Theory of Electric Circuits II
Lecture #2: Laplace Transform
Spring 2023
Department of Electrical Engineering
Amirkabir University of Technology
Hajar Atrianfar
Outline
1
Definition of Laplace transform
2
Properties of Laplace transform
3
Solution of general networks in s-domain
4
Network function
5
Convolution Theorem
6
6
Periodic functions Laplace transform
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Definition of Laplace transform
● Why Laplace transform?
● Laplace transform is a very effective tool for studying linear timeinvariant networks
● It is a systematic way to relate the time-domain behavior of circuits
to its frequency-domain behavior
o It reduces the solution of integrodifferential equations to the
solution of linear algebraic equations
● Laplace transform theory uses the concept of network function
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Definition of Laplace transform
● Definition:
● F(s) is the unilateral Laplace transform of f : [0, )

● Bilateral Laplace transform: F ( s )
− st
f
(
t
)
e
dt

● F (s) is the Laplace transform of f (t ) −
● The variable 𝑠 is called complex frequency
● If f (t ) includes an impulse at t = 0 , the defining integral includes the
impulse because of 0 − as the lower limit of integration
● The region of convergence (ROC) is defined as the set of points in the
s-plane for which the Laplace transform of function x(t) converges
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Definition of Laplace transform
● In LTI circuits, the input-output relation can be stated in the form of a
differential equation with constant coefficients
● The exponential functions can pass through LTI systems
● The Laplace transform shows a signal as an integral of exponential
functions
● Therefore, it reduces the solution of integrodifferential equations to the
solution of linear algebraic equations of the variable 𝑠
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Definition of Laplace transform
● Example: f (t ) = u(t ), unit step



0−
0−
0−
− st
− st
− ( + jw ) t
u
(
t
)
e
dt
=
e
dt
s
=

+
jw
e
dt =



− ( + jw ) t

e
1
 0
−( + jw) 0−
( + jw)
● Therefore,
1
●
is a rational function well-defined except at s = 0
s
f (t ) = e − at u (t )
● Example:


 e e dt =  e
0−
− at − st
ℒ 𝑒 −𝑎𝑡 𝑢(𝑡)
●
0−
− ( a + s )t
1
=
𝑠+𝑎
− ( s + a )t

e
1
dt =
Re( s + a )  0
− ( s + a ) 0−
( s + a)
𝑅𝑒 𝑠 ≻ −𝑎
1
is a rational function well-defined except at s = a
s+a
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Definition of Laplace transform
● Example: 𝑓 𝑡 = 𝑒 𝑡
2
is not Laplace transformable
● We shall implicitly assume that all considered functions have Laplace
transform
● This assumption imposes no restriction since we are interested in the
properties of response in finite time
t2
● Example: the response to e at some large time t1 is considered
● We can take the input as:
which has a well-defined Laplace transform
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Definition of Laplace transform
● Example: Exponential signals
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Definition of Laplace transform
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Definition of Laplace transform
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Definition of Laplace transform
● Example: Step signals and higher-order integrations
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Definition of Laplace transform
● Example: f (t ) =  (t ), impulse
t
  ( )d = u (t )
−

  ( )d = 1
−
f (t )  (t ) = f (0)  (t )

