7. AC Analysis
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Transcript of 7. AC Analysis
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7. AC ANALYSIS CIRCUITS by Ulaby & Maharbiz
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Overview
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Linear Circuits at acObjective: 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 tVtv cosm
f 2
fT 1
Useful relations
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Phase Lead/Lag
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Complex NumbersWe will find it is useful to represent sinusoids as complex numbers
jyxz jezzz
1j
Rectangular coordinatesPolar coordinates
sincos je j
Relations based on Euler’s Identity
yzxz
)Im(
Re
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Relations for Complex Numbers
Learn how to perform these with your calculator/computer
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Phasor Domain
1. The phasor-analysis technique transforms equationsfrom the time domain to the phasor domain.
2. Integro-differential equations get converted intolinear 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 domainprovides the same solution that would have been obtained had the original integro-differential equations been solved entirely in the time domain.
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Phasor Domain
Phasor counterpart of
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Time and Phasor Domain
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 .
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Phasor Relation for Resistors
Time Domain Frequency Domain tRIiRv cosm
Current through a resistor
tIi cosm
Time domain
Phasor Domain
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Phasor Relation for Inductors
Time Domain
Current through inductor in time domain
Time domain
Phasor Domain tIi cosm
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Phasor Relation for Capacitors
Time Domain
Voltage across capacitor in time domain is
Time domain
Phasor Domain
tVv cosm dtdvCi
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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 1-4: RL Circuit
Cont.
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Example 1-4: RL Circuit cont.
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Impedance and Admittance
R = resistance = Re(Z)
Impedance is voltage/current
X = reactance = Im(Z)
Resistor
Inductor
Capacitor
RZ
LjZ
Cj/1Z
R/1Y
Lj/1Y
CjY
G = conductance = Re(Y)
Admittance is current/voltage
B = susceptance = Im(Y)
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Impedance Transformation
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Voltage & Current Division
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Cont.
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Example 7-6: Input Impedance (cont.)
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Example 7-9: Thévenin Circuit
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Linear Circuit PropertiesThévenin/Norton and Source Transformation Also Valid
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Phasor Diagrams
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Phase-Shift Circuits
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Example 7-11: Cascaded Phase Shifter
Solution leads to:
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Node 1 Cont.
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(cont.)
Cont.
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(cont.)
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Example 7-14: Mesh Analysis by Inspection
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Example 7-16: Thévenin Approach
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Example 7-16: Thévenin Approach (Cont.)
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Example 7-16: Thévenin Approach (Cont.)
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Power Supply Circuit
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Ideal Transformer
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Half-Wave Rectifier
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Full-Wave RectifierCurrent flow during first half of cycle
Current flow during second half of cycle
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Smoothing RC Filter
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Complete Power Supply
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Example 7-20: Multisim Measurement of Phase Shift
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Example 7-20 (cont.)
Using Transient Analysis
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Summary