Short Notes: Transient Analysis | Short Notes for Electrical Engineering - Electrical Engineering (EE) PDF Download

 Page 2


 
 
 
 
 
 
General method of analysis 
 
? ? ? ? ? ? ? ?
? ?
? ? ?
?
??
??
??
? ? ? ? ? ?
o
tt
0
x t x x t x e , t 0 
If switching is done at t=t0 
? ? ? ?
?
? ?
00
 initial va x t x t at lue of tt   
? ? ? ?
?? ? ? final valu x x t f at t eo  
 
Algorithm 
1. Choose any voltage & current in the circuit which has to be determined. 
2. Assume circuit had reached steady state before switch was thrown at 
0
tt ? . Draw the circuit at 
0
tt
?
? with capacitor replaced by open circuit and inductor replaced by short circuit. Solve for 
? ?
?
C0
vt & 
? ?
?
L 0
i t . 
3. Voltage across capacitor and inductor current cannot change instantaneously.
? ? ? ? ? ?
??
??
C 0 C 0 C 0
V t V t V t 
? ? ? ? ? ?
??
??
L 0 L 0 L 0
i t i t i t 
4. Draw the circuit for tt
?
? with switches in new position. Replace a capacitor with a voltage source 
? ? ? ?
??
?
C 0 C 0
V t V t and inductor with a current source of value
? ? ? ?
??
?
L 0 L 0
i t i t . Solve for initial value 
of variable
? ?
?
0
xt . 
5. Draw the circuit for t ?? , in a similar manner as step-2 and calculate ? ? x ? . 
Calculate time constant of circuit 
 
6. t=Rth C  or t=L/Rth    
 
7. Substitute all value to calculate x(t). 
Example 
In the circuit shown below, ? ?
1
Vt for t > 0 will be given as  
 
 
 
 
 
 
Page 3


 
 
 
 
 
 
General method of analysis 
 
? ? ? ? ? ? ? ?
? ?
? ? ?
?
??
??
??
? ? ? ? ? ?
o
tt
0
x t x x t x e , t 0 
If switching is done at t=t0 
? ? ? ?
?
? ?
00
 initial va x t x t at lue of tt   
? ? ? ?
?? ? ? final valu x x t f at t eo  
 
Algorithm 
1. Choose any voltage & current in the circuit which has to be determined. 
2. Assume circuit had reached steady state before switch was thrown at 
0
tt ? . Draw the circuit at 
0
tt
?
? with capacitor replaced by open circuit and inductor replaced by short circuit. Solve for 
? ?
?
C0
vt & 
? ?
?
L 0
i t . 
3. Voltage across capacitor and inductor current cannot change instantaneously.
? ? ? ? ? ?
??
??
C 0 C 0 C 0
V t V t V t 
? ? ? ? ? ?
??
??
L 0 L 0 L 0
i t i t i t 
4. Draw the circuit for tt
?
? with switches in new position. Replace a capacitor with a voltage source 
? ? ? ?
??
?
C 0 C 0
V t V t and inductor with a current source of value
? ? ? ?
??
?
L 0 L 0
i t i t . Solve for initial value 
of variable
? ?
?
0
xt . 
5. Draw the circuit for t ?? , in a similar manner as step-2 and calculate ? ? x ? . 
Calculate time constant of circuit 
 
6. t=Rth C  or t=L/Rth    
 
7. Substitute all value to calculate x(t). 
Example 
In the circuit shown below, ? ?
1
Vt for t > 0 will be given as  
 
 
 
 
 
 
 
 
 
 
 
 
Solution 
Step 1 :  
For t < 0 
? ? ? ? 30u t 0 & 3u t 0 ??
 
? ?
?
?
1
V 0 0V 
?? For  t 
? ?
1
V 3mA 10k ? ? ? ? ?
 
 = -30 V 
 
Step 2 :   
?
? At t0 
? ? ? ?
11
V 0 30 V 0
3mA 0
20k 10k
??
?
? ? ?
 
? ?
1
3
V 0 1.5mA
20k
?
??
 
? ?
1
V 0 10V
?
??
 
? ? ? ?
t
1
V t 30 10 30 e
?
?
? ? ? ? ?
 
? ? ? ?
TH TH
R 30k ; R C 0.3s
 
? ? ? ?
?
??
??
??
? ? ?
t
0.3
1
V t 30 20e u t V
 
 
Series RLC circuit 
Without Source 
? ? ? ?
??
??
?
0
0
1
V 0 i t dt V
C
 
? ?
?
0
i o I 
By KVL 
? ?
? ?
? ?
??
? ? ?
?
t
di t
1
Ri t L i t dt 0
dt C
 
Difference both sides 
 
? ? ? ?
? ?
2
2
d i t di t
R1
i t 0
L dt LC dt
? ? ?
 
