Circuit Theory Chapter 14 Frequency Response Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 Frequency Response Chapter 14 14.1 Introduction 14.2 Transfer Function 14.7 Passive Filters 2 14.1 Introduction (1) What is FrequencyResponse of a Circuit? It is the variation in a circuits behavior with change in signal frequency and may also be considered as the variation of the gain and phase with frequency.

3 14.2 Transfer Function (1) The transfer function H() of a circuit is the ) of a circuit is the frequency-dependent ratio of a phasor output Y() of a circuit is the ) (an element voltage or current ) to a phasor input X() of a circuit is the ) (source voltage or current). Y( ) H( ) | H( ) | X( ) 4 14.2 Transfer Function (2) Four possible transfer functions: H ( ) Voltage gain Vo ( ) Vi ( ) H ( )

I o ( ) H( ) Current gain I i ( ) H( ) Transfer Impedance Vo ( ) Ii ( ) Y( ) | H( ) | X( ) I o ( ) H( ) Transfer Admittance Vi ( ) 5 14.2 Transfer Function (3) Example 1 For the RC circuit shown below, obtain the transfer function Vo/Vs and its frequency response.

Let vs = Vmcos) of a circuit is the t. 6 14.2 Transfer Function (4) Solution: The transfer function is 1 V 1 j C H ( ) o Vs R 1/ j C 1 j RC , The magnitude is H( ) The phase is tan 1

o o 1/RC 1 1 ( / o ) 2 Low Pass Filter 7 14.2 Transfer Function (5) Example 2 Obtain the transfer function Vo/Vs of the RL circuit shown below, assuming vs = Vmcos) of a circuit is the t. Sketch its frequency response. 8 14.2 Transfer Function (6) Solution: Vo

j L 1 H( ) Vs R j L 1 R j L The transfer function is High Pass Filter , The magnitude is H ( ) The phase is 90 tan 1 o R/L 1 1 ( o )2

o 9 14.5 Bandpass Filter() 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: 1 rad/s LC 1 fo Hz 2 LC o 1

Z R j ( L ) C or 10 14.3 Series Resonance (2) The features of series resonance: Z R j ( L 1 ) C The impedance is purely resistive, Z = R; The supply voltage Vs and the current I are in phase, so cos = 1; The magnitude of the transfer function H() of a circuit is the ) = Z() of a circuit is the ) is minimum; The inductor voltage and capacitor voltage can be much more than the source voltage.

11 14.3 Series Resonance (3) Bandwidth B The frequency response of the resonance circuit current is I | I | Z R j ( L 1 ) C Vm R 2 ( L 1 / C) 2 The average power absorbed by the RLC circuit is 1 P( ) I 2 R

2 The highest power dissipated occurs at resonance: 1 Vm2 P(o ) 2 R 12 14 3 Series Resonance (4) Half-power frequencies ) of a circuit is the 1 and ) of a circuit is the 2 are frequencies at which the dissipated power is half the maximum value: 1 (Vm / 2 ) 2 Vm2 P(1 ) P(2 ) 2 R 4R The half-power frequencies can be obtained by setting Z equal to 2 R.

11 RR RR 11 (( ))22 2L 2L 2L 2L LC LC Bandwidth B RR RR 11 22 (( ))22 2L 2L

2L 2L LC LC oo 11 22 B 2 1 13 14.3 Series Resonance (5) Quality factor, Q The relationship between the B, Q and ) of a circuit is the o:

Peak energy stored in the circuit o L 1 Energy dissipated by the circuit R o CR in one period at resonance B R o o2 CR L Q 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. 14

14.3 Series Resonance (6) Example 3 A series-connected circuit has R = 4 and L = 25 mH. a. Calculate the value of C that will produce a quality factor of 50. b. Find ) of a circuit is the 1 and ) of a circuit is the 2, and B. c. Determine the average power dissipated at ) of a circuit is the = ) of a circuit is the o, ) of a circuit is the 1, ) of a circuit is the 2. Take Vm= 100V. 15 14.4 Parallel Resonance (1) It occurs when imaginary part of Y is zero 1 1 Y j ( C ) R L Resonance frequency:

1 1 o rad/s or f o Hz LC 2 LC 16 14.4 Parallel Resonance (2) Summary of series and parallel resonance circuits: characteristic Series circuit Parallel circuit o 1 LC

1 LC Q o L 1 or R o RC R or o RC o L B 1, 2 Q 10, 1, 2 o Q o

Q o 1 ( 1 2 ) o 2Q 2Q o B 2 o 1 ( 1 2 o ) 2Q 2Q o

B 2 17 14.4 Parallel Resonance (3) Example 4 Calculate the resonant frequency of the circuit in the figure shown below. Answer: 19 2.179 rad/s 2 18 14.7 Passive Filters (1) A filter is a circuit that is designed to pass signals with desired frequencies and reject or attenuate others.

Low Pass High Pass Passive filter consists of only passive element R, L and C. Band Pass There are four types of filters. Band Stop 19 14.7 Passive Filters (2) Example 5 For the circuit in the figure below, obtain the transfer function Vo() of a circuit is the )/Vi() of a circuit is the ). Identify the type of filter the circuit represents and determine the corner frequency. Take R1=100 =R2 and L =2mH.

Answer: 25 krad/s HW18 Ch14: 47, 55, 57, 59 20