Spectrum Representations;
Frequency Response
Dr. Holbert
April 14, 2008
Lect20
EEE 202
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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)
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Frequency Response of a Resistor
• Consider the frequency dependent impedance of the
resistor, inductor and capacitor circuit elements
• Resistor (R):
ZR = R 0°
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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
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Frequency Response of an Inductor
• Inductor (L):
ZL = L 90°
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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
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Frequency Response of a Capacitor
• Capacitor (C): ZC = 1/(C) –90°
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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
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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) ejt = X(s) est
Y(j) ejt = Y(s) est
H(j) = H(s)
• The transfer function can be separated into
magnitude and phase angle information
H(j) = |H(j)| H(j)
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Poles and Zeros
• The transfer function is a ratio of polynomials
N( s) K ( s  z1 )( s  z 2 )  ( s  zm )
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)=
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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 (ωL=1/ωC)
• 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
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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
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Bandwidth (BW)
• The bandwidth (BW) is the difference
between the two half-power frequencies
BW = ωHI – ωLO = 0 / Q
• Hence, a high-Q circuit has a small
bandwidth
• Note that:
02 = ωLO ωHI
 LO &  HI
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 1
 0 

 2Q
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
1
 1
2
2Q  
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Quality Factor: RLC Circuits
• For a series RLC circuit the quality factor is
Q
0
BW
 Qseries 
0 L
R

1
1 L

 0 CR R C
• For a parallel RLC circuit, the quality factor is
Q
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0
BW
 Q parallel
R
C

  0 CR  R
0 L
L
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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; we’ll 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
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