BIBECHANA 18 (2) (2021) 9-17 

9 
 

 

 

Visualization, formulation and intuitive explanation of 

iterative methods for transient analysis of series RLC circuit  

 

Jeevan Kafle1, Bhogendra Kumar Thakur1, Indra Bahadur Bhandari2,3,* 

1Central Department of Mathematics, Tribhuvan University, Kirtipur, Nepal 
2Central Department of Physics, Tribhuvan University, Kirtipur, Nepal 
3Department of Applied Sciences, Purwanchal Campus, Tribhuvan University, Dharan, Nepal 

*E-mail: bhandari.indra@gmail.com 
 

 

Article Information: 

Received: September 14, 2020 

Accepted: November 08, 2020 

 

Keywords:  

Iterative methods 

Transient analysis 

Series RLC circuit 

Damping Factor 

Error analysis and DC source. 
 

 

 

 

 

 

 

 

 

ABSTRACT 

The time varying currents and voltages resulting from the sudden application of 

sources usually due to switching are transients. An RLC circuit is an electrical 

circuit consisting of a resistor, an inductor, and a capacitor, connected in series or 

in parallel. The transient response is dependent on the value of the different 

characteristics of damping factor (i.e., over damped, critically damped and under 

damped). We have computed the numerical solutions of second order differential 

equation with initial value problem (IVP) by using Explicit (Forward) Euler 

method, Third order Runge-Kutta (RK3) methods and Butchers fifth order Runge-

Kutta (BRK5). The observation compares this numerical solution of ODEs 

obtained by above-mentioned methods among them with necessary visualization 

and analysis of the error. These iterative methods will be extended and implement 

to analyze the transient analysis of an RLC circuit.  We have examined the 

superiority of those methods over one another. The Butchers fifth order Runge-

Kutta (BRK5) method is found to be the best numerical technique to solve the 

transient analysis due to its high accuracy of approximations. Moreover, we 

consider the possibility of discussion and analyze above mentioned iterative 

methods in the cases of different characteristics of damping factor.

 

DOI: https://doi.org/10.3126/bibechana.v18i2.31208  

This work is licensed under the Creative Commons CC  BY-NC License. https://creativecommons.org/licenses/by-nc/4.0/ 

  

1. Introduction 

 

While regarding the problem of mathematical 

modelling in the area of engineering, physics, 

aeronautics, medicine, environmental science, 

astronomy, chemistry, biology and many other 

applied fields, the role of differential equation is 

momentous [1, 2]. Meanwhile all of the differential 

model equation in real life appears to be nonlinear. 

On contrary, almost all differential equation 

BIBECHANA 
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Jeevan Kafle et al. / BIBECHANA 18 (2) (2021) 9-17  

10 
 

encountered in course of modelling to be solved 

analytical, is out of the question [1]. Planetary 

motion under gravity, motion of simple pendulum, 

etc. are typical example of second order differential 

equation that won’t solve analytically [2, 3]. To 

wriggle out from such confounding situation one has 

to device a more rigorous alternative approach that 

would be simple and more pragmatic. One of such 

method is numerical method [3]. 
 

Transients are the momentary fluctuation of energy 

induced upon the electrical circuit due to the time 

varying currents and voltages resulting from the 

sudden application of sources. It usually occurs at 

the period of switching [4]. Sensitive component of 

the electrical circuitry such as millions of inductors, 

capacitors, resistors, transistor and diodes embedded 

inside integrated circuits are most vulnerable to this 

brief increase in current in voltage. An electrical 

system under the influence of transient condition 

rendered the system to an unsteady state. Thus, a 

transient analysis is to analyze the response of an 

electrical circuit under these unsteady state 

condition [5]. If the variables do not vary with 

respect to time, then the state of system is said to be 

in a steady state otherwise it is said to be in an 

unsteady state. Performance of any electrical circuit 

is measured by employing the methodology of 

transient analysis [16]. Although transients do occur 

in the most apparent electrical equipment such as 

charger of an android phone, electric mixture, CFL 

bulb, iron etc. but it is not at threatening level 

because components are adjusted enough to handle 

such ailments [7]. But transient occurring in the 

microscopic level inside a microprocessor, 

motherboard and memory card could be lethal to the 

existence of the system. 
 