 f ( ) ( )d = f (0)
−
e


0−
0−
t2
− st

(
t
)
e
dt =   (t )dt = 1 → L[ (t )] = 1

t1
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Definition of Laplace transform
● Example: Higher-order derivatives of impulse
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Laplace transform of elementary functions
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Uniqueness
● Uniqueness:
● Except for those trivial differences, a time function
is uniquely specified by its Laplace transform
● Example:
● This property is fundamental to all applications of Laplace transform
● Therefore, given a Laplace transform F(s), there is a unique time
function f(t) on interval [0, ) such that
● f(t) is the inverse Laplace transform of F(s):
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Linearity
● Linearity
● the Laplace transform is a linear function
● Theorem :
● Let f1 and f 2 be any two time functions and c1 and c2 be two arbitrary
constants, then:
● Proof (??)
● Example: remember
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Linearity
● Linearity
● Proof
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Differentiation rule
● Differentiation rule
● Example:
● In this case, the Laplace transform of df / dt is equal to s times the
Laplace transform of f
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Differentiation rule
● Example: consider the function
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Differentiation rule
● Now consider:
??
● It shows that the Laplace transform depends on the magnitude of the
jump at t=0
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Differentiation rule
● Differentiation theorem :
● Proof
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Differentiation rule
● Example: calculate the impulse response
h(t ) : impulse response of i (t )
linearity
Differentiation rule
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Differentiation rule
● Repeating the application of differentiation rule, we obtain that:
● Example:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Integration rule
● Integration rule
● Example:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Translation in Time Domain
● Translation in time domain
● Proof:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Translation in Frequency Domain
● Translation in frequency domain
● Proof:
● Example:
• We know:
• Then,
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Translation in Frequency Domain
● Differentiation in frequency domain
● Proof:
● We can extend the rule as follows:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Properties of the Unilateral Laplace Transform
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Properties of the Unilateral Laplace Transform
•
Another elementary function Laplace transform
● One important property is the convolution property which will be
discussed later
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Properties of the Unilateral Laplace Transform
● Example:
● Example:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Properties of the Unilateral Laplace Transform
● Example:
● Example:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Partial fraction expansion
● For more complicated rational function, the corresponding time function
can be easily found by reducing the transform to simpler elements
● There is a general method for breaking up any rational function, called
as partial-fraction expansion
● Consider the rational function as:
where P(s) and Q(s) are polynomials in complex frequency variable s and
coefficients
are real numbers
●The alternative representation of F(s):
● zi ' s are zeros and pi ' s are poles of the rational function
● Each pole can be simple pole or multiple pole of order r
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Partial fraction expansion
● A rational function is strictly proper if the degree of the numerator
polynomial is less than the degree of the denominator polynomial
● Partial fraction expansion:
● S.1. put the rational function into a strictly proper form
● S.2. factor the denominator polynomial Q(s) and obtain the poles of the
rational function
● S.3. write the partial fraction expansion
● Case 1: simple poles
The residue K j of the pole p j is given by
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Partial fraction expansion
● Example:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Partial fraction expansion
● Example:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Partial fraction expansion
● Case 2: multiple poles
1
d n1 − j
n1

K1, j =
(
s
−
p
)
F ( s ) 
1
n1 − j 
(n1 − j )! ds
s= p
1
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Partial fraction expansion
● Example:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Partial fraction expansion
● Case 2: complex poles
● F (s) is the ration of polynomials with real coefficients
● Poles and zeros of F (s) are in complex conjugate pairs
● Assume F (s) has simple poles at   j
● Then the following term appears
● Residues are obtained as:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Partial fraction expansion
● Example:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Partial fraction expansion
● Example:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Partial fraction expansion
● Example: Find the solution of the given initial value problem
y = −20sinh(t − 3)u (t − 3) + cosh(t )u (t )
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Solution of electric circuits
● 3 approaches to solve the electric circuits:
❖ 1- Writing the circuit equations in time-domain
solving the
integrodifferential equations in time-domain
❖ 2- Writing the circuit equations in time-domain
transforming the
integrodifferential equations into 𝑠 −domain algebraic equations
using Laplace transform
solving 𝑠 −domain algebraic
equations for the unknown of the interest
❖ 3- Writing the circuit equations in Laplace domain (𝑠-domain)
solving the algebraic equations in 𝑠-domain (operational method)
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Solution of electric circuits
● Example: Approach 1 without Laplace transform
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Solution of electric circuits
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Solution of electric circuits
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Solution of electric circuits
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Solution of electric circuits
● Example: Approach 2 with Laplace transform
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Solution of electric circuits
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Solution of electric circuits
Approach 3:
● Using Laplace transform to solve electric circuits
● Transform the circuit from the time domain to the s-domain
● Writing the KVl and KCL equations in 𝑠-domain and solve the circuit
using nodal analysis/ mesh analysis/ source transformation/
superposition/ …
● Take the inverse transform of the solution and obtain the solution in the
time domain
● The advantages of Laplace transform tool:
▪ Transforms a set of linear, constant coefficient differential equations into a
set of linear polynomial equations
▪ Automatically introduces into the polynomial equations the initial values
of the variables
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Circuit elements in the 𝑠 −domain
● Need to find the equivalent of three main elements of circuits in
Laplace domain
● Once we have the equivalent, the circuit can easily be
transformed from the time domain to the s-domain
● The three main elements of concern are
▪ Resistors
▪ Inductors
▪ Capacitors
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Impedance & Admittance
● Impedance: ratio of the Laplace transform of the zero-state
voltage response to the Laplace transform of the current
V (s)
Z (s) =
I ( s ) no initial condition
● Admittance: ratio of the Laplace transform of the zero-state
current response to the Laplace transform of the voltage
I (s)
Y (s) =
V ( s ) no initial condition
Y ( s ) = Z −1 ( s )
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Resistor in the 𝑠 −domain
v(t ) = Ri (t )  V ( s ) = RI ( s )
R  Z R (s) = R
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Inductor in the 𝑠 −domain
di (t )
v(t ) = L
 V ( s ) = LsI ( s )
dt
L  Z L ( s ) = LS
● Initial condition exists:
V ( s ) = L ( sI ( s ) − i (0− ) )
1
1 −
I ( s ) = V ( s ) + i (0 )
Ls
s
● Based on node or loop analysis we select either the parallel (current source)
or the series (voltage source) connection
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Capacitor in the 𝑠 −domain
t
v(t ) =
1
i ( )d

