RLC Circuits
Natural Response
ECE 201 Circuit Theory I
1
Parallel RLC Circuit
iC
+
C
1uF
V0
iL
iR
L
1mH
I0
-
+
R
1kOhm
v
-
ECE 201 Circuit Theory I
2
Parallel RLC Circuit
iC
iL
+
C
1uF
iR
L
1mH
V0
I0
+
R
1kOhm
-
v
-
t
v 1
dv
  vd  I 0  C
0
R L0
dt
ECE 201 Circuit Theory I
3
t
v 1
dv
  vd  I 0  C
0
R L0
dt
1 dv v
d 2v
 C 2  0
R dt L
dt
dv 2
1 dv v


0
2
dt
RC dt LC
st
v  Ae
As st
A st
2 st
As e 
e 
e 0
RC
LC
s
1 
st  2
Ae  s 

0
RC LC 

ECE 201 Circuit Theory I
4
Characteristic Equation
s
1
s 

0
RC LC
2
Look at the roots
2
1
1
 1 
s1  
 
 
2 RC
 2 RC  LC
2
1
1
 1 
s2  
 
 
2 RC
 2 RC  LC
ECE 201 Circuit Theory I
5
d 2 v 1 dv v


0
2
dt
RC dt LC
Solved by
v1  A1e s1t
v2  A2 e s2t
v1  v2  A1e s1t  A2 e s2t
ECE 201 Circuit Theory I
6
The general solution is given by
v  A1e s1t  A2e s2t
The circuit behavior is determined by the
values of s1 and s2. Rewrite them as
s1     2  02
s2     2  02
1

2 RC
1
0 
LC
Neper Frequency
Resonant Radian Frequency
ECE 201 Circuit Theory I
7
s1      
2
2
0
s2      
2
2
0
s1 and s2 are complex frequencies
There are three possible outcomes for the roots –
Real, distinct roots when ω02 < α2 “overdamped”
Complex conjugate roots when ω02 > α2
“underdamped”
Real and equal roots when ω02 = α2 “critically
damped”
ECE 201 Circuit Theory I
8
Overdamped Response
• Real, distinct roots
• Solution has the form
v  A1e  A2e
s1t
s2t
• Where s1 and s2 are the roots of the
characteristic equation
• A1 and A2 are determined by initial
conditions
ECE 201 Circuit Theory I
9
The Solution
v  A1e  A2e
s1t
s2t
If s1 and s2 are known determine A1 and A2 from
v(0)  A1  A2
Initial Voltage on the Capacitor
dv(0)
 s1 A1  s2 A2
dt
Rate of change of the initial
Capacitor voltage
ECE 201 Circuit Theory I
10
Initial Value of dv/dt
dvC (t )
iC (t )  C
dt
dvC (t ) 1
 iC (t )
dt
C
dvC (0) 1
 iC (0)
dt
C
ECE 201 Circuit Theory I
11
Initial Value of Capacitor current
+
iC(0+)
C
1uF V0
L
1mH
I0
R
1kOhm
-
V0
iC (0 )  I 0   0
R
V0
iC (0 )    I 0
R
ECE 201 Circuit Theory I
12
Example 8.2 page 272
• For the circuit shown, v(0+) = 12 Volts,
and iL(0+) = 30 mA.
iC
0.2 μF
+
iR
iL
50 mH
200 Ω
v
-
ECE 201 Circuit Theory I
13
Find the initial current in each branch
• For the inductor, iL(0-) = iL(0+) = 30 mA
• For the resistor, iR(0+) = 12V/200Ω = 60 mA
• For the capacitor, iC(0+) = -iL(0+) – iR(0+), or
iC(0+) = -30 mA -60 mA = -90 mA
ECE 201 Circuit Theory I
14
Find the initial value of dv/dt
dv
iC  C
dt
dv 1
 iC
dt C
dv(0) 1
90 10 3
 iC (0) 
dt
C
0.2 106
dv(0)
 450 103V / s
dt
ECE 201 Circuit Theory I
15
Find the expression for v(t)
1
1
4



1.25

10
2 RC 2(200)(0.2 106 )
1
1
0 

 1 104
LC
(50 103 )(0.2 106 )
s1        12,500  7,500  5000rad / s
2
2
0
s2     2  02  12,500  7,500  20, 000rad / s
Roots are real and distinct, therefore overdamped
ECE 201 Circuit Theory I
16
v(0 )  A1  A2
dv(0 )
 s1 A1  s2 A2
dt
12  A1  A2
450  10  5, 000 A1  20, 000 A2
3
...
A1  14V
A2  26V
ECE 201 Circuit Theory I
17
v(t )  (14e
5000 t
 26e
20,000 t
)V , t  0
v(0)  14  26  12V
dv(0)
 (5, 000)(14)  (20, 000)(26)
dt
dv(0)
 70, 000  520, 000  450, 000
dt
checks
ECE 201 Circuit Theory I
18
Sketch v(t) for 0<= t <= 250μs
ECE 201 Circuit Theory I
19