Research

RESEARCH
Mr.K.P.H.H.PANDIKORALA
CIRCUIT THEORY

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1) Abstract
This report is a research based on RC, RL and RLC circuits, which contains Transient analysis in
RLC circuits, a thorough discussion of the Laplace Transform and the S-Plane, first and second
order systems, Circuit responses and the power factor and RMS values of periodic complex wave
forms.

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2) Table of Contents
Contents
1) Abstract.................................................................................................................................... 1
2) Acknowledgement..................................................................Error! Bookmark not defined.
3) Table of Contents..................................................................................................................... 2
4) List of Figures.......................................................................................................................... 4
5) List of Tables........................................................................................................................... 5
6) Introduction ............................................................................................................................. 6
7) What is Transient Analysis?.................................................................................................... 7
8) Laplace Transform................................................................................................................... 7
9) S - Plane................................................................................................................................... 8
Transfer function of a system ..................................................................................................... 9
Linear Time Invariant systems.................................................................................................... 9
Zeroes:....................................................................................................................................... 11
Poles:......................................................................................................................................... 11
Examples of pole/Zero Plots:.................................................................................................... 12
Applications for Pole-Zero Plots .............................................................................................. 14
10) Definition of Laplace Transform of f(t)............................................................................. 14
Linearity and Existence............................................................................................................. 15
Laplace Transforms of Derivatives and Integrals..................................................................... 16
Shifting on the t and s Axes...................................................................................................... 18
Differentiation and Integration of Transforms.......................................................................... 20
Convolution............................................................................................................................... 21
Inverse Laplace Transform ....................................................................................................... 22
11) Laplace Transformation Table........................................................................................... 23

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12) First Order Systems/Circuits.............................................................................................. 24
1) RC (Resistor and Capacitor) .............................................................................................. 25
2) RL (Resistor and Inductor) ................................................................................................ 27
13) Second Order Systems/Circuits ......................................................................................... 29
1) Series RLC (Resistor, Inductor and Capacitor) circuit ...................................................... 29
2) Parallel RLC (Resistor, Inductor and Capacitor) Circuit................................................... 30
Summary................................................................................................................................... 31
14) Reponses of RLC Circuits ................................................................................................. 33
Natural Response of a simple RC circuit.................................................................................. 33
Natural Response of a simple RL circuit .................................................................................. 34
Response of RLC circuits/Transient Response......................................................................... 34
Case 1 : Overdamped 𝑹𝟐 > 𝟒𝑳/𝑪......................................................................................... 38
Case 2 : Critically Damped 𝑹𝟐 = 𝟒𝑳/𝑪 ................................................................................. 38
Case 3 : Underdamped 𝑹𝟐 < 𝟒𝑳/𝑪 ....................................................................................... 39
Case 4 : Undamped 𝜻 = 𝟎 ....................................................................................................... 40
Properties of RLC networks...................................................................................................... 41
15) Power Factor and RMS value of Complex Periodic Waveforms ...................................... 42
Complex Periodic Waveforms.................................................................................................. 42
Analysis of complex periodic waveforms (Fourier Analysis) .................................................. 43
RMS value of complex periodic wave forms. .......................................................................... 43
Power Factor of complex periodic wave forms ........................................................................ 44
16) Conclusion ......................................................................................................................... 46
17) References.......................................................................................................................... 47

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3) List of Figures
FIGURE 1- BASIC PROCESS OF LAPLACE TRANSFORM..................................................................................................................8
FIGURE 2-LINEAR SYSTEM ....................................................................................................................................................9
FIGURE 3-TIME-INVARIANT SYSTEM .....................................................................................................................................10
FIGURE 4-TIME-INVARIANT SYSTEM .....................................................................................................................................10
FIGURE 5-A GENERAL SYSTEM.............................................................................................................................................10
FIGURE 6- THE S-PLANE.....................................................................................................................................................11
FIGURE 7-EXAMPLE 1 S-PLANE PLOT....................................................................................................................................12
FIGURE 8-EXAMPLE 2 S-PLANE PLOT....................................................................................................................................13
FIGURE 9- SIMPLE RC CIRCUIT ............................................................................................................................................33
FIGURE 10- SIMPLE RL CIRCUIT...........................................................................................................................................34
FIGURE 11-RLC SERIES......................................................................................................................................................34
FIGURE 12-RLC PARALLEL..................................................................................................................................................35
FIGURE 13-RLC DAMPING.................................................................................................................................................36
FIGURE 14-RLC DAMPING.................................................................................................................................................36
FIGURE 15-GRAPH OF OVERDAMPED CASE.............................................................................................................................38
FIGURE 16-GRAPH OF CRITICALLY DAMPED CASE.....................................................................................................................38
FIGURE 17-GRAPH OF UNDERDAMPED CASE...........................................................................................................................39
FIGURE 18-CRITICAL VS UNDAMPED....................................................................................................................................41
FIGURE 19 - BASIC SINUSOIDAL A.........................................................................................................................................42
FIGURE 20-BASIC SINUSOIDAL B ..........................................................................................................................................42
FIGURE 21-COMPLEX PERIODIC WAVE FORM..........................................................................................................................42
FIGURE 22-POWER TRIANGLE.............................................................................................................................................45

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4) List of Tables
TABLE 1-LAPLACE TRANSFORMATION TABLE..........................................................................................................................23
TABLE 2-FIRST ORDER CIRCUITS SUMMARY............................................................................................................................31
TABLE 3-SECOND ORDER CIRCUITS SUMMARY ........................................................................................................................32
TABLE 4-IMPORTANT RELATIONSHIPS ...................................................................................................................................32

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5) Introduction
This report is based on RL, RC and RLC circuits which are discussed in detail and Transient
analysis, (which is the analysis of a circuits which has undergone instantaneous events) of RL, RC
and RLC networks are explained clearly. Laplace Transform plays a major role in transient
analysis as it is a way of simplifying Differential equations through transforming them into
algebraic expressions (in frequency domain) which can then be solved easily and can in-tern be
transformed back to the original form (time-domain) .And the first and second order systems are
analyzed thoroughly which leads to circuit responses, and power factor and RMS value
calculations of complex periodic wave forms.

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6) What is Transient Analysis?
A Transient event is an instantaneous or a short lived burst of energy in a system caused by a
sudden change of state. Transient Analysis is the Analysis of a system which is in an unsteady
state, where by an unsteady state, it is meant that the variables involved in defining the state of the
system changes with respect to time.(Analysis is the process of breaking a complex topic into
smaller parts in order to gain a better understanding of it). With reference to Electrical Engineering
Transient Analysis is the Analysis of a system which includes some instant event such as a switch
opens or closes, the power supply turns on, a part is suddenly pulled out of the circuit, a fuse
blows…etc. The conclusion of the analysis can be either a formula for v(t) {The potential
difference cross two nodes} or i(t) {The current flow through the system or a branch}or a graph
of v {Voltage}or i {Current} on the vertical axis and time on the horizontal axis.
Differential equations play an important role in the transient analysis and the non-linearity
properties of both capacitors and inductors should be taken into consideration. Differential
equations can also be converted to the frequency domain by taking Laplace transform, which is
useful in controlling the behavior of the system.
7) Laplace Transform
The Laplace transform provides a useful method of solving certain types of differential equations
(Linear Differential equations) when certain initial conditions are given. Ultimately what the
Laplace transform does is reduce complex Differential equations to basic algebra problems in order
to make the solving process easier. The basic process of solving a differential equation through
Laplace transform is shown below in Figure 1.

