International Journal of Advanced Research and Publications

ISSN: 2456-9992

Variation Of Parameters: Application To Electric Circuit Analysis

Engr. Dindo T. Ani

Batangas State University, Electrical and Computer Engineering Department Alangilan, Batangas City, Philippines, PH-0063 950 394 5795 [email protected]

Abstract: This paper conducted a study on the application of variation of parameters as method in solving electric circuit analysis problems particularly the series combination of resistor, inductor and capacitor (RLC). The researcher wanted to explore the ideas of solving the second-order circuits using the method of variation of parameters as a user-friendly method which has been tested by the researcher. The paper tried to explain and discuss the method of variation of parameters in three distinct cases namely real and distinct roots, real and repeated roots, and complex and conjugate roots. The researcher exposed and expounded formulas and present variation of parameters in solving linear differential equation of second-order. The generalization of principles and concepts were established through examples. This study was able to derive the variation of parameters formula for electric circuit analysis specifically series resistor, inductor and capacitor. Results showed that the variation of parameters can be used as a method for solving series RLC electric circuit.

Keywords: differential equation, electric circuit, variation of parameters

1. Introduction focused on the method of variation of parameters, as well as Electric circuit analysis is one of the foundations of electrical the applications of these methods to series RLC circuit engineering. Almost all branches of electrical engineering, (resistor-inductor-capacitor). This study hopes to stimulate such as power systems, electrical machines, electronics, the interest of students toward solving electric circuit control systems, are based on circuit theory. Thus, it can be analysis in different way. This can be used by instructors in said that basic circuit theory is considered as one of the most engineering mathematics and electrical engineering as an important subjects in electrical engineering curriculum. An additional input study for their lectures. This can also be electric circuit is determined by the type of elements it used by professionals as an alternative technique in solving contains and the manner in which the elements are connected differential equations and some circuit analysis. This study [1]. The relationships of different variables related to the may also be helpful for future researchers as basis to study, analysis of circuits are expressed in the describing equations. analyze and expound the techniques and formulas under The describing equations for resistive circuits are simple study. This paper may be helpful not only to the students and algebraic equations. However, the circuits are not only educators but also to those with knowledge and interest in resistive but inductive or capacitive in nature. The describing calculus especially engineering professionals. equations for these types of circuits are differential rather than algebraic. The application of Kirchhoff’s Laws to these 2. Methodology networks gives rise to differential equations that, in general, This study is descriptive and expository in nature. The paper are more difficult to solve than algebraic equations [2,4]. tried to explain and discuss the method of variation of Most of the electrical engineering problems, especially the parameters. The researcher exposed and expounded formulas circuit analysis is governed or characterized by differential and present variation of parameters in solving linear equations. This equation can be solved using the method of differential equation of second-order. The generalization of undetermined coefficients. For the differential equation principles and concepts were established through examples. shown in equation 1, the method of undetermined coefficient It also discussed the related theorems and basic concepts to can be used [3]. aid understanding of the main concept. This research relied ( ) ( ) ( ) ( ) ( ) on books dealing with differential equations, differential and integral calculus, engineering mathematics and electric The method of undetermined coefficients works only when circuit analysis. The derivation of the general formula for the coefficients a, b, and c are constants and the right-hand finding the relationships between the variables such as term f(x) is of special form, i.e. exponential, polynomial, current, voltage and time was shown. The electric circuit sine, cosine, or combinations. If these restrictions do not problems were then solved using the method of variation of apply to a given nonhomogeneous linear differential parameters. The researcher then recorded, compiled, equation, then a more powerful method of determining synthesized and analyzed the gathered information as the particular solution is needed, the method known as variation highlights of the study. of parameters [3,5]. In electrical engineering, the variation of parameters is not given much attention on books and 3. Results and Discussion researches. This paper conducted a study on the application This section discusses the results of the implemented of variation of parameters as method in solving electric methodology of the study. As shown in figure 1, the circuit is circuit analysis problems particularly the series combination composed of the voltage source, resistor, inductor and of resistor, inductor and capacitor. The researcher wanted to capacitor. explore the ideas of solving the second-order circuits using the method of variation of parameters as a user-friendly method which has been tested by the researcher.. This study

