Problem

Interchange the Rs and Cs in the circuit in Fig. 1; that is, replace R1 with C1, R2 with C...

Interchange the Rs and Cs in the circuit in Fig. 1; that is, replace R1 with C1, R2 with C2, R3 with C3, C1 with R1, and C2 with R2.

a) Describe the type of filter implemented as a result of the interchange.
b) Confirm the filter type described in (a) by deriving the transfer function Vo /Vi. Write the transfer function in a form that makes it compatible with Table 15.1.
c) Set C2 = C3 = 1 F and derive the expressions for C1, R1, and R2 in terms of K, b1, and b0. (See Problem 1 for the definition of b1, and b0.)
d) Assume the filter described in (a) is used in the same type of third-order Butterworth filter that has a passband gain of 8. With C2 1 F, calculate the prototype values of C1, R1, and R2in the second order section of the filter.

Problem 1

The purpose of this problem is to guide you problem through the analysis necessary to establish a design procedure for determining the circuit components in a filter circuit. The circuit to be analyzed is shown in Fig. 1.

a) Analyze the circuit qualitatively and convince yourself that the circuit is a low-pass filter with a passbahd gain of R2/R1.
b) Support your qualitative analysis by deriving the transfer function Vo/ Vi. To make the transfer function useful in terms of the entries in Table 15.1, put it in the form
c) Now observe that we have five circuit components − R1, R2,R3, C1, and C2 − and three transfer function constraints−K, b1, and bo. At first glance, it appears we have two free choices among the five components. However, when we investigate the relationships between the circuit components and the transfer function constraints, we see that if C2 is chosen, there is an upper limit on C1 in order for R2(G2) to be realizable. With this in mind, show that if C2− 1 F, the three conductances arc given by the expressions
For G2 to be realizable:
d) Based on the results obtained in (c), outline the design procedure for selecting the circuit components once K, bo, and b1 are known.