Frequency Response This is an extremely important topic in EE. Up until this point we have analyzed circuits without considering the effect on the answer over a wide range of frequencies. Many circuits have frequency limitations that are very important. Example: Discuss the frequency limitations on the following items. 1) An audio amplifier 2) An op amp circuit Read: Ch. 14, Sect. 1-5 in Electric Circuits, 9 th Edition by Nilsson 1 Chapter 14 EGR 272 – Circuit Theory II

Example: Discuss the frequency limitations on the following items (continued) 3) A voltmeter (% error vs frequency) 4) The tuner on a radio (band-pass filter) 2 Chapter 14 EGR 272 – Circuit Theory II

Filters A filter is a circuit designed to have a particular frequency response, perhaps to alter the frequency characteristics of some signal. It is often used to filter out, or block, frequencies in certain ranges, much like a mechanical filter might be used to filter out sediment in a water line. Basic Filter Types Low-pass filter (LPF) - passes frequencies below some cutoff frequency, w C High-pass filter (HPF) - passes frequencies above some cutoff frequency, w C Band-pass filter (BPF) - passes frequencies between two cutoff frequency, w C1 and w C2 Band-stop filter (BSF) or band-reject filter (BRF) - blocks frequencies between two cutoff frequency, w C1 and w C2 3 Chapter 14 EGR 272 – Circuit Theory II

Ideal filters An ideal filter will completely block signals with certain frequencies and pass (with no attenuation) other frequencies. (To attenuate a signal means to decrease the signal strength. Attenuation is the opposite of amplification.) LM w wCwC Ideal LPF LM w wCwC Ideal HPF LM w w C1 Ideal BPF w C2 LM w w C1 Ideal BSF w C2 4 Chapter 14 EGR 272 – Circuit Theory II

Filter order Unfortunately, we can’t build ideal filters. However, the higher the order of a filter, the more closely it will approximate an ideal filter. The order of a filter is equal to the degree of the denominator of H(s). (Of course, H(s) must also have the correct form.) LM w wCwC Ideal LPF 3rd-order LPF 2nd-order LPF 1st-order LPF 4th-order LPF 5 Chapter 14 EGR 272 – Circuit Theory II

Defining frequency response Recall that a transfer function H(s) is defined as: Where Y(s) = some specified output and X(s) = some specified input In general, s =  + jw. For frequency applications we use s = jw (so  = 0). So now we define: Since H(jw) can be thought of as a complex number that is a function of frequency, it can be placed into polar form as follows: 6 Chapter 14 EGR 272 – Circuit Theory II

When we use the term "frequency response", we are generally referring to information that is conveyed using the following graphs: 7 Chapter 14 EGR 272 – Circuit Theory II Example: Find H(jw) for H(s) below. Also write H(jw) in polar form.

Example: A) Find H(s) = V o (s)/V i (s) B) Find H(jw) 8 Chapter 14 EGR 272 – Circuit Theory II

Example: (continued) C) Sketch the magnitude response, |H(jw)| versus w D) Sketch the phase response,  (w) versus w E) The circuit represents what type of filter? 9 Chapter 14 EGR 272 – Circuit Theory II

Example: A) Find H(s) = V(s)/I(s) B) Find H(jw) 10 Chapter 14 EGR 272 – Circuit Theory II

Example: (continued) C) Sketch the magnitude response, |H(jw)| versus w D) Sketch the phase response,  (w) versus w E) The circuit represents what type of filter? 11 Chapter 14 EGR 272 – Circuit Theory II

Example: A) Find H(s) = V o (s)/V i (s) B) Find H(jw) 12 Chapter 14 EGR 272 – Circuit Theory II

Example: (continued) C) Sketch the magnitude response, |H(jw)| versus w D) Sketch the phase response,  (w) versus w E) The circuit represents what type of filter? 13 Chapter 14 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  s + w o 2 Various types of 2 nd order filters can be formed using a second order circuit, including: 14 Chapter 14 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  s + w o 2 ). Also show that: 15 Chapter 14 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  s + w o 2 ). Also show that: 16 Chapter 14 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) 17 Chapter 14 EGR 272 – Circuit Theory II

Determining H max Find H(jw) and then  H(jw) . Show that 18 Chapter 14 EGR 272 – Circuit Theory II

Determining w c1 and w c2 : leads to Show that 19 Chapter 14 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. 20 Chapter 14 EGR 272 – Circuit Theory II

Example: A parallel RLC circuit has 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  21 Chapter 14 EGR 272 – Circuit Theory II

Example: A parallel RLC circuit has 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  22 Chapter 14 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)| 23 Chapter 14 EGR 272 – Circuit Theory II