Magnetically Coupled Circuits XIII.Frequency Response. XIV.The Laplace Transform XV.[r]
(1)Electric Circuit Theory
(2)Second-Order Circuits - sites.google.com/site/ncpdhbkhn
Contents
I. Basic Elements Of Electrical Circuits II. Basic Laws
III Electrical Circuit Analysis IV Circuit Theorems
V Active Circuits
VI Capacitor And Inductor VII First Order Circuits
VIII.Second Order Circuits
IX Sinusoidal Steady State Analysis X AC Power Analysis
XI Three-phase Circuits
XII Magnetically Coupled Circuits XIII.Frequency Response
(3)Second-Order Circuits 1 The Source-Free Series RLC Circuit 2 Initial Conditions
3 The Characteristic Equation 4 The Classical Method
5 Second-Order Op Amp Circuits
(4)The Source-Free Series RLC Circuit (1)
Second-Order Circuits - sites.google.com/site/ncpdhbkhn
0
0
0
(0) R ; (0) E
v V E i I
R R R R
= = = = + + 1 0 t di
Ri L idt
dt C −∞
+ + ∫ =
2
2 0
d i R di i
dt L dt LC
→ + + =
st
i = Ae
2
0 st R st A st
As e A se e
L LC
→ + + = 1
0 st R
Ae s s
L LC → + + = 0 st
i = Ae ≠
2 1 0 R s s L LC → + + = 2 2 2 1 2 2 1 2 2 R R s
L L LC
R R
s
L L LC
α α ω α α ω = − + − = − + − → = − − − = − − − 1 2
( ) s t s t
i t A e A e
→ = +
t = 0 A
B E
R L C
R0 i
+ v –
(5)The Source-Free Series RLC Circuit (2)
Second-Order Circuits - sites.google.com/site/ncpdhbkhn
0
0
0
(0) R ; (0) E
v V E i I
R R R R
= = = =
+ +
1
0 t
di
Ri L idt
dt C −∞
+ + ∫ =
2 1
0
R
s s
L LC
→ + + =
1
1
( ) s t s t
i t A e A e
→ = +
( )
1 2 1 2
0
'(0) s t s t
t
i A s e A s e A s A s
=
= + = +
0
1
(0)
i = A e + A e = I
1
A A
→
t = 0 A
B E
R L C
R0 i
+ v –
(6)The Source-Free Series RLC Circuit (3)
Second-Order Circuits - sites.google.com/site/ncpdhbkhn
1
1
( ) s t s t
i t = A e + A e
2 2 1,2 0 1 2 2 1 , 2 R R s
L L LC
R L LC α α ω α ω = − ± − = − ± − = = 2 :
( ) s t s t
i t A e A e
α ω>
= +
0
1
:
( ) ( ) st
i t A A t e
α ω=
= +
0 1,2
1
, :
( ) ( cos sin ) d
t
d d
s j
i t A t A t e α
α ω α ω
ω ω −
< = − ±
= + ( ) i t t 0 ( ) i t t 0 ( ) i t t 0
t = 0 A
B E
R L C
R0 i
+ v –
(7)Second-Order Circuits - sites.google.com/site/ncpdhbkhn
0
0
0
(0) R ; (0) E
v V E i I
R R R R
= = = =
+ +
1
0 t
di
Ri L idt
dt C −∞
+ + ∫ =
2 1
0
R
s s
L LC
→ + + =
1
1
( ) s t s t
i t A e A e
→ = +
( )
1 2 1 2
0
'(0) s t s t
t
i A s e A s e A s A s
=
= + = +
0
1
(0)
i = A e + A e = I
1
A A
→
Initial Conditions
t = 0 A
B E
R L C
R0 i
+ v –
(8)Second-Order Circuits
1 The Source-Free Series RLC Circuit
2 Initial Conditions
3 The Characteristic Equation 4 The Classical Method
5 Second-Order Op Amp Circuits
(9)Initial Conditions (1)
Second-Order Circuits - sites.google.com/site/ncpdhbkhn
Ex 1
Find i(0), (0),v di(0) , dv(0) ? dt dt 30
(0) (0 ) 1A
25 5
i = i − = =
+
5
(0) (0 ) 30 5 V
25 5 v = v − = =
+
25i 5 di v 0 dt
+ + = 25 (0) 5i di(0) v(0) 0 dt
→ + + =
(0) 25 5
6 A/s 5
di dt
× +
→ = − = −
3
50 10 dv i
dt
−
= × (0)
(0) 50 10 dv i
dt
−
→ = × (0) 1 3 20 V/s
50 10 dv
dt −
→ = =
×
t = 0 A
B 30V 25Ω 5H
50mF 5Ω i
+ v –
(10)Initial Conditions (2)
Second-Order Circuits - sites.google.com/site/ncpdhbkhn 10
Ex 2
Find i(0), (0),v di(0) , dv(0) ?
dt dt +
–
6Ω
4Ω
0.1F
0.5H 12V
t = 0 A B
+
–
v
i
12
(0) (0 ) 3A
4
i = i − = =
(0) (0 ) 12 V
v = v − =
(0)
0.5 di v 0.5 di v(0)
dt = → dt =
(0) 12
24 A/s 0.5
di dt
→ = =
0.1 0
6
v dv i dt
+ + = (0) 0.1 (0) (0) 0
6
v dv
i dt
→ + + =
12 3
(0) 6
50 V/s 0.1
dv dt
+