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Electric Circuits
Fall, 2014
Chapter 12
Phasors and Impedance
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12.1 Introduction
12.1 Introduction
This chapter will cover alternating current.
A discussion of complex numbers is included prior
to introducing phasors.
Applications of phasors and frequency domain
analysis for circuits including resistors, capacitors,
and inductors will be covered.
The concept of impedance and admittance is also
introduced.
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12.2 Phasors and Complex Number
12.2 Phasors and Complex Number
Complex Numbers
A powerful method for representing sinusoids is the phasor.
But in order to understand how they work, we need to cover
some complex numbers first.
A complex number z can be represented in rectangular form
as:
z  x  jy
It can also be written in polar or exponential form as:
z  r  re j
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12.2 Phasors and Complex Number
Complex Numbers II
The different forms can be
interconverted.
Starting with rectangular form,
one can go to polar:
r  x2  y 2
  tan 1
y
x
Likewise, from polar to
rectangular form goes as
follows:
x  r cos 
y  r sin 
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12.2 Phasors and Complex Number
Complex Numbers III
The following mathematical operations are important
Addition
Subtraction
Multiplication
z1  z2   x1  x2   j  y1  y2  z1  z2   x1  x2   j  y1  y2  z1 z2  r1r2 1  2 
Division
z1 r1
  1  2 
z2 r2
Reciprocal
1 1
    
z r
Square Root
z  r   / 2 
Complex Conjugate
z*  x  jy  r    re j
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12.2 Phasors and Complex Number
Phasors
The idea of a phasor representation is based on
Euler’s identity:
e j  cos   j sin 
From this we can represent a sinusoid as the real
component of a vector in the complex plane.
The length of the vector is the amplitude of the
sinusoid.
The vector,V, in polar form, is at an angle  with
respect to the positive real axis.
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12.2 Phasors and Complex Number
Phasors II
Phasors are typically represented at t=0.
As such, the transformation between time domain to
phasor domain is:
v  t   Vm cos t     V  Vm
(Time-domain
representation)
(Phasor-domain
representation)
They can be graphically represented as shown here.
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12.2 Phasors and Complex Number
Phasor Diagram
A convenient way to
compare sinusoids of the
same frequency is to use
a phasor diagram.
Here the magnitude is
represented as the length
of the arrow and the
phase is represented by
the angle.
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12.2 Phasors and Complex Number
Example and Problem
Example 12.2
Example 12.3
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12.3 Phasor Relationships for Circuit Elements
12.3 Phasor Relationships for Circuit Elements
Here is a handy table for transforming various time
domain sinusoids into phasor domain:
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12.3 Phasor Relationships for Circuit Elements
Sinusoids-Phasor Transformation
Note that the frequency of the phasor is not explicitly
shown in the phasor diagram
For this reason phasor domain is also known as
frequency domain.
Applying a derivative to a phasor yields:
dv
dt

jV
(Phasor domain)
(Time domain)
Applying an integral to a phasor yeilds:
 vdt
(Time domain)
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
V
j
(Phasor domain)
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12.3 Phasor Relationships for Circuit Elements
Phasor Relationships for Resistors
Each circuit element has a relationship
between its current and voltage.
These can be mapped into phasor
relationships very simply for resistors
capacitors and inductor.
For the resistor, the voltage and current
are related via Ohm’s law.
As such, the voltage and current are in
phase with each other.
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12.3 Phasor Relationships for Circuit Elements
Phasor Relationships for Inductors
Inductors on the other hand have
a phase shift between the voltage
and current.
In this case, the voltage leads the
current by 90°.
Or one says the current lags the
voltage, which is the standard
convention.
This is represented on the phasor
diagram by a positive phase angle
between the voltage and current.
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12.3 Phasor Relationships for Circuit Elements
Voltage current relationships
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12.3 Phasor Relationships for Circuit Elements
Example and Problem
Example 12.6
v(t )  12 cos(60t  45)
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12.4 Impedance and Admittance
12.4 Impedance and Admittance
It is possible to expand Ohm’s law to capacitors and
inductors.
In time domain, this would be tricky as the ratios of voltage
and current and always changing.
But in frequency domain it is straightforward
The impedance of a circuit element is the ratio of the phasor
voltage to the phasor current.
Z
V
I
or V  ZI
Admittance is simply the inverse of impedance.
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12.4 Impedance and Admittance
Impedance and Admittance II
It is important to realize that in
frequency domain, the values
obtained for impedance are only
valid at that frequency.
Changing to a new frequency will
require recalculating the values.
The impedance of capacitors and
inductors are shown here:
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12.4 Impedance and Admittance
Impedance and Admittance III
As a complex quantity, the impedance may be
expressed in rectangular form.
The separation of the real and imaginary components
is useful.
The real part is the resistance.
The imaginary component is called the reactance, X.
When it is positive, we say the impedance is
inductive, and capacitive when it is negative.
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12.4 Impedance and Admittance
Impedance and Admittance IV
Admittance, being the reciprocal of the impedance, is also a
complex number.
It is measured in units of Siemens
The real part of the admittance is called the conductance, G
The imaginary part is called the susceptance, B
These are all expressed in Siemens or (mhos)
The impedance and admittance components can be related to
each other:
G
R
R2  X 2
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B
X
R2  X 2
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12.4 Impedance and Admittance
Impedance and Admittance V
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12.4 Impedance and Admittance
Kirchoff’s Laws in Frequency Domain
A powerful aspect of phasors is that Kirchoff’s laws apply to
them as well.
This means that a circuit transformed to frequency domain
can be evaluated by the same methodology developed for
KVL and KCL.
One consequence is that there will likely be complex values.
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12.4 Impedance and Admittance
Kirchoff’s Laws in Frequency Domain
A powerful aspect of phasors is that Kirchoff’s laws apply to
them as well.
This means that a circuit transformed to frequency domain
can be evaluated by the same methodology developed for
KVL and KCL.
One consequence is that there will likely be complex values.
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12.4 Impedance and Admittance
Example and Problem
Example 12.7
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12.5 Impedance Combinations
12.5 Impedance Combinations
Once in frequency domain, the impedance elements are
generalized.
Combinations will follow the rules for resistors:
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12.5 Impedance Combinations
Seires Combinations
Series combinations will result in a sum of the
impedance elements:
Z eq  Z1  Z 2  Z3 
 ZN
Here then two elements in series can act like a
voltage divider
Z1
V1 
V
Z1  Z 2
Z2
V2 
V
Z1  Z 2
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12.5 Impedance Combinations
Parallel Combinations
Likewise, elements combined in parallel will
combine in the same fashion as resistors in parallel:
1
1
1
1
 
 
Zeq Z1 Z 2 Z3
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
1
ZN
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12.5 Impedance Combinations
Delta-Wye transformation
Zb Zc
Z1 
Z a  Zb  Zc
Z1Z 2  Z 2 Z 3  Z 3 Z1
Za 
Z1
Zc Za
Z2 
Z a  Zb  Zc
Z1Z 2  Z 2 Z 3  Z 3 Z1
Zb 
Z2
Z a Zb
Z3 
Z a  Zb  Zc
Z1Z 2  Z 2 Z 3  Z 3 Z1
Zc 
Z3
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12.5 Impedance Combinations
Example and Problem
Example 12.8
Example 12.9
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Homework
Homework
Read Text Chapter 13. Thevenin and Norton
Equivalent Circuits pp. 341-362
Prepare Presentation
Solve Problems
12.7, 12.20, 12.24, 12.35, 12.47
Computer Problem
12.53
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