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However to understand this method the knowledge about S – plane is essential.
8) S - Plane
In Mathematics and Engineering the S-plane is the complex plane(Geometrical representation of
a complex number, Real axis as the vertical axis and the Imaginary axis as the horizontal axis) on
which Laplace transforms are graphed. It is a mathematical domain where instead of viewing
processes in the time domain modelled with time-based functions they are viewed as equations
in the frequency domain.( A Domain of a function is the set of all possible input values (commonly
the "x" variable), which produce a valid output from a particular function. It is the set of all real
numbers for which a function is mathematically defined.) It is used as a graphical analysis tool in
Engineering and physics.
Once the Laplace transform of a system has been determined, one can use the information
contained in function’s polynomials (an expression consisting of variables and coefficients which
only employs the operation of addition, subtraction, multiplication and non-negative integer
exponents.) to graphically represent the function and easily observe many defining characteristics.
Differential equation
𝑑2
𝑦
𝑑𝑥2
− 4𝑦 = 𝑒−3𝑥
sin2𝑥
Laplace Transform
𝐹 𝑠 =
1
𝑠 − 5 𝑆 − 3
=
1
2
1
𝑠 − 5
−
1
𝑠 − 3
Inverse Laplace Transform
𝐿−1
𝐹 𝑠
Solution
𝐿−1
𝐹 𝑠 =
1
2
𝑒5𝑡
− 𝑒3𝑡
Figure 1- Basic process of Laplace Transform

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The Laplace Transform will have the below structure, based on Rational functions (For any two
polynomials A and B, there quotient is called a rational function; i.e. if f(x) = A/B, F(x) is a rational
function),
𝐻 𝑠 =
𝑃 𝑠
𝑄 𝑠
The equation above is called The Transfer function of a system. This can be derived using
Differential equations. Before moving forward, the concept of The Transfer Function must be
clarified.
Transfer function of a system
In engineering a transfer function is a mathematical representation which describes inputs and
outputs of black box models which are devices, systems or objects which can be viewed in terms
of its inputs and outputs, which are Linear Time-Invariant in nature. It is a Method to represent
system dynamics, via s(s-plane) representation from Laplace transforms. Transfer functions show
flow of signal through a system, from input to output.
Linear Time Invariant systems
For the system to be Linear Time-Invariant,
𝑥1 𝑛
𝑥2 𝑛
𝑦1 𝑛
𝑦2 𝑛
H
H
𝑎1 𝑥1 𝑛 + 𝑎2 𝑥2 𝑛 𝑎1 𝑦1 𝑛 + 𝑎2 𝑦2 𝑛
H
𝑥1(𝑛)
Figure 2-Linear System
1

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And
The transfer Function of a Linear Time-Invariant system can be obtain as shown below,
Transfer Function,
𝐻 𝑛 =
𝑌 𝑛
𝑋 𝑛
𝑇𝑟𝑎𝑠𝑓𝑒𝑟 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 {𝐻 𝑛 } =
𝑆𝑡𝑒𝑎𝑑𝑦 𝑆𝑡𝑎𝑡𝑒 𝑂𝑢𝑡𝑝𝑢𝑡 𝑆𝑆𝑂
𝑆𝑡𝑒𝑎𝑑𝑦 𝑆𝑡𝑎𝑡𝑒 𝐼𝑛𝑝𝑢𝑡 𝑆𝑆𝐼
(SSO and SSI are the input and output measured when the system is stabilized)
𝐻 𝑠 =
𝑃 𝑠
𝑄 𝑠
The two Polynomials P(s) and Q(s), allow us to find the poles and the Zeroes of Laplace Transform
which can be used plot the Laplace transform onto the S-plane.
H
𝑥1(𝑛)
𝑥1 𝑛 − 𝑘 𝑦1 𝑛 − 𝑘
Figure 3-Time-Invariant system K = Time Delay
H
𝑥1(𝑛)
𝑥2 𝑛 − 𝑘 𝑦2 𝑛 − 𝑘
K = Time Delay
H(n)
𝑥1(𝑛)
X(n) Y(n)
Figure 5-A general system
Figure 4-Time-Invariant system
1

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Zeroes:
The value(s) for s where P(s) = 0.
The complex frequencies that make the overall gain of the filter Transfer Function zero.
Poles:
The value(s) for s where Q(s) = 0.
The complex frequencies that make the overall gain of the filter transfer function infinite.
Once the poles and zeros have been found for a given Laplace Transform, they can be plotted onto
the S-Plane. The S-plane is a complex plane with an imaginary and real axis referring to the
complex-valued variable z. The position on the complex plane is given by 𝑟𝑒 𝑖𝜃
and the angle from
the positive, real axis around the plane is denoted by θ. When mapping poles and zeros onto the
plane, poles are denoted by an "x" and zeros by an "o". The below figure shows the S-Plane, and
examples of plotting zeros and poles onto the plane can be found in the following section.
Figure 6- The S-plane

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Examples of pole/Zero Plots:
Example 1: simple pole/Zero Plot.
𝐻 𝑠 =
𝑠
𝑠 −
1
2
𝑠 +
3
4
The Zeroes: {0}
The Poles: {
1
2
,
−3
4
}
Figure 7-Example 1 S-plane Plot
Example 2: Complex Pole/Zero Plot.
𝐻 𝑠 =
𝑠 − 𝑖 𝑠 + 𝑖
𝑠 − (
1
2
−
1
2
𝑖) 𝑠 −
1
2
+
1
2
𝑖
The Zeroes: {i, -i}
The Poles: {-1,
1
2
+
1
2
𝑖,
1
2
−
1
2
𝑖}

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Figure 8-Example 2 S-plane Plot
Example 3: Pole-Zero Cancellation.
𝐻 𝑠 =
𝑠 + 3 𝑠 − 1
𝑠 − 1
One might come to conclusion that this is the same as H(s) = (s+3), although this conclusion is
theoretically correct where they cancel out in what is known as Pole-Zero cancellation. However
think that this is a Transfer Function of a system which is created with physical circuits, it is very
unlikely that the Pole and the Zero would remain in exactly the same place, for instance a slight
change in temperature could cause one of them to move just slightly If this were to occur a
tremendous amount of volatility is created in that area, since there is a change from infinity at the
pole to zero at the zero in a very small range of signals. This is generally a very bad way to try to
eliminate a pole. A much better way is to use control theory to move the pole to a better place.
Repeated Poles and Zeroes:
It is possible to have more than one pole or zero at any given point. For instance, the discrete-time
transfer function 𝐻 𝑧 = 𝑧2
will have two zeros at the origin and the continuous-time function
𝐻 𝑠 =
1
𝑠25
will have 25 poles at the origin.

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Applications for Pole-Zero Plots
 Stability and Control Theory: Basically what we can gather from this is that the
magnitude of the transfer function will be larger when it is closer to the poles and smaller
when it is closer to the zeros. This provides us with a qualitative understanding of what the
system does at various frequencies and is crucial to the discussion of stability.
 Frequency Response and Pole/Zero Plots: The reason it is helpful to understand and
create these pole/zero plots is due to their ability to help us easily design a filter. Based on
the location of the poles and zeros, the magnitude response of the filter can be quickly
understood. Also, by starting with the pole/zero plot, one can design a filter and obtain its
transfer function very easily.
Pole-Zero Plots are clearly quite useful in the study of the Laplace and Z transform, affording us
a method of visualizing the at times confusing mathematical functions.
9) Definition of Laplace Transform of f(t).
The Laplace transform ℒ, of a function f (t) for t > 0 is defined by the following integral over 0 to
∞:
𝐿{𝑓 𝑡 } = ∫ 𝑒−𝑠𝑡
∞
0
𝑓 𝑡 𝑑𝑡
The resulting expression is a function of s, it means that the resulting expression is the s-plane
representation of f (t), which we write as F(s).
L{f(t)} = F(s).
Similarly the Laplace Transform of a function g(t) would be written as,
L{g(t)} = G(s).

