Short Notes: Sinusoidal Steady State Analysis | Short Notes for Electrical Engineering - Electrical Engineering (EE) PDF Download

 Page 2


 
 
 
 
 
? ?
?
1
I V t A f  sinwt 
    
? ? ? ?
? ? ? ? ?
0
2
V t Bsin wt ; 0 
    
? ? ? ?
?
21
V t leads v t by an angle 
    
? ? ? ?
?
12
V t lags v t by an angle 
 
PHASORS 
A phasor is complex number that represents the amplitude & phase angle information of a 
sinusoidal function. 
? ? ? ?
? ? ?
m
v t V sin wt 
? ? ?
rms
Phasor representation, VV 
rms
magnitude V ?
 
phase ??
 
 
Networks Elements 
 
1. Resistor 
V RI ?
 
 
2. Inductor 
? ?
?? V j l I 
? ? ? ? 2 f ; f frequency of source 
 
3. Capacitor 
? ?
?? I j c V 
Impedance & Admittance 
? ?
? ? ? ? ? ?
rms
vi
rms
Impedance Z ,
V V
I I
 
? ?
? ? Unit of impedan e ohm c 
Z R jX Z ? ? ? ? ?
 
R = resistive component 
X = relative component 
Page 3


 
 
 
 
 
? ?
?
1
I V t A f  sinwt 
    
? ? ? ?
? ? ? ? ?
0
2
V t Bsin wt ; 0 
    
? ? ? ?
?
21
V t leads v t by an angle 
    
? ? ? ?
?
12
V t lags v t by an angle 
 
PHASORS 
A phasor is complex number that represents the amplitude & phase angle information of a 
sinusoidal function. 
? ? ? ?
? ? ?
m
v t V sin wt 
? ? ?
rms
Phasor representation, VV 
rms
magnitude V ?
 
phase ??
 
 
Networks Elements 
 
1. Resistor 
V RI ?
 
 
2. Inductor 
? ?
?? V j l I 
? ? ? ? 2 f ; f frequency of source 
 
3. Capacitor 
? ?
?? I j c V 
Impedance & Admittance 
? ?
? ? ? ? ? ?
rms
vi
rms
Impedance Z ,
V V
I I
 
? ?
? ? Unit of impedan e ohm c 
Z R jX Z ? ? ? ? ?
 
R = resistive component 
X = relative component 
 
 
 
 
 
? ?
?
??
1
Phase angle, tan 
X
R
 
             
22
Z R X ??
 
 
??
L
For Induct Z r j o L
 
?
??
?
?
C
For Capacitor 
1
j
Z
c
jL
 
??
L
Inductive reac X tance, L
 
?
?
?
L
Capacitive reactance
1
X ,
c
 
If X = 0, impedance is resistive; current & voltage are in same phase. 
If X > 0, impedance is inductive; current lags voltage. 
If X < 0, impedance is capacitive; current leads voltage. 
? ? ? Admittance,
1
YG
Z
 jB 
G = Capacitance 
B = Susceptance 
Impedance in AC circuits behave like resistance in DC circuits and all the laws remain same like 
? Series combination 
 
eq 1 2 n
Z Z Z ................ Z ? ? ? ?
 
 
? Parallel combination 
eq 1 2 n
1 1 1 1
......................
Z Z Z Z
? ? ? ?
 
eq 1 2 n
Y Y Y ......... Y ? ? ? ?
 
 
? Star-delta conversion also remain same here. 
 
Circuits analysis in AC domain 
 
1. Identify the sinusoidal source & note the excitation frequency. 
2. Convert source to phasor form. 
3. Represent each circuit element by its impedance. 
4. Solve circuits using circuit techniques (nodal analysis mesh analysis etc.) 
5. All circuits’ theorems are applicable here as well 
 
?
R
For Resis r Z to R
Page 4


 
 
 
 
 
? ?
?
1
I V t A f  sinwt 
    
? ? ? ?
? ? ? ? ?
0
2
V t Bsin wt ; 0 
    
? ? ? ?
?
21
V t leads v t by an angle 
    
? ? ? ?
?
12
V t lags v t by an angle 
 
PHASORS 
A phasor is complex number that represents the amplitude & phase angle information of a 
sinusoidal function. 
? ? ? ?
? ? ?
m
v t V sin wt 
? ? ?
rms
Phasor representation, VV 
rms
magnitude V ?
 
phase ??
 
 
Networks Elements 
 
1. Resistor 
V RI ?
 
 
2. Inductor 
? ?
?? V j l I 
? ? ? ? 2 f ; f frequency of source 
 
3. Capacitor 
? ?
?? I j c V 
Impedance & Admittance 
? ?
? ? ? ? ? ?
rms
vi
rms
Impedance Z ,
V V
I I
 
? ?
? ? Unit of impedan e ohm c 
Z R jX Z ? ? ? ? ?
 
R = resistive component 
X = relative component 
 
 
 
 
 
? ?
?
??
1
Phase angle, tan 
X
R
 
             
22
Z R X ??
 
 
??
L
For Induct Z r j o L
 
?
??
?
?
C
For Capacitor 
1
j
Z
c
jL
 
??
L
Inductive reac X tance, L
 
?
?
?
L
Capacitive reactance
1
X ,
c
 
If X = 0, impedance is resistive; current & voltage are in same phase. 
If X > 0, impedance is inductive; current lags voltage. 
If X < 0, impedance is capacitive; current leads voltage. 
? ? ? Admittance,
1
YG
Z
 jB 
G = Capacitance 
B = Susceptance 
Impedance in AC circuits behave like resistance in DC circuits and all the laws remain same like 
? Series combination 
 
eq 1 2 n
Z Z Z ................ Z ? ? ? ?
 
 
? Parallel combination 
eq 1 2 n
1 1 1 1
......................
Z Z Z Z
? ? ? ?
 
eq 1 2 n
Y Y Y ......... Y ? ? ? ?
 
 
? Star-delta conversion also remain same here. 
 
Circuits analysis in AC domain 
 
1. Identify the sinusoidal source & note the excitation frequency. 
2. Convert source to phasor form. 
3. Represent each circuit element by its impedance. 
4. Solve circuits using circuit techniques (nodal analysis mesh analysis etc.) 
5. All circuits’ theorems are applicable here as well 
 
?
R
For Resis r Z to R
 
 
 
 
 
 
Power analysis 
Real Power 
? ?
? ? ? ?
rms rms v i
P V I cos 
In a resistance 
2
2 rms
rms
V
P I R
R
??
 
Complex Power 
?
rms rms
*
S V I 
   
? ?
? ? ? ? ?
rms rms v i
VI 
? ?
? ? ? ? ?
rms rms v i
Real part of S P V I cos 
? ?
? ? ? ? ?
rms rms v i
Real part of S Q V I sin 
Reactive Power = Q 
Q = 0 for resistive loads. 
Q < 0 for capacitive loads ? ?
vi
? ? ?
. 
Q > 0 for inductive loads ? ?
vi
? ? ?
 
 
? If the current goes into an element, then it absorbs power and if current comes out it delivers 
power. 
? Hence, a capacitor absorbs leading reactive power. We can also say it delivers lagging reactive 
power.  
? Same way, inductor absorbs lagging reactive power & delivers leading reactive power. 
 
Power Factor 
 
? ?
vi
P
pf cos
S
? ? ? ? ?
 
? ? ? ? ? ?
vi
 power factor angle
  
 
If 0 ?? , power factor is lagging 
If 0 ?? , power factor is leading 
If 0 ?? , power factor is unity