IEEE TRANSACTIONS ON EDUCATION, VOL. 48, NO. 1, FEBRUARY 2005 53 Simplified Analysis of Feedback Amplifiers José Luis Rodríguez Marrero Abstract—A very simple and general method for the analysis of feedback amplifiers with large-loop gain is presented in this paper. The general properties of feedback amplifiers, such as gain and input and output resistances, are obtained using an open-loop circuit where the loading effect of the feedback network is easily taken into account. Emphasis is placed on quick, intuitive, and reliable calculations, useful for both the analysis and design of feedback amplifiers. Index Terms—Negative feedback, practical feedback amplifiers. I. INTRODUCTION T HE CONTENTS of most analog electronics textbooks are well suited for a sequence of two semester courses. A first course is usually devoted to devices and basic circuits, using operational amplifiers, diodes, and transistors. More advanced topics, such as differential amplifiers, multistage design, frequency response, and feedback are covered in a second course. The study of negative feedback is useful because it produces many important benefits for amplifiers, such as gain insensitivity against parameter changes and precise control of input and output impedances and bandwidth. A typical chapter on feedback in most textbooks starts with the analysis of the properties of the ideal feedback configuration [1]–[5]. The feedback amplifier signal gain is obtained; it is independent of the basic amplifier characteristics when the loop gain is large. Since the input and output signals can be voltages or currents, one needs to introduce four basic topologies in the study of amplifiers with negative feedback. Up to this point, the analysis of feedback amplifiers is straightforward. However, the analysis of practical feedback amplifiers becomes complicated because the feedback network is not unidirectional and because it loads the basic amplifier stage. The simplest approaches are often seen by the students as a collection of recipes [3], [4], whereas more formal approaches, such as the two-port representations of the basic amplifier and the feedback network, lack insight into the advantages and properties of these amplifiers and make their analysis a very complicated task [5]. Some books offer an alternative method, the return-ratio method [4]–[6], for the calculation of the loop gain. Although this method is not very intuitive, it looks simpler. Although the two-port analysis and the return-ratio method may be used to study feedback amplifiers, important differences may be present in the closed-loop formulas because the loop gain obtained by means of two-port analysis does not always Manuscript received July 15, 2003; revised December 15, 2003. This work was supported in part by the Ministerio de Ciencia y Tecnología of Spain under Grant MAT2002-04246-C05-02. The author is with the Universidad Pontificia Comillas, 28015 Madrid, Spain (e-mail: marrero@dea.icai.upco.es). Digital Object Identifier 10.1109/TE.2004.832878 agree with the return ratio [7], [8]. The reason for the disagreement is that some forward signal transfers through the feedback network. Since this network is usually a passive one, this forward signal transfer can be ignored at low frequencies, but it may become important at high frequencies where the gain of the basic amplifier falls [9], [10]. All feedback amplifiers can be analyzed as circuits. However, such treatment becomes very tedious and difficult in most practical cases, and, most important, the key aspects of circuit performance are not transparent. A different approach to analyze feedback amplifiers, where the feedback loop plays an important role, is presented in this paper. This approach is appropriate for the analysis of feedback amplifiers where it is easy to recognize the kind of connection at both the input and output of the amplifier. It is based on the calculation of the return ratio, and it gives accurate results when the loop gain is very large, which should be the case for a well-designed feedback amplifier [5]. Emphasis is placed on general properties common to all feedback topologies, using a straightforward and informal approach suitable for quick-hand analysis. This method has proven very successful in introductory courses where motivation and insight to circuit design is very important and where detailed analysis is not needed. There are, however, feedback circuits where one cannot easily identify the kind of connection at both the input and output of the amplifier or where the circuits do not fit into the ideal feedback amplifier configuration. In some current generators, such as the bipolar Wilson current mirror, the return-ratio method seems the appropriate choice for the analysis [5]. The topic will be presented as it appears in class notes, omitting important aspects of feedback, such as a detailed analysis of gain sensitivity, bandwidth extension, or effect of feedback on distortion. These topics and the frequency response of feedback amplifiers can be studied after these basic ideas have been presented. Section II is devoted to the review of the analysis of the ideal feedback amplifier, where the loading effect of the feedback network is ignored. This effect is studied in Section III, where a general and simple method for the analysis and design of 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 assumes that 1) the feedback network does not load the basic amplifier and that 2) each block is unidirectional. In this figure and are the input and output signals. The feedback network samples the output signal and produces a signal that 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. 