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RC & RL Circuits
Differential Equation
Algebraic Equation
Differential Equation
1
2 1
0.5
x x
x
x
0dx
x x xdt
Candidates
x x
2 , 2
cos , sin
,t t
x t x t
x t x t
x Ke x K e
Solution
( 1)
1
(0)
( ) (0)
t t
t t t
t
t
x x
x K e x Ke
K e Ke K e
x Ke
x K
x t x e
Another Example
1.5
2 3 0
,
2 3 0
(2 3) 0
1.5
(0)
t t
t t
t
t
x x
x Ke x K e
K e Ke
K e
x Ke
x K
Characteristic Equation
0
0
t
ax bx
a b
be
a
Non-zero input
1 2
1 1 2
1 2
2
1
1
( )
1
( 1) 1
1, 1
( ) 1
lim ( ) 1
t
t t
t
t
t
x x
x t K e K
x x K e K e K
K e K
K
x t K e
x t
1
1
(0) 1
(0) 1
( ) (0) 1 1
if : (0) 0
( ) 1
t
t
x K
K x
x t x e
x
x t e
Complete response = natural response + forced response
Natural response(homogeneous solution):
Forced response(particular solution), steady state:
Resistor
Capacitor
dq dvi C
dt dt
Charge:
Series and Parallel Capacitors
Inductor
d div L
dt dt
Series and Parallel Inductors
RC Circuit
:
,
, (0) 0
KVL
dvRi v e i C
dt
dvRC v e v
dt
Complete response = natural response + forced response
Natural response(homogeneous solution):
Forced response(particular solution), steady state:
0
11 0
( )t
RC
dvRC v
dt
RCs sRC
v t Ke
1 1
ss
ssss ss
v V
dVRC V V
dt
Complete response = natural response + forced response
( ) 1
(0) 1 0 1
( ) 1
t
RC
t
RC
v t Ke
v K K
v t e
RL Circuit
( ) ,
( ), (0) 0
di ve t L v i
dt R
L dvv e t v
R dt
Complete response = natural response + forced response
Natural response(homogeneous solution):
Forced response(particular solution), steady state:
0
1 0
( )L
tR
L dvv
R dt
L Ls s
R R
v t Ke
1 1
ss
ssss ss
v V
dVLV V
R dt
Complete response = natural response + forced response
( ) 1
(0) 1 0 1
( ) 1
Lt
R
Lt
R
v t Ke
v K K
v t e