Suhag (2013) [4] conducted the transient analysis of 

second order RLC circuit and observed the response 

of the system by changing the conditions from one 

steady state value to another. He concluded RK 

method was very efficient in solving second order 

differential equations. Kee and Ranom (2018) [5] 

studied the transient analysis of series RLC circuit 

under underdamped, critically damped and 

overdamped conditions using RK-4 with different 

time step size h. The accuracy of RK-4 method is 

determined by gradually as the step size decimates. 

Thus, the improved RK-4 method by decreasing the 

step size ℎ or increasing the number of steps 𝑛 was 

a favorably suited method for solving transient 

analysis of electric circuit due to its high degree of 

accuracy and efficiency. Henry et al. (2018) [7] 

undertook the task of transient analysis of a second-

order RLC circuit by employing two numerical 

methods viz. Heun’s method and Runge-Kutta 

method. They observed that the Heun’s method 

reaches the stable limit first, thus, converges faster 

but the Runge-Kutta 4th order method proved to be 

more accurate. Thus Runge-Kutta method was 

recommended for transient analysis of complex 

electric circuits.  
 

Ogbuka et al. (2008) [8] in their paper reiterated the 

need of a sound mathematical background and 

knowledge of basic laws of circuit analysis before 

handling the task of circuit analysis and they 

programmed the state equations of a sample 

complex circuit in two simulation software 

MATLAB and SIMPLORER to generate transient 

response. Deshpande (2014) [5] stated the source dc 

voltage applied in any circuit (RLC) constituting of 

energy storage device gives rise to transients and 

therefore estimation of the voltage was of significant 

importance for instrumentation and control purpose. 

Deshpande carried out the transient analysis to this 

circuit by employing Classical and Laplace 

Transformation method. Also, in cases of over 

damped, underdamped and critically damped 

conditions were evaluated significant for natural 

response of RLC series circuit. Laplace transform 

method had proved to be a better tool with respect to 

initialization condition in comparison to classical 

method. Ahamad and Charan (2019) [6] in order to 

solve initial value problem for fourth order ODEs 

presented fifth order Runge-Kutta method (RK5). 

Kamruzzaman and Nath (2018) [11] in their paper 

employed the Modified Euler and Runge-Kutta 

Methods to discuss the numerical solution of an 

ODE with IVP. They carried out the numerical 



Jeevan Kafle et al. / BIBECHANA 18 (2) (2021) 9-17  

11 
 

simulation and compared the numerical solution 

with exact solution. Maffezzoni et al. (2007) [12] 

presented a novel approach to the accurate time-

domain simulation of non-linear circuits that 

employs a class of high-order implicit Runge-Kutta 

(RK) method. The properties of stability and 

accuracy of these RK methods were briefly reviewed 

while the implementation in the flow of an analog 

simulator was described in detail. 
 

In this work, we take a series RLC circuit with DC 

source which can be modelled with a second order 

ODE. For solving IVP of ODE and transient 

analysis, we first compared the three iterative 

method viz. Explicit Euler, Third order Runge-Kutta 

(RK3) and Butcher fifth order Runge-Kutta (BRK5). 

BRK5 method is observed to be the best method as 

it gives the least error compared to other two 

methods. BRK5 method is then chosen to analyze 

the three-damping condition of RLC circuit. 
 

The approach of this paper is primarily analytical 

and comparative among the various numerical 

method for solving differential equation. Section-2 

comprises of the theoretical approach and three 

iterative methods with the necessary numerical 

formulation. Section-3 depicts the simulation result 

and comparison of above-mentioned method. 

Section-4 culminate the conclusion and finding of 

this work.  
 

2. Theory 
 

 

A RLC circuit constitute of three electrical elements 

viz. a resistor, an inductor and a capacitor. It is a 

second order circuit. These circuits are the most 

popular as they are applied to construct oscillator 

and tuners of radio or audio receiver [4]. In the 

adjacent Fig. 1, the electrical elements resistor (R), 

inductor (L) and capacitor (Ϲ) are connected in 

series with the D.C. source (𝑉𝑠), which we call RLC 

series circuit. A circuit equation is drawn to 

undertake the analytical solution part of a transient 

analysis [5]. An ordinary electrical circuitry could 

contain thousands of components. So, any analytical 

solution in such RLC circuits is virtually impossible. 

