order to ob-tain its gain . From Fig. 5 and (11), one can see thatso that

11 (14)

The stability of the bias point of is possible because theoverall circuit negative feedback guarantees that the quiescentoutput voltage is zero if has no dc component. This conditionis a result of the error signal 0for large-loop gains. Therefore, if the dc component of the inputvoltage is zero, the dc component of the output voltage is alsozero.

In general, one can infer the gain of the feedback amplifiereasily on the assumption that the loop gain is very large. How-ever, computing the loop gain directly to obtain the input andoutput resistances of the feedback amplifier is useful. The cal-culation of the loop gain follows the lines explained previouslyin (10) and Fig. 4. The loading effect of the feedback network(its input impedance is not infinite) is taken into account whenopening the loop by adding a resistor in series with the col-lector of , as shown in Fig. 6. Then, can be obtained fromthis circuit by means of (10). From Fig. 6, one can observe that

1 k 10 k 11 k , where 100 hasbeen used.

The loop gain is given by

295 (15)

Fig. 6. Open-loop circuit for the analysis of the feedback amplifier of Fig. 5.

where 1 k. As indicated before, the loopgain is very large so that (13) holds true. As for any amplifierwith negative feedback, is positive.

To calculate the input and output feedback amplifier resis-tances, one needs to compute the basic amplifier correspondingresistances. In order to take into account the loading effect, thecircuit of Fig. 6 is used with . The basic amplifier inputresistance is given by

1 k 10 k 11 k (16)

and its output resistance is

11 k (17)

The feedback amplifier input resistance is obtained using (16)and (6)

3.3 M (18)

while its output resistance is obtained from (17) and (7)

37 (19)

These results show that, if the input voltage source has an in-ternal resistance of up to 330 k , the output voltageis essentially unchanged, and it is delivered amplified by a factorof 11 at a very low output resistance of 37 .

C. ShuntShunt FeedbackThe inverting amplifier configuration of Fig. 7 will be ana-

lyzed as an example of a practical shuntshunt feedback ampli-fier. The shuntshunt configuration becomes clear if the voltagesource is converted into a current source using

(20)

as shown in Fig. 8. Therefore, the transresistance gain willbe given by

(21)


Fig. 7. Inverting amplifier.

Fig. 8. Shuntshunt feedback amplifier.


Fig. 10. Feedback network for the inverting amplifier.

Fig. 9 shows the circuit appropriate for feedback analysis,and Fig. 10 shows the feedback network. Typical operationalamplifiers have both very large differential input resistance andgain. Hence, the terminating resistor of Fig. 9 will be

. Since the loop gain will also be very large

(22)

Finally, the gain of the inverting amplifier is obtained by in-serting (20) in (22)

(23)

Fig. 11. Seriesseries feedback amplifier. The component values are R =9 k, R = 5 k, R = 600 , R = R = 100 , and R = 640 .

Fig. 12. Feedback network of circuit of Fig. 11.

Another quantity of interest is the input resistance of theinverting amplifier. From Fig. 7, it is seen that

(24)

In order to obtain , one needs to compute the feedback am-plifier input resistance, which is given by (Fig. 9)

0 (25)

because is very large. From Fig. 8, one can see that, which means that 0 and, therefore, .

D. SeriesSeries Feedback

Fig. 11 shows a circuit amplifier with this configuration. It ispart of the commercial integrated circuit MC 1553. The circuitomits the biasing of the input transistor needed to bias the restof the circuit. In this example, one can assume that the inputvoltage has a dc component needed to bias the circuit properly,so that 0.6 mA, 1 mA, and 4 mA.

The sampled signal that is fed back to the input is the emittercurrent of . The feedback network gain is obtained from thecircuit shown in Fig. 12.

11.9 (26)



Again, inspection of the circuit (two common emitters and anemitter follower) indicates that the loop gain is large. If it islarger than 10, then

(27)

The feedback amplifier voltage gain can be obtained as follows

50.4 (28)

In order to calculate the feedback amplifier resistances, theloop gain must be obtained. Fig. 13 shows the circuit used tocalculate and the characteristics of the basic amplifier . Theloading effect of the feedback network has been taken into ac-count by adding the resistor in series with the emitter of tran-sistor , where

87 (29)The loop gain is given by

247 (30)where 29 . Thus, the approximation of(27) is justified.The feedback amplifier input resistance is given by

, where is the basic amplifier inputresistance, obtained from Fig. 13 with . The result is

13 k (31)Then, 3.2 M .

