1. 1 Lecture #20 EGR 272 – Circuit Theory II General 2 nd Order Transfer Function For 2 nd order circuits, the denominator of any transfer function will take on the following form: s 2 + 2  s + w o 2 Various types of 2 nd order filters can be formed using a second order circuit, including: Read: Chapter 14 in Electric Circuits, 6 th Edition by Nilsson

2. 2 Lecture #20 EGR 272 – Circuit Theory II Series RLC Circuit (2 nd Order Circuit) Draw a series RLC circuit and find transfer functions for LPF, BPF, and HPF. Note that the denominator is the same in each case (s 2 + 2  s + w o 2 ). Also show that:

3. 3 Lecture #20 EGR 272 – Circuit Theory II Parallel RLC Circuit (2 nd Order Circuit) Draw a parallel RLC circuit and find transfer functions for LPF, BPF, and HPF. Note that the denominator is the same in each case (s 2 + 2  s + w o 2 ). Also show that:

4. 4 Lecture #20 EGR 272 – Circuit Theory II 2 nd Order Bandpass Filter A 2nd order BPF will now be examined in more detail. The transfer function, H(s), will have the following form: Magnitude response Show a general sketch of the magnitude response for H(s) above Define w o, w c1, w c2, H max, BW, and Q Sketch the magnitude response for various values of Q (in general)

5. 5 Lecture #20 EGR 272 – Circuit Theory II Determining H max Find H(jw) and then  H(jw) . Show that

6. 6 Lecture #20 EGR 272 – Circuit Theory II Determining w c1 and w c2 : leads to Show that

7. 7 Lecture #20 EGR 272 – Circuit Theory II Determining w o, BW, and Q: Show that w o is the geometric mean of the cutoff frequencies, not the arithmetic mean. Also find BW and Q. Specifically, show that: Damping ratio is simply defined here. Its significance will be seen later in this course and in other courses (such as Control Theory). Circuits with similar values of  have similar types of responses.

8. 8 Lecture #20 EGR 272 – Circuit Theory II Example: A parallel RLC circuit has components R, L = 100 mH, and C = 0.1 uF 1) Find w o, , H max, w c1, w c2, H max, BW, Q, and  2) Show that w o is the geometric mean of the w c1 and w c2, not the arithmetic mean. A) Use R = 1 k 

9. 9 Lecture #20 EGR 272 – Circuit Theory II Example: A parallel RLC circuit has components R, L = 100 mH, and C = 0.1 uF 1) Find w o, , H max, w c1, w c2, H max, BW, Q, and  2) Show that w o is the geometric mean of the w c1 and w c2, not the arithmetic mean. B) Use R 20 k 

10. 10 Lecture #20 EGR 272 – Circuit Theory II Example: Plot the magnitude response, |H(jw)|, for parts A and B in the last example. (Note that a curve with a geometric mean will appear symmetrical on a log scale and a curve with an arithmetic mean will appear symmetrical on a linear scale.) 5k6k7k8k9k10k20k w (log scale) |H(jw)|