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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

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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

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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

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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

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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

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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

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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.

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

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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

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

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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

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

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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

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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: 14 Chapter 14 EGR 272 – Circuit Theory II

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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: 15 Chapter 14 EGR 272 – Circuit Theory II

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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: 16 Chapter 14 EGR 272 – Circuit Theory II

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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

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

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

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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

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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

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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

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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

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