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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 jt )
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
 jCV
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  jC
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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  jC
jC

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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