Transcript EEE 302 Lecture 19 - Universitas Udayana

Complex Waveforms as Input
 When complex waveforms are used as inputs to the circuit
(for example, as a voltage source), then we
(1) must Laplace transform the inputs
(2) determine the transfer function
(3) feed the input through the transfer function
 The transfer function, H(s), is the ratio of some output
variable to some input variable
Y ( s) Output
H (s) 

X( s )
Input
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Lecture 19
Transfer Function
 The transfer function, H(s), is
Y ( s) Output
H (s) 

X( s )
Input
 All initial conditions are zero (makes transformation step easy)
 Can use transfer function to find output to an arbitrary input
Y(s) = H(s) X(s)
 The impulse response is the inverse Laplace transform of transfer function
h(t) = L-1[H(s)]
with knowledge of the transfer function or impulse response, we can
find response of circuit to any input
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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
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Frequency
0°
Frequency
Frequency Response of an Inductor
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Phase of ZL (°)
Magnitude of ZL ()
 Inductor (L): ZL = L 90°
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 at dc)
Frequency
90°
Frequency
Frequency Response of a Capacitor
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Phase of ZC (°)
Magnitude of ZC ()
 Capacitor (C):
ZC = 1/(C) –90°
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
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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Lecture 19
Common Transfer Functions
 Since the transfer function, H(j), is the ratio of some output variable
to some input variable,
H( j ) 
Y( j ) Output

X( j )
Input
 We may define any number of transfer functions
 ratio of output voltage to input current, i.e., transimpedance, Z(jω)
 ratio of output current to input voltage, i.e., transadmittance, Y(jω)
 ratio of output voltage to input voltage, i.e., voltage gain, GV(jω)
 ratio of output current to input current, i.e., current gain, GI(jω)
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Lecture 19
Poles and Zeros
 The transfer function is a ratio of polynomials
N ( s) K ( s  z1 )(s  z2 ) ( 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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Lecture 19