In such scenario numerical methods could offer a 

great relief to the solution of system. Here, we used 

possible iterative methods to solve the circuit 

equation under the terms of transient analysis in an 

RLC circuit and compare different methods. The 

Kirchhoff’s Voltage Law [5] is applied around the 

loop for 𝑡 > 0 in which the voltage source of the 

circuit in Fig. 1 is calculated by the following:  
 

𝑽𝑺 = 𝑽𝑹 + 𝑽𝑳 + 𝑽𝑪 
      

 

 

Fig. 1: RLC  circuit [7]. 

 

The differential equations of the RLC in Fig. 1 is 

based on method of loop currents where the 

fundamental relationship between the current, and 

the individual circuit elements (Capacitance, 

Inductance, Resistance) are given by   

𝑹𝒆𝒔𝒊𝒔𝒕𝒂𝒏𝒄𝒆 : 𝒗(𝒕) = 𝑹𝒊(𝒕), 

𝑪𝒂𝒑𝒂𝒄𝒊𝒕𝒂𝒏𝒄𝒆 : 𝒊(𝒕) = 𝑪
𝒅𝒗

𝒅𝒕
 , 

𝑰𝒏𝒅𝒖𝒄𝒕𝒂𝒏𝒄𝒆 ∶ 𝒗(𝒕) = 𝑳
𝒅𝒊

𝒅𝒕
 

The second-order differential equation of the RLC 

circuit with constant coefficients is written as [17] 

𝐿
ⅆ2𝑉

ⅆ𝑡2
+ 𝑅

ⅆ𝑉

ⅆ𝑡
+

1

𝐶
𝑉 =

𝑉𝑠

𝐶
         … … …      (1) 

The series RLC circuit is analyzed in order to 

determine its transient characteristics once the 

switch is closed. Equation (1) can be solved using 

different iterative methods [7]. 
 

The damping factor is responsible for amount by 

which the oscillation of a system gradually 

decreases with the time (𝒕).  The transient response 



Jeevan Kafle et al. / BIBECHANA 18 (2) (2021) 9-17  

12 
 

is dependent on the value of the damping factor (𝜻) 

[7]. In RLC circuit, the damping factor is given by  

𝜁 =  
𝛼

𝜔𝑜
 

where, 𝜶 =  
𝑹

𝟐𝑳
 (damping coefficient) and 𝝎𝟎 =

 
𝟏

√𝑳𝑪
  (resonant frequency).  Then 

𝜻 =  
𝑹

𝟐
√

𝑪

𝑳
 

The system is (i) over damped when  𝜻 > 1, (ii) 

critically damped when 𝜻 =1, and (iii) under 

damped when 𝜻 < 1 [4]. 
 

Numerical methodologies 
 

In this section, we discuss different iterative method 

to solve initial value problems (IVP) for ordinary 

differential equations (ODE). First, we introduce 

Euler’s method, the simplest method. We will 

provide details on algorithm development using the 

Euler method (explicit), Third order Ruge-Kutta 

Method (RK3) and Butcher’s fifth order Runge-

Kutta (BRK5) method [12, 13]. 
 

Euler’s method 
 

Consider one dimensional initial value problem 

(IVP)  

𝒚′(𝒕) = 𝒇(𝒕, 𝒚(𝒕)), 𝒚(𝒕𝟎) = 𝒚𝟎  … … … (𝟐) 

    The slope at the point  (𝒕𝒏, 𝒚𝒏) is 

 

𝒚𝒏+𝟏 − 𝒚𝒏 

𝒉
 ≈  

𝒅𝒚

𝒅𝒕
= 𝒇(𝒕𝒏, 𝒚𝒏)  

              ⇒ 𝒚𝒏+𝟏 = 𝒚𝒏

+ 𝒉𝒇(𝒕𝒏, 𝒚𝒏) … … … (𝟑) 

 

This formula (3) is Forward Euler’s Method. 
 