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ISSN: 2456-9992

The first step is to construct the two equations needed to solve for A’(t) and B’(t) using equations 5 and 6. The two equations are ( ) ( ) ( )

( ) ( ) ( )

Figure 1: Series RLC Circuit Then find A’(t) and B’(t) by solving equations 10 and 11 By applying Kirchhoff’s Voltage Law [2], the total using Cramer’s rule, voltage is equal to the sum of the voltages across the resistor | | R, inductor L, and capacitor C. In equation form,

( ) ( ) ( ) ( ) ( )

| | By substituting the voltages in terms of their characteristics, the equation becomes

( ) . /

( ) ∫ ( ) ( ) ( )

( )

Differentiating and rearranging, the equation becomes

( ) ( ) . / ( ) ( ) ( ) ( )

This equation (4) is a second-order linear ordinary ( ) differential equation with constant coefficients. First solve ( ) for the complimentary solution of the homogeneous differential equation The next step is to solve for the value of A(t) by integration.

( ) ( ) The equation is ( ) ( )

( ) ∫ ( ) Using the quadratic formula to solve for the roots, Since Vm, ω, m1, m2, and L are constants, the equation will √ ( ) yield,

( ) ∫ ( ) For simplicity of solution, let ( ) Use integration by parts to solve for A(t). The solution is ∫ ∫

√

From these roots, we have 3 possible cases.

3.1 Real and Distinct Roots (Case 1) ∫ The roots are real and unequal if ( ) ( )

∫ ( ) ( ) The roots are

√ ∫

√

∫ ( ) The complementary solution of the homogeneous equation is

( ) ( ) Solving for

Now, solve for the particular solution, ip(t), using variation of ∫ parameters. The assumed value of ip(t) is using integration by parts,

( ) ( ) ( ) ( )

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ISSN: 2456-9992

( )

( ) 6 7 ( ) ( ) ∫ Then solve for B’(t) using Cramer’s rule,

( ) ( ) | |

∫ ( ) ( ) ( )

| |

∫

( ) . / ( )

∫ ( ) ( )

( ) . / Then substitute equation 13 to equation 12, ( ) ∫ ( )( )

( ) ( ) [

Then solve for B(t) using integration by parts. The equation

∫ ] is

( ) ∫

∫ ( )

Since V , ω, m , m , and L are constants, the equation is now m 1 2

( ) ∫ ( ) ( )

Solving for ∫ ( ) ∫ Combining similar terms, the equation will yield

using integration by parts, ∫ ∫ ( )

( )

∫ ( ) ( ) 6 7 ∫ ( ) ∫ ( ) ( )

∫ ( )

∫ ∫ ( )

( ) Solving for

[ ] ∫ ( )

Simplifying, the equation becomes using integration by parts,

( ) ∫ ( )

Therefore,

( ) ∫ ( )

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ISSN: 2456-9992

∫ Combining similar terms and simplifying, ( ) ( ) ( ) ( ) 6 ( ) ( ) ( ) ∫ ( ) ( ) 7 ( ) ∫ Since

( ) ( ) ( )

∫ ( ) then ( )

( ) Substituting equation 15 to equation 14, 6 ( ) ( ) ∫ ( )

7 ( ) ( )

[

∫ ] √

∫ √ ( )

∫ ( ) To verify the validity of this equation, let us use this equation in an example. Combining similar terms,

Example 1. Suppose the given circuit parameters are: ∫ ∫ ( ) R = 10 Ω, L = 1 H, C = 1/16 F, Vm = 10 V, ω = 2 rad/s

The first step is to determine in what case the circuit belongs.

( )

( )

6 7 ∫ ( ) ( ) . /

( )

then this falls under case 1.

( ) ∫ Therefore, equation 16 must be used as shown below.