2. Ideal series–shunt feedback amplifier. Fig. 1. Ideal feedback amplifier. this signal to produce the output signal . Hence, the feedback amplifier signal gain is easily obtained as (1) has been introduced. For largewhere the loop gain , the feedback amplifier signal gain depends loop gains only on the feedback network (2) , which with For the feedback amplifier to have gain, implies that . Equation (2) shows that feedback with large-loop gain makes the feedback gain insensitive to changes of the basic amplifier gain . These changes may be a result of temperature dependence, aging, or bias conditions of the active devices of the basic amplifier. When the loop gain is higher, the feedback amplifier becomes more insensitive and has smaller gain. This result is the penalty paid for a well-defined gain and is one disadvantage of feedback (the other one being potential instability). Using (1), one sees that Fig. 3. Diagram to calculate the feedback amplifier output resistance. output signals are voltage signals with the feedback network connected in series with the basic amplifier input (voltages are added at the input) and in parallel at the output (both the basic amplifier and the feedback network share the same output are voltage). The properties of the ideal feedback network easily obtained because it does not load the basic amplifier and has zero output resistance (because it is connected in series with the basic amplifier input) and infinite input resistance (because it is connected in parallel with the basic amplifier output). Since the feedback and output signals are voltage signals, the gain of the feedback amplifier is obtained from (1) by replacing and by and , respectively. (3) is much smaller than For large-loop gains, the error signal the input signal , and since , then . In , one obtains the result of (2) this case, inverting (4) If , the error signal becomes zero, which means that and are equal. This result is the main objective of both the feedback amplifier, an important result that will be used to establish very general procedures to determine bias conditions and to compute the gain of feedback amplifiers. The feedback network samples the output signal to produce a signal that is fed back to the amplifier input. Since these signals can be voltages or currents, four basic configurations are possible. The properties of the feedback amplifier are closely related to these configurations and, hence, to the type of feedback network. As an example, the properties of the series–shunt feedback configuration are reviewed subsequently. The properties of the other configurations are easily obtained from these considerations. Fig. 2 shows the ideal configuration of a series–shunt feedback amplifier. In this configuration, both the feedback and (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 resistance has been added to the figure, where an ideal zero-output resistance basic amplifier is considered. To calculate the output re, an external output voltage is applied with the sistance input voltage turned off. Since the feedback network is ideal and it is connected in parallel at the output of the basic amplifier, it draws no current at its input and hence (7) where voltage has been used, since the input . MARRERO: SIMPLIFIED ANALYSIS OF FEEDBACK AMPLIFIERS 55 These results are very general. If is the basic amplifier resistance at the port being considered, a series connection of the feedback network increases the resistance at that port by a factor of (8) while a shunt connection of the feedback network reduces the resistance by the same factor. (9) III. PRACTICAL FEEDBACK AMPLIFIERS In this section, the small-signal analysis of electronic circuits model [4], will be carried out using the bipolar transistor [11]. A similar model can be used for the field-effect transistors (FETs). The small-signal voltages and currents will be represented by lower case letters with lower case subscripts. For exrepresents the small-signal collector voltage of tranample, sistor , while represents the small-signal base current of . In the model, the transistor collector current transistor is modeled by a voltage-controlled current source with the control voltage being , the small-signal base-emitter voltage. The small-signal collector current is given by , is the transconductance of the bipolar tranwhere 25 mV at room temperature). One sistor at its bias point ( ( will will assume that is very large so that be used hereafter). The resistance represents the small-signal resistance seen looking into the emitter, for the base held at a constant voltage. The resistance seen looking into the base, for , because the the emitter held at a constant voltage, is simply emitter current is times larger than the base current. A very elegant use of this model appears in [11] and [12]. For a well-designed feedback amplifier, the feedback amplifier gain depends only on the characteristics of the feedback network. Hence, two important steps should be followed to start the analysis of a feedback amplifier: 1) check that the circuit has negative feedback and 2) find the feedback network. Negative feedback is required to have a well-defined gain, although large-loop gains may lead to oscillations. In order to find the feedback network, one must recognize the kind of connection at both the input and output of the feedback amplifier. This information is not always easy to obtain, particularly for output series connections involving bipolar transistor circuits. Once the feedback network has been identified, many properties of the feedback amplifier are easily obtained, including its gain as follows from (2). A. The Loading Effect The division between basic amplifier and feedback network is not always obvious when analyzing practical feedback amplifiers. This analysis becomes more complicated because 1) the feedback network loads the basic amplifier and 2) the feedback network is not unidirectional. The effect of loading is usually the most important effect and must be taken into account. The parameters that are needed to compute the characteristics 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 by opening the feedback loop. In practical feedback amplifiers, the loading effect of the feedback network must be considered when opening the loop. This task is accomplished by terminating the basic amplifier output with a resistance equal to that seen looking to the left of the feedback network [4], as shown in Fig. 4, which shows the feedback amplifier of Fig. 1 with the loop open. From Fig. 4, the loop gain is obtained as (10) In fact, this result coincides with the return ratio [10] and is independent of the signals chosen when opening the loop, as will be shown in some examples hereafter. The feedback network gain is easily obtained by disconnecting 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, including the loading effect of the feedback network, can be computed 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. Series–Shunt Feedback Fig. 5 shows a circuit with series–shunt feedback. The resistive voltage divider (inside the dashed box) connecting the amforms the feedback plifier output with the base of transistor network. It samples the output voltage and produces a feed. The divider is the simplest circuit that back voltage accomplishes this task with passive components. Several points are worth discussing: bias points of the transistors, voltage gain and input, and output resistances of the feedand form a differential pair back amplifier. Transistors biased by the left current source, with the common emitter amplifying its output. These transistors and the biasing components make up the basic amplifier. Since both the differential 56 IEEE TRANSACTIONS ON EDUCATION, VOL. 48, NO. 1, FEBRUARY 2005 Fig. 6. Open-loop circuit for the analysis of the feedback amplifier of Fig. 5. Fig. 5. Series–shunt feedback amplifier. pair and the common emitter have high gains, one can see that the open-loop gain is very large. Then, (5) shows that (13) The feedback network must be disconnected in order to obtain its gain . From Fig. 5 and (11), one can see that so that 11 (14) is possible because the The stability of the bias point of overall circuit negative feedback guarantees that the quiescent output voltage is zero if has no dc component. This condition is a result of the error signal 0 for large-loop gains. Therefore, if the dc component of the input voltage is zero, the dc component of the output voltage is also zero. In general, one can infer the gain of the feedback amplifier easily on the assumption that the loop gain is very large. However, computing the loop gain directly to obtain the input and output resistances of the feedback amplifier is useful. The calculation of the loop gain follows the lines explained previously in (10) and Fig. 4. The loading effect of the feedback network (its input impedance is not infinite) is taken into account when opening the loop by adding a resistor in series with the col, as shown in Fig. 6. Then, can be obtained from lector of this circuit by means of (10). From Fig. 6, one can observe that 1k 10 k 11 k , where 100 has been used. The loop gain is given by 1 k. As indicated before, the loop where gain is very large so that (13) holds true. As for any amplifier with negative feedback, is positive. To calculate the input and output feedback amplifier resistances, one needs to compute the basic amplifier corresponding resistances. In order to take into account the loading effect, the . The basic amplifier input circuit of Fig. 6 is used with resistance 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 internal resistance of up to 330 k , the output voltage is essentially unchanged, and it is delivered amplified by a factor of 11 at a very low output resistance of 37 . C. Shunt–Shunt Feedback The inverting amplifier configuration of Fig. 7 will be analyzed as an example of a practical shunt–shunt feedback amplifier. The shunt–shunt configuration becomes clear if the voltage source is converted into a current source using (20) as shown in Fig. 8. Therefore, the transresistance gain be given by 295 (15) will (21) MARRERO: SIMPLIFIED ANALYSIS OF FEEDBACK AMPLIFIERS Fig. 7. 57 Inverting amplifier. Fig. 11. Series–series feedback amplifier. The component values are R = 9 k , R = 5 k , R = 600 , R = R = 100 , and R = 640 . Fig. 8. Shunt–shunt feedback amplifier. Fig. 12. Feedback network of circuit of Fig. 11. Another quantity of interest is the input resistance inverting amplifier. From Fig. 7, it is seen that of the (24) Fig. 9. Open-loop circuit for the analysis of the feedback amplifier of Fig. 8. In order to obtain , one needs to compute the feedback amplifier input resistance, which is given by (Fig. 9) 0 Fig. 10. (25) because is very large. From Fig. 8, one can see that , which means that 0 and, therefore, Feedback network for the inverting amplifier. Fig. 9 shows the circuit appropriate for feedback analysis, and Fig. 10 shows the feedback network. Typical operational amplifiers have both very large differential input resistance and gain. 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 inserting (20) in (22) (23) . D. Series–Series Feedback Fig. 11 shows a circuit amplifier with this configuration. It is part of the commercial integrated circuit MC 1553. The circuit omits the biasing of the input transistor needed to bias the rest of the circuit. In this example, one can assume that the input voltage 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 emitter . The feedback network gain is obtained from the current of circuit shown in Fig. 12. 