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Basic Engineering Circuit Analysis-II HiCSLHiCSL1 http://hicsl.kw.ac.kr
Analysis of First- and Second-Order
Transient Circuits
< Chapter-7 >
Jongsik Kim
High-Speed Integrated Circuits and Systems Lab.
Department of Wireless Communications Engineering
Kwangwoon University
Basic Engineering Circuit Analysis-II HiCSLHiCSL2 http://hicsl.kw.ac.kr
Outline
• First- and Second-Order Circuits
• Charge and Discharge Time
• General Form of the Response Equations
• Examples
Basic Engineering Circuit Analysis-II HiCSLHiCSL3 http://hicsl.kw.ac.kr
1st- and 2nd- Order Circuits
• First-order circuit : single storage element
First-order differential equation
• Circuit’s time constant – Transient analysis
How fast the circuit will respond to changes
• Second-order circuit : an inductor and a capacitor are present
simultaneously
• Laplace transform
• PSPICE
Basic Engineering Circuit Analysis-II HiCSLHiCSL4 http://hicsl.kw.ac.kr
Charge and Discharge Time
free-body diagram
ex : flash circuit in a camera
( ) ( )0
( ) 1( ) 0
C C
CC
d t tC
dt R
d tt
dt RC
KCL
0( ) t RC
C t V e
Decaying exponentially
with function of RC
Basic Engineering Circuit Analysis-II HiCSLHiCSL5 http://hicsl.kw.ac.kr
General Form of the Response Equations (1/4)
• Solution of the first-order circuitsFirst-order differential equation
( )( ) ( )
dx tax t f t
dt
If x(t)=xp(t) and x(t)=xc(t)
( )( ) 0,
dx tax t
dt ( ) ( ) ( )p cx t x t x t
( ) ( )( ) , ( ) 0
p cp c
dx t dx tax t A ax t
dt dt
xp(t) : particular integral solution
xc(t) : complementary solution, natural response
f(t)=A, constant
Since A is constant, xp(t) must also be a constant
1 1( ) , p
Ax t K K
a
2
( ), ln ( )
( )
ln ( )
( )
cc
c
c
at
c
dx t dt da x t a
x t dt
x t at c
x t K e
Hence,
2
( ) ( ) ( )
p c
at
x t x t x t
AK e
a
Basic Engineering Circuit Analysis-II HiCSLHiCSL6 http://hicsl.kw.ac.kr
General Form of the Response Equations (2/4)
1 2( ) tx t K K e
• General form of the solution of first-order transient circuits
K1 : steady-state solution for t infinite
: time constant
Basic Engineering Circuit Analysis-II HiCSLHiCSL7 http://hicsl.kw.ac.kr
General Form of the Response Equations (3/4)
( )( )0
( ) ( )
S
S
t Vd tC
dt R
Vd t t
dt RC RC
For time t>0 2
( )( )
( )
S
Rt
LS
di tL Ri t V
dt
Vi t K e
R
For time t>0
Using same manner
Basic Engineering Circuit Analysis-II HiCSLHiCSL8 http://hicsl.kw.ac.kr
General Form of the Response Equations (4/4)
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Example 7.1 (1/2)
• Calculate i(t) for t>0
At t = 0-
*Capacitor is open for DC
Basic Engineering Circuit Analysis-II HiCSLHiCSL10 http://hicsl.kw.ac.kr
Example 7.1 (2/2)
Basic Engineering Circuit Analysis-II HiCSLHiCSL11 http://hicsl.kw.ac.kr
Example 7.2 (1/2)
• Find the output voltage v0(t) for t>0*Circuit modification to Thevenin equvalent
Basic Engineering Circuit Analysis-II HiCSLHiCSL12 http://hicsl.kw.ac.kr
Example 7.2 (2/2)