The equivalent output circuit of the seriesseries feedbackamplifier is shown inside the dashed box of Fig. 14, where

Fig. 14. Equivalent output circuit of the feedback amplifier of Fig. 11.

Fig. 15. Seriesshunt feedback amplifier. The component values are the sameas for the circuit of Fig. 11.

is the resistance seen looking into the collector of , as shownin Fig. 11. Since , is the resis-tance seen when opening the loop at the emitter of [4]. Thebasic amplifier output resistance is obtained as the resis-tance between points and of Fig. 13, and it is given by

143 . An approximate value1for can be computed by assuming that the effect of feed-back is to place at the emitter of [4]. If the Early effectof the transistors is ignored, is infinite. With finite Earlyeffect, is very large because it represents the output resis-tance of a common emitter amplifier with emitter degenerationand because of the presence of the resistor 35.5 k atthe emitter of . Therefore, it should be expected that isvery large. As a result, the output voltage is supplied at a re-sistance 600 .

E. Another Look at the Loop GainA given circuit can realize different feedback functions if the

output is taken from different nodes. As an example, considerthe circuit of Fig. 11, but assume that the output signal is takenas the voltage at the emitter of , as shown in Fig. 15. In thiscase, the resistor is not needed for the actual operation ofthe circuit but is kept here to show that both circuits are thesame, another example of seriesshunt feedback. Fig. 16 shows

1A detailed analysis of the circuit shows that this result is very accurate.



the small-signal analysis circuit used to compute the loop gain,where and 29 .

The loop gain is given by

247 (32)This result coincides with the one obtained in (30) for the se-riesseries feedback amplifier of Fig. 11. As one would expect,the loop gain is unchanged by the choice of the output portif the circuit is unchanged by this choice. Since the loop gain isvery large, the voltage gain of the feedback circuit is given by

7.4 (33)

The feedback amplifier input resistance is obtained as

(34)where is the basic amplifier input resistance, obtained fromFig. 16 with . The result is

13 k (35)Then, 3.2 M , which is the same as the result of the se-riesseries amplifier of Fig. 11. A closer look reveals that thereare minor differences between (31) and (35) because of the dif-ferences in the loading effect of the feedback network at theamplifier output, a consequence of the level of approximationinvolved in this analysis.

IV. CONCLUSIONThe traditional treatment of practical feedback amplifiers

often relies on two ports to convert the feedback circuit into uni-

lateral basic amplifier and feedback networks. This technique isdifficult to use and does not provide insight into the propertiesand advantages of the feedback amplifier. Other techniques,like the return-ratio method, are often simpler to use but do notoffer greater insight as to the circuit performance.

In this paper, a very simple technique for the analysis of feed-back amplifiers with large-loop gain has been shown to give veryreliable results with minor effort. General properties of feed-back amplifiers, such as gain and input and output resistances,are obtained using an open-loop circuit where the loading effectof the feedback network is easily taken into account. Emphasisis placed on quick, intuitive, and reliable calculations, useful forboth the analysis and design of feedback amplifiers.

This technique is appropriate for the analysis of feedback am-plifiers where the kind of connection at both the input and outputof the amplifier is easy to recognize. However, this informationmay be somewhat difficult to obtain for some cases, especiallyfor some current mirrors such as the bipolar Wilson mirror,where the return-ratio method seems the appropriate choice forthe analysis.

ACKNOWLEDGMENT

The author would like to thank the Ministerio de Ciencia yTecnologa of Spain for support and the anonymous reviewersof the paper for their very helpful comments.