C0
 V (s) =
11
I ( s)
Cs
1
C  ZC (s) =
Cs
● Initial condition exists:
t
1
v(t ) =  i ( )d + v(0− )
C0
1
v(0− )
V (s) =
I (s) +
Cs
s
I ( s ) = CsV ( s ) − Cv(0− )
● Based on node or loop analysis we select either the parallel (current source)
or the series (voltage source) connection
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Solution of electric circuits
● How to transform a circuit from time domain to frequency
domain:
1- Replace all elements including resistors, inductors and
capacitors with equivalent impedance
2- Consider the initial conditions as series voltage source
(for loop analysis) or parallel current source (for node
analysis)
3- Transform all independent sources to Laplace domain
4- Transfer all dependent sources identically
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Solution of electric circuits
● Example: Approach 3 with frequency domain (𝑠 −domain)
● Closing the switch results in the
voltage 12 sin 5𝑡 𝑢(𝑡) applied to the
circuit
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Solution of electric circuits
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Solution of electric circuits
● The rules for combining impedances and admittances in the 𝑠
−domain are the same as phasor domain
● The delta-to-wye conversions are applicable to 𝑠 −domain
● The Kirchhoff’s laws apply to 𝑠 −domain currents and voltages
● All the techniques of circuit analysis developed for pure
resistive networks may be used in 𝑠 − domain analysis:
▪ Node voltages
▪ Mesh currents
▪ Source transformations
▪ Thevenin-Norton equivalents
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Series impedances
• What is the equivalent impedance for an RLC circuit in
series?
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Parallel admittances
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Source transformation
• Converting a voltage source to an equivalent current source &
vice versa
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Norton equivalent
• Conversion of Norton to Thevenin equivalent circuit and vice
versa
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Mesh analysis in s-domain
• Example: the initial energy stored in the circuit is zero. Find i1 & i2
i1
i2
• Two mesh current equations:
• Using Cramer’s method to solve:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Mesh analysis in s-domain
• Expanding into a sum of partial fractions:
• Next, we test the solutions to see whether they make sense in
terms of the circuit
• No initial energy at the instant when the switched is closed:
i1 (0− ) = i2 (0− ) = 0
• After a long time, inductors appears as short circuits
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Thevenin equivalent in s-domain
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Thevenin equivalent in s-domain
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Thevenin equivalent in s-domain
• Example: the initial energy before closing the switch is zero.
find the capacity current
• Thevenin voltage is the open circuit voltage at terminals:
• Thevenin impedance:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Thevenin equivalent in s-domain
• Example: the initial energy before closing the switch is zero.
find the capacity current
• Thevenin voltage is the open circuit voltage at terminals:
• Thevenin impedance:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Thevenin equivalent in s-domain
• Thevenin equivalent circuit:
• Inverse Laplace transform:
• Test the solution to see whether it makes sense
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Mutual inductance in s-domain
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Mutual inductance in s-domain
• Example: derive the time domain expression of i2 i1
i2
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Impulse function in circuit analysis
• Impulse functions occur either because of a switching operation or excitation
•
•
by an impulsive source
Example:
Initial conditions at 𝑡 = 0−
• Opening the switch forces an instantaneous change in the current of 𝐿 = 2𝐻
which causes 𝑣𝑜 to contain an impulsive component
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Thevenin equivalent in s-domain
• Current of mesh in a clock-wise direction:
• Inverse transforming:
• After the switch is opened, the current in 𝐿1 and 𝐿2 is 6A at 𝑡 = 0+
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
S-domain application of superposition
● The superposition principle says that we can divide the response into
components that can be identified with particular driving sources and initial
conditions
● Example: compute the voltage of R2
iL (0− ) = 
vC (0− ) = 
● S-domain equivalent circuit:
● Define:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
S-domain application of superposition
● 𝑉𝑔 acting alone:
● 𝐼𝑔 acting alone:
●
𝜌
acting alone:
𝑠
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
S-domain application of superposition
● 𝛾𝐶 acting alone:
● Complete response:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Solution of simple circuits
● As we mentioned before, one principal use of Laplace transform is
solving linear integrodifferential equations with constant coefficients
● Example: calculate the impulse response for the output v
● Branch Eqs.:
● KVL:
● By definition of impulse response, we have:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Solution of simple circuits
● Let h be the impulse response, we have :
linearity
Diff. & integ. rules
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Solution of simple circuits
● For calculating the inverse Laplace of H (s) , define
● Then, we have:
● and
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Solution of simple circuits
● Now, we want to calculate the sinusoidal steady state response of the
circuit to e(t ) = Em cos(wt +  )
● Applying the phasor method, we have:
L
1 1
jwV + V +
V =E
R
RC jw
V R
jw
=
E L ( jw) 2 + R jw + 1
L
LC
V
= H ( jw)
E
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Zero-state response
● Consider the LTI circuit shown in the figure
● We want to calculate the zero-state response to an arbitrary input e(.)
● Put all initial conditions zero
● It can be rewritten as:
V
= H ( jw)
● Recall the sinusoidal steady-state analysis and network function:
E
● We formally define “the ratio of the Laplace transform of zero-state
response to the Laplace transform of the input” as the network function
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Network function
● Since the impulse response is the zero-state response and the Laplace
transform of a unit impulse is unity, we have:
● The Laplace transform of the impulse response is the network function
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Network function
● The transfer function 𝐻(𝑠) is the ratio of the output 𝑌(𝑠) to the
input 𝑋(𝑠), assuming all the initial conditions are zero
Y (s)
H (s) =
X (s)
X ( s ) : input
● Four types of transfer functions :
Vout ( s )
H (s) =
voltage gain
Vin ( s )
I out ( s )
H (s) =
I in ( s )
V (s)
H (s) =
I (s)
I (s)
H (s) =
V (s)
current gain
impedance
admittance
Y ( s ) : output
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Applications of network function
● We know three facts from our previous knowledge of circuit theory:
1- The zero-state response of an LTI network can be calculated by the
convolution integral
The response of an LTI system to any given input the convolution of the
impulse response with the input
2- The Laplace transform of the impulse response is the network function
3- The Laplace transform of the zero-state response is the product of the
Laplace transform of the input times the network function
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Applications of network function
● Example:
● Driving point impedance:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Applications of network function
● Example:
● Driving point impedance:
● Transfer function:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Applications of network function
● Example:
● Driving point impedance:
● Transfer function:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Applications of network function
● Impulse response:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Applications of network function
● Impulse response:
● Step response:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Applications of network function
● Impulse response:
● Step response:
● Zero state response to input 𝑣𝑖 𝑡 = 8cos(2𝑡)
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Applications of network function
● Impulse response:
● Step response:
● Zero state response to input 𝑣𝑖 𝑡 = 8cos(2𝑡)
1
8s
Y (s) =
→
2
2s + 3 s + 4
24
24
64
8
−
s+
s+
24
−1
25
25
25
Y ( s) =
+ 2
= (
+ 2 3)
s +3/ 2
s +4
25 s + 3 / 2 s + 4
24  −1.5t
4

