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Complex Impedances
Sinusoidal Steady State Analysis
ELEC 308
Elements of Electrical Engineering
Dr. Ron Hayne
Images Courtesy of Allan Hambley and Prentice-Hall
Complex Impedances
 Inductance and Capacitance represented as
Complex Numbers
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Inductance
Consider an inductance in which the current is a sinusoid given by
iL t   Im sin t     IL  Im  90 o
The voltage across an inductance is
diL t 
v L t   L
 LIm cost     VL  LIm  Vm
dt
Note : The current LAGS the voltage for a pure inductance.
The voltage can be written as


VL  LIm  L90 o Im  90 o   jLIL 
So we have Ohm' s Law in phasor form : VL  Z L IL
where Z L  jL  L90 o is the impedance of the inductance.
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Inductance
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Impedance
 Ohm’s Law in phasor form:

Phasor voltage equals impedances times the
phasor current
 Impedance is COMPLEX, in general

Can be strictly REAL


Impedance = Resistance
Can be strictly IMAGINARY
Impedance = Reactance
 Both inductances and capacitances

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Capacitance
Consider an capacitance where the voltage across it is given by
vC t   Vm sin t     VC  Vm  90 o
The current through the capacitance is
dvC t 
iC t   C
 CVm cost     IC  CVm  Im
dt
Note : The current LEADS the voltage for a pure capacitance.
The voltage can be written as
I
I 
IC
1
VC  m   90 o m



j
IC
o
C
C90
jC
C
So we have Ohm' s Law in phasor form : VC  ZC IC
1
1
where ZC 

  90 o is the impedance of the capacitance.
jC C
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Capacitance
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Resistance
 The phasors are related by
VR = RIR
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Exercise 5.7
A voltage vC t   100 cos200t  is applied
to a 100 - F capacitanc e.
Find the impedance of the capacitanc e,
phasor current, phasor vol tage.
Draw the phasor diagram.
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Exercise 5.8
A voltage vR t   100 cos200t  is applied
to a 50 -  resistance .
Find the impedance of the resistance ,
phasor current, phasor vol tage.
Draw the phasor diagram.
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Steady-State Circuit Analysis
Circuit Analysis Using Phasors and Impedances
1. Replace the time descriptions of voltage and
current sources with corresponding phasors.
All of the sources must have the SAME frequency!
2. Replace inductances, capacitances, and resistances
with their corresponding impedances.
3. Analyze the circuit using any of the techniques
from Chapters 1 and 2 by performing the
calculations with complex arithmetic.
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Example 5.4
 Find the steady-state current for the circuit shown below.
Also, find the phasor voltage across each element and
construct a phasor diagram.
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Phasor Diagram
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Example 5.5
 Series and Parallel Combinations of Complex Impedances
 Find the voltage vc(t) in steady state. Find the phasor current
through each element, and construct a phasor diagram showing
the currents and source voltage.
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Phasor Diagram
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Exercise 5.9
 Find i(t) in the circuit below. What is the
phase relationship between vs(t) and i(t)?
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Exercise 5.9
 Find i(t) in the circuit below. What is the
phase relationship between vs(t) and i(t)?
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Exercise 5.10
 Find the phasor voltage and current for each
circuit element.
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Summary
 Complex Impedances



Inductance
Capacitance
Resistance
 Sinusoidal Steady State Analysis



Ohm’s Law
KVL (Mesh-Current Analysis)
KCL (Node-Voltage Analysis)
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