American Journal of Circuits, Systems and Signal Processing

Vol. 5, No. 1, 2019, pp. 1-8

http://www.aiscience.org/journal/ajcssp
ISSN: 2381-7194 (Print); ISSN: 2381-7208 (Online)

Application of Linear Differential Equation in an

Analysis Transient and Steady Response for
Second Order RLC Closed Series Circuit
Ahammodullah Hasan1, *, Md. Abdul Halim2, Md. Al-Amin Meia3
1
Department of Mathematics, Faculty of Applied Science and Technology, Islamic University, Kushtia, Bangladesh
2
Department of Electrical and Electronic engineering, World University of Bangladesh, Dhaka, Bangladesh
3
Department of Mathematics, Govt. Adamjeenagar Merchant Worker (M. W) College, Narayanganj, Bangladesh

Abstract
In this Paper, this work investigates the application of RLC diagrams in the catena study of linear RLC closed series electric
circuits. The Relevant second order ordinary differential equations were solved by Kirchhoff’s Voltage law. This solution
obtained was employed to procedure RLC diagram simulated by MATLAB and Mathematica 9.0. A circuit containing an
inductance L or capacitor C and resistor R with current and voltage variable given by differential equation. The general
solution of differential equation has two parts complementary function (C. F) and particular integral (P. I) in which C. F.
represents transient response and P. I. represents steady response. The general solution of differential equation represents the
complete response of network. In this connection, this paper includes RLC circuit and ordinary differential equation of second
order and its solution.

Keywords
Circuit Analysis, RLC Circuit, Ordinary Differential Equation, Transient Response, Steady Response, Kirchhoff’s Law

Received: November 9, 2018 / Accepted: January 7, 2019 / Published online: January 27, 2019
@ 2018 The Authors. Published by American Institute of Science. This Open Access article is under the CC BY license.
http://creativecommons.org/licenses/by/4.0/

1. Introduction
Nonlinear dynamic study of systems governed by equations attractive response than RC or RL circuit. An equation which
in which a small change in one variable can induce a large involves differential coefficient is called differential equation
systematic change is known as chaos. Dislike a linear system, [2-5]. A differential equation involving derivatives with
in which a small change in one variable produces a small and respect to single independent variable is called ordinary
easily quantifiable systematic change, a nonlinear system differential equation and involving partial derivatives with
exhibits a sensitive dependence on initial conditions. Chaotic respect to more than one independent variable is called
behaviour has been observed in nonlinear electric circuits, partial differential equation [6]. The inter-connection of
oscillating system, chemical reaction, magneto-mechanical simple electric device in which there is at least one closed
devices, weather and climate etc. Ajide et al., (2011) [1]. path for current to flow is called electric circuit. The circuit is
RLC Circuit is extensively used in a diversity of applications switch from one condition to another by change in the
such as filters in communications systems, surmise systems applied source or a change in the circuit elements there is a
in automobiles, defibrillator circuits in biomedical transition period during which the branch current and voltage
applications etc. The RLC circuit has much portly and changes from their former values to new ones. This period is

* Corresponding author
E-mail address:
2 Ahammodullah Hasan et al.: Application of Linear Differential Equation in an Analysis Transient and Steady
Response for Second Order RLC Closed Series Circuit

called transient [7-10]. After the transient has passed the (3)
circuit is said to be steady state. The linear differential
equation that describes the circuit will have two parts to its Where , , a constant of proportionality is called the
solution the complementary function corresponds to the resistance, inductance, capacitance, and is the current. Since
transient and the particular solution corresponds to steady , this often can written
state. The v-i relation for an inductor or capacitor is a
differential [11-13]. A circuit containing an inductance L or a ! (4)
capacitor C and resistor R with current and voltage variable
given by differential equation of the same form. It is a linear Table 1. Elements symbol and units of measurements.
second order differential equation with constant coefficient S. No Element Symbol Unit
when the value of R, L, C are constant. L and C are storage 1. Voltage V volt
2. Current I ampere
elements. Circuit has two storage elements like one L and
3. Charge Q coulomb
one C are referred to as second order circuit. Therefore, the 4. Resistance R ohm
series or parallel combination of R and L or R and C are first 5. Inductance L henry
order circuit and LRC in series or parallel are second order 6. Capacitance C farad