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Example: find the Laplace Transform of f(t) = 𝒆 𝒂𝒕
, s > a.
L{f(t)} = ∫ 𝑒−𝑠𝑡
𝑒 𝑎𝑡
𝑑𝑡
∞
0
= ∫ 𝑒 𝑎−𝑠 𝑡
𝑑𝑡
∞
0
=
1
𝑎−𝑠
𝑒 𝑎−𝑠 𝑡
|∞
0
The integral is divergent whenever s ≤ a. However, when s > a, it converges to,
=
1
𝑎−𝑠
0 − 𝑒0
F(s) =
1
𝑠−𝑎
Linearity and Existence
Linearity
Laplace Transform is a linear operation, that is if the Laplace Transform of f(t) and g(t) are F(s)
and G(s) respectively, then the Laplace transform of αf(t) + βg(t) is αF(s) + βG(s), for any constants
α and β.
Proof:
𝐿 𝛼𝑓 𝑡 + 𝛽𝑔 𝑡 = ∫ 𝛼𝑓 𝑡 + 𝛽𝑔 𝑡 𝑒−𝑠𝑡
𝑑𝑡
∞
0
= 𝛼 ∫ 𝑓 𝑡 𝑒−𝑠𝑡∞
0
𝑑𝑡 + 𝛽 ∫ 𝑔 𝑡 𝑒−𝑠𝑡
𝑑𝑡
∞
0
= 𝛼𝐿 𝑓 𝑡 + 𝛽𝐿 𝑔 𝑡
= 𝛼𝐹 𝑠 + 𝛽𝐺 𝑠
Existence
Let f(t) be a function which is piece wise continuous on every finite interval in the range t ≥ 0, and
satisfies,
|𝑓 𝑡 | ≤ 𝑀𝑒 𝑟𝑡

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For some M and r.
Then the Laplace Transform of f(t) exists if Re(s) > r. [Re(s) denotes the Real part of s].
Proof:
𝐿{𝑓 𝑡 } = ∫ 𝑓 𝑡 𝑒−𝑠𝑡
𝑑𝑡
∞
0
|𝐿{𝑓 𝑡 }| = |∫ 𝑓 𝑡 𝑒−𝑠𝑡
𝑑𝑡
∞
0
|
= ∫ 𝑒−𝑠𝑡|𝑓 𝑡 |𝑑𝑡
∞
0
= ∫ 𝑒−𝑠𝑡
𝑀𝑒 𝑟𝑡
𝑑𝑡
∞
0
= 𝑀 ∫ 𝑒 𝑟−𝑠 𝑡
𝑑𝑡
∞
0
𝐿|{𝑓 𝑡 }| =
𝑀
𝑠−𝑟
; If Re(s) > r.
Therefore L[f(t)] exists if Re(s) > r.
Laplace Transforms of Derivatives and Integrals
Theorem 1
Let f(t) be a continuous function for all t ≥ 0, and |𝑓 𝑡 | ≤ 𝑀𝑒 𝑟𝑡
for some M and r. If the derivative
of f(t), is piece-wise continuous in every finite interval in the range t ≥ 0, then the Laplace
Transform of 𝑓′
(t) exists when Re(s) > r and L{𝑓′
𝑡 } = s.L{F(t)-f(0)}.
Proof:
𝐿{𝑓′
𝑡 } = ∫ 𝑓′
𝑡 𝑒−𝑠𝑡
𝑑𝑡
∞
0
= 𝑓 𝑡 𝑒−𝑠𝑡
|∞
0
+ 𝑠 ∫ 𝑒−𝑠𝑡
𝑓 𝑡 𝑑𝑡
∞
0
= −𝑓 0 + 𝑠𝐿{𝑓 𝑡 }
= 𝑠𝐿{𝑓 𝑡 } − 𝑓 0

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Theorem 1 can be extended to higher order derivatives,
𝐿{𝑓′′
𝑡 } = ∫ 𝑓′′
𝑒−𝑠𝑡
𝑑𝑡
∞
0
= 𝑓′
𝑡 𝑒−𝑠𝑡
|∞
0
+ 𝑠 ∫ 𝑒−𝑠𝑡
𝑓′
𝑡 𝑑𝑡
∞
0
= −𝑓′
0 + 𝑠𝐿{𝑓′
𝑡 }
= −𝑓′
0 + 𝑠(𝑠𝐿{𝑓 𝑡 )} − 𝑓 0
= −𝑓′
0 + 𝑠2
𝐿{𝑓 𝑡 } − 𝑠𝑓 0
= 𝑠1
𝐿{𝑓 𝑡 } − 𝑠𝑓 0 − 𝑓′
0
Similarly,
𝐿{𝑓′′′
𝑡 } = 𝑠3
𝐿{𝑓 𝑡 } − 𝑠2
𝑓 0 − 𝑠𝑓′ 0
− 𝑓′′
0
Theorem 2
Let f(t) and its derivatives 𝑓′
𝑡 , 𝑓′′
𝑡 , … , 𝑓 𝑛−1
𝑡 be continuous functions in the range t ≥ 0, and
|𝑓 𝑡 | ≤ 𝑀𝑒 𝑟𝑡
for some M and r. let 𝑓 𝑛
𝑡 be a piecewise continuous function in the range t ≥ 0.
Then the Laplace Transform of 𝑓 𝑛
𝑡 exists when Re(s) > r and is given by,
𝐿{𝑓 𝑛
𝑡 } = 𝑠 𝑛
𝐿{𝑓 𝑡 } − 𝑠 𝑛−1
𝑓 0 − 𝑠 𝑛−2
𝑓′
0 − ⋯ − 𝑓 𝑛−1
0
Theorem 3
Let f(t) be a piece wise continuous function for t ≥ 0 and |𝑓 𝑡 | < 𝑀𝑒 𝑟𝑡
for some M and r. Then,
𝐿[∫ 𝑓 𝑡 𝑑𝑡
∞
0
] =
1
𝑠
. 𝐿{𝑓 𝑡 }; For s > 0 and s > r.
Proof:
Let g(t) = ∫ 𝑓 𝑡 𝑑𝑡
𝑡
0
|𝑔 𝑡 | = |∫ 𝑓 𝑡 𝑑𝑡
𝑡
0
|

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|𝑔 𝑡 | = ∫ |𝑓 𝑡 |𝑑𝑡
𝑡
0
= ∫ 𝑀𝑒 𝑟𝑡
𝑑𝑡
𝑡
0
= 𝑀 ∫ 𝑒 𝑟𝑡
𝑑𝑡
𝑡
0
= 𝑀
𝑒 𝑟𝑡
𝑟
| 𝑟
0
=
𝑀
𝑟
𝑒 𝑟𝑡
− 1
Therefore,
|𝑔 𝑡 | =
𝑀
𝑟
𝑒 𝑟𝑡
− 1
|𝑔 𝑡 | =
𝑀
𝑟
𝑒 𝑟𝑡
if r > 0.
Since 𝑔′
𝑡 = f(t) ,
g(t) is also piecewise continuous,
𝐿{𝑓 𝑡 } = 𝐿{𝑔′
𝑡 } = 𝑠. 𝐿{𝑔 𝑡 } − 𝑔 0
Since g(0) = 0,
𝐿{𝑓 𝑡 } = 𝑠. 𝐿{𝑔 𝑡 } − 0
Therefore,
𝐿{𝑔 𝑡 } =
1
𝑠
𝐿{𝑓 𝑡 }
Therefore,
𝐿 {∫ 𝑓 𝑡 𝑑𝑡
𝑡
0
} =
1
𝑠
. 𝐿{𝑓 𝑡 }
Shifting on the t and s Axes

