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Chapter 9
Sinusoids and Phasors
Phasor Relationships for circuit Elements.
Impedance and Admittance.
Kirchoff’s Laws in the Frequency Domain.
Impedance Combinations.
Applications.
Huseyin Bilgekul
EENG224 Circuit Theory II
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
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Phasor Relationships for Circuit Elements
After we know how to convert RLC components
from time to phasor domain, we can transform a time
domain circuit into a phasor/frequency domain
circuit.
Hence, we can apply the KCL laws and other
theorems to directly set up phasor equations
involving our target variable(s) for solving.
Next we find the phasor or frequency domain
equivalent of the element equations for RLC
elements.
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Phasor Relationships for Circuit Elements
i (t ) I m cos(t ) Re(Ie jt )
v(t ) i (t ) R RI m cos(t )
V RI m =RI
Phasor voltage and current of a
resistor are in phase
Time Domain
Frequency Domain
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Phasor Relationship for Resistor
v(t ) i(t ) R RI m cos(t )
Frequency Domain
V RI m =RI
Voltage and current of a resistor
are in phase
Time Domain
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Phasor Relationships for Inductor
di
d
L I m cos(t ) LI m sin(t ) LI m cos(t 90)
dt
dt
V LI m( 90)= LI me j e j 90 j LI
v(t ) L
Phasor current of an inductor
LAGS the voltage by 90 degrees.
Time Domain
Frequency Domain
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Phasor Relationships for Inductor
Frequency Domain
Phasor current of an inductor
LAGS the voltage by 90 degrees.
Time Domain
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Phasor Relationships for Capacitor
dv
d
i (t ) C
C Vm cos(t ) CVm sin(t ) CVm cos(t 90)
dt
dt
I
j j 90
I CVm ( 90)=CVm e e
jCV
V=
j C
Time Domain
Phasor current of a capacitor LEADS
Frequency Domain the voltage by 90 degrees.
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Phasor Relationships for Capacitor
Frequency Domain
Phasor current of a capacitor
LEADS the voltage by 90
degrees.
Time Domain
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Phasor Relationships for Circuit Elements
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Phasor Relationships for Circuit Elements
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Impedance and Admittance
The Impedance Z of a circuit is the ratio of phasor voltage V to the phasor
current I.
V
Z
I
or V =ZI
The Admitance Y of a circuit is the reciprocal of impedance measured in
Simens (S).
I 1
Y
or I =YV
V Z
Impedances and Admitances of passive elements.
Element Impedance Admitance
R
Z=R
L
Z j L
C
Z=
1
j C
1
R
1
Y=
j L
Y=
Y jC
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Impedance as a Function of Frequency
The Impedance Z of a circuit is a function of the frequency.
Element Impedance Admitance
1
L
Z j L
Y=
j L
1
C
Z=
Y jC
jC
Inductor is SHORT CIRCUIT at DC and OPEN CIRCUIT at high frequencies.
Capacitor is OPEN CIRCUIT at DC and SHORT CIRCUIT at high frequencies.
Z L j L
0 (Short at DC)
Z L (Open as )
ZL 0
0
1
j C
Z C 0 (Open at DC)
ZC =
ZC 0
(Open as )
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Impedance of Joint Elements
The Impedance Z represents the opposition of the circuit to the flow of
sinusoidal current.
V
Z R jX
I
=Resistance + j Reactance
= Z
Z R X
2
R Z cos
2
Z
+
V
I
-
X
tan
R
X Z sin
1
The Reactance is Inductive if X is positive and it is Capacitive if X is negative.
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Impedance as a Function of Frequency
As the applied frequency increases, the resistance of a resistor remains
constant, the reactance of an inductor increases linearly, and the reactance of a
capacitor decreases nonlinearly.
Reactance of inductor versus
frequency
Reactance of capacitor versus
frequency
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Z
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Admittance of Joint Elements
The Admittance Y represents the admittance of the circuit to the flow of
sinusoidal current.
The admittance is measured in Siemens (s)
Y
+
V
I
-
1 I
Y G jB
Z V
Conductance + j Suseptance= Y
1 R jX
R jX
Y G jB
2
R jX R jX R X 2
R
X
G 2
B 2
2
R X
R X2
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Application of KVL for Phasors
The Kirchoff”s Voltage Law (KVL) holds in the frequency domain. For series
connected impedances:
Z eq
V
Z1 Z 2
I
Z N (Equivalent Impedance)
The Voltage Division for two elements in series is:
Z1
V1
V
Z1 Z 2
V2
Z2
V
Z1 Z 2
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Parallel Combination for Phasors
The Kirchoff”s Voltage Law (KVL) holds in the frequency domain. For series
connected impedances:
1
I
Yeq
Y1 Y2
Zeq V
1 1
YN
Z1 Z 2
1
(Eqiv. Admitance)
ZN
The Current Division for two elements is:
I1
Z2
I
Z1 Z 2
I2
Z1
I
Z1 Z 2
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Z3
Z1
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Application of Current Division for Phasors
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Application of Current Division for Phasors
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Example
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Z1
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