circuit. The circuit changes are assumed to occur at time t=0 The fundamental law in the study of electric circuits is the
and represented by a switch [14-16]. The switch may be following:
supposed to closed (on) and open (off) at t=0. The order of
differential equation represent derivatives involve and is 2.1. Kirchhoff’s Voltage Law (Form 1)
equal to the number of energy storing elements and The algebraic sum of the instantaneous voltage drops around
differential equation considered as ordinary. The differential a close circuit in a specific direction is zero.
equation that formed for transient analysis will be linear
Since voltage drops across resistors, inductors and capacitors
ordinary differential equation with constant coefficient. The
have the opposite sign from voltages arising from
value of voltage and current during the transient period are
electromotive force, we may state this law in the following
known as transient response. The C. F. of differential
alternative from.
equation represents the transient response. The value of
voltage and current after the transient has died out are known 2.2. Kirchhoff’s Voltage Law (Form 2)
as steady state response [17-20]. The P. I. of differential
equation represents the steady state response. The complete The sum of the voltage drops across resistors, indicators and
or total response of network is the sum of the transient capacitors is equal to the total electromotive force in a closed
response and steady state response which is represented by circuit.
general solution of differential equation. The value of voltage
and current that result from initial conditions when input
function is zero are called zero input response [21]. The
value of voltage and current for the input function which is
applied when all initial condition are zero called zero state
response. In this Study, we sketch the entire figure by using
and 9.0.

2. Data and Methods

The formation of differential equation for an electric circuit
depends upon the following laws.
The voltage drop across a resistor is given by

(1) Figure 1. Initial RLC Circuit Diagram.

The voltage drop across an inductor is given by
Applying by Kirchhoff’s law to the circuit of Figure 1.
Letting " denoted the electromotive force and using the law
(2)
(1), (2) & (3) for voltage drops that were given above, then it
The voltage drop across a capacitor is given by becomes the equation of the form is
 American Journal of Circuits, Systems and Signal Processing Vol. 5, No. 1, 2019, pp. 1-8 3

# # (5) This is a second-order linear differential equation in the

single dependent variable . Thus it have the two second-
This equation contains two dependent variable $! . order linear differential equations (7) and (8) for the charge
However, it recall that these two variables are related to each and current respectively. Further observe that in two very
other by the equation simple cases the problem reduces to first-order liner
differential equation. If the circuit contains no capacitor,
(6) equation (5) itself reduce directly to RL series circuit.

#
Using equation (6) may eliminate from equation (5) and
(9)
write it in the from
% If it solves this equation (9) then getting the solution is of the
% # #
&
(7) form
( ),-
)1 + /0
Equation (7) is a second-order linear differential equation in .
the single dependent variable . On the other hand, if it
(10)

differentiates equation (5) and write Here the graph of the equation (10) is
% '
% # # (8)
&

( ),-
Figure 2. The graph of )1 + . /0 .

The plot shows the transition period during which the current ! 1
( #
adjusts from its initial value of zero to final value , which is !
steady state. 1 # ! (11)
And while if no inductor is present, Equation (7) reduce to
RC series circuit. Differentiating equation (11) with respect to then it will be
obtain
4 Ahammodullah Hasan et al.: Application of Linear Differential Equation in an Analysis Transient and Steady
Response for Second Order RLC Closed Series Circuit

# 0 (12) The function has an exponential decay shape as shown in the

graph. The current stops flowing as the capacitor becomes
Solving this equation then the equation is fully charged.
( 0
,-2
(13)

( 0
,-2
Figure 3. The graph of , an exponential decay curve.