[ ] ( ) ( ) ( ) 6 Simplifying, ( ) ( )

( ) ( ) ∫ 7 ( ) ( ) ( )

Therefore, The next step is to determine the roots,

( ) ∫ ( ) √ √ ( ) ( ) ( ). /

( ) ( ) 6 7 ( ) ( ) then plug-in the values to equation 16 to get

After solving for A(t) and B(t), substitute to ip(t), ( ) ( )( ) ( ) ( ) ( ) ( ) 6 ( )( ) ( ) ( )

( ) ( ) 7 8 6 79 ( ) ( ) ( ) ( )

( ) 8 6 79 ( ) ( ) Combining similar terms and simplifying,

( )

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ISSN: 2456-9992

3.2 Real and Repeated Roots (Case 2) ∫ ∫ ( ) ∫ ( ) The roots are real and equal if

( ) ( ) The roots are

The roots are always negative for case 2 since there will be no negative value of R and L. Therefore, use Therefore,

∫ The complementary solution of the homogeneous equation is ( ) 4 5 ( )

Now, solve for the particular solution, i (t), using variation of p ∫ 4 5 ( ) parameters.

The assumed value of ip(t) is

( ) ( ) ( ) ∫

Aside from using equations 5 and 6, we can also use the derived variation of parameters formula as shown in equations 7 and 8. Solving for A(t) using equation 7, ∫ ( ) ( ) ( ) ∫ ∫ ( ) ( )

Solving for

∫

using integration by parts,

( ). /

( ) ∫ ,( )( ) ( )( )-

( ) ∫ ∫ ( ) ( )

∫ ( ) ( )

( ) ∫

∫ Since m, ω, V , and L are constants, m

( ) ∫ ∫ ( )

Solving for Solving for

∫ ∫ using integration by parts, using integration by parts,

Solving for ∫ ∫ ∫ ( ) 4 5 using integration by parts, ∫ 4 5 ( )

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ISSN: 2456-9992

∫ ∫

∫ ( )

∫ Substituting equation 18 to equation 21,

∫ ( ) ∫

[ Substituting equations 18 and 19 to equation 17 will yield

∫ ∫ ]

∫

6

∫ ∫

Combining similar terms and simplifying,

∫ 7

4 5 ∫

[

∫ ]

∫ ∫

∫

∫ ( )

∫ Substituting equation 22 to equation 20,

4 5 ∫ ∫

Combining similar terms and simplifying,

4 5 ∫

6 7

Combining similar terms and simplifying,

∫ ( ) 4 5 ∫

6 Solving for

∫ 7 ( )

using integration by parts, 6

7 ( )

∫ ( ) ( ) ∫ ( ) ( ) [ ] [ ] ( ) ( )

∫ ( ) ( ) ( ) [ ]

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ISSN: 2456-9992

( ) 0 1 0 1 ∫ ( ) ( )

( ) 0 1 0 1 8 6 7 ( ) ( ) ( )

( ) ( ) 6 79 ( ) 2 0 1 0 13 ( ) ( ) ( )

Since Substituting this equation to A(t), ( ) ( ) ( ) ( ) then ( )

4 8 6 7 ( ) ( ) ( ) ( ) 8 6 7 ( ) 6 795 ( ) [ ]9 ( ) ( )

( ) *, ( ) ( )

-

,( ) - + Example 2. Suppose the given circuit parameters are: R = 10 Ω, L = 1 H, C = 1/25 F, Vm = 12 V, ω = 3 rad/s Solving for B(t) using variation of parameters formula, ( ) ( ) The first step is to determine in what case the circuit belongs. ( ) ∫

( ) ( )

( ). /

( ) ∫ ( ) . / ,( )( ) ( )( )-

then this falls under case 2. ( ) ∫

Therefore, equation 23 must be used as shown below.

