# Now suppose one pole is much farther from the origin

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Now suppose one pole is much farther from the origin than the other: . (This is called the “dominant pole” approximation to emphasize that dominates the frequency response). Then, , i.e., (11.76) and from (11.72), (11.77) How does this result compare with that obtained using the Miller approximation? Equa- tion (11.77) does reveal the Miller effect of but it also contains the additional term [which is close to the output time constant predicted by (11.59)]. To determine the “nondominant” pole, , we recognize from (11.75) and (11.76) that (11.78) (11.79) Example 11.17 Using the dominant-pole approximation, compute the poles of the circuit shown in Fig. 11.31(a). Assume both transistors operate in saturation and . Solution Noting that , , and do not affect the circuit (why?), we add the remaining capacitances as depicted in Fig. 11.31(b), simplifying the result as illustrated in Fig. 11.31(c), where (11.80) (11.81) (11.82) As explained in more advanced courses, this zero does become problematic in the internal circuitry of op amps.

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BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 564 (1) 564 Chap. 11 Frequency Response M 1 V DD in V R S M 1 C C C out V V b M 2 in V R S M 2 C GD2 C DB2 M 1 C C C in V R S out XY in r O1 r O2 (c) (a) (b) out V out V GD1 GS1 DB1 C SB2 Figure 11.31 It follows from (11.77) and (11.79) that (11.83) (11.84) Exercise Repeat the above example if . Example 11.18 In the CS stage of Fig. 11.29(a), we have fF, fF, fF and k . Plot the frequency response with the aid of (a) Miller’s approximation, (b) the exact transfer function, (c) the dominant-pole approximation. Solution (a) With , Eqs. (11.58) and (11.59) yield (11.85) (11.86) (b) The transfer function in Eq. (11.70) gives a zero at GHz). Also, s and s. Thus, (11.87) (11.88)
BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 565 (1) Sec. 11.4 Frequency Response of CE and CS Stages 565 Note the large error in the values predicted by Miller’s approximation. This error arises be- cause we have multiplied by the midband gain rather than the gain at high frequencies. (c) The results obtained in part (b) predict that the dominant-pole approximation produces rel- atively accurate results as the two poles are quite far apart. From Eqs. (11.77) and (11.79), we have (11.89) (11.90) Figure 11.32 plots the results. The low-frequency gain is equal to 22 dB and the -dB bandwidth predicted by the exact equation is around 250 MHz. 10 7 10 8 10 9 10 10 30 20 10 0 10 20 30 Frequency (Hz) Magnitude of Transfer Function (dB) Dominant Pole Appr. Miller’s Approx. Exact Eq. Figure 11.32 Exercise Repeat the above example if the device width (and hence its capacitances) and the bias current are halved. 11.4.5 Input Impedance The high-frequencyinput impedances of the CE and CS amplifiers determine the ease with which these circuits can be driven by other stages. Our foregoing analysis of the frequency response and particularly the Miller approximation readily yield this impedance.

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