Third order Runge-Kutta (RK3) method 
 

The third order Runge-Kutta method (RK3) is 

broadly used for solving initial value problems (IVP) 

for ordinary differential equation (ODE). The 

general formula for Runge-Kutta third order (RK3) 

method is shown below [15] 

𝒚𝒏+𝟏 =  𝒚𝒏 + 𝒉 [
𝒌𝟏

𝟔
+

𝟐𝒌𝟐

𝟑
+

𝒌𝟑

𝟔
]   … … …  (𝟒) 

 where 

𝑘1 = 𝑓(𝑡𝑛, 𝑦𝑛),     𝑘2 = 𝑓 (𝑡𝑛 +
ℎ

2
, 𝑦𝑛 +

ℎ

2
𝑘1), 

 𝑘3 =  𝑓(𝑡𝑛 + ℎ, 𝑦𝑛 − ℎ𝑘1 + 2ℎ𝑘2) 

 

 

Fig. 2: Visualization of Euler method [14]. 

 

Butchers fifth order Runge-Kutta (BRK5) 

method 

 

This method is distinguished by their order in the 

sense that agrees with Taylors series solution up to 

terms of  ℎ𝑟 where r is the order of the method [7]. 

In this method, we consider the IVP (2). The method 

is shown below [13]. 

𝑦𝑛+1 =  𝑦𝑛 +
ℎ

90
(7𝑘1 + 32𝑘3 + 12𝑘4 + 32𝑘5

+ 7𝑘6)       … … …                     (5) 

where,     

 𝑡𝑛+1 =  𝑡𝑛 + ℎ,   𝑘1 = 𝑓(𝑡𝑛, 𝑦𝑛), 𝑘2 = 𝑓 (𝑡𝑛 +

ℎ

4
, 𝑦𝑛 +

ℎ

4
𝑘1), 

       𝑘3 =  𝑓(𝑡𝑛 +
ℎ

4
, 𝑦𝑛 +

1

8
ℎ𝑘1 +

1

8
ℎ𝑘2),  𝑘4 =

 𝑓(𝑡𝑛 +
ℎ

2
, 𝑦𝑛 −

1

2
ℎ𝑘2 + ℎ𝑘3), 

𝑘5 =  𝑓(𝑡𝑛 +
3ℎ

4
, 𝑦𝑛 +

3

16
ℎ𝑘1 +

9

16
ℎ𝑘4),  𝑘6 =

 𝑓(𝑡𝑛 + ℎ, 𝑦𝑛 −
3

7
ℎ𝑘1 +

2

7
ℎ𝑘2 +

12

7
ℎ𝑘3 −

12

7
ℎ𝑘4 +

8

7
ℎ𝑘5) 

We can extend above discussed iterative methods 

for the solution of higher order initial value 



Jeevan Kafle et al. / BIBECHANA 18 (2) (2021) 9-17  

13 
 

problems (IVP) for ordinary differential equations 

(ODE) and system of initial value problems (IVP) 

for ordinary differential equations (ODE) [3, 16]. 
 

Numerical formulation of RLC circuit 

Let,  

𝑣 = 𝑥 and  
𝑑𝑣

𝑑𝑡
=

𝑦

𝐶
   … … …                 (6) 

then equation (1) becomes 

ⅆ𝑦

ⅆ𝑡
=

𝑉𝑠 − 𝑥 − 𝑅 × 𝑦

𝐿
   … … …          (7) 

Hence equation (6) and (7) form a system of first 

order differential equation [7]. 

    Let,  

𝑓 =  
𝑑𝑣

𝑑𝑡
=

𝑦

𝐶
   and 𝑔 =

𝑑𝑦

𝑑𝑡
=

𝑉𝑠−𝑥−𝑅×𝑦

𝐿
 

 
Euler’s method for RLC circuit 

𝑥𝑖+1 = 𝑥𝑖 +  ℎ 𝑓(𝑡𝑖 , 𝑥𝑖, 𝑦𝑖) 

𝑦𝑖+1 = 𝑦𝑖 + ℎ 𝑔(𝑡𝑖 , 𝑥𝑖 , 𝑦𝑖) 

     where h is the time step size.  
 

