58 IEEE TRANSACTIONS ON EDUCATION, VOL. 49, NO. 1, FEBRUARY 2006

High-Frequency Transistor Amplifier AnalysisChetan S. Mandayam Nayaka

Abstract—In this paper, some modifications are suggested to thestandard procedure for high-frequency analysis of transistor am-plifiers using Miller’s theorem. These suggestions remove a majorstumbling block in the application of this theorem and extend therange of applicability of this method.

Index Terms—Inverting amplifiers, Miller’s theorem, negativecapacitance, upper cutoff frequency.

THE CURRICULUM content relevant to this paper is pre-sented in a first course in electronic circuits for undergrad-

uates. It is concerned with the calculation of the upper cutofffrequency of transistor amplifiers using Miller’s theorem. Themethod used until now in all textbooks [1]–[6] contains seriousdeficiencies in that there is a major flaw in the procedure that istotally unnecessary and completely avoidable, introducing se-rious unacceptable errors into calculations and greatly reducingthe validity of the analysis. In fact, for quite a long time, thisproblem has proved to be a major stumbling block in the anal-ysis and design of transistor amplifiers whose importance cannever be overemphasized. In addition, a major, totally unnec-essary constraint has been assumed to exist, which severely re-stricts the range of applicability of this method. The suggestionsin this paper eliminate all errors and do away with all unneces-sary constraints, greatly extending the scope and magnitude ofthe method of analysis referenced earlier.

In a transistor amplifier, the high-frequency gain is reducedbecause of parasitic shunt capacitances [1]–[6]. One of thesecapacitances is modeled as being connected between one of theinput terminals and one terminal of the output port [1]–[6]. Thissetup makes analysis of the circuit inconvenient, and the ca-pacitance is, therefore, replaced by one equivalent capacitanceacross the input terminals and another across the output termi-nals [1]–[6] as shown in Fig. 1. If is the original capacitanceand is the voltage gain, the equivalent capacitance across theinput terminals is , and the equivalent outputcapacitance is [1]–[6]. The “dilemma”[1] of the above equations (as implied in all books on the subject[1]–[6]) is that for calculating the upper cutoff frequency, thevalues of the capacitances are needed, which, in turn, require aknowledge of the gain. In standard practice (as in all textbooks),this “problem” is “solved” by taking to be equal to the mid-band gain of the amplifier [1]–[6]. This solution is obviouslywrong and unnecessary. It leads to very erroneous calculationsand to results that are of very limited value. This error has proved

Manuscript received December 1, 2003; revised June 20, 2005.The author is with the Department of Electrical Engineering, Indian Institute

of Technology, Bangalore, Madras 560070, India (e-mail: mn_chetan@yahoo.com).

Digital Object Identifier 10.1109/TE.2005.856151

Fig. 1. Definition of Miller equivalent capacitances.

to be a major handicap in high-frequency amplifier analysis anddesign. However, there exists a way by which the problem canbe eliminated completely. Clearly, can be taken to be equal to

in all calculations, where is the midband gain,since the upper cutoff frequency is, by definition, the frequencyat which the gain is equal to . In other words, for thepurpose of calculating the upper cutoff frequency, can besubstituted by , and can be substitutedby . Obviously, this suggestion is correct be-cause, by its very definition, the upper cutoff frequency is thefrequency at which the amplifier gain drops to times itsmidband value. No other justification need be or can be given forthis claim because it follows in a very direct and straightforwardmanner from the very definition of the upper cutoff frequencygiven above. If is large, as it often is, the difference be-tween and can be quite large; and in case theMiller effect capacitance plays a dominant role in determiningthe upper cutoff frequency, which it often does, using in-stead of for the amplifier gain can lead to large errorsin the calculated value of the upper cutoff frequency. Therefore,the importance of this result cannot be overemphasized. Thefollowing numerical example will clearly demonstrate the dif-ferences between the results predicted by the existing approachfound in textbooks and the approach presented in this paper. Inthis example, the upper cutoff frequency will be calculated forthe circuit of Fig. 2 [1]. The values of the various parametersare as follows: 1 , 40 , 10 ,

2 , 4 , 2.2 , 10 ,1 , 20 , 36 , 4 ,

1 , 6 , 8 , 100, ,20 V. The high-frequency ac equivalent model for the

circuit is shown in Fig. 3.In this figure, , and

. The Thevenin equivalent circuits for the input and outputnetworks of the network of Fig. 3 are shown in Fig. 4(a) and

0018-9359/$20.00 © 2006 IEEE

MANDAYAM NAYAKA: HIGH-FREQUENCY TRANSISTOR AMPLIFIER ANALYSIS 59

Fig. 2. Common emitter amplifier.

