Lecture on concept of Impedance and Admittance_Chpt09_Alexander Sadiku.pptx

 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 KVL/ KCL laws and other
theorems to directly set up phasor equations
involving our target variable(s) for solving.
 Hence, we need to find the phasor or frequency
domain equivalent of the property for RLC elements.
Phasor Relationships for Circuit Elements

Phasor Relationships for Circuit Elements-Resistor
Time Domain Frequency Domain
( ) cos( ) Re( )
( ) ( ) cos( )
= I
j t
m
m
m
i t I t e
v t i t R RI t
RI R

 
 

  
  
 
I
V

Phasor Relationship for Resistor
Frequency Domain
Time Domain
Phasor voltage and current of a resistor are in phase

Response of Basic R, L and C Elements to a
Sinusoidal Voltage or Current- Resistor
 Resistor
 For power-line frequencies and
frequencies up to a few hundred
kilohertz, resistance is, for all
practical purposes, unaffected
by the frequency of the applied
sinusoidal voltage or current.
 For a purely resistive element,
the voltage across and the
current through the element are
in phase, with their peak values
related by Ohm’s law.

Phasor Relationships for Inductor
90
( ) cos( ) sin( ) cos( 90 )
( 90 )= I
m m m
j j
m m
di d
v t L L I t LI t LI t
dt dt
LI LI e e j L

       
   

        
    
V

Phasor Relationships for Inductor
Phasor current of an inductor LAGS the voltage by 90 degrees.
Frequency Domain
Time Domain

Response of Basic R, L and C Elements to a
Sinusoidal Voltage or Current- Inductor
 Inductor
 The magnitude of the voltage
across the element is determined
by the opposition of the element to
the flow of charge, or current i.
 The inductive voltage, therefore,
is directly related to the frequency
(or, more specifically, the angular
velocity of the sinusoidal ac
current through the coil) and the
inductance of the coil.
 For an inductor, vL leads iL by 90°,
or iL lags vL by 90°.

Phasor Relationships for Capacitor
90
( ) cos( ) s
I
in( ) cos( 90 )
( 90 )= V V=
j C
m m m
j j
m m
dv d
i t C C V t CV t CV t
dt dt
CV CV e e j C

       
   


        
    
I

Phasor Relationships for Capacitor
Phasor current of a capacitor LEADS the voltage by 90 degrees.
Frequency Domain
Time Domain

Response of C to a Sinusoidal Voltage or
Current- Capacitor
 Capacitor
 Since capacitive current is a
measure of the rate at which a
capacitor will store charge on its
plates, for a particular change in
voltage across the capacitor, the
greater the value of capacitance,
the greater will be the resulting
capacitive current.
 The fundamental equation relating
the voltage across a capacitor to
the current of a capacitor [i =
C(dv/dt)] indicates that for a
particular capacitance, the greater
the rate of change of voltage
across the capacitor, the greater
the capacitive current.

Phasor Relationships for Circuit Elements

or =Z
Z 
V
V I
I
1
or =Y
Y
Z
 
I
I V
V
Impedance and Admittance
 The Impedance Z of a circuit is the ratio of phasor voltage V to the phasor
current I.
 Impedance represents the opposition that an element/ element combination
exhibits to the flow of sinusoidal current.
 Impedance is not a true phasor as it does not correspond to a time varying
quantity.
 The Admitance Y of a circuit is the reciprocal of impedance measured in
Siemens (S).
1
R Z=R Y=
R
1
Y=
j L
1
=
j
L Z j L
C Z Y j C
C






Element Impedance Admitance
 Impedances and Admitances of passive elements.

1
Y=
j L
1
=
j
L Z j L
C Z Y j C
C






Element Impedance Admitance
0
 
Impedance as a Function of Frequency
 The Impedance Z of a circuit is a function of the frequency.
 Inductor is SHORT CIRCUIT at DC and OPEN CIRCUIT at high frequencies.
Capacitor is OPEN CIRCUIT at DC and SHORT CIRCUIT at high frequencies.
0 0 (Short at DC)
(Open as )
1
=
j
0 (Open at DC)
0 (Open as )
C
C
L
L
L
C
Z j L
Z
Z
C
Z
Z
Z





 


 
 
  

 

 
  

Impedance of Joint Elements (R and L)
 The Impedance Z represents the opposition of the circuit to the flow of
sinusoidal current.
The quantity wL, called the reactance
(from the word reaction) of an inductor,
is symbolically represented by XL and is
measured in ohms; that is,
Inductive reactance is the opposition to
the flow of current, which results in the
continual interchange of energy between
the source and the magnetic field of the
inductor.

Impedance of Joint Elements (R and C)
 The Impedance Z represents the opposition of the circuit to the flow of
sinusoidal current.
 An increase in frequency corresponds to
an increase in the rate of change of
voltage across the capacitor and to an
increase in the current of the capacitor.
 The quantity 1/ wC, called the reactance
of a capacitor, is symbolically represented
by Xc and is measured in ohms; that is,
 Capacitive reactance is the opposition to
the flow of charge, which results in the
continual interchange of energy between
the source and the electric field of the
capacitor.
)
(ohms,
1


C
C
X


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

Lecture on concept of Impedance and Admittance_Chpt09_Alexander Sadiku.pptx

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)
1
Conductance + j Suseptance= Y
I
Y G jB
Z V

   
  
Y
I
+
V
-

Application of KVL for Phasors
 The Kirchoff”s Voltage Law (KVL) holds in the frequency domain. For series
connected impedances:
1 2 (Equivalent Impedance)
eq N
V
Z Z Z Z
I
    

 The Voltage Division for two elements in series is:
1
1
1 2
2
2
1 2
Z
V V
Z Z
Z
V V
Z Z





Parallel Combination for Phasors
 The Kirchoff”s Voltage Law (KVL) holds in the frequency domain. For series
connected impedances:
 The Current Division for two elements is:
1 2
1 2
1 1 1 1
(Eqiv. Admitance)
eq N
eq N
I
Y Y Y Y
Z V Z Z Z
         
 
2
1
1 2
1
2
1 2
Z
I I
Z Z
Z
I I
Z Z





Application of Current Division for Phasors

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