RK3 method for RLC circuit 
 

𝑥𝑖+1 = 𝑥𝑖 + (𝑓1 + 4𝑓2 + 𝑓3)/6 

𝑦𝑖+1 = 𝑦𝑖 + (𝑔1 + 4𝑔2 + 𝑔3)/6 

   where 

𝑓1 = ℎ 𝑓(𝑡𝑖  , 𝑥𝑖, 𝑦𝑖),   𝑔1 = ℎ𝑔(𝑡𝑖  , 𝑥𝑖, 𝑦𝑖),  

𝑓2 = ℎ𝑓 ((𝑡𝑖 +
ℎ

2
) , (𝑥𝑖 +

𝑓1

2
) , (𝑦𝑖 +

𝑔1

2
)),                                    

   𝑔2 = ℎ𝑔 ((𝑡𝑖 +
ℎ

2
) , (𝑥𝑖 +

𝑓1

2
) , (𝑦𝑖 +

𝑔1

2
)), 

𝑓3 = ℎ𝑓((𝑡𝑖 + ℎ), (𝑥𝑖 − 𝑓1 + 2𝑓2), (𝑦𝑖 − 𝑔1 +

2𝑔2)) and  

𝑔3 = ℎ𝑔((𝑡𝑖 + ℎ), (𝑥𝑖 − 𝑓1 + 2𝑓2), (𝑦𝑖 − 𝑔1

+ 2𝑔2)) 
 

Formulating BRK5 method for RLC circuit 
 

𝑥𝑖+1 = 𝑥𝑖 + (7𝑓1 + 32𝑓3 + 12𝑓4 + 32𝑓5

+ 7𝑓6)/90 

𝑦𝑖+1 = 𝑦𝑖 + (7𝑔1 + 32𝑔3 + 12𝑔4 + 32𝑔5

+ 7𝑔6)/90 

where    

𝑓1 = ℎ 𝑓(𝑡𝑖  , 𝑥𝑖, 𝑦𝑖), 𝑔1 = ℎ𝑔(𝑡𝑖  , 𝑥𝑖, 𝑦𝑖), 𝑓2 =

ℎ𝑓 ((𝑡𝑖 +
ℎ

4
) , (𝑥𝑖 +

𝑓1

4
) , (𝑦𝑖 +

𝑔1

4
)), 

𝑔2 = ℎ𝑔 ((𝑡𝑖 +
ℎ

4
) , (𝑥𝑖 +

𝑓1

4
) , (𝑦𝑖 +

𝑔1

4
)) , 𝑓3

= ℎ𝑓 ((𝑡𝑖 +
ℎ

4
) , (𝑥𝑖 +

𝑓1 + 𝑓2

8
) , (𝑦𝑖

+
𝑔1 + 𝑔2

8
)) 

