Short Circuit Time-Constant Approximation (SCTCA)

Equation (4.13) is accurate, but it's not clear that it is very useful. We could program it and evaluate it numerically. That would give us the behavior. Another, probably more useful approach would be to find the roots of all the polynomials on the numerator and denominator. That would give all the poles and zeros. Then a Bode plot could be performed to give the response of the circuit. However, this would still require more algebra than you may want to do. Another way is to realize that all the filters in the circuit are of the high-pass type, and tend to yield increasing gain with frequency. It's also worth noticing that there are the same number of poles as there are zeros. Under these circumstances, it's expedient to use the Short-Circuit Time Constant Approximation (SCTCA).

With the SCTCA we approximate the combined effect of all the poles and zeros to make a single first order high pass filter. In other words, we approximate the gain of the overall circuit as

$$\frac{V_{out}}{V_{in}}=A_m\frac{s}{s-p_o}$$ (79)

Where A_m is the midband voltage gain, and p_o is the single pole that approximates the combined effects of the actual poles and zeros of the circuit. To determine p_o, you determine the resistance seen by each capacitor, while all the other capacitors are shorted. We then multiply the resistance seen by the value of the capacitor of interest to obtain the short-circuit time constant for that capacitor. This procedure is then repeated for all capacitors which govern the low frequency response. Finally, p_o is then given by the sum of the reciprocal of each time constant. Mathematically, this is expressed as:

$$p_o \approx \sum_i\frac{-1}{R_iC_i}$$ (80)

Where R_i is the resistance seen by the i'th capacitor with all other capacitors shorted.