Investigating Student Learning
of the Voltage and Potential Concepts
in Introductory Electrical Engineering
Dion Timmermann
Christian Kautz
Engineering Education Research Group
Hamburg University of Technology
Email: dion.timmermann@tuhh.de
Professor for Engineering Education
Hamburg University of Technology
Email: kautz@tuhh.de
Abstract—Voltage is one of the most fundamental concepts
in electrical engineering, but nevertheless has been shown to
be a difficult concept for many students. To help address those
difficulties, we designed a tutorial worksheet similar to those
published by the Physics Education Group at the University of
Washington. In this activity, students were introduced to electric
potential and compared and contrasted it to voltage. As electric
potential and Kirchhoff’s Voltage Law are closely related, we
assumed learning about potential would help students gain a better qualitative understanding of voltage. Post-test and exam data
from an introductory circuits course show that after the tutorial
many students still had difficulties with voltage and potential.
More than three quarters of the responses concerning voltage
and potential were inconsistent. This suggests that most students
were unable to link those two concepts. We therefore conclude
that potential and voltage remain conceptually very different for
students in their learning process. As tutorial worksheets have
proved to be very effective in students overcoming conceptual
difficulties, it is likely that there are specific difficulties with these
concepts that have not yet been identified.
Keywords—student understanding, voltage, potential, tutorial
worksheet, concept, student learning.
I.
I NTRODUCTION
Voltage is one of the most important concepts in electrical
engineering. The concept of voltage is closely related to
the concept of electric potential, with any voltage being the
difference between two potentials: VXY = φX − φY . Thus,
wherever one of these concepts occurs, it can be replaced by
the other one. It can be argued, that both concepts are just
different descriptions of the same underlying physical entity.
While there is virtually no course on electrical engineering,
in which voltage is not used, many lectures and curricula
spend very little time on the electric potential. Often, it is
simply defined by its relation to voltage and not examined
further. However, potential has the advantage that its theoretical description is much easier than that of voltage. It is a
scalar defined for every point in the circuit instead of a scalar
defined between every pair of points. This makes potential
easier to visualize in a diagram. For example, it is possible to
use color coding to indicate different potentials in a circuit, as
for example students are asked to do in task 2.1.c. in Fig. 1.
This is impossible for voltages. Additionally, it is rather easy
to compare the electric potential to other potential fields, e. g.
the gravitational potential, which is familiar to many students.
Many students have difficulties with the conceptual understanding of voltage [1]. Investigations at our institution
have confirmed these difficulties to be present also among
our students [2], [3]. In a course where potential is usually
not introduced at all, we decided to spend considerable time
on that concept. We hoped that through its close relation
to voltage, an understanding of potential would also further
students’ understanding of voltage.
For the introduction of potential, we chose to use tutorial
worksheets, as introduced by the Physics Education Group
at the University of Washington [4]. These are structured
worksheets for collaborative group work that focus on conceptional questions and are designed to address known student
difficulties. In our group, many tutorial worksheets have been
developed based on our own research. In addition, we have
gained considerable experience in implementing such tutorial
worksheets [2], [5].
In the following section, we will describe the course
in which this investigation was conducted, as well as our
methodology. Afterwards, the specific maerials used (i. e. the
tutorial worksheet, post-test, and relevant exam question) will
be described. We will give an overview over the students’
answers in the post-test and exam and highlight important
results. Finally, we will explain our interpretation of these
results and draw conclusions.
II.
C ONTEXT OF THE I NVESTIGATION
The investigation was carried out in an introductory level
electrical engineering course for mechanical, process, and
logistics engineering students, most of whom were in their
first semester. The course is organized and given by faculty
from another department at our university. It is taught in a
traditional manner with 90 minutes of lecture and 45 minutes
of recitation sessions per week. The former is attended by
about 350 students, while the latter are split into groups of
about 20 students. The course covers the subject of circuit
theory with direct and alternating, as well as three-phase
current. Potential is not part of the curriculum.
