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Proceedings of the 2012 ASEE North-Central Section Conference

Copyright 2012, American Society for Engineering Education

Effective Method for Solving Equivalent Elements in Electric Circuits

Khalid S. Al-Olimat

Ohio Northern University, Ada, OH 45810

[email protected]

Abstract

Most engineering students have difficulty in solving circuits for equivalent circuit elements.

Circuit elements include resistances, capacitances, and inductances. The difficulty arises

whenever any two or more nodes in a given circuit are shorted together. This type of circuits

confuses students in such a way that they will not be able to tell if the elements are connected in

series or in parallel. This paper shows a method to solve such problem. The method has shown

its effectiveness through the assessment of students performance and by the feedback received

from students.

1. Introduction

In electrical engineering, we are often interested in communicating or transferring energy from

one point to another. To do this requires an interconnection of electrical devices. Such

interconnection is referred to as an electric circuit. An electrical engineer must acquire many

skills, one of which is knowledge of electric circuit analysis. If a student has already entered or

intend to enter an electrical engineering program, then circuit analysis may represent the

introductory course in his/her chosen field. Many branches of electrical engineering, such as

power, electric machines, control, electronics, communications, and instrumentation, are based

on electric circuit theory. Circuit theory is also valuable to students specializing in other

branches of engineering because it is plain that every engineer of whatever discipline will be

faced with using and operating electrical equipment and systems in his or her own practice.

Mechanical engineers will need motors to drive their machines. Chemical engineers apply heat

and drive pumps. Civil engineers operate construction sites and apply electronic surveying

devices. Further, all these activities need instrumentation and control equipment that is largely

electrical.

Engineering students learn the electric circuits concepts and analysis skills in Electric Circuitscourse which nowadays is one combined course in most colleges. That course combines both

direct current and alternating current circuits, and covers various topics. Typical topics include

resistive circuits, nodal analysis, loop analysis, superposition theorem, Thevenins and Nortons

theorems, operational amplifiers, capacitance and inductance, first-order circuits, ac steady-state

analysis, steady-state power analysis, ideal transformers, and three phase circuits. When students

perform analysis in these topics, they must have the ability to solve for equivalent resistance,
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equivalent capacitance, or equivalent inductance. Solving for such equivalency is very important

step especially in resistances which is required in circuit analysis for Thevenins equivalent

circuits, maximum power transfer, and first-order circuits. Most engineering students have

difficulty in solving for such equivalency. The difficulty arises whenever any two or more nodes

in a given circuit are shorted together. This type of circuits confuses students in such a way that

they will not be able to tell if the elements are connected in series or in parallel. This paper

presents an effective method to overcome this difficulty, and reports on the assessment of

students performance before and after this method introduced to them. In addition students

feedback is included.

2. Topic Coverage

The Electric Circuits course covers the topics of series and parallel connections of resistances,

capacitances and inductances. The coverage includes the derivation of the equations for series

and parallel connections of the aforementioned circuit elements which results in formulas to be

used in the calculations.

Series Resistors

The equivalent or total resistance of nresistors connected in series is given by

Series Capacitor s

The equivalent or total capacitance of ncapacitors connected in series is given by

Seri es I nductorsThe equivalent or total inductance of ninductors connected in series is given by

Parall el Resistors

The equivalent or total resistance of nresistors connected in parallel is given by

Paral lel Capacitors

The equivalent or total capacitance of ncapacitors connected in parallel is given by

Parallel I nductors

The equivalent or total inductance of ninductors connected in parallel is given by

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After these formulas are derived, some examples are covered for students to show the application

of these formulas. Figure 1 shows a circuit where students can apply the resistance formulas to

obtain the total resistance across terminals A-B. Students usually dont have any problems in

utilizing the formulas in such circuits because all the connections are so clear and in turn students

start reducing the circuit through the series-parallel combination till they end up with one

resistance value.

Figure 1 Series-parallel combination of resistors

This single resistance value is the equivalent or the total resistance of that given circuit.

However, when there a short exists between two nodes or more, the circuit will throw students

off because they are unable to determine if the elements are connected in series or connected in

parallel. To overcome this problem, students are encouraged to use the following method which

simplifies the circuit, and the connections of elements will be so clear to them.

