First Order RL Circuits
Switch S is closed for t<0, and open for t ≥ 0
When the switch S is closed for t<0, the
inductor behaves as a short circuit to dc.
The voltage across the inductor v=0,
Hence the voltage across the R is also
zero
zero.
+
i
v
-
The current component i (t ) 
V
Rg
Dr. Jagadish Nayak ,BITS Pilani, Dubai Campus
First Order RL Circuits
When the switch S is open at t=0s and remains open for t>0. Since the
current through the inductor does not change instaneously.
V
i (0) 
Rg
For t≥0, applying KVL, we get
di
L  Ri  0
dt
di R
 i0
dt L
di
R
 i
d
dt
L
................((2)
Dr. Jagadish Nayak ,BITS Pilani, Dubai Campus
First Order RL Circuits
Equation (2) can be solved as follows,
Divide both side of (2) by i and integrate both side w.r.t t , we get
1 di
R
 i dt dt    L dt
1
R
di

i
 L dt
R
ln i   t  K
L
take exp onential both side , we get
i (t )  e
R
 tK
L
 e
R
 t
L
e K , at t  0, i (0)  e K
Dr. Jagadish Nayak ,BITS Pilani, Dubai Campus
First Order RL Circuits
Substituting
i (0)  e K in i (t )  e
i (t )  i (0) e
R
 t
L
R
 t
L
e K , we get
for t  0 s
Dr. Jagadish Nayak ,BITS Pilani, Dubai Campus
First Order RL Circuits
Example : In the following circuit at t<0 , the switch is closed and at t≥0, the
switch is open. Find current through inductor iL at tt<0
0 , tt>0
0 and at tt=1ms
1ms
Dr. Jagadish Nayak ,BITS Pilani, Dubai Campus
First Order RL Circuits
When t<0 switch is closed and voltage across the inductor is zero and it acts
like a short circuit.
circuit
The current iL=2A
2A
Dr. Jagadish Nayak ,BITS Pilani, Dubai Campus
First Order RL Circuits
For t ≥ 0, switch is open and the current through the inductor iL (t>0) does not
change instantaneously from iL (t<0)
At t=1ms
We know that ,
i (t )  i (0) e
i (t )  2 e
R
 t
L
for t  0 s
20 x 1 x 10 3

4
 2 x 0.995  1.99 A
Dr. Jagadish Nayak ,BITS Pilani, Dubai Campus
First Order RL Circuits
Example :
Dr. Jagadish Nayak ,BITS Pilani, Dubai Campus
First Order RC Circuits
Dr. Jagadish Nayak ,BITS Pilani, Dubai Campus
First Order RC Circuits
For t<0 switch is closed and capacitor behaves as a open circuit
By voltage division we get ,
v
V R
R  Rg
for t  0
Dr. Jagadish Nayak ,BITS Pilani, Dubai Campus
First Order RC Circuits
At t=0 and t>0 switch is open
Since the voltage across the
capacitors does not change
instantaneously, voltage v
iR remains same at t=0s
V R
v(0) 
for t  0
R  Rg
iC
Applying KCL at RC circuit , we get,
iC  iR  0
dv v
C  0
dt R
dv
1

v    (2)
dt
RC
Dr. Jagadish Nayak ,BITS Pilani, Dubai Campus
First Order RC Circuits
Equation (2) can be solved as follows,
Divide both side of (2) by v and integrate both side w.r.t t , we get
1 dv
1
 v dt dt    RC dt
1
1
d
dv


v
 RC dt
1
ln v(t )  
tK
RC
take exp onential both side , we get
v(t )  e

t
K
RC
 e

t
RC
e K , at t  0, v(0)  e K
Dr. Jagadish Nayak ,BITS Pilani, Dubai Campus
First Order RC Circuits
Substituting
v(0)  e K in
i v(t )  e
v(t )  v(0) e

t
RC

t
RC
e K , we gett
for t  0 s
Dr. Jagadish Nayak ,BITS Pilani, Dubai Campus
First Order RC Circuits
Given following circuit, Find vc and v0 for
I
I.
t<0
II. t>0
III. t=1.3ms
Dr. Jagadish Nayak ,BITS Pilani, Dubai Campus
First Order RC Circuits
For t < 0
Dr. Jagadish Nayak ,BITS Pilani, Dubai Campus
First Order RC Circuits
120 x 1250
vC 
 100 V
1250  250
Dr. Jagadish Nayak ,BITS Pilani, Dubai Campus
First Order RC Circuits
v ?
What is 0
Current through the 400 Ω resistance = 0.096A
g across the 400 Ω resistance i. e v0 = 38.4 V
Voltage
Dr. Jagadish Nayak ,BITS Pilani, Dubai Campus
First Order RC Circuits
For t > 0 , vc , does not change instantaneously so vc =100V
Let us redraw the circuit for t ≥ 0
+
100V
-
Find voltage between these two points using voltage division rule
Dr. Jagadish Nayak ,BITS Pilani, Dubai Campus
First Order RC Circuits
+
+
100V
v’’
-
-
Once you get this voltage say v’ , then get the find vo y applying voltage division
v’=32V and v0 at t=0 is 25.6 V
Dr. Jagadish Nayak ,BITS Pilani, Dubai Campus
First Order RC Circuits
At t=1.3ms, we know that
vc (t )  vc (0) e
t

RC
for t  0s
vc (t  1.3ms )  59.5 V
vc (0)  100 V and
d C  4 H
Req  R  ?
Req=R= 625Ω
Req=R=
Dr. Jagadish Nayak ,BITS Pilani, Dubai Campus