Fundamentals-of-Microelectronics-Behzad-Razavi.pdf

Exercise repeat the above example if the capacitor

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Exercise Repeat the above example if the capacitor and the inductor in Fig. 14.12(c) are swapped. Example 14.5 Explain why the poles of the circuits in Fig. 14.12 must lie in the right half plane. Solution Recall that the impulse response of a system contains terms such as . If , these terms grow indefinitely with time while oscillating at a frequency of [Fig. 14.14(a)]. If , such terms still introduce oscillation at [Fig. t t t σ k > 0 σ k = 0 σ k < 0 (c) (a) (b) Figure 14.14 14.14(b)]. Thus, we require for the system to remain stable [Fig. 14.14(c)].
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BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 729 (1) Sec. 14.1 General Considerations 729 Exercise Redraw the above waveforms if is doubled. It is instructive to make several observations in regards to Eq. (14.1). (1) The order of the numerator, , cannot exceed that of the denominator; otherwise, as , an unrealistic situation. (2) For a physically-realizable transfer function, complex zeros or poles must occur in conjugate pairs, e.g., and . (3) If a zero is located on the axis, , then drops to zero at a sinusoidal input frequency of (Fig. 14.15). This is because the numerator contains a product such as , ( ) ω H ω ω 1 Figure 14.15 Effect of imaginary zero on the frequency response. which vanishes at . In other words, imaginary zeros force to zero, thereby providing significant attenuation in their vicinity. For this reason, imaginary zeros are placed only in the stopband . 14.1.4 Problem of Sensitivity The frequency response of analog filters naturally depends on the values of their constituent com- ponents. In the simple filter of Fig. 14.10(a), for example, the -dB corner frequency is given by . Such dependencies lead to errors in the cut-off frequency and other parameters in two situations: (a) the value of components varies with process and temperature (in integrated circuits), or (b) the available values of components deviate from those required by the design (in discrete implementations). We must therefore determine the change in each filter parameter in terms of a given change (tolerance) in each component value. Example 14.6 In the low-pass filter of Fig. 14.10(a), resistor experiences a (small) change of . Deter- mine the error in the corner frequency, . Solution For small changes, we can utilize derivatives: (14.7) For example, a particular design requires a 1.15-k resistor but the closest available value is 1.2 k .
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BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 730 (1) 730 Chap. 14 Analog Filters Since we are usually interested in the relative (percentage) error in in terms of the relative change in , we write (14.7) as (14.8) (14.9) (14.10) For example, a change in translates to a error in . Exercise Repeat the above example if experiences a small change of . The above example leads to the concept of “sensitivity,” i.e., how sensitive each filter param- eter is with respect to the value of each component. Since in the first-order circuit, , we say the sensitivity of with respect to is unity in this example. More formally, the sensitivity of parameter with respect to the component value is defined as (14.11)
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