Problem

The fourth-order low-pass unity-gain Bulterworth filter in Problem 15.55 is used in a syst...

The fourth-order low-pass unity-gain Bulterworth filter in Problem 15.55 is used in a system where the cutoff frequency is 3 kHz. The filter has 4.7 nF capacitors.

a) Specify the numerical values of R1 and R2 in each section of the filter.
b) Draw a circuit diagram of the filter and label all the components.

Problem 1

The purpose of this problem is to develop the problem design equations for the circuit in Fig. 1. (See Problem 1 for suggestions 011 the development of design equations.)

a) Based on a qualitative analysis, describe the type of filter implemented by the circuit.
b) Verify the conclusion reached in (a) by deriving the transfer function Vo / Vi . Write the transfer function in a form that makes it compatible with the entries in Table 15.1.
c) How many free choices are there in the selection of the circuit components?
d) Derive the expressions for the conductances G1 = 1/R1 and G2 = 1/R2 in terms of C1 C2, and the coefficients bo and b1. (See Problem 1 for the definition of bo and b1.)
e) Are there any restrictions on C1 or C2?
f) Assume the circuit in Fig. 1 is used to design a fourth-order low-pass unity-gain Butterworth filter. Specify the prototype values of R1 and R2 in each second-order section if 1 F capacitors are used in the prototype circuit.

Figure 1

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.