Question

Find the transfer function H(s) = Vo/Vs for the circuit in Fig. 1,2 and 3. Describe the filter characteristics.

Fig. 1. An active filter Fig. 2. Another active filter. Fig. 3. Yet another active filter. R2 R1

Answer 1

Apply K C L at node V_1 V_1 - V_s/R + V_1/R + V_1 - V_0/(1/SC) + V_1 - 0/1/sc = 0 implies 2 V_1 - V_S/R + SC (2 V_1 - V_0) = 0 implies 1/R (2 V_1 - V_s) + S C(2 V_1 - V_0) = 0 implies 2 (1/R + sc) v_1 = v_S/R + S C V_0 (1) By virtual ground concept V^- = V^+ = 0 Apply K C L at negative terminal of op - amp 0 - V_1/(1/sc) + 0 - V_0/R = 0 implies S C V_1 = -V_0/R implies V_1 = -V_0/S C R (2) from equations (1) and (2) 2 (1/R + SC) (- V_0/S C R) = V_S/R + SC V_0 implies - 2 (1/S C R^2 + 1/R) V_0 - S C V_0 = V_s/R implies [- 2/S C R^2 + - 2/R - S C] V_0 = V_s/R implies [- 2/S C R - 2 - S C R] V_0 = V_s implies [- 2- 2 S C R -
The first filter is second order low pass filter, second is second order high pass filter and third is first order high pass filter.