3. The Method

The best way to present the method is through some examples. It consists of three steps:

1. Assign circuits nodes. If there is a short between two nodes then the voltages at the two

nodes are equal since there is no voltage drop across a short circuit. This leads us to the

fact that both nodes are the same node.

2. Redraw the circuit based on the assigned nodes. Connect the elements between the nodes

per the original circuit.

3. Solve for the required equivalent circuit element using series-parallel combination.

Example 1: Find the total (equivalent) resistance in the circuit shown in figure 2.

Figure 2: Resistive circuit for example 1

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Solution:

Step 1: Assign circuits nodes. There are three nodes assigned A, B, and C.

Step 2: Redraw the circuit

Step 3: Solve for the required equivalent circuit element using series-parallel combination.

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So the equivalent resistance is

Example 2: Find the total (equivalent) capacitance in the circuit shown in figure 2.

Figure 2 Capacitive circuit for example 2

Solution:

Step 1: Assign circuits nodes. There are two nodes assigned A and B.

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Step 2: Redraw the circuit

Step 3: Solve for the required equivalent circuit element using series-parallel combination.

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Since the top and bottom capacitors are shorted out, the circuit results as shown

This leads us to the value as

4. Data Collection and Assessment Method

The pool of participants includes the population of the students in the class (24 students). There

were 6 electrical engineering students, 13 mechanical engineering students and 5 computer

engineering students. The instructor introduced the aforementioned method to students for the

first time during a review session at the end of the term. There were 21 students from the class

who attended that session.

Measur es

The primary assessment goal was to measure the performance of students before and after the

method introduced to them. In order to perform this assessment, a problem was given in the first

exam asking students to solve for the total resistance in a given circuit. In addition, a problem

was given in the final exam asking for the total inductance in a given circuit.

Statistical Resul ts

The performance of students in determining the total resistance was assessed through grading the

solution provided by students. The assigned full score for that problem was 20 points. The data

obtained is shown in Table 1.

Table 1 Students scores before method introduced

Score

20 15 10 5 0

Electrical Engineering 2 1 0 2 1

Mechanical Engineering 4 2 3 1 3

Computer Engineering 1 3 0 1 0

Total 7 6 3 4 4

As shown from the table, only 29% of the students got full score on that problem. Around 17%

of the students got that problem completely wrong. The rest of the students had various mistakes

and some points were deducted.

The instructor included a problem in the final exam but asked the students to determine the total

inductance of a given circuit. As explained earlier, the formulas for series connection and

parallel connection of inductances are the same form. But this time students were already

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introduced to the aforementioned method on how to calculate the total resistance, capacitance or

inductance. The results of grading that problem are shown in Table 2.

Table 2 Students scores after method introduced

Score

20 15 10 5 0

Electrical Engineering 5 1 0 0 0

Mechanical Engineering 10 2 0 0 1

Computer Engineering 4 0 0 1 0

Total 19 3 0 1 1

From the table, the results show a huge improvement in the performance of students. Around

79% of the class solved the problem correctly and got a full score. Only 1 student in the class

got the solution completely wrong. Based on these results, one can see the effectiveness of this

method in solving such circuit problem.

Students Comments

At the end of the review session, many students expressed their appreciation to the instructor for

introducing this method and jokingly blamed him for getting bad scores on the problem of the

first exam.

Conclusion

A method of solving electric equivalent circuit was presented. It is believed that the method can

be followed easily by any student and will improve the student skills in solving such problems.

Finally, since the role of instruction is not to distribute facts but to grant students with ways toassemble knowledge, educators must find favored strategies that build students confidence and

enhanced course relevance. This can be achieved through the continual investigation of

appropriate ways to introduce different methods in solving problems.

References

[1] C. Alexander and M. Sadiku, Fundamentals of Electric Circuits, 4th

Edition, McGraw

Hill, 2009.

[2] J. Irwin and R. Nelms, Basic Engineering Circuit Analysis, 8th

Edition, Wiley, 2005.

[3] R. Dorf and J. Svoboda, Introduction to Electric Circuits, 8thEdition, Wiley, 2010.

[4] R. Thomas and A. Rosa, The Analysis and Design of Linear Circuits, Prentice Hall, 1994.

[5] W. Hayt, Jr, J. Kemmerly and S. Durbin, Engineering Circuit Analysis, 6th Edition,

McGraw Hill, 2002.

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