For several years, we have implemented four tutorial worksheets in this course, two of which are done in the recitation
sessions and two during lecture time with all students and
about 15 teaching assistants present. The two worksheets in
the recitation sessions are based on hands-on observations
that students make in the context of batteries and light bulbs.
The course mostly focuses on quantitative analysis of circuits,
while the worksheets focus on qualitative understanding. On
each exam, at least one question focuses on the concepts
in qualitative reasoning developed in the tutorials. All exam
questions are in the multiple choice format.
In our research, we mainly investigate conceptual understanding of students, as in our opinion, this is required to reliably apply one’s knowledge. To reach this kind of understanding, students have to develop concepts for themselves, ideally
guided by an instructor and other learning aids. We believe
that tutorial worksheets are a very good method to accomplish
this aim. Whenever possible, we base the development of our
instructional materials on empirical evidence and quantitative
data.
III.
60
2
BASIC CIRCUIT ANALYSIS - Tutorial
Analysis of circuits using electric potential
In order to determine the voltage between two distant points in a circuit more easily, we
introduce the term electric potential. The potential difference between two arbitrary points
within a circuit is, by definition, equal to the voltage between those two points (e. g. VP Q =
P ≠ Q ).
The potential
at any arbitrary point is then defined by assigning an arbitrary value
to a certain point within the circuit. Often the negative terminal of a battery is assigned a
potential of = 0 V. Consequently, the positive terminal of that battery would be at potential
= Vbatt .
2.1 Reference directions and network equations
The circuit from section 1 is shown again at right.
b. Are there points that have equal potential? If so, how did
you identify them? If not, explain why there are no points
with equal potential.
During the 90-minute tutorial session, two and a half
pages of the tutorial were covered. The students worked
through all exercises concerning potential, but only started on
those concerning the relation between circuit diagrams and
equations.
Seven weeks later, at the beginning of the next tutorial
session, a two-page post-test was given. The first page of
this post-test asked questions about voltage and potential. An
English translation of the first two questions is shown in Fig.
2. Students were given 12 minutes for the post-test. The first
two questions ask students to rank voltages and potentials in
a given circuit and to give an explanation for their ranking in
each case.
Q
S
X
3
4
Y
5
Z
c. To highlight the different potential values that occur in the circuit, color coding
can be used. Mark all points on the wires (entire sections of the circuit) that have
equal potential with the same color (e. g., red for the highest occurring potential,
blue for the lowest, etc.).
This section illustrates three of our contributions to the
course described above: the third tutorial worksheet, which
introduced the concept of potential; the post-test to evaluate
this tutorial; and one exam question that we designed and
administered.
This tutorial was used in a large lecture hall during lecture
time. To allow teaching assistants to reach everyone, students
were asked to only occupy every other row of seats. Also,
students were asked to work in groups of three and work was
structured in segments of about fifteen minutes. After each
segment, a lecturer summarized some but not all answers to
questions of the current segment. This structure ensured that
all students could be made aware of the main points of the
worksheet.
1
2
M ATERIALS U SED
The goal of the third tutorial worksheet was to improve
students’ understanding of voltage and Kirchhoff’s Voltage
Law (KVL) through the introduction of electric potential. Most
parts of the worksheet were adapted from two other tutorials.
The first page revisits the properties of parallel and series
connections and their implications for current and voltage. It
is based on questions from a tutorial that introduces circuit
analysis [6]. The next one and a half pages define and introduce
the electric potential and investigate its properties. They are
based on a whole worksheet on potential difference [7]. The
rest of the tutorial examines the relationship between elements
in a circuit and equations used to describe that circuit. This
part is again based on the worksheet about circuit analysis [6].
Fig. 1 shows an English translation of the second page of the
tutorial.
P
a. Order points P , Q, S, X, Y , and Z according to their
electric potential. Explain.
d. Explain, how you identified the sections with equal potential.
e. How can you use color coding to identify single circuit elements or groups of circuit
elements that are connected in parallel?
2.2 Potential and voltage
a. Is it possible that the potential difference across two circuit elements is equal while
all terminals have different potentials? If so, try to give an example based on the
circuit above. If not, explain why not.
b. Is it possible for two circuit elements to have the same potential at the terminal with
higher potential, respectively, while the potential difference across both elements is
different? If so, try to give an example based on the circuit above. If not, explain
why not.
Fig. 1.
Translation of the second of four pages of the tutorial.
About two months later, the exam was given. All questions
c
Christian H. Kautz, Tutorien zur Elektrotechnik, •Pearson
Studium, 2011
were multiple
choice questions and students were
not asked to
explain their reasoning. A circuit from the first of eight exam
questions is shown in Fig. 3.
IV.
R ESULTS
In the post-test and the exam, we observed two things: The
answers concerning voltage and potential often contradicted
each other. Also, the majority of students gave answers that
violated KVL.
The post-test shown in Fig. 2 was administered to 311
students. Among their responses were 27 distinct rankings of
voltages and 47 distinct rankings of potentials. Table I gives an
overview of the most prominent rankings. The correct answers,
printed in bold face, were given by only 3 % and 1 % of the
students, respectively.
Of the 219 students who ranked both quantities, 84 %
gave inconsistent answers. Their ranking of the potentials
contradicted their voltage ranking. For example, many students
correctly stated VAD > VAB , but then answered that potentials
D and B were equal (φD = φB ).
Fundamentals of EE I
QUIZ
Course of Study: _____________
Matriculation Number: _____________
The circuit at right contains two identical batteries, which can be
treated as ideal voltage sources. The short line indicates the
negative clamp of the battery, the long line the positive clamp.
The symbol at node D indicates, that the potential there is zero.
Bulbs 1 and 2 are also identical, switch S is open at first.
When the switch is open:
1.
Rank the three voltages VAB, VAC VAD by their absolute value. Please use the relational
operators >, < and =. Please indicate, if two voltages are equal or a voltage is equal to zero.
Answer:
X
I3
I2
I1
A
C
B
D
Y
E
Z
Fig. 3. Circuit given in the exam with translation of the accompanying text.
TABLE II.
A NSWERS TO SELECTED QUESTIONS IN THE EXAM ABOUT
THE CHANGES TO QUANTITIES AFTER REMOVING BULB D.
Explanation:
Answer2
2. Rank nodes A through D by their electric potential ϕA through ϕD.
Answer:
a
Task
Explanation:
The switch is now closed:
Fig. 2.
I
The battery in the circuit at right can
be treated as an ideal voltage source.
The long line indicates the positive
terminal. The five bulbs are identical.
All occurring voltages are within operating range of the bulbs.
Translation of the first part of the post-test.
3. After the switch is closed, does bulb 1 glow brighter, equally bright or less bright than before
the switch was closed?
¨ equally bright
less bright AND CORRECT
¨ I am unsure
TABLE¨ I.brighter OVERVIEW
OF MOST ¨COMMON
ANSWERS TO
Explanation:
THE POST- TEST SHOWN IN F IG . 2.
4
5
6
7
8
Quantity
|VXY |
|VYZ |
|I|
|I1 |
|I3 |
TABLE III.
(b) Part 2 of post-test.
(a) Part 1 of post-test.
increases
b
stays
the same
c
decreases
to 0
d
decreases,
but not to 0
21 %
12 %
29 %
49 %
53 %
18 %
16 %
41 %
42 %
33 %
51 %
62 %
1%
1%
2%
11 %
10 %
30 %
11 %
11 %
R ELATIVE DISTRIBUTION FOR PAIRS OF ANSWERS TO
QUESTIONS FROM THE TASKS IN TABLE II.
4. After the switch is closed, is the absolute of voltage VAC larger, equal to or less than before the
12
switch
was closed?
