2. 2
METHODOLOGY
Examples of nonlinear
circuits:
logic circuits, digital circuits,
or any circuits where the
output is not linearly
proportional to the input.
Examples of linear circuits:
amplifiers, lots of OPM
circuits, circuits made of
passive components (RLCs).
If the circuit is a linear circuit
Laplace transform of the sources
of excitation: s(t) → S(s)
Laplace transform of the all the
elements in the circuit
Find the output O(s) in the
Laplace freq. domain
Obtain the time response O(t) by
taking the inverse Laplace
transform
Stop or approximate
the circuit into a linear
circuit and continue
NO
YES
3. 3
THE s-DOMAIN CIRCUITS
Equation of circuit analysis:
integrodifferential equations.
Convert to phasor circuits for AC
steady state.
Convert to s-domain using Laplace
transform.
KVL, KCL, Thevenin,etc.
4. 4
KIRCHHOFF’S VOLTAGE LAW
Consider the KVL in time domain:
Apply the Laplace transform:
0)()()()( 4321 =++++ tvtvtvtv
0)()()()( 4321 =++++ sVsVsVsV
5. 5
KIRCHHOFF’S CURRENT LAW
Consider the KCL in time domain:
Apply the Laplace transform:
0)()()()( 4321 =++++ tItItItI
0)()()()( 4321 =++++ titititi
6. 6
OHM’S LAW
Consider the
Ohm’s Law in
time domain
Apply the
Laplace
transform
RsIsV RR )()( =
Rtitv RR )()( =
11. 11
RLC VOLTAGE
The voltage across the RLC elements
in the s-domain is the sum of a term
proportional to its current I(s) and a
term that depends on its initial
condition.
)]0()([)( −
−= LLL issILsV
( ) ( ) )(v
s
(s)I
sC
(s)V ccc
−
+= 011
13. 13
IMPEDANCE
If we set all initial conditions to zero,
the impedance is defined as:
[all initial conditions=0]
)(
)()(
sI
sVsZ =
14. 14
IMPEDANCE & ADMITANCE
The impedances in
the s-domain are
The admittance is
defined as:
sC
sZ
sLsZ
RsZ
C
L
R
1
)(
)(
)(
=
=
=
sCsY
sL
sY
R
sY
C
L
R
=
=
=
)(
1
)(
1
)(
43. 43
TIME DOMAIN TO s-DOMAIN
CIRCUITS
s replaced t in the unknown currents
and voltages.
Independent source functions are
replaced by their s-domain transform
pair.
The initial condition serves as a
second element, the initial condition
generator.
44. 44
THE ELEMENTS LAW OF s-
DOMAIN
)0(
1
)(
1
)(
)0()()(
)()(
−
−
+=⇒
−=⇒
=⇒
CCC
LLL
RR
v
sC
sI
sC
sV
LissLIsV
sRIsV
45. 45
THE ELEMENTS LAW OF s-
DOMAIN
)0()()(
)0()(
)(
)(
)(
−
−
−=⇒
+=⇒
=⇒
CC
LL
L
R
R
CvssCVsI
s
i
sL
sV
sI
R
sV
sI
46. 46
TRANSFORM OF CIRCUITS-
RESISTOR
In the time
domain:
In the s-domain:
i ( t ) + v ( t ) -
R
v ( t ) = i ( t ) R
I ( s ) + V ( s ) -
R
V ( s ) = I ( s ) R