EE22 Circuits II Chapter 4 Frequency Response
Frequency Response Chapter 4 4. Introduction 4.2 Transfer Function 4.3 Bode Plots 4.4 Series Resonance 4.5 Parallel Resonance 4.6 Passive Filters 4.7 Active filters 2
4. Introduction What is Frequency Response 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 phase with frequency. 3
4.2 Transfer Function The transfer function H(ω) of a circuit is the frequencydependent ratio of a phasor output Y(ω) (voltage or current ) to a phasor input X(ω) (voltage or current). H( ω) Y( ω) = = H( ω) X( ω) φ 4
4.2 Transfer Function Four possible transfer functions: H( ω ) = Voltage gain = V o( ω) V ( ω) i H( ω ) = Transfer Impedance = V o( ω) I ( ω) i H( ω) Y( ω) = = H( ω) φ X( ω) H( ω ) = Current gain = Io( ω) I ( ω) i H( ω ) = Transfer Admittance = Io( ω) V ( ω) i 5
4.2 Transfer Function Example For the RC circuit shown below, obtain the transfer function Vo/Vs and its frequency response. Let v s = V m cosωt. 6
4.2 Transfer Function Solution: The transfer function is H( ω) = V V o s jωc = = R + / jω C + jω RC The magnitude is H( ω) = 2, + ( ω / ω o ) The phase is ω φ = tan ω o ω o =/RC 7 Low Pass Filter
4.2 Transfer Function Example 2 Obtain the transfer function Vo/Vs of the RL circuit shown below, assuming v s = V m cosωt. Sketch its frequency response. 8
4.2 Transfer Function Solution: The transfer function is H( ω) = V V o s jω L = = R + jω L R + jω L, H( ω ) = The magnitude is ω 2 + ( o ) ω High Pass Filter The phase is φ = 90 tan ω o = R/L ω ω o 9
4.4 Bode Plots Bode Plots are semilog plots of the magnitude (in db) and phase (in deg.) of the transfer function versus frequency. H = He jφ H db = 20 log 0 H 0
Bode Plot of Gain K
Bode Plot of a zero (jω) 2
Bode plot of a zero 3
Bode Plot of a quadratic pole 4
Summary 5
Summary
Example 7
Example 2 8
Example 3
4.4 Series Resonance 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: Z = R + j ( ω L ω C ) ωo = rad/s or LC fo = Hz 2π LC 20
4.4 Series Resonance The features of series resonance: Z= R+ j( ωl ) ωc The impedance is purely resistive, Z = R; The supply voltage Vs and the current I are in phase, so cos θ = ; 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. 2
4.4 Series Resonance The frequency response of the resonance circuit current is I = I = R 2 V m + ( ω L / ω C) 2 Z = R + j ( ω L ) ω C The average power absorbed by the RLC circuit is P( ω) = The highest power dissipated occurs at resonance: P( 2 I ω o 2 ) R = 2 V R 2 m 22
4 4 Series Resonance Half-power frequencies ω and ω 2 are frequencies at which the dissipated power is half the maximum value: P( ω ) = P( ω ) = 2 (V / 2) R 2 m 2 = 2 Vm 4R The half-power frequencies are obtained by setting Z equal to 2 R. R 2 + ω = + 2L R ( ) 2L LC ω R 2 2 = + ω o = ω ω2 + 2L R ( ) 2L LC Bandwidth B B = ω ω 2 23
4.4 Series Resonance Quality factor, Q = ω o L R = ω CR o The relationship between the B, Q and ωo: The quality factor is the ratio Rof B ω= o the resonant frequency to its 2 o ω QL bandwidth. If the bandwidth is narrow, the quality factor of the resonant circuit is high. If the band of frequencies is wide, the quality factor is low. 24
Example: 25
4.5 Parallel Resonance It occurs when imaginary part of Y is zero Y = R + j ( ω C ) ω L Resonance frequency: ω o = rad/s or fo = LC 2π LC Hz 26
4.5 Series Parallel Resonance Summary of series and parallel resonance circuits: characteristic Series circuit Parallel circuit ω o LC LC Q B ωol R or ω o Q ω RC o R ω L o or ω RC ω o Q o ω, ω 2 Q 0, ω, ω 2 ω o + ( ) ± 2Q B ω o ± 2 ω 2Q 2 o 2 ωo ω o + ( ) ± 2Q 2Q B ω o ± 2 27
4.5 Resonance Example 4 Calculate the resonant frequency of the circuit in the figure shown below. Answer: ω = 9 = 2 2.79rad/s 28
4.6 Passive Filters 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 29
Low-pass and high-pass filters
Band-pass and band-reject filters
Magnitude and Frequency Scaling Example: 4 th order low-pass filter Corner Frequency: rad/sec Resistance: Ω Corner Frequency: 00π krad/sec Resistance: 0 kω 32
Low Pass Filter (Active) 33
High Pass Filter (Active) 34
Band Pass Filter (Active) 35
Band Reject Filter (Active) 36