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Theorem 4
If F(s) is the Transform of f(t) with Re(s) > r, then F(s-a) is the Laplace Transform of 𝑒 𝑎𝑡
𝑓 𝑡 with
Re(s) > a + r.
Proof:
By definition F(s) = ∫ 𝑒−𝑠𝑡
𝑓 𝑡 𝑑𝑡
∞
0
𝐿{𝑒 𝑎𝑡
𝑓 𝑡 } = ∫ 𝑒−𝑠𝑡
𝑒 𝑎𝑡
𝑓 𝑡 𝑑𝑡 = ∫ 𝑒− 𝑠−𝑎 𝑡∞
0
∞
0
𝑓 𝑡 𝑑𝑡 = 𝐹 𝑠 − 𝑎
Theorem 5
If F(s) is the Laplace Transform of f(t), then the Laplace Transform of the function,
g(t) = 0 ; if t < a
= f(t-a) ; if t > a for a > 0
Is 𝑒−𝑎𝑠
𝐹 𝑠 .
Proof:
𝐿{𝑔 𝑡 } = ∫ 𝑒−𝑠𝑡
𝑔 𝑡 𝑑𝑡
∞
0
= ∫ 𝑒−𝑠𝑡
𝑓 𝑡 − 𝑎 𝑑𝑡
∞
0
Let i = t-a ,
Therefore, 𝐿{𝑔 𝑡 } = ∫ 𝑒−𝑠 𝑖+𝑎
𝑓 𝑖 𝑑𝑖
∞
0
= 𝑒−𝑎𝑠
∫ 𝑒−𝑠𝑖
𝑓 𝑖 𝑑𝑖 = 𝑒−𝑎𝑠
𝐹 𝑠
∞
0
Therefore, 𝐿{𝑔 𝑡 } = 𝑒−𝑎𝑠
𝐹 𝑠

20 | P a g e
Differentiation and Integration of Transforms
Theorem 6
If F(s) is the Laplace Transform of the function f(t), then 𝐹′
𝑠 is the Laplace transform of – t f(t).
Proof:
If F(s) is the Laplace transform of f(t) then,
𝐹 𝑠 = ∫ 𝑒−𝑠𝑡
𝑓 𝑡 𝑑𝑡
∞
0
By differentiating with respect to s we obtain,
𝐹′
𝑠 = − ∫ 𝑒−𝑠𝑡
. 𝑡. 𝑓 𝑡 𝑑𝑡 = −𝐿{𝑡𝑓 𝑡 }
∞
0
Therefore, if 𝐹 𝑠 = 𝐿{𝑓 𝑡 } then 𝐹′
𝑠 = −𝐿{𝑡𝑓 𝑡 }
Theorem 7
If F(s) is the Laplace Transform of f(t) then, ∫ 𝐹 𝑠 𝑑𝑠 = 𝐿 [
𝑓 𝑡
𝑡
]
∞
0
Proof:
If F(s) is the Laplace Transform of f(t) then,
𝐹 𝑠 = ∫ 𝑒−𝑠𝑡
𝑓 𝑡 𝑑𝑡
∞
0
∫ 𝐹 𝑠 𝑑𝑠 = ∫ [∫ 𝑒−𝑠𝑡
𝑓 𝑡 𝑑𝑡
∞
0
] 𝑑𝑠
∞
0
∞
0
= ∫ 𝑑𝑠 ∫ 𝑑𝑡 𝑒−𝑠𝑡
𝑓 𝑡
∞
0
∞
0
= ∫ 𝑑𝑡 𝑓 𝑡 [∫ 𝑒−𝑠𝑡
𝑑𝑠
∞
0
]
∞
0
= ∫ 𝑑𝑡 𝑓 𝑡
∞
0
[−
𝑒−𝑠𝑡
𝑡
] ∞
𝑠

21 | P a g e
= ∫
𝑓 𝑡
𝑡
𝑒−𝑠𝑡
𝑑𝑡
∞
0
Since lim
𝑠→∞
= 0
Therefore,
∫ 𝐹 𝑠 𝑑𝑠 = 𝐿 [
𝑓 𝑡
𝑡
]
∞
0
Convolution
Theorem 8
Let F(s) and G(s) be the Laplace Transforms of the functions f(t) and g(t). If H(s) = F(s)G(s)
is the Laplace Transform of the function h(t), then,
ℎ 𝑡 = ∫ 𝑓 𝑖 𝑔 𝑡 − 𝑖 𝑑𝑖
∞
0
Proof:
Since G(s) is the Laplace Transform of g(t),
𝐺 𝑠 = ∫ 𝑒−𝑠𝑡
𝑔 𝑡′ 𝑑𝑡′
∞
0
(Note that 𝑡′ is a dummy variable)
If I > 0, 𝑒−𝑠𝑖
𝐺 𝑠 = ∫ 𝑒−𝑠𝑡 𝑡+𝑖
𝑔 𝑡′ 𝑑𝑡′
∞
0
Let t = 𝑡′
+ 𝑖,
Therefore,
𝑒−𝑠𝑖
𝐺 𝑠 = ∫ 𝑒−𝑠𝑡
𝑔 𝑡 − 𝑖 𝑑𝑡
∞
0
Since F(s) is the La[place Transform of f(t),
𝐹 𝑡 = ∫ 𝑒−𝑠𝑖
𝑓 𝑖 𝑑𝑖
∞
0
𝐺 𝑠 𝐹 𝑠 = ∫ 𝑒−𝑠𝑖
∞
0
𝐺 𝑠 𝑓 𝑖 𝑑𝑖

22 | P a g e
= ∫ 𝑑𝑖 ∫ 𝑑𝑡 𝑒−𝑠𝑡
𝑔 𝑡 − 𝑖 𝑓 𝑖
∞
0
∞
0
By changing the order of the Integration, we obtain,
𝐺 𝑠 𝐹 𝑠 = ∫ 𝑑𝑡 ∫ 𝑑𝑖 𝑒−𝑠𝑡
𝑔 𝑡 − 𝑖 𝑓 𝑖
∞
0
∞
0
Therefore, 𝐻 𝑠 = ∫ 𝑑𝑡
∞
0
𝑒−𝑠𝑡
∫ 𝑑𝑖 𝑔 𝑡 − 𝑖 𝑓 𝑖
∞
0
Since H(s) is the Laplace Transform of h(t),
𝐻 𝑠 = ∫ 𝑑𝑡 𝑒−𝑠𝑡
ℎ 𝑡
∞
0
Therefore, ℎ 𝑡 = ∫ 𝑔 𝑡 − 𝑖 𝑓 𝑖 𝑑𝑖
∞
0
Inverse Laplace Transform
Definition:
If F(s) is the Laplace Transform of a function f(t), f(t) is called the Inverse Laplace Transform of
F(s) and is denoted by 𝐿−1
{𝐹 𝑠 }.
Finding the Inverse Laplace Transform is merely a use of algebra (mostly partial fractions) to
convert F(s) into a known more familiar function and then by using the Laplace transformation
table one can obtain f(t).
Example:
Determine the Inverse Transform of, 𝐹 𝑠 =
𝑠+5
𝑠3+5𝑠2+6𝑠
.
𝐹 𝑠 =
𝑠 + 5
𝑠3 + 5𝑠2 + 6𝑠
=
𝑠 + 5
𝑠 𝑠 + 2 𝑠 + 3
𝐿𝑒𝑡
𝑠+5
𝑠 𝑠+2 𝑠+3
=
𝐴
𝑠
+
𝐵
𝑠+2
+
𝐶
𝑠+3
: Partial fractions.
→ 𝐴 =
5
6
, 𝐵 = −
3
2
, 𝐶 =
2
3
∴ 𝐹 𝑠 =
1
6
(
5
𝑠
−
9
𝑠+2
+
4
𝑠+3
)
𝐿−1{𝐹 𝑠 } =
1
6
5 − 9𝑒−2𝑡
+ 4𝑒−3𝑡
: By using the Table.

