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)=

)())((

)())((

)(

)()(

21

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