( ) ∫ ( )

( ) 8 6 7 ( ) Since ω, V , and L are constants, m

( ) ∫ [ ]9 ( ) ( )

From previous computation, The next step is to determine the roots,

∫ ( ) therefore, then plug-in the values to equation 23 to get

( ) 6 7 ( ) ( ) ( ) ( ) 8 6 7 ( ) Substituting A(t) and B(t) to ip(t), ( )( )( ) ( ) ( ) ( ) 6 79 ( )

( ) Simplifying, ( ) 8 6 7 ( ) ( ) ( ) 3.3 Conjugate and Complex Roots (Case 3) 6 79 ( ) The roots are conjugate and complex if

6 7

Combining similar terms and simplifying, The roots are

( ) ( ) ( ) 0 1 0 1 ( ) ( )

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ISSN: 2456-9992

( )

√

( )

The complementary solution of the homogeneous equation is

( ) Substituting B’(t) to A’(t),

( ) ( ) Then solve the particular solution using variation of parameters. The assumed value of ip(t) is

( ) ( ) ( ) ( ) 6 7

By using equations below, ( )

Since ω, Vm, and L are constants the equation becomes,

( ) ( ) ∫

we have the two simultaneous equations, The next step is to find the values of A(t) and B(t). ( ) ( ) ( ) Solve for A(t) using product formula,

( ) ( ) ( ),( )( )( ) ( )( )- ( ),( )( )( ) ( ) ( ) ( )( )-

( ) Substituting the above equation to A(t),

( ) ∫ [ ( ) Simplifying equation 25 will yield ( ) , ( )- ( )] ( ) , ( )-

( ) ( ) {∫ ( )

Use substitution and elimination to solve for A’(t) and B’(t). Substituting equation 24 to equation 26 will yield ∫ ( ) }

( ) ( ) Solving for

∫ ( )

( ) , ( )-

( ) , ( )- using integration by parts, ( )

( ) ( )

Simplifying,

( ) 6 7

∫ ( )

, ( )- [ ]

( ) ∫ [ ] ,( ) ( ) -

∫ ( )

( ) ( )

( ) ∫ (

) ( )

( ) ( ) Solving for

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ISSN: 2456-9992

∫ ( ) ∫ ( ) ( ) ( ) ( ) using integration by parts, ( ) ( ) ( ) ( )

∫ ( )

( ) ( ) ( ) ∫ ( ) ( ) ( )

, ( )- [ ]

Using the same procedure, ∫ [ ] , ( ∫ ( ) ) ( ) - ( ) ( ) ( ) ( ) ∫ ( ) ( )

( ) Substituting equations 29 and 30 to A(t),

( ) ( ) ( ) ( ) ∫ ( ( ) 0 ( ) ( ) ( ) ( ) ) ( ) 1 ( ) Substituting equation 28 to equation 27, The equation for B(t) can be solved using integration as ∫ ( ) shown below, ( ) ∫ ( ) ( )

( )

6 ( ) ( ) ∫

( ) ∫ ( ) 7 Using product formula, B(t) becomes

( ) ( ) ∫ ( )

( ) ( ) ( )

( ) ( ) ( ) ∫ [ ( ) ( )]

( ) ∫ ( )

( ) [∫ ( )

( ) 6 7 ∫ ( ) ∫ ( ) ]

( ) Substituting equation 27 to equation 28, ( ) ( ) ∫ ( )

( )

∫ ( ) ( ) 6 ( ) ( ) ( ) ( )

( ) ( ) ∫ ( ) 7 [ ]

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ISSN: 2456-9992

∫ ( ) ( ) 8 6 ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ( ) ( ) ( )7 ( )

Combining similar terms and simplifying, 6 ( ) ( ) ( ) ∫ ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) [ ] ( ) ( ) ( )79 ∫ ( ) ( ) ( ) ( ) ( ) Since

( ) ( ) ( )

∫ ( ) then the equation becomes, ( ) ( ) ( ) ( ) ( ) ( ) 2 0 ( ( )

Using the same procedure, ( ) ) ( ∫ ( ) ( ) ( ) ( ) ( ) ) ( ) ( )

( ) ( ) Substituting equations 31 and 32 to B(t), ( ) ( ) ( )1 ( ) ( ) ( ) ( ) 6 ( )

( ) ( ) ( ) 0 ( ) 7 ( ) ( ) ( ) ( ) ( ) Substituting A(t) and B(t) to ip(t),

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) 0 ( ) ( ) ( ) ( ) ( ) ( )13 1 ( ) ( ) ( ) ( ) ( ) 0 ( )