y (t ) =  −e
+ cos(2t ) + sin(2t )  u (t )
25 
3

Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
The complete response
● We start by an example
● Example: consider the LTI circuit in the figure by initial conditions as
vc (0− ) = 1, iL (0− ) = 5 and the input e(t ) = 12 sin 5t , t  0 . Calculate the
current i(t )
● KVL:
−
v
(0
)
1 

−
C
 Ls + R +
 I ( s ) = E ( s ) + Li:L (0 ) −
Cs 
s

Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
The complete response
● We obtain:
● What is the network function?
● For the zero-state response, we have:
● Using partial fraction expansion:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
The complete response
● Zero-state response:
● Zero-input response:
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
The complete response
● Zero-state response:
● Zero-input response:
● Define the transient response and sinusoidal steady-state
response
● The transient that exists in a total response may be comprised of two parts
– Transient due to initial conditions
– Transient due to input
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
The convolution theorem
● Convolution theorem
● Let F1 ( s ) and F2 ( s ) be the Laplace transforms of f1 (t ) and f 2 (t ) ,
respectively. Let f 3 (t ) be the convolution of f1 (t ) and f 2 (t ) ; that is,
Then
● The limits of integral are indicated as 0 − and t + , to include the possible
impulses at the origin
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
The convolution theorem
● Proof
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
The convolution theorem
● The Laplace transform of the zero-state response is the product of the
Laplace transform of the input times the network function
Y ( s ) = H ( s )U ( s )
+
+
−
−
y (t ) =  u ( )h(t −  )d =  h( )u (t −  )d
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
The convolution theorem
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
The convolution theorem
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
The convolution theorem
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
The convolution theorem
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
The convolution theorem
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
The convolution theorem
t+
t+
0−
0−
y (t ) =  u ( )h(t −  )d =  h( )u (t −  )d
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
The convolution theorem
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
The convolution theorem
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
The convolution theorem
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
The convolution theorem
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
The convolution theorem
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
The convolution theorem
● Next, we solve the previous example using Laplace transform
3 2 s 3 −2 s
g (t ) = 3u (t + 2) − 3u (t − 2) → G ( s ) = e − e
s
s
2 1 1 −2 s
f (t ) = (2 − t )u (t ) + (t − 2)u (t − 2) → F ( s ) = − 2 + 2 e
s s
s
6 2 s −2 s
3
 F ( s )G ( s ) = 2 ( e − e ) + 3 (1 − e 2 s + e −2 s − e −4 s )
s
s
3 2
f (t ) * g (t ) = 6(t + 2)u (t + 2) − 6(t − 2)u (t − 2) + t u (t )
2
3
3
3
2
2
− (t + 2) u (t + 2) + (t − 2) u (t − 2) − (t − 4) 2 u (t − 4)
2
2
2
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Periodic functions Laplace Transform
● Assume that f (t ) is a causal periodic signal as:

f (t ) = f1 (t ) + f1 (t − T ) + f1 (t − 2T ) + ... =  f1 (t − kT )
k =0
● Laplace transform gives:

F ( s) = F1 ( s ) + e −Ts F1 ( s) + e − 2Ts F1 ( s) + ... = F1 ( s) e − kTs = F1 ( s )
k =0
1
1 − e −Ts
e −Ts  1 → ROC : Re( s)  0
● A noncausal periodic signal has not Laplace transform
F ( s ) = ... + e 2Ts F1 ( s ) + eTs F1 ( s ) + F1 ( s ) + e −Ts F1 ( s ) + e −2Ts F1 ( s ) + ...

= F1 ( s )  e
k =−
− kTs
1
1
= F1 ( s )(
+
− 1)
−Ts
+ Ts
1− e
1− e
e sT  1, e − sT  1
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Periodic functions Laplace Transform
● Assume that f (t ) is a periodic function as:

f (t ) = f1 (t ) + f1 (t − T ) + f1 (t − 2T ) + ... =  f1 (t − kT )
k =0
● Laplace transform gives:
F ( s) = F1 ( s ) + e
−Ts
F1 ( s) + e
− 2Ts
e −Ts  1 → ROC : Re( s)  0

F1 ( s) + ... = F1 ( s) e
k =0
− kTs
1
= F1 ( s )
1 − e −Ts
● Example: find the Laplace transform of the shown periodic function
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Periodic functions Laplace Transform
● Consider a circuit with periodic input 𝑥(𝑡) and transfer function
𝐻 𝑠 :

X ( s ) = X 1 ( s ) + e −Ts X 1 ( s ) + e −2Ts X 1 ( s ) + ... = X 1 ( s ) e − kTs
k =0

Y ( s ) = H ( s ) X ( s ) = H ( s ) X 1 ( s )  e − kTs
Y1 ( s )
k =0
y (t ) = y1 (t ) + y1 (t − T ) + y1 (t − 2T ) + ...
● The output of an LTI system to a periodic input is a periodic
output with the same period T
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Periodic functions Laplace Transform
● Next we apply the convolution method to obtain the response
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Periodic functions Laplace Transform
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Periodic functions Laplace Transform
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Periodic functions Laplace Transform
Definition of Laplace transform
Properties of Laplace transform
Solution of general networks in s-domain
Network function
Convolution Theorem
Periodic functions Laplace transform
Periodic functions Laplace Transform