Page 4


 
 
 
 
 
 
General method of analysis 
 
? ? ? ? ? ? ? ?
? ?
? ? ?
?
??
??
??
? ? ? ? ? ?
o
tt
0
x t x x t x e , t 0 
If switching is done at t=t0 
? ? ? ?
?
? ?
00
 initial va x t x t at lue of tt   
? ? ? ?
?? ? ? final valu x x t f at t eo  
 
Algorithm 
1. Choose any voltage & current in the circuit which has to be determined. 
2. Assume circuit had reached steady state before switch was thrown at 
0
tt ? . Draw the circuit at 
0
tt
?
? with capacitor replaced by open circuit and inductor replaced by short circuit. Solve for 
? ?
?
C0
vt & 
? ?
?
L 0
i t . 
3. Voltage across capacitor and inductor current cannot change instantaneously.
? ? ? ? ? ?
??
??
C 0 C 0 C 0
V t V t V t 
? ? ? ? ? ?
??
??
L 0 L 0 L 0
i t i t i t 
4. Draw the circuit for tt
?
? with switches in new position. Replace a capacitor with a voltage source 
? ? ? ?
??
?
C 0 C 0
V t V t and inductor with a current source of value
? ? ? ?
??
?
L 0 L 0
i t i t . Solve for initial value 
of variable
? ?
?
0
xt . 
5. Draw the circuit for t ?? , in a similar manner as step-2 and calculate ? ? x ? . 
Calculate time constant of circuit 
 
6. t=Rth C  or t=L/Rth    
 
7. Substitute all value to calculate x(t). 
Example 
In the circuit shown below, ? ?
1
Vt for t > 0 will be given as  
 
 
 
 
 
 
 
 
 
 
 
 
Solution 
Step 1 :  
For t < 0 
? ? ? ? 30u t 0 & 3u t 0 ??
 
? ?
?
?
1
V 0 0V 
?? For  t 
? ?
1
V 3mA 10k ? ? ? ? ?
 
 = -30 V 
 
Step 2 :   
?
? At t0 
? ? ? ?
11
V 0 30 V 0
3mA 0
20k 10k
??
?
? ? ?
 
? ?
1
3
V 0 1.5mA
20k
?
??
 
? ?
1
V 0 10V
?
??
 
? ? ? ?
t
1
V t 30 10 30 e
?
?
? ? ? ? ?
 
? ? ? ?
TH TH
R 30k ; R C 0.3s
 
? ? ? ?
?
??
??
??
? ? ?
t
0.3
1
V t 30 20e u t V
 
 
Series RLC circuit 
Without Source 
? ? ? ?
??
??
?
0
0
1
V 0 i t dt V
C
 
? ?
?
0
i o I 
By KVL 
? ?
? ?
? ?
??
? ? ?
?
t
di t
1
Ri t L i t dt 0
dt C
 
Difference both sides 
 
? ? ? ?
? ?
2
2
d i t di t
R1
i t 0
L dt LC dt
? ? ?
 
 
 
 
 
 
? ?
?
st
Substitute tA i e 
? ?
? ? ? ? ? ??
2 st 2
R1
A 0        S s 0
L LC
R1
e S s
L LC
 
??
??
??
? ? ? ?
2
1
R
R1
S
2L LC
2L
 ,       
??
??
??
?
? ? ?
2
2
RR
1
S
LC
2L 2L
 
22
1 2 0 0
1
R
S ,S w ; ; w
2L
LC
? ? ? ? ? ? ? ? ?
 
1. If 
0
w ?? roots are real & unequal (over-damped) 
? ?
??
12
s t s t
i t Ae Be 
2. If 
0
w ?? , roots are real & equal (critically damped) 
? ? ? ?
??
??
t
i t A Bt e 
3. If 
0
w ?? , roots are complex conjugate (under-damped) 
? ? ? ?
??
??
t
dd
i t e Acosw t Bsinw t 
? ? ?
22
d0
ww 
Calculate A & B using initial conditions. 
 
With a Source 
? ?
? ? ?
12
S
s t s t
v t V Ae Be (Over-damped) 
? ? ? ?
??
??
S
t
v t V A Bt e (Critically damped) 
? ? ? ? ? ?
??
?? ?
t
S d d
v t under damped V Acosw t Bsinw t e 
 
Parallel RCL Circuit 
Without Source 
? ? ? ?
0
1
i 0 v t dt
L
??
?
?
 
? ?
0
v 0 V ?
 
By KCL 
? ? ? ?
? ?
??
? ? ? ? ?
?
t
dv t
11
v t v d C 0
R L dt
 
 
Page 5


 
 
 
 
 
 
General method of analysis 
 
? ? ? ? ? ? ? ?
? ?
? ? ?
?
??
??
??
? ? ? ? ? ?
o
tt
0
x t x x t x e , t 0 
If switching is done at t=t0 
? ? ? ?
?
? ?
00
 initial va x t x t at lue of tt   
? ? ? ?
?? ? ? final valu x x t f at t eo  
 