( ) ( ) ( ) ( ) 2 ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) 0 ( ) ( ) ( ) ( ) ( ) ( ) 13 ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

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ISSN: 2456-9992

( ) Since ( ) ( ) ( ) ( ) ( ) ( ) ( ) then the equation becomes, ( )

( ) ( ) 8[ ( ) ( )

( ) ( ) ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) 6 ( )3 ( ) ( ) ( ) 7 ( ) 9 Combining similar terms and simplifying, ( )

( ) 8 ( ) ( ) Since ( ) ( ) ( ) ( )

( ) then ( ) ( ) ( ) ( ) 8[ ( ) ( ) ( ) ( ) ] ( ) ( )

( ) ( ) ( ) 6 ( ) ( ) ( ) ( ) ( ) 7 ( ) 9 ( ) ( ) ( ) ( ) ( ) Example 3. Suppose the given circuit parameters are: ( )9 ( ) R = 20 Ω, L = 1 H, C = 0.005 F, Vm = 10 V, ω = 5 rad/s

The first step is to determine in what case the circuit belongs.

( ) 8[ ( ) ( )

] ( ) ( ) ( ) ( )( )

6 ( ) ( ) then this falls under case 3. 7 ( ) ( )

[ Therefore, equation 33 must be used as shown below. ( ) ( )

] ( ) ( ) 8[ ( ) ( ) 6 ] ( ) ( ) ( ) ( ) ( ) 7 ( )9 6 ( ) ( ) ( ) 7 ( ) 9 ( ) ( ) ( ) 8[ ( )

] ( ) ( The next step is to determine the roots, ( ) ) √ ( ) 6 ( )

( ) √ 7 ( ) ( ( ) ( ) ( )( ) ( )

)9

Volume 3 Issue 6, June 2019 138 www.ijarp.org International Journal of Advanced Research and Publications

ISSN: 2456-9992 then plug-in the values to eq. 33 to get Philippines Los Baños in 2003, MS Mathematics at University of Batangas in 2012, and MS Electrical Engineering major in Power ( ) ( )( ) System at Mapua Institute of Technology in 2016. He is currently pursuing his Doctor of Technology degree at Batangas State 8[ ( ) ( ) ( ) ( ) University.

] ( ) ( ) ( ) ( ) 6 ( ) ( ) ( ) 7 ( ) 9 ( ) ( )

Simplifying,

( )

4. Conclusion and Recommendations This study was able to derive the variation of parameters formula for electric circuit analysis specifically series resistor, inductor and capacitor. Results showed that the variation of parameters can be used as a method for solving series RLC electric circuit. The researcher strongly recommend to conduct study on the application of variation of parameters to other engineering discipline such as mechanical, chemical and civil engineering. Furthermore, the researcher recommends to compare and to relate the variation of parameters to other method such as Laplace transformation and power series method.

References [1] Johnson, D. E., et al., Electric Circuit Analysis, 2nd edition. Prentice-Hall Inc., New Jersey, 1992

[2] Hayt, W. H. Jr., et al., Engineering Circuit Analysis, International edition. McGraw-Hill, USA, 2002

[3] Rainville, E. D., et al., Elementary Differential Equations, 8th edition. Prentice-Hall Inc., New Jersey, 2001

[4] Monier, C. J., Electric Circuit Analysis, Prentice-Hall, Inc., New Jersey, 2001

[5] Abell, M. L. and J. P. Braselton, Modern Differential Equations: Theory Application, Technology, Harcourt Brace & Company, USA, 1995

Author Profile

Engr. Dindo T. Ani is a full-time instructor at the College of Engineering, Architecture and Fine Arts of Batangas State University under the Electrical and Computer Engineering Department. He is currently the Program Chair of the Electrical Engineering program of the department and serves as a university researcher of the institution. He has more than 14 years of academic experience, served as research adviser of undergraduate thesis, presented research papers in international research conferences, and published paper in a refereed engineering journal. He graduated BS Electrical Engineering major in Power Engineering at the University of the

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