Voltage
Ranking
¨ brighter
¨ equal
VACExplanation:
> VAD > VAB = 0
VAC > VAD > VAB
Quantity
2%
1%
¨ less
Potential Ranking2
¨ I am unsure
φA = φB > φD > φC
Quantity
1%
φA > φB = φC = φD
18 %
φA > φB = φC > φD
9%
VAD > VAB = VAC = 0
40 %
5. State for every node, if the electric potential at this node does change or does not change when
φA > φB > φC = φD
5%
VADthe
>switch
VAB is
=closed.
VAC
19 %
φA > φ B > φ C > φ D
5%
VAB = VAD >Potential
VAC =changes/
0
5%
φA > φExplanation
5%
D > φB = φC
does not change
blank Node
2%
A
blank
15 %
B
C
D
The first three tasks in the exam asked students to compare
the brightness of certain bulbs in the circuit in Fig. 3. In
dion.timmermann@tuhh.de
Winter
2013/2014
the
subsequent five tasks, students were asked if the
absolute
values of certain quantities would (a) increase, (b) stay the
same, (c) decrease to 0, or (d) decrease, but not to 0, when
bulb D was removed from its socket. The quantities considered
were the voltage between X and Y (VXY ), the voltage between
Y and Z (VYZ ), and the currents I, I1 , and I3 . The students’
answers to these five questions can be found in Table II with
correct answers indicated in bold.
Instead of looking at individual answers to tasks in the
exam, we will examine the consistency of students’ answers
with KVL: One closed loop in the circuit from Fig. 3 is across
bulb C, bulb D, and the battery. The text in the exam states
that the battery can be treated as an ideal voltage source. Table
III(a) shows the distribution of answers to the questions about
the changes to |VXY | (task 4) and |VYZ | (task 5) when bulb D is
removed. The battery voltage stays the same when the bulb is
removed. From KVL, it follows that the sum of the voltages
VXY and VYZ also has to stay the same. For the underlined
combinations of answers, VXY + VYZ cannot stay the same, as
e. g. both quantities decrease. 70 % of the students gave such
an answer that is inconsistent with KVL.
We will now contrast this by examining the consistency
of students’ answers with Kirchhoff’s Current Law (KCL).
1 Most
students did not use absolute value signs in their responses. As all
three voltages are positive, the absolute value signs are not printed here.
2 Correct answers indicated by bold font.
(a) For all 483 students who answered both tasks. Distractors as
shown in Table II.2
(b) For all 196 students who answered tasks 7 and 8 and chose b
in task 6. Distractors as shown in
Table II.
Task 5 (VYZ )
Task 4 (VXY )
a
a
b
c
d
2%
2%
8%
0%
b
c
Task 8 (I3 )
d
2 % 14 % 2 %
5% 7% 3%
7 % 33 % 2 %
1% 7% 2%
Combinations conflicting KVL: 70 %
Task 7 (I1 )
a
b
c
d
a
b
c
59 % 2 % 2 %
12 % 13 % 1 %
0% 0% 1%
2% 1% 1%
d
2%
2%
0%
4%
Combinations conflicting KCL: 21 %
Specificially, we will look at those 41 % of the students that
indicated in task 6 that the current I through the battery
remains the same. While this answer is incorrect and indicates
a previously identified misconception [8], it may or may not
lead to answers consistent with KCL. Table III(b) shows the
distribution of these students’ answers to the questions about
I1 (task 7) and I3 (task 8). During the exam, a clarifying
announcement was made that removing bulb D would result
in the wire between points Y and Z to be disconnected. As
these students indicated that I remains the same and I2 has
to decrease to zero because of the wire being disconnected,
the sum of I1 and I3 has to increase. For the underlined
combinations of answers in III(a) this cannot be true, as neither
of the two currents is indicated to increase. In all, only 21 %
of the students that assumed the battery current to be constant
gave answers that definitely contradict KCL.
