Chapter 14 Frequency Response 電路學(二) Chapter 14 Frequency Response

Frequency Response Chapter 14 14.1 Introduction 14.2 Transfer Function 14.3 The Decible Scale 14.4 Bode Plots 14.5 Series Resonance 14.6 Parallel Resonance 14.7 Passive Filters 14.8 Active Filters 14.9 Scaling

What is Frequency Response of a Circuit? 14.1 Introduction (1) 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.

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) Four possible transfer functions:

14.2 Transfer Function (3) The transfer function can be expressed in terms of its numerator polynomial N() and denominator polynomial D() zero 零點: a root of the numerator polynomial. pole 極點: a root of the denominator polynomial.

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 = Vmcosωt.

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

14.2 Transfer Function (5) Example 2 Obtain the transfer function Vo/Vs of the RL circuit shown below, assuming vs = Vmcosωt. Sketch its frequency response.

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

14.3 The Decibel Scale (1) bel 為功率增益的單位 dB (decibel) 為bel 的1/10 log P1P2 = log P1 + log P2 log P1/P2 = log P1 - log P2 log Pn = n log P log 1 = 0

14.4 Bode Plots (1) Bode plots are semilog plots of the magnitude (in dB) and phase (in degree) of a transfer function vs frequency.

14.4 Bode Plots (2)

14.4 Bode Plots (2) A gain K A pole (j)-1 or zero (j) at origin

14.4 Bode Plots (3) A simple pole 1/(1+j/p1) or zero (1+j/z1)

14.4 Bode Plots (4) A quadratic pole 1/[1+j22/n+(j/n)2] or zero [1+j21/k+(j/k)2]

14.4 Bode Plots (5)

14.4 Bode Plots (6)

14.4 Bode Plots (7) Example 3 Construct the Bode plots for the transfer function

14.4 Bode Plots (8) Example 4 Draw the Bode plots for the transfer function

14.4 Bode Plots (9) Example 5 Obtain the Bode plots for

14.4 Bode Plots (10) Example 6 Draw the Bode plots for

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) 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) 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) Half-power frequencies ω1 and ω2 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) Quality factor, The relationship between the B, Q and o: 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.

14.3 Series Resonance (6) Example 7 A series-connected circuit has R = 2 Ω, L = 1 mH, and C = 0.4 F. Find the resonant frequency and the half-power frequencies. Calculate the quality factor and bandwidth. Determine the amplitude of the current at 0 , 1 and 2.

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

14.4 Parallel Resonance (2) Summary of series and parallel resonance circuits: characteristic Series circuit Parallel circuit ωo Q B ω1, ω2 Q ≥ 10, ω1, ω2

14.4 Parallel Resonance (3) Example 8 In a parallel RLC circuit, R = 8 kΩ, L = 0.2 mH, and C = 8 F (a) Calculate 0, Q, and B. (b) Find 1 and 2 (c) Determine the power dissipated at 0, 1 and 2

14.4 Parallel Resonance (4) Example 9 Calculate the resonant frequency of the circuit in the figure shown below. Answer:

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 Judi: the eq is still blurry at 100%. cb

14.5 Passive Filters (2) Example 10 For the circuit in the figure below, obtain the transfer function Vo()/Vi(). Identify the type of filter the circuit represents and determine the corner frequency. Take R1 = 100 W = R2 and L =2 mH. Answer: