# Spectrum Representations; Frequency Response

Spectrum Representations; Frequency Response Dr. Holbert April 14, 2008 Lect20 EEE 202 1 Variable-Frequency Response Analysis As an extension of AC analysis, we now vary the frequency and observe the circuit behavior Graphical display of frequency dependent circuit behavior can be very useful; however, quantities such as the impedance are complex valued such that we will tend to graph the magnitude of the impedance versus frequency (i.e., |Z(j)| v. f) and the phase angle versus frequency (i.e., Z(j) v. f) Lect20

EEE 202 2 Frequency Response of a Resistor Consider the frequency dependent impedance of the resistor, inductor and capacitor circuit elements Resistor (R): ZR = R 0 Lect20 Phase of ZR () Magnitude of ZR () So the magnitude and phase angle of the resistor impedance are constant, such that plotting them versus frequency yields R

Frequency 0 Frequency EEE 202 3 Frequency Response of an Inductor Inductor (L): ZL = L 90 Lect20 Phase of ZL () Magnitude of ZL () The phase angle of the inductor impedance is a constant 90, but the magnitude of the inductor

impedance is directly proportional to the frequency. Plotting them vs. frequency yields (note that the inductor appears as a short circuit at dc) Frequency 90 Frequency EEE 202 4 Frequency Response of a Capacitor Capacitor (C): ZC = 1/(C) 90 Lect20 Phase of ZC () Magnitude of ZC () The phase angle of the capacitor impedance is 90,

but the magnitude of the inductor impedance is inversely proportional to the frequency. Plotting both vs. frequency yields (note that the capacitor acts as an open circuit at dc) -90 Frequency Frequency EEE 202 5 Transfer Function Recall that the transfer function, H(s), is Y ( s ) Output H( s) X( s) Input The transfer function can be shown in a block diagram

as X(j) ejt = X(s) est H(j) = H(s) Y(j) ejt = Y(s) est The transfer function can be separated into magnitude and phase angle information H(j) = |H(j)| H(j) Lect20 EEE 202 6 Poles and Zeros The transfer function is a ratio of polynomials N( s) K ( s z1 )( s z 2 ) ( s z m ) H( s) D( s) ( s p1 )( s p2 ) ( s pn ) The roots of the numerator, N(s), are called the

zeros since they cause the transfer function H(s) to become zero, i.e., H(zi)=0 The roots of the denominator, D(s), are called the poles and they cause the transfer function H(s) to become infinity, i.e., H(pi)= Lect20 EEE 202 7 Resonant Circuits Resonant frequency: the frequency at which the impedance of a series RLC circuit or the admittance of a parallel RLC circuit is purely real, i.e., the imaginary term is zero (L=1/C)L=1/L=1/C)C) For both series and parallel RLC circuits, the resonance frequency is 1 0 LC

At resonance the voltage and current are in phase, (i.e., zero phase angle) and the power factor is unity Lect20 EEE 202 8 Quality Factor (Q) An energy analysis of a RLC circuit provides a basic definition of the quality factor (Q) that is used across engineering disciplines, specifically: WS Max Energy Stored at 0 Q 2 2 WD Energy Dissipated per Cycle The quality factor is a measure of the sharpness of the resonance peak; the larger the Q value, the sharper the peak

0 Q where BW=bandwidth BW Lect20 EEE 202 9 Bandwidth (BW) The bandwidth (BW) is the difference between the two half-power frequencies BW = L=1/C)HI L=1/C)LO = 0 / Q Hence, a high-Q circuit has a small bandwidth Note that: 02 = L=1/C)LO L=1/C)HI LO & HI Lect20

1 0 2Q EEE 202 1 1 2 2Q 10 Quality Factor: RLC Circuits For a series RLC circuit the quality factor is 0 0 L 1 1 L Q Qseries

BW R 0 CR R C For a parallel RLC circuit, the quality factor is 0 R C Q Q parallel 0 CR R BW 0 L L Lect20 EEE 202 11 Class Examples

Drill Problems P9-3, P9-4, P9-5 Use MATLAB or Excel to create the Bode plots (both magnitude and phase) for the above; well make hand plots next time Start Excel and open the file BodePlot.xls from the class webpage, -or Start MATLAB and open the file EEE202BodePlt.m from the class webpage Lect20 EEE 202 12

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