23 | P a g e
10) Laplace Transformation Table
Table 1-Laplace Transformation Table
𝑓 𝑡 = 𝐿−1
{𝐹 𝑠 } 𝐹 𝑠 = 𝐿{𝑓 𝑡 }
1 1 1
𝑠
2 𝑒 𝑎𝑡 1
𝑠 − 𝑎
3 𝑡 𝑛
, 𝑛 = 1,2,3, … 𝑛!
𝑠 𝑛+1
4 𝑡 𝑝
, 𝑝 > −1 Γ p + 1
𝑠 𝑝+1
5 √ 𝑡 √ 𝜋
2𝑠3/2
6
𝑡 𝑛−
1
2, 𝑛 = 1,2,3, … 1.3.5 … 2𝑛 − 1 √ 𝜋
2 𝑛 𝑠 𝑛+
1
2
7 sin 𝑎𝑡 𝑎
𝑠2 + 𝑎2
8 cos 𝑎𝑡 𝑠
𝑠2 + 𝑎2
9 𝑡 sin 𝑎𝑡 2𝑎𝑠
𝑠2 + 𝑎2 2
10 𝑡 cos 𝑎𝑡 𝑠2
− 𝑎2
𝑠2 + 𝑎2 2
11 sin 𝑎𝑡 − 𝑎𝑡 cos 𝑎𝑡 2𝑎3
𝑠2 + 𝑎2 2
12 sin 𝑎𝑡 + 𝑎𝑡 cos 𝑎𝑡 2𝑎𝑠2
𝑠2 + 𝑎2 2
13 cos 𝑎𝑡 − 𝑎𝑡 sin 𝑎𝑡 𝑠 𝑠2
− 𝑎2
𝑠2 + 𝑎2 2
14 cos 𝑎𝑡 + 𝑎𝑡 sin 𝑎𝑡 𝑠 𝑠2
+ 3𝑎2
𝑠2 + 𝑎2 2
15 sin 𝑎𝑡 + 𝑏 𝑠 sin 𝑏 + 𝑎 cos 𝑏
𝑠2 + 𝑎2
16 cos 𝑎𝑡 + 𝑏 𝑠 cos 𝑏 − 𝑎 sin 𝑏
𝑠2 + 𝑎2
17 sinh 𝑎𝑡 𝑎
𝑠2 − 𝑎2
18 cosh 𝑎𝑡 𝑠
𝑠2 − 𝑎2
19 𝑒 𝑎𝑡
sin 𝑏𝑡 𝑏
𝑠 − 𝑎 2 + 𝑏2
20 𝑒 𝑎𝑡
cos 𝑏𝑡 𝑠 − 𝑎
𝑠 − 𝑎 2 + 𝑏2
21 𝑒 𝑎𝑡
sinh 𝑏𝑡 𝑏
𝑠 − 𝑎 2 − 𝑏2
22 𝑒 𝑎𝑡
cosh 𝑏𝑡 𝑠 − 𝑎
𝑠 − 𝑎 2 − 𝑏2
23 𝑡 𝑛
𝑒 𝑎𝑡
, 𝑛 = 1,2,3, … 𝑛!
𝑠 − 𝑎 𝑛+1
24 𝑓 𝑐𝑡 1
𝑐
𝐹 (
𝑠
𝑐
)
25 𝑢 𝑐 𝑡 = 𝑢 𝑡 − 𝑐 𝑒−𝑐𝑠
𝑠
26 𝛿 𝑡 − 𝑐 𝑒−𝑡𝑠
27 𝑢 𝑐 𝑡 𝑓 𝑡 − 𝑐
Heaviside Function
𝑒−𝑐𝑠
𝐹 𝑠 28 𝑢 𝑐 𝑡 𝑔 𝑡
Dirac Delta Function
𝑒 𝑐𝑠
𝐿{𝑔 𝑡 + 𝑐 }
29 𝑒 𝑐𝑡
𝑓 𝑡 𝐹 𝑠 − 𝑐 30 𝑡 𝑛
𝑓 𝑡 , 𝑛 = 1,2,3, … −1 𝑛
𝐹 𝑛
𝑠
31 1
𝑡
𝑓 𝑡 ∫ 𝑓 𝑢 𝑑𝑢
∞
𝑠
32
∫ 𝑓 𝑣 𝑑𝑣
𝑡
0
𝐹 𝑠
𝑠

24 | P a g e
33
∫ 𝑓 𝑡 − 𝜏 𝑔 𝜏 𝑑𝜏
𝑡
0
𝐹 𝑠 𝐺 𝑠 34 𝑓 𝑡 + 𝑇 = 𝑓 𝑡 ∫ 𝑒−𝑠𝑡
𝑓 𝑡 𝑑𝑡
𝑇
0
1 − 𝑒−𝑠𝑇
35 𝑓′
𝑡 𝑠𝐹 𝑠 − 𝑓 0 36 𝑓′′
𝑡 𝑠2
𝐹 𝑠 − 𝑠𝑓 0 − 𝑓′
0
37 𝑓 𝑛
𝑡 𝑠 𝑛
𝐹 𝑠 − 𝑠 𝑛−1
𝑓 0 − 𝑠 𝑛−2
𝑓′
0 − ⋯ − 𝑠𝑓 𝑛−2
0 − 𝑓 𝑛−1
0
NOTE:
 This is not the complete list of Laplace Transforms, to create a complete version of this
table, which will span a few hundred pages as there are many uses for Laplace Transforms
in Varies fields, is not ethical. This table only contains some of the more commonly used
Laplace Transforms and formulas.
 Formula #4 uses the Gamma function which is defined as,
Γ 𝑡 = ∫ 𝑒−𝑥
𝑥 𝑡−1
𝑑𝑥
∞
0
If n is appositive Integer then,
Γ 𝑛 + 1 = 𝑛!
The Gamma function is an extension of the normal factorial function. Here are a couple of
quick facts for the Gamma function.
Γ 𝑝 + 1 = 𝑝Γ p
𝑝 𝑝 + 1 𝑝 + 2 … 𝑝 + 𝑛 − 1 =
Γ 𝑝+𝑛
Γ 𝑝
Γ (
1
2
) = √ 𝜋
11) First Order Systems/Circuits
First order Circuits are circuits which contain only one energy storage element such as a
capacitor(C) or an inductor (L).
There are two possible types of First Order Circuits,
1) RC (Resistor and Capacitor)
2) RL (Resistor and Inductor)

25 | P a g e
1) RC (Resistor and Capacitor)
Series:
i
R
C𝑉𝑠
By applying KVL,
𝑉𝑅 + 𝑉𝐶 = 𝑉𝑠 : Equation 1
By using Om’s Law, 𝑉𝑅 = 𝑅𝑖
𝑉𝑅 = 𝑅 (𝐶
𝑑𝑉 𝐶
𝑑𝑡
) : Equation 2
Equation 1 and 2 gives,
𝑅𝐶
𝑑𝑉 𝐶
𝑑𝑡
+ 𝑉𝐶 = 𝑉𝑠
Since Time Constant (𝜏 𝑅𝐶) = RC,
𝜏 𝑅𝐶
𝑑𝑉𝐶
𝑑𝑡

26 | P a g e
Parallel:
By applying KCL,
𝑖 𝑅 + 𝑖 𝐶 = 𝑖 𝑠 : Equation 3
Since this a Parallel circuit, 𝑉𝑅 = 𝑉𝐶 = 𝑉
By applying Ohm’s Law, 𝑖 𝑅 =
1
𝑅
𝑉 : Equation 4
𝑖 𝐶 = 𝐶
𝑑𝑣
𝑑𝑡
: Equation 5
Equation 3,4 and 5 gives,
𝐶
𝑑𝑣
𝑑𝑡
+
1
𝑅
𝑉 = 𝑖 𝑠
𝑑𝑣
𝑑𝑡
+
1
𝑅𝐶
𝑉 =
1
𝐶
𝑖 𝑠
𝑑𝑣
𝑑𝑡
+
1
𝜏 𝑅𝐶
𝑉 =
1
𝐶
𝑖 𝑠
𝑖 𝑅
𝑖 𝐶
𝑖 𝑆
𝑡 𝑠
R C