11.9 (26) 58 IEEE TRANSACTIONS ON EDUCATION, VOL. 48, NO. 1, FEBRUARY 2005 Fig. 14. Fig. 13. Equivalent output circuit of the feedback amplifier of Fig. 11. Open-loop circuit for the analysis of the feedback amplifier of Fig. 11. Again, inspection of the circuit (two common emitters and an emitter follower) indicates that the loop gain is large. If it is larger 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, the loop gain must be obtained. Fig. 13 shows the circuit used to calculate and the characteristics of the basic amplifier . The loading effect of the feedback network has been taken into account by adding the resistor in series with the emitter of tran, where sistor 87 (29) The loop gain is given by 247 Fig. 15. Series–shunt feedback amplifier. The component values are the same as for the circuit of Fig. 11. is the resistance seen looking into the collector of , as shown , is the resisin Fig. 11. Since [4]. The tance seen when opening the loop at the emitter of is obtained as the resisbasic amplifier output resistance tance between points and of Fig. 13, and it is given by 143 . An approximate value1 for can be computed by assuming that the effect of feedat the emitter of [4]. If the Early effect back is to place of the transistors is ignored, is infinite. With finite Early is very large because it represents the output resiseffect, tance of a common emitter amplifier with emitter degeneration 35.5 k at and because of the presence of the resistor the emitter of . Therefore, it should be expected that is very large. As a result, the output voltage is supplied at a re600 . sistance (30) where 29 . Thus, the approximation of (27) is justified. The feedback amplifier input resistance is given by , where is the basic amplifier input resistance, obtained from Fig. 13 with . The result is 13 k (31) Then, 3.2 M . The equivalent output circuit of the series–series feedback amplifier is shown inside the dashed box of Fig. 14, where E. Another Look at the Loop Gain A given circuit can realize different feedback functions if the output is taken from different nodes. As an example, consider the circuit of Fig. 11, but assume that the output signal is taken as the voltage at the emitter of , as shown in Fig. 15. In this is not needed for the actual operation of case, the resistor the circuit but is kept here to show that both circuits are the same, another example of series–shunt feedback. Fig. 16 shows 1A detailed analysis of the circuit shows that this result is very accurate. MARRERO: SIMPLIFIED ANALYSIS OF FEEDBACK AMPLIFIERS Fig. 16. 59 Open-loop circuit for the analysis of the feedback amplifier of Fig. 15. the small-signal analysis circuit used to compute the loop gain, and 29 . where The loop gain is given by lateral basic amplifier and feedback networks. This technique is difficult to use and does not provide insight into the properties and advantages of the feedback amplifier. Other techniques, like the return-ratio method, are often simpler to use but do not offer greater insight as to the circuit performance. In this paper, a very simple technique for the analysis of feedback amplifiers with large-loop gain has been shown to give very reliable results with minor effort. General properties of feedback amplifiers, such as gain and input and output resistances, are obtained using an open-loop circuit where the loading effect of the feedback network is easily taken into account. Emphasis is placed on quick, intuitive, and reliable calculations, useful for both the analysis and design of feedback amplifiers. This technique is appropriate for the analysis of feedback amplifiers where the kind of connection at both the input and output of the amplifier is easy to recognize. However, this information may be somewhat difficult to obtain for some cases, especially for some current mirrors such as the bipolar Wilson mirror, where the return-ratio method seems the appropriate choice for the analysis. ACKNOWLEDGMENT The author would like to thank the Ministerio de Ciencia y Tecnología of Spain for support and the anonymous reviewers of the paper for their very helpful comments. REFERENCES 247 (32) This result coincides with the one obtained in (30) for the series–series feedback amplifier of Fig. 11. As one would expect, the loop gain is unchanged by the choice of the output port if the circuit is unchanged by this choice. Since the loop gain is very large, the voltage gain of the feedback circuit is given by 7.4 (33) The feedback amplifier input resistance is obtained as (34) is the basic amplifier input resistance, obtained from where . The result is Fig. 16 with 13 k (35) 3.2 M , which is the same as the result of the seThen, ries–series amplifier of Fig. 11. A closer look reveals that there are minor differences between (31) and (35) because of the differences in the loading effect of the feedback network at the amplifier output, a consequence of the level of approximation involved in this analysis. IV. CONCLUSION The traditional treatment of practical feedback amplifiers often relies on two ports to convert the feedback circuit into uni- [1] D. L. Schilling and C. Belove, Electronic Circuits: Discrete and Integrated. New York: McGraw-Hill, 1968. [2] J. Millman and C. C. Halkias, Integrated Electronics. New York: McGraw-Hill, 1972. [3] M. N. Horenstein, Microelectronic Circuits and Devices, 2nd ed. Englewood 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. 1382–1389, Nov. 1991. [8] , “A comparison of two approaches to feedback circuit analysis,” IEEE Trans. Educ., vol. 35, pp. 253–261, Aug. 1992. [9] S. Ben-Yaakov, “A unified approach to teaching feedback in electronic circuits courses,” IEEE Trans. Educ., vol. 34, pp. 310–316, 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. 415–418. [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 Electronics. Cambridge, U.K.: Cambridge Univ. Press, 1989. José Luis Rodríguez Marrero received the Ingeniero de Telecomunicaciones degree from Universidad Politécnica de Cataluña, Barcelona, Spain, and the M.S. degree in electrical and computer engineering and the Ph.D. degree in physics 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 are in the areas of electronic instrumentation, nonlinear dynamics, and engineering education.