Fig. 3. High-frequency small-signal equivalent circuit.

Fig. 4.(a) Thevenin equivalent circuit of input network. (b) Thevenin equivalentcircuit of output network.

(b), respectively. In these figures, , and. One can easily show that 1.32 ,

90, and 0.531 . Using the textbookapproach

406

1.419

and

13.04

Therefore, 738.24 kHz, and8.6 MHz. Using the approach presented in

this paper, the only differences are in the values of and .

301

and

13.06

Therefore, 995.77 kHz, and 8.6 MHz. Sinceis about a decade higher than in both the approaches, the

upper cutoff frequencies are mainly determined by the values ofin both cases. Thus, the values of upper cutoff frequency

obtained using the textbook approach and the approach in thispaper are approximately 738.24 and 995.77 kHz, respectively, asignificant difference.

In addition, textbooks state that this method is applicableonly to inverting amplifiers since, otherwise, one of the two

60 IEEE TRANSACTIONS ON EDUCATION, VOL. 49, NO. 1, FEBRUARY 2006

Miller equivalent capacitances will be negative [1]–[6]. Thisconstraint is equally unnecessary. The author’s suggestion, notfound in textbooks, is that if the Miller equivalent capacitanceturns out to be negative, one should take it as negative andcombine it with all the other capacitances present in the net-work to find the equivalent capacitance. If now a positive ca-pacitance is obtained, no problem exists, but if the equivalentcapacitance turns out to be negative, the suggestion is to takethe absolute value of the capacitance and use it to calculatethe high frequency cutoff. Now, considering the R–C combina-tion [1] that determines the upper cutoff frequency

, implying that ,which implies that , which means ;hence, the upper cutoff frequency is . This trickof using the absolute value of the capacitance works because themagnitude of the transfer function of the amplifier, i.e., the gainof the amplifier, depends on the square of the capacitance and,hence, only on its absolute value, as is evident from the relevantexpression. Thus, a completely convincing justification of thetechnical validity of this suggestion has been given. With thismodification, the method can be applied to any kind of ampli-fier, no matter what its gain.

ACKNOWLEDGMENT

The author would like to thank his brother ChinmoyVenkatesh M. N. for his help while preparing this manuscript.

REFERENCES

[1] R. L. Boylestad and L. Nashelsky, Electronic Devices and Circuit Theory,6th ed. Englewood Cliffs, NJ: Prentice-Hall, 1996, pp. 536–539.

[2] A. Mottershead, Electronic Devices and Circuits: An Introduction.Englewood Cliffs, NJ: Prentice-Hall, 1991, pp. 270–272, 585, 586.

[3] M. S. Roden, G. L. Carpenter, and W. R. Wieserman, Electronic Design,4th ed. Los Angeles, CA: Discovery Press, 2002, pp. 529–541.

[4] J. Millman and C. C. Halkias, Integrated Electronics: Analog andDigital Circuits and Systems. New York: McGraw-Hill, 1991, pp.362–365.

[5] D. L. Schilling and C. Belove, Electronic Circuits: Discrete and Inte-grated, 3rd ed. New York: McGraw-Hill, 1989, pp. 457–462.

[6] M. H. Jones, A Practical Introduction to Electronic Circuits, 3rd ed.Cambridge, MA: Cambridge University Press, 1995, pp. 151–155.

Chetan S. Mandayam Nayaka was born in Bangalore, India, in August 1980.He is currently working toward the B.S. and M.S. degrees at the Department ofElectrical Engineering, Indian Institute of Technology, Madras, India.

He has been actively engaged in research in quantum physics, the physics(thermodynamics) of intelligence, and electronics. His research interests alsoinclude the elementary foundations of physics and mathematics.