27 | P a g e
2) RL (Resistor and Inductor)
Series:
By applying KVL,
𝑉𝑅 + 𝑉𝐿 = 𝑉𝑠 : Equation 6
BY using Om’s Law,
𝑉𝑅 = 𝑅𝑖 : Equation 7
𝑉𝐿 = (𝐿
𝑑𝑖 𝐿
𝑑𝑡
) : Equation 8
Equation 6,7 and 8 gives,
𝐿
𝑑𝑖 𝐿
𝑑𝑡
+ 𝑅𝑖 = 𝑉𝑠
𝑑𝑖 𝐿
𝑑𝑡
+
𝑅
𝐿
𝑖 =
𝑉𝑠
𝐿
Since Time Constant (𝜏 𝑅𝐿) =
𝐿
𝑅
,
𝑑𝑖 𝐿
𝑑𝑡
+
1
𝜏 𝑅𝐿
𝑖 =
1
𝐿
𝑉𝑠
𝑉𝑠
i
R
L

28 | P a g e
Parallel:
By applying KCL, 𝑖 𝑅 + 𝑖 𝐶 = 𝑖 𝑠 : Equation 9
By applying Ohm’s Law, 𝑖 𝑅 =
1
𝑅
𝑉𝑅 : Equation 10
Since this is a Parallel Circuit, 𝑉𝑅 = 𝑉𝐿 = 𝑉′ : Equation 11
Equation 8, 10 and 11 gives, 𝑖 𝑅 =
1
𝑅
𝑉𝐿 =
1
𝑅
(𝐿
𝑑𝑖 𝐿
𝑑𝑡
)
Equation 9 gives, 𝑖 𝐿 +
𝐿
𝑅
(
𝑑𝑖 𝐿
𝑑𝑡
) = 𝑖 𝑠
𝑖 𝐿 + 𝜏 𝑅𝐿 (
𝑑𝑖 𝐿
𝑑𝑡
) = 𝑖 𝑠
𝑖 𝑅 𝑖 𝐿
𝑖 𝑆
𝑡 𝑠 R L

29 | P a g e
12) Second Order Systems/Circuits
Second order Circuits are circuits which contain only two energy storage elements such as a
capacitor(C) and an inductor (L).
There are two possible types of Second Order Circuits,
1). Series RLC (Resistor, Inductor and Capacitor) circuit.
2).Parallel RLC (Resistor, Inductor and Capacitor) circuit.
1) Series RLC (Resistor, Inductor and Capacitor) circuit
By applying KVL, 𝑉𝑅 + 𝑉𝐶 + 𝑉𝐿 = 𝑉𝑠 : Equation 12
By rewriting for voltages in terms of loop current,
𝑉𝐿 = 𝐿
𝑑𝑖
𝑑𝑡
𝑉𝑅 = 𝑅𝑖 ; 𝑂ℎ𝑚′
𝑠 𝐿𝑎𝑤
𝑉𝐶 =
1
𝐶
∫ 𝑖 𝜏 𝑑𝜏
𝑡
−∞
By substituting corresponding values for Equation 12,
𝑉𝑠
R
C
L
i

30 | P a g e
𝑅𝑖 +
1
𝐶
∫ 𝑖 𝜏 𝑑𝜏
𝑡
−∞
+ 𝐿
𝑑𝑖
𝑑𝑡
= 𝑉𝑠
The integral is a problem, so by taking the time derivative of every term,
𝑅
𝑑𝑖
𝑑𝑡
+
1
𝐶
𝑖 + 𝐿
𝑑2
𝑖
𝑑𝑡2
=
𝑑𝑉𝑠
𝑑𝑡
𝑑2
𝑖
𝑑𝑡2
+
𝑅
𝐿
𝑑𝑖
𝑑𝑡
+
1
𝐿𝐶
𝑖 =
1
𝐿
𝑑𝑉𝑠
𝑑𝑡
𝑆𝑖𝑛𝑐𝑒 𝜏 𝑅𝐿 =
𝐿
𝑅
𝑎𝑛𝑑 𝜔0
2
=
1
𝐿𝐶
,
𝑑2
𝑖
𝑑𝑡2
+
1
𝜏 𝑅𝐿
𝑑𝑖
𝑑𝑡
+ 𝜔0
2
𝑖 =
1
𝐿
𝑑𝑉𝑠
𝑑𝑡
2) Parallel RLC (Resistor, Inductor and Capacitor) Circuit
By applying KCL, 𝑖 𝐿𝐶 + 𝑖 𝑅 = 𝑖 𝑆 𝑎𝑛𝑑 𝑖 𝐿 + 𝑖 𝐶 = 𝑖 𝐿𝐶 , combined give 𝑖 𝐿 + 𝑖 𝑅 + 𝑖 𝐶 = 𝑖 𝑠 ,
By writing Currents in terms of Voltage,
𝑖 𝐿 =
1
𝐿
∫ 𝑉𝐿 𝜏 𝑑𝜏 , 𝑖 𝑅 =
1
𝑅
𝑉𝑅 , 𝑖 𝐶 = 𝐶
𝑑𝑉𝐶
𝑑𝑡
𝑡
−∞
By substituting corresponding relationships to the derived equation,
1
𝐿
∫ 𝑉𝐿 𝜏 𝑑𝜏 +
1
𝑅
𝑉𝑅 + 𝐶
𝑑𝑉𝐶
𝑑𝑡
𝑡
−∞
= 𝑖 𝑠
𝑖 𝐿
𝑖 𝐶𝑖 𝐿𝐶
𝑖 𝑅
𝑖 𝑠
𝑡 𝑠 R L C

31 | P a g e
Since the circuit is Parallel, 𝑉𝐿 = 𝑉𝑅 = 𝑉𝐶 = 𝑉′′
The Integral is a problem, therefore by taking the time derivative of every term,
𝐶
𝑑2
𝑉′′
𝑑𝑡2
+
1
𝑅
𝑑𝑉′′
𝑑𝑡
+
1
𝐿
𝑉′′ =
𝑑𝑖 𝑠
𝑑𝑡
𝑑2
𝑉′′
𝑑𝑡2
+
1
𝑅𝐶
𝑑𝑉′′
𝑑𝑡
+
1
𝐿𝐶
𝑉′′
=
1
𝐶
𝑑𝑖 𝑠
𝑑𝑡
𝑆𝑖𝑛𝑐𝑒 𝜏 𝑅𝐶 = 𝑅𝐶 𝑎𝑛𝑑 𝜔0
2
=
1
𝐿𝐶
,
𝑑2
𝑉′′
𝑑𝑡2
+
1
𝜏 𝑅𝐶
𝑑𝑉′′
𝑑𝑡
+ 𝜔0
2
𝑉′′
=
1
𝐶
𝑑𝑖 𝑠
𝑑𝑡
Summary
Table 2-First order circuits Summary
Circuit Differential Equation form
First order series RC circuit
𝜏 𝑅𝐶
𝑑𝑉𝐶
𝑑𝑡
First order Parallel RC circuit 𝑑𝑣
𝑑𝑡
+
1
𝜏 𝑅𝐶
𝑉 =
1
𝐶
𝑖 𝑠
First order Series RL circuit 𝑑𝑖 𝐿
𝑑𝑡
+
1
𝜏 𝑅𝐿
𝑖 =
1
𝐿
𝑉𝑠
First order Parallel RL circuit
𝑖 𝐿 + 𝜏 𝑅𝐿
𝑑𝑖 𝐿
𝑑𝑡
= 𝑖 𝑠
First Order General Form 𝑑𝑥
𝑑𝑡
+
1
𝜏
𝑥 = 𝑓 𝑡

32 | P a g e
Table 3-Second order Circuits summary
Circuit Differential Equation form
Second order Series RLC circuit 𝑑2
𝑖
𝑑𝑡2
+
1
𝜏 𝑅𝐿
𝑑𝑖
𝑑𝑡
+ 𝜔0
2
𝑖 =
1
𝐿
𝑑𝑉𝑠
𝑑𝑡
Second order Parallel RLC circuit 𝑑2
𝑉′′
𝑑𝑡2
+
1
𝜏 𝑅𝐶
𝑑𝑉′′
𝑑𝑡
+ 𝜔0
2
𝑉′′
=
1
𝐶
𝑑𝑖 𝑠
𝑑𝑡
Second Order General Form 𝑑2
𝑥
𝑑𝑡2
+
1
𝜏
𝑑𝑥
𝑑𝑡
+ 𝜔0
2
𝑥 = 𝑓 𝑡
Table 4-Important relationships
Time constants
𝜏 𝑅𝐿 =
𝐿
𝑅
𝜏 𝑅𝐶 = 𝑅𝐶
Natural frequency
𝜔0
2
=
1
𝐿𝐶

