complex waveforms as input lecture 19 1 when complex waveforms are used as inputs to the circuit...
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Complex Waveforms as Input
Lecture 191
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 functionThe transfer function, H(s), is the ratio of
some output variable to some input variable
Input
Output
s
ss
)(
)()(
XY
H
Transfer Function
Lecture 192
The transfer function, H(s), is
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 functionh(t) = L-1[H(s)]
with knowledge of the transfer function or impulse response, we can find response of circuit to any input
Input
Output
s
ss
)(
)()(
XY
H
Variable-Frequency Response Analysis
Lecture 193
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)
Frequency Response of a Resistor
Lecture 194
Consider the frequency dependent impedance of the resistor, inductor and capacitor circuit elements
Resistor (R): ZR = R 0°So the magnitude and phase angle of the resistor impedance are constant, such that plotting them versus frequency yields
Mag
nitu
de o
f Z
R (
)
Frequency
RP
hase
of
ZR (
°)
Frequency
0°
Frequency Response of an Inductor
Lecture 195
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)
Mag
nitu
de o
f Z
L (
)
Frequency
Pha
se o
f Z
L (
°)
Frequency
90°
Frequency Response of a Capacitor
Lecture 196
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)
Mag
nitu
de o
f Z
C (
)
Frequency
Pha
se o
f Z
C (
°)
Frequency
-90°
Transfer Function
Lecture 197
Recall that the transfer function, H(s), is
The transfer function can be shown in a block diagram as
The transfer function can be separated into magnitude and phase angle information, H(j) = |H(j)| H(j)
Input
Output
s
ss
)(
)()(
XY
H
H(j) = H(s)X(j) ejt = X(s) est Y(j) ejt = Y(s) est
Common Transfer Functions
Lecture 198
Since the transfer function, H(j), is the ratio of some output variable to some input variable,
We may define any number of transfer functionsratio 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)
Input
Output
j
jj
)(
)()(
X
YH
Poles and Zeros
Lecture 199
The transfer function is a ratio of polynomials
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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