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Circuit Theory:

1 Circuit Theory Chapter 14 Frequency Response Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Frequency Response Chapter 14:

2 Frequency Response Chapter 14 14.1 Introduction 14.2 Transfer Function 14.3 Series Resonance 14.4 Parallel Resonance 14.5 Passive Filters

What is FrequencyResponse of a Circuit?:

3 What is FrequencyResponse of a Circuit? It is the variation in a circuit’s behavior with change in signal frequency and may also be considered as the variation of the gain and p hase with frequency . 14.1 Introduction (1)

14.2 Transfer Function (1):

4 14.2 Transfer Function (1) The transfer function H(ω) of a circuit is the frequency-dependent ratio of a phasor output Y ( ω ) (an element voltage or current ) to a phasor input X ( ω ) (source voltage or current).

14.2 Transfer Function (2):

5 14.2 Transfer Function (2) Four possible transfer functions:

14.2 Transfer Function (3):

6 14.2 Transfer Function (3) Example 1 For the RC circuit shown below, obtain the transfer function Vo/Vs and its frequency response. Let v s = V m cosωt.

14.2 Transfer Function (4):

7 14.2 Transfer Function (4) Solution : The transfer function is , The magnitude is The phase is Low Pass Filter

14.2 Transfer Function (5):

8 14.2 Transfer Function (5) Example 2 Obtain the transfer function Vo/Vs of the RL circuit shown below, assuming v s = V m cosωt. Sketch its frequency response.

14.2 Transfer Function (6):

9 14.2 Transfer Function (6) Solution : The transfer function is , The magnitude is The phase is High Pass Filter

14.3 Series Resonance (1):

10 14.3 Series Resonance (1) Resonance is a condition in an RLC circuit in which the capacitive and inductive reactance are equal in magnitude, thereby resulting in purely resistive impedance. Resonance frequency:

14.3 Series Resonance (2):

11 14.3 Series Resonance (2) The features of series resonance: The impedance is purely resistive, Z = R; The supply voltage Vs and the current I are in phase, so cos q = 1; The magnitude of the transfer function H(ω) = Z(ω) is minimum; The inductor voltage and capacitor voltage can be much more than the source voltage.

14.3 Series Resonance (3):

12 14.3 Series Resonance (3) Bandwidth B The frequency response of the resonance circuit current is The average power absorbed by the RLC circuit is The highest power dissipated occurs at resonance:

14 3 Series Resonance (4):

13 14 3 Series Resonance (4) Half-power frequencies ω 1 and ω 2 are frequencies at which the dissipated power is half the maximum value: The half-power frequencies can be obtained by setting Z equal to √2 R. Bandwidth B

14.3 Series Resonance (5):

14 14.3 Series Resonance (5) Quality factor , The quality factor is the ratio of its resonant frequency to its bandwidth . If the bandwidth is narrow , the quality factor of the resonant circuit must be high . If the band of frequencies is wide , the quality factor must be low . The relationship between the B, Q and ω o :

14.3 Series Resonance (6):

15 14.3 Series Resonance (6) Example 3 A series-connected circuit has R = 4 Ω and L = 25 mH. a. Calculate the value of C that will produce a quality factor of 50. b. Find ω 1 and ω 2 , and B. c. Determine the average power dissipated at ω = ω o , ω 1 , ω 2 . Take V m = 100V.

14.4 Parallel Resonance (1):

16 14.4 Parallel Resonance (1) Resonance frequency: It occurs when imaginary part of Y is zero

14.4 Parallel Resonance (2):

17 Summary of series and parallel resonance circuits: 14.4 Parallel Resonance (2) characteristic Series circuit Parallel circuit ω o Q B ω 1, ω 2 Q ≥ 10, ω 1, ω 2

14.4 Parallel Resonance (3):

18 14.4 Parallel Resonance (3) Example 4 Calculate the resonant frequency of the circuit in the figure shown below. Answer :

14.5 Passive Filters (1):

19 14.5 Passive Filters (1) A filter is a circuit that is designed to pass signals with desired frequencies and reject or attenuate others. Passive filter consists of only passive element R, L and C. There are four types of filters. Low Pass High Pass Band Pass Band Stop

14.5 Passive Filters (2):

20 14.5 Passive Filters (2) Example 5 For the circuit in the figure below, obtain the transfer function Vo(ω)/Vi(ω). Identify the type of filter the circuit represents and determine the corner frequency. Take R1=100 W =R2 and L =2mH. Answer : HW16 Ch14: 47, 55, 57, 59

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