33 | P a g e
13) Reponses of RLC Circuits
The Response of a RLC circuit is of utmost importance and is, particularly a very interesting
matter. The response of the circuit depends on a number of factors which will be discussed below.A
Resistor-Inductor-Capacitor circuit is an electric circuit composed of a set of resistors, inductors,
or capacitors and driven by a voltage or current. This type of circuit forms a harmonic oscillator
for current, meaning that when the circuit is displaced from its steady-state or equilibrium position
it experiences a resulting force proportional to the displacement. As such the circuit resonates
between two frequencies which slowly decay as a result of the resistance in the circuit. Many
engineering applications calls for a configuration which quickly settles at an equilibrium current
or quickly reaches the so called Steady-State or system stability. System stability is a very
important factor because system stability is directly related with the efficiency of a system.
Natural Response of a simple RC circuit
Figure 9- Simple RC circuit
The differential equation of the above system, 𝐶
𝑑𝑉
𝑑𝑡
+
𝑉
𝑅
= 0.
The time constant of the system, 𝜏 = 𝑅𝐶.
When an RC or RL circuit has reached a constant voltage and current and is disconnected from a
power source it reaches a state called zero input response or natural response. The resulting
reaction of the system for an RC circuit is the function:

34 | P a g e
𝑉 𝑡 = 𝑉0 𝑒−
𝑡
𝑅𝐶
Natural Response of a simple RL circuit
Figure 10- Simple RL circuit
The differential equation of the above system, 𝐿
𝑑𝑖
𝑑𝑡
+ 𝑅𝑖 = 0.
The time constant of the system, 𝜏 =
𝐿
𝑅
.
When an RC or RL circuit has reached a constant voltage and current and is disconnected from a
power source it reaches a state called zero input response or natural response. The resulting
reaction of the system for an RL circuit:
𝑖 𝑡 = 𝑖0 𝑒−
𝑅
𝐿
𝑡
: 𝑖0 = 𝑖𝑛𝑡𝑖𝑎𝑙 𝑖𝑛𝑑𝑢𝑐𝑡𝑜𝑟 𝑐𝑢𝑟𝑟𝑒𝑛𝑡
Response of RLC circuits/Transient Response
Transient response is the response of a system to a change in its equilibrium or steady state. The
primary factor in determining how a circuit will react to this change is called the damping factor,
which is represented by the Greek letter zeta ( 𝜁 ). In series RLC circuits the damping factor is
defined mathematically by:
Figure 11-RLC series

35 | P a g e
𝜁 =
𝑅
2
√
𝐶
𝐿
And in parallel RLC circuits the damping factor is defined mathematically by:
Figure 12-RLC parallel
𝜁 =
1
2𝑅
√
𝐿
𝐶
Values of the damping factor can be broken down into four key classifications:
 Undamped : 𝜻 = 𝟎
 Underdamped : 𝜻 < 𝟏
 Overdamped : 𝜻 > 𝟏
 Critically Damped : 𝜻 = 𝟏
As seen in the graph shown below, these different damping factors have varying effects on the
circuit's current over time. When designing a circuit, many engineering applications call for a
critically damped damping factor due to its fast oscillation correction, only essentially containing
one current peak before quickly settling at an equilibrium current.

36 | P a g e
Figure 13-RLC Damping
Detailed explanation on Damping:
Figure 14-RLC Damping

37 | P a g e
Consider a series RLC circuit (one that has a resistor, an inductor and a capacitor) with a constant
driving electro-motive force (emf) E. The current equation for the circuit is:
𝐿
𝑑𝑖
𝑑𝑡
+ 𝑅𝑖 +
1
𝐶
∫ 𝑖 𝑑𝑡 = 𝐸
𝐿
𝑑𝑖
𝑑𝑡
+ 𝑅𝑖 +
1
𝐶
𝑞 = 𝐸 :{∫ 𝑖 𝑑𝑡 = 𝑞 ∶ 𝑤ℎ𝑒𝑟𝑒 𝑖 = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑎𝑛𝑑 𝑞 = 𝑐ℎ𝑎𝑟𝑔𝑒 }
By Differentiating with respect to t,
𝐿
𝑑2
𝑖
𝑑𝑡2
+ 𝑅
𝑑𝑖
𝑑𝑡
+
1
𝐶
𝑖 = 0
This is a second order linear homogeneous equation.
Its corresponding auxiliary equation is:
𝐿𝑚2
+ 𝑅𝑚 +
1
𝐶
= 0
With Roots ;
𝑚1 =
−𝑅
2𝐿
+
√ 𝑅2 −
4𝐿
𝐶
2𝐿
= −𝛼 + √𝛼2 − 𝜔0
2
𝑚2 =
−𝑅
2𝐿
−
√ 𝑅2 −
4𝐿
𝐶
2𝐿
= −𝛼 − √𝛼2 − 𝜔0
2
Where,
𝛼 =
𝑅
2𝐿
Is called the damping coefficient of the circuit.
𝜔0 = √
1
𝐿𝐶
Is the Resonant frequency of the circuit.
𝑚1 And 𝑚2 are called the natural frequencies of the circuit.
The nature of the current will depend on the relationship between R, L and C.

38 | P a g e
Case 1 : Overdamped 𝑹 𝟐
> 𝟒𝑳/𝑪
Figure 15-graph of overdamped case
Here both m1 and m2 are real, distinct and negative. The general solution is given by,
𝑖 𝑡 = 𝐴𝑒 𝑚1 𝑡
+ 𝐵𝑒 𝑚2 𝑡
The motion (current) is not oscillatory, and the vibration returns to equilibrium.
Case 2 : Critically Damped 𝑹 𝟐
= 𝟒𝑳/𝑪
Figure 16-graph of critically damped case
Here the roots are negative, real and equal,
𝑖. 𝑒. 𝑚1 = 𝑚2 =
−𝑅
2𝐿

39 | P a g e
The general solution is given by,
𝑖 𝑡 = 𝐴 + 𝐵𝑡 𝑒
−𝑅𝑡
2𝐿
The vibration (current) returns to equilibrium in the minimum time and there is just enough
damping to prevent oscillation.
Case 3 : Underdamped 𝑹 𝟐
< 𝟒𝑳/𝑪
Figure 17-graph of underdamped case
Here the roots are complex where,
𝑚1 = 𝛼 + 𝑗𝜔, 𝑎𝑛𝑑 𝑚2 = 𝛼 − 𝑗𝜔.
The general solution is given by,
𝑖 𝑡 = 𝑒−𝛼𝑡
𝐴 cos 𝜔𝑡 + 𝐵 sin 𝜔𝑡
Where,
𝛼 =
𝑅
2𝐿
is called the Damping coefficient, and 𝜔 is given by,
𝜔 = √
1
𝐿𝐶
−
𝑅2
4𝐿2
In this case, the motion (current) is oscillatory and the amplitude decreases exponentially, bounded
by,

40 | P a g e
𝑖 = ±√ 𝐴2 + 𝐵2 𝑒
−𝑅𝑡
2𝐿
as we can see in Figure 17 .
When R = 0, the circuit displays its natural or resonant frequency, 𝜔0 = √
1
𝐿𝐶
.
Case 4 : Undamped 𝜻 = 𝟎
An 'undamped' capacitive-discharge is an RLC circuit where resistance R = 0. Although it's not
possible to build one, it is very useful because it represents the limiting case of a very high-Q
(charge) circuit.
Interesting comparison:
The equation for maximum current peak in a critically damped case can be shown as,
𝑖max 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 =
𝑉0
𝑒
√
𝐶
𝐿
The equation for maximum current peak in an undamped case can be shown as,
𝑖max 𝑢𝑛𝑑𝑎𝑚𝑝𝑒𝑑 = 𝑉0√
𝐶
𝐿
We can compare the results by finding the ratio of peak current from each case:
𝑖max 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙
𝑖max 𝑢𝑛𝑑𝑎𝑚𝑝𝑒𝑑
=
𝑉0
𝑒
√ 𝐶
𝐿
𝑉0√ 𝐶
𝐿
=
1
𝑒
= 36.8%
There are two interesting results:
 A critically-damped RLC circuit will deliver a current peak that is only 36% of the
maximum possible peak from an undamped circuit. This is a very heavy penalty for the
convenience of a simple circuit.
 It is somewhat amazing to see a simple result from a complicated analysis. The ratio of 1/e
was completely unexpected.