Algorithm 
1. Choose any voltage & current in the circuit which has to be determined. 
2. Assume circuit had reached steady state before switch was thrown at 
0
tt ? . Draw the circuit at 
0
tt
?
? with capacitor replaced by open circuit and inductor replaced by short circuit. Solve for 
? ?
?
C0
vt & 
? ?
?
L 0
i t . 
3. Voltage across capacitor and inductor current cannot change instantaneously.
? ? ? ? ? ?
??
??
C 0 C 0 C 0
V t V t V t 
? ? ? ? ? ?
??
??
L 0 L 0 L 0
i t i t i t 
4. Draw the circuit for tt
?
? with switches in new position. Replace a capacitor with a voltage source 
? ? ? ?
??
?
C 0 C 0
V t V t and inductor with a current source of value
? ? ? ?
??
?
L 0 L 0
i t i t . Solve for initial value 
of variable
? ?
?
0
xt . 
5. Draw the circuit for t ?? , in a similar manner as step-2 and calculate ? ? x ? . 
Calculate time constant of circuit 
 
6. t=Rth C  or t=L/Rth    
 
7. Substitute all value to calculate x(t). 
Example 
In the circuit shown below, ? ?
1
Vt for t > 0 will be given as  
 
 
 
 
 
 
 
 
 
 
 
 
Solution 
Step 1 :  
For t < 0 
? ? ? ? 30u t 0 & 3u t 0 ??
 
? ?
?
?
1
V 0 0V 
?? For  t 
? ?
1
V 3mA 10k ? ? ? ? ?
 
 = -30 V 
 
Step 2 :   
?
? At t0 
? ? ? ?
11
V 0 30 V 0
3mA 0
20k 10k
??
?
? ? ?
 
? ?
1
3
V 0 1.5mA
20k
?
??
 
? ?
1
V 0 10V
?
??
 
? ? ? ?
t
1
V t 30 10 30 e
?
?
? ? ? ? ?
 
? ? ? ?
TH TH
R 30k ; R C 0.3s
 
? ? ? ?
?
??
??
??
? ? ?
t
0.3
1
V t 30 20e u t V
 
 
Series RLC circuit 
Without Source 
? ? ? ?
??
??
?
0
0
1
V 0 i t dt V
C
 
? ?
?
0
i o I 
By KVL 
? ?
? ?
? ?
??
? ? ?
?
t
di t
1
Ri t L i t dt 0
dt C
 
Difference both sides 
 
? ? ? ?
? ?
2
2
d i t di t
R1
i t 0
L dt LC dt
? ? ?
 
 
 
 
 
 
? ?
?
st
Substitute tA i e 
? ?
? ? ? ? ? ??
2 st 2
R1
A 0        S s 0
L LC
R1
e S s
L LC
 
??
??
??
? ? ? ?
2
1
R
R1
S
2L LC
2L
 ,       
??
??
??
?
? ? ?
2
2
RR
1
S
LC
2L 2L
 
22
1 2 0 0
1
R
S ,S w ; ; w
2L
LC
? ? ? ? ? ? ? ? ?
 
1. If 
0
w ?? roots are real & unequal (over-damped) 
? ?
??
12
s t s t
i t Ae Be 
2. If 
0
w ?? , roots are real & equal (critically damped) 
? ? ? ?
??
??
t
i t A Bt e 
3. If 
0
w ?? , roots are complex conjugate (under-damped) 
? ? ? ?
??
??
t
dd
i t e Acosw t Bsinw t 
? ? ?
22
d0
ww 
Calculate A & B using initial conditions. 
 
With a Source 
? ?
? ? ?
12
S
s t s t
v t V Ae Be (Over-damped) 
? ? ? ?
??
??
S
t
v t V A Bt e (Critically damped) 
? ? ? ? ? ?
??
?? ?
t
S d d
v t under damped V Acosw t Bsinw t e 
 
Parallel RCL Circuit 
Without Source 
? ? ? ?
0
1
i 0 v t dt
L
??
?
?
 
? ?
0
v 0 V ?
 
By KCL 
? ? ? ?
? ?
??
? ? ? ? ?
?
t
dv t
11
v t v d C 0
R L dt
 
 
 
 
 
 
 
 
Characteristics equation 
2
0
1 1 1 1
s s 0 ; , w
RC LC 2RC LC
? ? ? ? ? ?
 
? ?? ? ? ?
22
1 2 0
S ,S w 
 
? ? ? ?
?? ?
12
s t S t
v t over damped Ae Be 
? ? ? ? ? ?
??
??
t
v t A Bt critically damped e 
? ? ? ? ? ?
??
? ? ?
dd
t
v t e Acosw t Bsinw t under damped 
 
With a step input 
? ?
? ?
? ? ? ?
12
s
s t S t
Over it d I amped Ae Be
 
? ? ? ?
??
? ? ?
t
s
Critically i t I A B da d t mpe e 
? ? ? ?
??
? ? ??
s d d
t
i t I Acosw t Bsinw t Under d e amped 
 
Steps: 
1. Write differential equation that describe the circuit. 
2. From differential equation model, construct characteristics equation & find roots. 
3. Roots of characteristics equation determine the type of response over-damped, critically damped 
& under-damped. 
4. Obtain the constant using initial conditions.