REFERENCES[1] D. L. Schilling and C. Belove, Electronic Circuits: Discrete and Inte-

grated. New York: McGraw-Hill, 1968.[2] J. Millman and C. C. Halkias, Integrated Electronics. New York: Mc-

Graw-Hill, 1972.[3] M. N. Horenstein, Microelectronic Circuits and Devices, 2nd ed. En-

glewood Cliffs, NJ: Prentice-Hall, 1996.[4] A. S. Sedra and K. C. Smith, Microelectronic Circuits, 4th ed. New

York: Oxford Univ. Press, 1998.[5] P. R. Gray, P. J. Hurst, S. H. Lewis, and R. G. Meyer, Analysis and Design

of Analog Integrated Circuits, 4th ed. New York: Wiley, 2001.[6] S. Rosenstark, Feedback Amplifier Principles. New York: MacMillan,

1986.[7] P. J. Hurst, Exact simulation of feedback circuit parameters, IEEE

Trans. Circuits Syst., vol. 38, pp. 13821389, Nov. 1991.[8] , A comparison of two approaches to feedback circuit analysis,

IEEE Trans. Educ., vol. 35, pp. 253261, Aug. 1992.[9] S. Ben-Yaakov, A unified approach to teaching feedback in electronic

circuits courses, IEEE Trans. Educ., vol. 34, pp. 310316, Nov. 1991.[10] B. Nikolic and S. Marjanovic, A general method of feedback amplifier

analysis, in IEEE Int. Symp. Circuits Systems, Monterey, CA, 1998, pp.415418.

[11] P. Horowitz and W. Hill, The Art of Electronics, 2nd ed. Cambridge,U.K.: Cambridge Univ. Press, 1989.

[12] T. C. Hayes and P. Horowitz, Student Manual for The Art of Elec-tronics. Cambridge, U.K.: Cambridge Univ. Press, 1989.

Jos Luis Rodrguez Marrero received the Ingeniero de Telecomunicacionesdegree from Universidad Politcnica de Catalua, Barcelona, Spain, and theM.S. degree in electrical and computer engineering and the Ph.D. degree inphysics from the University of Massachusetts Amherst.

He joined the faculty of the Universidad Pontificia Comillas de Madrid,Spain, in 1987, where he is now Professor in the School of Engineering,teaching courses on electronics and signal processing. His research interests arein the areas of electronic instrumentation, nonlinear dynamics, and engineeringeducation.

tocSimplified Analysis of Feedback AmplifiersJos Luis Rodrguez MarreroI. I NTRODUCTIONII. I DEAL F EEDBACK A MPLIFIERS

Fig.1. Ideal feedback amplifier.Fig.2. Ideal series shunt feedback amplifier.Fig.3. Diagram to calculate the feedback amplifier output resisIII. P RACTICAL F EEDBACK A MPLIFIERSA. The Loading Effect

Fig.4. Open-loop amplifier with loading effect.B. Series Shunt Feedback

Fig.5. Series shunt feedback amplifier.Fig.6. Open-loop circuit for the analysis of the feedback ampliC. Shunt Shunt Feedback

Fig.7. Inverting amplifier.Fig.8. Shunt shunt feedback amplifier.Fig.9. Open-loop circuit for the analysis of the feedback ampliFig.10. Feedback network for the inverting amplifier.Fig.11. Series series feedback amplifier. The component values Fig.12. Feedback network of circuit of Fig.11 .D. Series Series Feedback

Fig.13. Open-loop circuit for the analysis of the feedback amplFig.14. Equivalent output circuit of the feedback amplifier of Fig.15. Series shunt feedback amplifier. The component values aE. Another Look at the Loop Gain

Fig.16. Open-loop circuit for the analysis of the feedback amplIV. C ONCLUSIOND. L. Schilling and C. Belove, Electronic Circuits: Discrete andJ. Millman and C. C. Halkias, Integrated Electronics . New York:M. N. Horenstein, Microelectronic Circuits and Devices, 2nd ed. A. S. Sedra and K. C. Smith, Microelectronic Circuits, 4th ed. NP. R. Gray, P. J. Hurst, S. H. Lewis, and R. G. Meyer, Analysis S. Rosenstark, Feedback Amplifier Principles . New York: MacMillP. J. Hurst, Exact simulation of feedback circuit parameters, IES. Ben-Yaakov, A unified approach to teaching feedback in electrB. Nikolic and S. Marjanovic, A general method of feedback ampliP. Horowitz and W. Hill, The Art of Electronics, 2nd ed. CambridT. C. Hayes and P. Horowitz, Student Manual for The Art of Elect

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