Not many students gave explanations to their answers in the
post-test, especially on the later questions. However, several
questions in the post-test and exam have been given in similar
form in tests and exams in other courses. The answers from
students in this course were in general similar to the answers
given in previous years. Thus, we do believe the students in
this year’s course were fairly typical students.
We did expect many students to have trouble with ranking
the voltages and potentials in the post-test because of the open
circuit (open switch) between points B and C. Several faculty
members at our university have mentioned open circuits to be
problematic for students, and data from different publications
support this assumption. In the DIRECT test, a concept inventory on DC circuits, one of the most difficult questions is
question 28 (version 1.0), which asks about the voltage across
an open switch [9]. Hussain also reports that less than 5 % of
students were able to correctly describe the voltage across an
open switch [10]. For the same reason, we expected task 5
(|VYZ |) in the exam to be troublesome for many students. This
proved to be true.
V.
C ONCLUSION
The results from the post-test and exam show that after
instruction the majority of students in this course did not
correctly apply KVL in qualitative questions. As KVL is a
direct consequence of the electric potential, these students also
cannot have a functional understanding of the electric potential.
Of those students that incorrectly treated the battery in
the exam as a current source, about four fifths gave answers
that are consistent with KCL. From the fact that so many
students with a common misconception did not give answers
that contradicted KCL, we conclude that the application of
a conservation law was not problematic for students. Instead,
the recognition that such laws hold for the specific quantities
of voltage and/or potential proved to be difficult. If they had
understood “the physics”, they would have been able to do
“the math”.
The post-test showed that virtually no student had a functional understanding of potential in a test that most instructors
would judge fair. Additionally, the conflicting answers to the
current and voltage ranking questions in the post-test indicated
that students did not see the relationship between voltage
and electric potential. Instead of recognizing both concepts
as different representations of the same underlying physical
entity, and thus getting a better understanding of it, potential
was just another independent concept they had to learn.
In papers by our group and others, such as the Physics
Education Group at the University of Washington, the tutorial
approach has been shown to be quite successful in a variety of
topics and settings [2], [5], [11]. Members of our group have
been involved in the development and use of tutorial worksheets for more than ten years. The fact that our attempt to help
students with their understanding of voltage and potential has
failed so dramatically therefore requires a different explanation
than worksheets being an ineffective method for instruction.
Compared to other courses and curricula, which sometimes
do not introduce electric potential at all or only by definition,
our introduction was quite extensive. Our tutorial worksheet
not only defined the electric potential but also highlighted
its connection to, its similarities to, and differences from
voltage. As almost no student had a functional understanding
of potential in the post-test and exam, we conclude that
much more time or a different approach for teaching the
electric potential is required. Smith and van Kampen presented
one possible approach. They described their curriculum as
successful, although not many details about it were published.
Interestingly, their worksheet introduces potential in a circuit
that closely resembles that from Fig. 2 [12].
Questions concerning an open switch might be a good task
to evaluate students’ understanding of voltage and potential.
While it is commonly known to be troublesome for students,
there seems not to be much research on student understanding of open switches. It has been suggested, that students
incorrectly apply Ohm’s Law for switches [10], [13], but the
problem might also be that students see voltage (or potential)
as a property of current. Thus, without current, there could be
no voltage (or potential).
As there were almost no students that correctly ranked the
potentials in the post-test, we could not see if a functional
understanding of potential also helped the students in understanding the concept of voltage. Besides a better worksheet,
a less challenging set of tasks about the two quantities would
help to investigate this further. Moreover, our results suggest,
that there are specific difficulties with these concepts that have
not yet been identified. An in-depth investigation possibly using student interviews might help to uncover such difficulties.
ACKNOWLEDGMENT
We would like to thank MacKenzie Stetzer for his valuable
comments as well as Prof. G. Ackermann and his staff for
being supportive of this project by allowing us access to their
students.
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[7]
[8]
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[10]
[11]
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