Effective Method for Solving Equivalent Elements in Electric Circuits
Khalid S. Al-Olimat
Ohio Northern University, Ada, OH 45810
k-al-olimat@onu.edu
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 Circuits
course 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, Thevenin’s and Norton’s
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,
Proceedings of the 2012 ASEE North-Central Section Conference
Copyright © 2012, American Society for Engineering Education
equivalent capacitance, or equivalent inductance. Solving for such equivalency is very important
step especially in resistances which is required in circuit analysis for Thevenin’s 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 n resistors connected in series is given by
Series Capacitors
The equivalent or total capacitance of n capacitors connected in series is given by
Series Inductors
The equivalent or total inductance of n inductors connected in series is given by
Parallel Resistors
The equivalent or total resistance of n resistors connected in parallel is given by
Parallel Capacitors
The equivalent or total capacitance of n capacitors connected in parallel is given by
Parallel Inductors
The equivalent or total inductance of n inductors connected in parallel is given by
Proceedings of the 2012 ASEE North-Central Section Conference
Copyright © 2012, American Society for Engineering Education
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 don’t 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 circuit’s 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
Proceedings of the 2012 ASEE North-Central Section Conference
Copyright © 2012, American Society for Engineering Education
Solution:
Step 1: Assign circuit’s 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.
Proceedings of the 2012 ASEE North-Central Section Conference
Copyright © 2012, American Society for Engineering Education
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 circuit’s nodes. There are two nodes assigned A and B.
Proceedings of the 2012 ASEE North-Central Section Conference
Copyright © 2012, American Society for Engineering Education
Step 2: Redraw the circuit
Step 3: Solve for the required equivalent circuit element using series-parallel combination.
Proceedings of the 2012 ASEE North-Central Section Conference
Copyright © 2012, American Society for Engineering Education
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.
Measures
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 Results
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
Electrical Engineering
2
1
0
2
Mechanical Engineering
4
2
3
1
Computer Engineering
1
3
0
1
Total
7
6
3
4
0
1
3
0
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
Proceedings of the 2012 ASEE North-Central Section Conference
Copyright © 2012, American Society for Engineering Education
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
Electrical Engineering
5
1
0
0
Mechanical Engineering
10
2
0
0
Computer Engineering
4
0
0
1
Total
19
3
0
1
0
0
1
0
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 to
assemble 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, 4 th 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, 8th Edition, 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, 6 th Edition,
McGraw Hill, 2002.
Proceedings of the 2012 ASEE North-Central Section Conference
Copyright © 2012, American Society for Engineering Education