𝒈𝟑 = 𝒉𝒈 ((𝒕𝒊 +
𝒉

𝟒
) , (𝒙𝒊 +

𝒇𝟏+𝒇𝟐

𝟖
) , (𝒚𝒊 +

𝒈𝟏+𝒈𝟐

𝟖
)),   

𝒇𝟒 = 𝒉𝒇 ((𝒕𝒊 +
𝒉

𝟐
) , (𝒙𝒊 −

𝒇𝟐

𝟐
+ 𝒇𝟑) , (𝒚𝒊 −

𝒈𝟐

𝟐
+ 𝒈𝟑)),  

𝒈𝟒 = 𝒉𝒈 ((𝒕𝒊 +
𝒉

𝟐
) , (𝒙𝒊 −

𝒇𝟐

𝟐
+ 𝒇𝟑) , (𝒚𝒊 −

𝒈𝟐

𝟐
+ 𝒈𝟑)), 

  𝒇𝟓 = 𝒉𝒇 ((𝒕𝒊 +
𝟑𝒉

𝟒
) , (𝒙𝒊 +

𝟑𝒇𝟏

𝟏𝟔
+

𝟗𝒇𝟒

𝟏𝟔
) , (𝒚𝒊 +

𝟑𝒈𝟏

𝟏𝟔
+

𝟗𝒈𝟒

𝟏𝟔
)),   

𝒈𝟓 = 𝒉𝒈 ((𝒕𝒊 +
𝟑𝒉

𝟒
) , (𝒙𝒊 +

𝟑𝒇𝟏

𝟏𝟔
+

𝟗𝒇𝟒

𝟏𝟔
) , (𝒚𝒊 +

𝟑𝒈𝟏

𝟏𝟔
+

𝟗𝒈𝟒

𝟏𝟔
)),  

𝒇𝟔 = 𝒉𝒇 ((𝒕𝒊 + 𝒉), (𝒙𝒊 −
𝟑𝒇𝟏−𝟐𝒇𝟐−𝟏𝟐𝒇𝟑+𝟏𝟐𝒇𝟒−𝟖𝒇𝟓

𝟕
) , (𝒚𝒊 −

𝟑𝒈𝟏−𝟐𝒈𝟐−𝟏𝟐𝒈𝟑+𝟏𝟐𝒈𝟒−𝟖𝒈𝟓

𝟕
)) , 𝒈𝟔 = 𝒉𝒈 ((𝒕𝒊 + 𝒉), (𝒙𝒊 −

𝟑𝒇𝟏−𝟐𝒇𝟐−𝟏𝟐𝒇𝟑+𝟏𝟐𝒇𝟒−𝟖𝒇𝟓

𝟕
) , (𝒚𝒊 −

𝟑𝒈𝟏−𝟐𝒈𝟐−𝟏𝟐𝒈𝟑+𝟏𝟐𝒈𝟒−𝟖𝒈𝟓

𝟕
)) 

 

3. Results and Discussion 
 

In the following sub section, we compare numerical 

solution of RLC circuits by above discussed 

numerical methods with exact solution and analyze 

the different characteristics of damping factor by 

using best superior iterative method. 
 

3.1 Comparison between analytical solution and 

numerical methods 
 

In this experiment, the simulation time for all the 

conditions is from 0 s to 20 s with the step size of 

ℎ = 0.1 s.  Moreover, we consider source voltage 

(𝑉𝑠) = 6 𝑉, resistance (𝑅)=1 Ω, inductance (𝐿) = 1 

𝐻 and capacitance (𝐶) = 0.25 𝐹 [7] for the simulation 

result of RLC circuit. Fig. 3 describe simulation 

results of numerical solution of RLC circuit and 

exact solution under above conditions. In order to 

compare the accuracy of the numerical methods, the 

computed points of the three numerical methods are 

taken at the specific time. The computed point of 

each method is compared with the analytic solution. 

From the above Fig. 3, we observed that the 

approximate solution curve obtained from the 

Butcher fifth order Runge-Kutta (BRK5) method is 

converging faster towards the analytical solution in 



Jeevan Kafle et al. / BIBECHANA 18 (2) (2021) 9-17  

14 
 

comparison with the other two methods. This shows 

that there is less error between BRK5 and analytical 

solution in comparison with other two methods 

(Table 1). Table 1 contain details of simulated 

results obtained for Iterative methods and exact 

solution with Error in different time slices.  Hence, 

we concluded that the BRK5 method is more 

efficient method to approximate the solution of RLC 

circuit. 

So, in the following section, we consider three 

damping cases (i.e., underdamped, critically damped 

and over damped) for Butcher fifth order Runge-

Kutta (BRK5) method. In Fig. 4, we simulate the 

results under discussed above condition with 

different time slices. 
 

3.2 Characteristics of Damping Factor by using 

BRK5 method 
 

The different damping condition of RLC circuit and 

their characteristics are as discussed below:   
 

Condition 1: Underdamped (𝐢. 𝐞., 𝛇 < 𝟏 )   

In this condition, the resistance, R must be less than 

4Ω in order to achieve underdamped response. In 

order to obtain the best numerical method for 

transient analysis of RLC circuit, three numerical 

methods are compared with the analytical solution. 

The three numerical methods mentioned are Euler 

method, RK3 method and BRK5 method. The 

experiment is conducted by taking 1Ω as the value 

of resistor, R for the circuit [4, 5, 7]. 
 

Condition 2: Critically damped (𝐢. 𝐞. , 𝛇 = 𝟏) 

In critically damped response, the resistance, R must 

be equal to 4Ω in order to achieve the critically 

damped condition.   
 

Condition 3: Overdamped (𝐢. 𝐞. , 𝛇 > 𝟏) 

In condition 3, the resistance, R must be more than 

4Ω in order to achieve overdamped response. The 

experiment is conducted by taking 6Ω as the value 

of resistor, R for the circuit. Table 2 contain all the 

condition for different damping factor. By the 

variation of the resistance (R), we can get three 

different damping conditions. 

In Fig. 4, we observed that the under damped 

responses are a decaying oscillation decays at a rate 

determined by the attenuation (α) by using BRK5 

                                           Fig. 3: Comparison of Euler, RK3, BRK5 and Analytical Solution. 