Before considering example, we observe an interesting and The second order RLC circuits drew using linear
useful analogy. The differential equation (7) for the charge is technology spice software as shown in figure 4. The
exactly the same as the differential equation (8). resistor value is selected at 2 ohm, capacitor at ,
345
inductor at 0.1 Henry and electromotive force
3. An Example 100 sin 60 . If the initial current and initial charge on
the capacitor is both zero then find the charge on the
capacitor at any time 1 0.
Formulation 1: By directly applying Kirchhoff’s law; Let
denoted the current and the charge on the capacitor at time
t. the total electromotive force is100 : $ 60 . Using the
voltage drop laws (1), (2) and (3) then the equation is of the
form:

1 !
# 2 # 260 100 sin 60
10 !

since , this reduce to

%
% #2 # 260 100 sin 60 (14)
Figure 4. RLC Circuit Diagram. 5
 American Journal of Circuits, Systems and Signal Processing Vol. 5, No. 1, 2019, pp. 1-8 5

Formulation 2: Applying equation (7) for the charge it auxiliary equation

becomes , 2, $! 100 sin 60 . 3
# 20 # 2600 0
5 345

The roots of this equation are -10 ±50 and so the

substituting these values directly into equation (7) it again
obtain equation (14) at once. Multiplying equation (14)
complementary function of equation (15) is
through by 10, now considering the differential equation of
the form is &
, 5
) sin 50 # 3 cos 50 / (18)
%
# 20 # 2600 1000 sin 60
Employing the method of undermined coefficients to find a
% (15)
particular integral of (15), then it becomes

) sin 60 # cos 60 /
Since the charge is initially zero, so the first initial
@ (19)
condition is

)0/ 0
Differentiating twice and substituting into equation (15), this
(16)
gives
Since the current is also initially zero and , it takes the +25 +30
$!
second initial condition in the form 61 61
′)0/ 0 (17) And so the general solution of equation (15) is

The homogeneous equation corresponding to (15) has the

3B C5
, 5
) sin 50 # 3 cos 50 / + sin 60 + cos 60 (20)
4 4
DE
FG HIJ0 [), 5 I ,B5 % / LMN B5 O)B5 I , 5 % / PQL B5 ]
D0
HISJJ IUJJ V (21)
PQL 45 O LMN 45
TI TI

Applying condition (16) to equation (18) and condition (17) to equation (19), this gives

36 30
$! 3
61 61
Thus the solution of the problem is
4G HIJ0 B
)6 sin 50 # 5 cos 50 / + )5 sin 60 # 6 cos 60 / (22)
4 4

Or,
4√4 B√4
, 5
X: )50 + ∅/ + cos)60 + Z/ (23)
4 4

B 4 4 B
Where cos ∅ , sin ∅ $! cos Z , sin Z . From these equations we determine
√4 √4 √4 √4
∅ ≈ 0.88 )] ! $:/ $! Z ≈ 0.69 )] ! $:/. Thus our solution is given approximately by

0.77 , 5
cos)50 + 0.88/ + 0.64 cos)60 + 0.69/ (24)
0.88 ] ! indicates that the electromotive force leads the
4. Result and Discussion steady state current by approximately Xb c X$ )24/.
This term mention equation (19), and its indicates the
The first term in the above solution clearly becomes
Figure 6. The Figure 6 and Figure 7 are represent actual and
negligible after a relatively short time, it is the transient
approximation sketch of steady part, and finally we see that
term. The transient term is equation (18), its indicates the
the two graphs are both matching. The graphs of these two
Figure 5. After a sufficient time essentially all that remains
components and that of their sum (the complete solution)
is the periodic second term; this is the steady-state term.
Observe that its period `/20 is the same as that of
are shown in figure bellow:
electromotive force. However, the phase angle ∅ ≈
6 Ahammodullah Hasan et al.: Application of Linear Differential Equation in an Analysis Transient and Steady
Response for Second Order RLC Closed Series Circuit

Figure 5. The charge & when time t=0 and t=0.5.

Figure 6. The charge @ when time t=0 and t=0.5.

American Journal of Circuits, Systems and Signal Processing Vol. 5, No. 1, 2019, pp. 1-8 7

Figure 7. The charge actual result when time t=0 and t=0.5.

Figure 8. The charge approximate result when time t=0 and t=-0.5.

Figure 9. The charge Comparison between actual and approximate result when time t=0 & t=0.5.
8 Ahammodullah Hasan et al.: Application of Linear Differential Equation in an Analysis Transient and Steady
Response for Second Order RLC Closed Series Circuit

pp. 321-322, 2012.

5. Conclusion
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[9] J. D. Irwin, “Basic Engineering Circuit Analysis” 5th edition,
somewhat familiar with these items. We have been simply
Prentice Hall, Upper Saddle River, NJ, 1996
recalled that the electromotive force produces a flow of
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