41 | P a g e
Figure 18-Critical Vs Undamped
Properties of RLC networks
 Behavior of RLC network is described as damping, which is a gradual loss of the initial
stored energy
 The resistor R causes the loss
 𝛼 Determined the rate of the damping response.
 If R = 0, the circuit is loss-less and energy is shifted back and forth between the inductor
and capacitor forever at the natural frequency.
 Oscillatory response of a loss causing RLC network is possible because the energy in the
inductor and capacitor can be transferred from one component to the other.
 Underdamped response is a damped oscillation, which is called ringing.
 Critically damped circuits reach the final steady state in the shortest amount of time as
compared to overdamped and underdamped circuits.
 However, the initial change of an overdamped or underdamped circuit may be greater than
that obtained using a critically damped circuit.

42 | P a g e
14) Power Factor and RMS value of Complex Periodic Waveforms
Complex Periodic Waveforms
Complex Periodic Waveforms are a combination of sine waves of various frequencies, amplitude,
and phase.
Figure 19 - Basic sinusoidal a
Figure 20-Basic sinusoidal b
The combination of the wave forms in Figure 19 and 20 forms the complex periodic wave form in
Figure 21.
Figure 21-Complex periodic wave form

43 | P a g e
Analysis of complex periodic waveforms (Fourier Analysis)
Fourier Analysis:
Fourier Analysis, named after the nineteenth century French mathematician Jean Baptiste Fourier,
enables one to break down complex periodic waveforms into their basic components, which
happen to be sine waves of various frequencies, amplitudes, and phases. The opposite method,
combining sine waves of various frequencies, amplitude, and phase to create complex periodic
waveforms, is Fourier Synthesis.
The Fourier theorem states that any periodic waveform can be synthesized by adding appropriately
weighted sine and cosine terms of various frequencies.
Let our periodic function be 𝑓 𝑡 = 𝑓 𝑡 ± 𝑛𝑇 where T is the time of one period and n is an
integer. And the Fundamental angular frequency: 𝜔0 =
2𝜋
𝑇
By the Fourier theorem, the periodic function can be written as the following sum:
𝑓 𝑡 = 𝐴0 + ∑ 𝐴 𝑛 sin 𝑛𝜔0 𝑡 + 𝐵𝑛 cos 𝑛𝜔0 𝑡
𝑛
Where,
𝐴0 =
1
𝑇
∫ 𝑓 𝑡 𝑑𝑡, 𝐴 𝑛 =
𝑡1+𝑇
𝑡1
2
𝑇
∫ 𝑓 𝑡
𝑡1+𝑇
𝑡1
sin 𝑛𝜔0 𝑡 𝑑𝑡 𝑎𝑛𝑑 𝐵𝑛 =
2
𝑇
∫ 𝑓 𝑡 cos 𝑛𝜔0 𝑡 𝑑𝑡
𝑡1+𝑇
𝑡1
𝐴 𝑛 and 𝐵𝑛 are the Fourier coefficients and the sum is the Fourier series.
𝑨 𝟎 in Fourier Analysis is the Average of any periodic waveform and can be used directly.
RMS value of complex periodic wave forms.
The RMS or the Root Mean Square value for a complex periodic wave form is given by,
𝑋𝑟𝑚𝑠 = √
1
𝑇
∫ 𝑋 𝑡 2 𝑑𝑡.
𝑇
0

44 | P a g e
Power Factor of complex periodic wave forms
What is power Factor?
For an easy explanation of the physical meaning of power factor (PF), let's consider a simple
example. Suppose you connect an ideal capacitor "C" across an AC line. Since AC voltage is
continuously changing its polarity, half of the time the cap will be charging from the input and half
of the time it will be returning the stored energy back to the source. As a result, certain current will
be continuously circulating in the line, but there will be no net energy transfer. So, we will have
both volts and amps, but not useful power.
The volt-amp product "VA" is called apparent power because it is just a mathematical quantity
which has no real physical meaning. A single ideal inductor will result in PF=0, except its current
will lag the voltage. Let's now consider the opposite extreme case of a resistive load. In such a
case, all the electric energy entering the load is consumed and converted to other forms of energy,
such as heat. This is an example of PF=1. All real circuits operate somewhere between these two
extremes.
We see that PF is a quantity that basically tells us how effectively your device utilizes electricity.
In electrical terms, it's defined as a ratio between working power P (watts) and apparent power S
(volt-amps):
𝑃𝑜𝑤𝑒𝑟 𝐹𝑎𝑐𝑡𝑜𝑟 =
𝑃
𝑆
=
𝑊𝑎𝑡𝑡𝑠
𝑉𝑟𝑚𝑠. 𝐼𝑟𝑚𝑠
In circuits analysis a sine wave signal can be represented by a complex number (called a phasor)
whose modulus is proportional to the magnitude of the signal and angle equals to its phase relative
to some reference. In linear circuits PF= cosφ, where φ is angle between voltage and current
phasors. These vectors and corresponding ones for active and reactive components of power can
be presented by a triangle (see the diagram below). Of course, voltage is an electric field and
current is a flow of electrons, so the so-called angle between their phasors is nothing more than
another mathematical quantity. By convention, inductive load create positive volt-amp reactive

45 | P a g e
(VARs). It is associated with so-called "lagging" power factor because current is behind the
voltage. Likewise, capacitive load create negative VARs and "leading" power factor.
Figure 22-Power Triangle
Inductors and caps are not the only reasons of low PF. Except for ideal R, L and C, all practical
electrical circuits are non-linear, particularly because of the presence of active components, such
as rectifiers. In such circuits current I(t) is not proportional to the voltage V(t), i.e. even if V(t) is
a pure sinusoid, I(t) will be periodic but non-sinusoidal. According to Fourier theorem, any
periodic function is a sum of sine waves with frequencies that are whole multiples of the original
one. These waves are called harmonics. It can be shown that they don't contribute to net energy
transfer, but increase Irms and reduce PF. When voltage is sinusoidal, only first harmonic I1rms
can deliver real power. However its amount depends on the phase shift between I1rms and V. These
facts are reflected by the general formula for PF:
𝑃𝐹 =
𝐼1𝑟𝑚𝑠
𝐼𝑟𝑚𝑠
× cos 𝜑
Another detailed formula can be obtained by substituting corresponding relationships in the
generic power factor formula PF=P/S,
𝑃𝐹 =
∫ 𝑉 𝑡
𝑇
0
𝑖 𝑡 𝑑𝑡
√∫ 𝑉 𝑡 2 𝑑𝑡
𝑇
0
× √∫ 𝐼 𝑡 2 𝑑𝑡
𝑇
0
By simplifying this formula using Fourier Analysis we can obtain the above simplified version of
the formula,
𝑃𝐹 =
𝐼1𝑟𝑚𝑠
𝐼𝑟𝑚𝑠
× cos 𝜑

46 | P a g e
15) Conclusion
This research provides a vast knowledge on RL, RC and RLC circuit analysis. This proves the
importance of Laplace transforms in Circuit analysis. And the circuit responses which by proper
analysis can be used to optimize systems to be more stable and efficient which is the main output
of this research.

47 | P a g e
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50 | P a g e
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