Jeevan Kafle et al. / BIBECHANA 18 (2) (2021) 9-17  

15 
 

method. The exponential decay describes the 

envelope of the oscillation. Here, the oscillation is  

sinusoidal with exponentially decaying amplitude. 

The critically damped response represents the circuit 

response that decays in the fastest possible time 

without going into oscillation. This consideration is 

important in control systems where it is required to 

reach the desired state as quickly as possible without 

overshooting. The overdamped response is a decay 

of the transient current without oscillation, which 

reach the stable state slower than the critically 

damped case.  The quantitative description of 

numerical of solution under three damping condition 

by using BRK5 method is detailed in Table 3. 

 

Table 1: Simulated results obtained for Iterative methods and exact solution with error. 

 

Table 2: The values of electrical elements of series RLC circuit for three conditions [5]. 

 

 

Time (t) 

(seconds) 

Euler 𝑅𝐾3 𝐵𝑅𝐾5 Exact 

Voltage(V) Voltage(V) 𝐸𝑅 (10−3) Voltage(V) 𝐸𝑅 (10−3) Voltage(V) 𝐸𝑅 (10−3) 

0.0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 

0.1 0.000000 4.378000 0.116000 1.622000 0.115714 1.336000 0.114378 

0.2 0.240000 3.654000 0.444083 0.429000 0.443657 3.000000 0.443654 

0.3 0.696000 6.189000 0.951609 0.580000 0.951190 9.990000 0.952189 

0.4 1.336800 5.764000 1.602239 0.325000 1.601968 5.960000 1.612564 

0.5 2.125680 1.791000 2.357671 8.800000 2.357671 8.800000 2.366471 

0.6 3.022200 7.630000 3.179275 0.555000 3.179643 1.870000 3.189830 

0.7 3.984041 1.851000 4.029593 4.701000 4.030393 5.501000 4.004892 

0.8 4.968810 4.868000 4.873643 0.299000 4.874908 9.034000 4.883942 

0.9 5.935740 3.419000 5.680016 2.305000 5.681740 5.810000 5.692321 

1.0 6.847225 2.573000 6.421720 2.932000 6.423867 7.850000 6.424652 

Element                      Value 

Condition 1 Condition 2 Condition 3 

Resistor Underdamped Critically damped Overdamped 

DC (Vs)      6V      6V      6V 

Resistor  R    <4Ω      4Ω     >4Ω 

Inductor L      1H     1H       1H 

Capacitor C      0.25F     0.25F     0.25F 



Jeevan Kafle et al. / BIBECHANA 18 (2) (2021) 9-17  

16 
 

 

 

Table 3: Values of three conditions of BRK5 method. 

 

4. Conclusion 
 

The transient analysis of electrical circuit is 

analyzed using analytical method, Euler (Explicit) 

Method, RK3 Methods and BRK5 Method. With 

the usage of computational software, the process of 

obtaining results of transient analysis is done 

systematically and conveniently. It has been 

observed that the BRK5 method is very efficient in 

solving second order differential equations. Thus, 

we can conclude that by carrying out the transient 

Time (s) Values of BRK5 Method 

Underdamped Critically damped Overdamped 

0.0 0.000000 0.000000 0.000000 

0.1 0.115714 0.105139 0.098897 

0.2 0.443657 0.369312 0.330084 

0.3 0.951190 0.731409 0.626944 

0.4 1.601968 1.147248 0.950945 

0.5 2.357643 1.585448 1.280130 

0.6 3.179643 2.024237 1.602290 

0.7 4.030393 2.449004 1.910937 

0.8 4.874908 2.850415 2.202915 

0.9 5.681740 3.222979 2.476992 

1.0 6.423867 3.563965 2.733027 

                           Fig. 4 : Comparison of three different damping condition by using BRK5. 



Jeevan Kafle et al. / BIBECHANA 18 (2) (2021) 9-17  

17 
 

analysis of a system, we can find out the response 

of the system by changing the conditions from one 

steady state value to another. Our observation 

clearly shows that the under damped decay is 

oscillatory and exponential. However, the other two 

are non-oscillatory exponential decay. The decay in 

critically damped case is observed faster than in 

over damped case.  This response helps in designing 

a system, which meets our requirements, and we can 

further optimize the time domain parameters of the 

system. 
 

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