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7. AC ANALYSIS
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CIRCUITS by Ulaby & Maharbiz
Overview
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Linear Circuits at ac
Objective: To determine the steady state response of
a linear circuit to ac signals



Sinusoidal input is common in electronic circuits
Any time-varying periodic signal can be represented
by a series of sinusoids (Fourier Series)
Time-domain solution method can be cumbersome
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Sinusoidal Signals
v t   V m cos  t   
Useful relations
  2 f
T 
1
f
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Phase Lead/Lag
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Phasor Domain
1. The phasor-analysis technique transforms equations
from the time domain to the phasor domain.
2. Integro-differential equations get converted into
linear equations with no sinusoidal functions.
3. After solving for the desired variable--such as a particular voltage or
current-- in the phasor domain, conversion back to the time domain
provides the same solution that would have been obtained had
the original integro-differential equations been solved entirely in the
time domain.
Phasor Domain
Phasor counterpart of
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Time and Phasor Domain
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It is much easier to deal
with exponentials in the
phasor domain than
sinusoidal relations in
the time domain.
You just need to track
magnitude/phase,
knowing that everything
is at frequency .
Phasor Relation for Resistors
Current through a resistor
Time Domain
Frequency Domain
Time domain
i  I m cos  t   
v  iR  RI m cos  t   
Phasor Domain
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Phasor Relation for Inductors
Current through inductor in time
domain
i  I cos  t   
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Time domain
m
Phasor Domain
Time Domain
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Phasor Relation for Capacitors
Voltage across capacitor in
time domain is v  V cos  t   
m
Time domain
iC
dv
dt
Time Domain
Phasor Domain
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Summary of R, L, C
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ac Phasor Analysis General Procedure
Using this procedure, we can apply our techniques from
dc analysis
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Example 7-4: RL Circuit
Cont.
Example 7-4: RL Circuit cont.
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Impedance and Admittance
Impedance is
voltage/current
Admittance is
current/voltage
R = resistance = Re(Z)
X = reactance = Im(Z)
G = conductance = Re(Y)
B = susceptance = Im(Y)
Resistor
Z  R
Y  1/ R
Inductor
Z  j L
Y  1 / j L
Z  1 / j C
Y  j C
Capacitor
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Impedance Transformation
Voltage & Current Division
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Press
Cont.
Example 7-5: Input Impedance (cont.)
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Example 7-8: Thévenin Circuit
Linear Circuit Properties
Thévenin/Norton and Source Transformation Also Valid
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Phasor Diagrams
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Phase-Shift Circuits
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Example 7-10: Cascaded Phase Shifter
Choose R such that output is 1200 ahead of input
Solution leads to:
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Node 1
Cont.
(cont.)
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Cont.
(cont.)
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Example 7-13: Mesh Analysis by Inspection
Power Supply Circuit
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Ideal Transformer
Half-Wave Rectifier
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Full-Wave Rectifier
Current flow
during first half
of cycle
Current flow
during second
half of cycle
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Smoothing RC Filter
Complete Power Supply
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Example 7-19:
Multisim Measurement of Phase Shift
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Example 7-19 (cont.)
Using Transient Analysis
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Summary
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