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Investigating student understanding of operational-amplifier circuits
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DOI: 10.1119/1.4934600
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Investigating student understanding of operational-amplifier circuits
Christos P. Papanikolaou, George S. Tombras, Kevin L. Van De Bogart, and MacKenzie R. Stetzer
Citation: American Journal of Physics 83, 1039 (2015); doi: 10.1119/1.4934600
View online: https://doi.org/10.1119/1.4934600
View Table of Contents: https://aapt.scitation.org/toc/ajp/83/12
Published by the American Association of Physics Teachers
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PHYSICS EDUCATION RESEARCH SECTION
The Physics Education Research Section (PERS) publishes articles describing important results from the
field of physics education research. Manuscripts should be submitted using the web-based system that can
be accessed via the American Journal of Physics home page, http://ajp.dickinson.edu, and will be forwarded
to the PERS editor for consideration.
Investigating student understanding of operational-amplifier circuits
Christos P. Papanikolaou and George S. Tombras
Faculty of Physics, National and Kapodistrian University of Athens, Athens 15784, Greece
Kevin L. Van De Bogart and MacKenzie R. Stetzer
Department of Physics and Astronomy and Maine Center for Research in STEM Education,
University of Maine, Orono, Maine 04469
(Received 15 June 2015; accepted 2 October 2015)
The research reported in this article represents a systematic, multi-year investigation of student
understanding of the behavior of basic operational-amplifier (op-amp) circuits. The participants in
this study were undergraduates enrolled in upper-division physics courses on analog electronics at
three different institutions, as well as undergraduates in introductory and upper-division electrical
engineering courses at one of the institutions. The findings indicate that many students complete
these courses without developing a functional understanding of the behavior of op-amp circuits.
This article describes the most prevalent conceptual and reasoning difficulties identified (typically
after lecture and hands-on laboratory experience) as well as several implications for electronics
instruction that have emerged from this investigation. VC 2015 American Association of Physics Teachers.
[http://dx.doi.org/10.1119/1.4934600]
I. INTRODUCTION
Physics education researchers have only recently begun to
focus their attention on the upper-division and advanced laboratory courses that are a hallmark of most undergraduate
physics programs.1 To date, upper-division physics courses
on analog (or hybrid analog and digital) electronics have not
been studied in sufficient depth to inform systematic,
research-based instructional improvements.
While there are many important learning goals associated
with upper-division laboratory instruction both in electronics
and in general (e.g., development of experimental design
abilities and troubleshooting expertise), students are
expected to leave electronics courses with a functional
conceptual understanding of the behavior of electronic circuits so that they may design and construct practical circuits
for both research and real-world applications. Research on
student understanding of circuits, however, indicates that
undergraduates struggle with basic dc circuits in their introductory physics courses,2–4 and some of these difficulties
have been shown to persist both during and after upperdivision electronics courses.5 For all of these reasons, we
have conducted a multi-institutional investigation of student
conceptual understanding of analog electronics, with an
emphasis on both canonical electronics topics (e.g., transistor
and op-amp circuits) and fundamental circuits concepts (e.g.,
Kirchhoff’s rules).
While there has been considerable work on student understanding of basic electric circuits in the PER literature, very
little work has been conducted in upper-division electronics
courses. The work that has been conducted in such courses
in physics has either exclusively focused on student understanding of fundamental circuits concepts5,6 or simply used
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Am. J. Phys. 83 (12), December 2015
http://aapt.org/ajp
electronics as a context in which to study other science education phenomena.7
The engineering education literature includes studies of
both fundamental circuits concepts8 and more advanced
circuits topics such as phase relationships in ac circuits9 and
the behavior of RC filters.10 Much of the work on canonical
electronics topics described in the engineering education
literature has focused on pedagogical approaches and
instructional resources.11 There has been relatively little
work on student understanding of these topics.12 The only
work related to student learning of operational-amplifier
(op-amp) circuits was conducted by Mazzolini et al. and
published in 2011.13 Mazzolini and colleagues developed
and administered a multiple-choice conceptual test on
op-amp circuits in order to assess the impact of a sequence
of interactive lecture demonstrations on student learning in a
first-year unit on electronics. Though the identification of
specific difficulties was not a goal, the authors noted that
students tended to employ “‘shallow learning’ approaches”
in which they memorized standard op-amp “circuit configurations and the gain formulas that apply to these particular
configurations,” leading to errors when, for example, resistor
labels (e.g., R1 and R2) were swapped in a circuit diagram or
circuits were drawn in a non-traditional manner.13
In this paper, we describe an in-depth, multi-institutional
empirical study of student understanding of basic op-amp
circuits guided by the following research questions:
1. To what extent do students develop a functional understanding of basic op-amp circuits after relevant instruction
in an electronics course? In particular:
(a) To what extent are students able to reason productively and/or correctly about op-amp circuits that
C 2015 American Association of Physics Teachers
V
1039
correspond to “perturbations” of canonical op-amp
circuits covered in the course?
(b) To what extent are students able to correctly
describe (qualitatively and/or quantitatively) the
currents and voltages in a canonical op-amp circuit
such as the inverting amplifier?
2. What ideas and approaches, both correct and incorrect, do
students employ when analyzing op-amp circuits?
but the format was modified during the investigation so that
the course now consists of two 50-min lectures and one 2-h
laboratory each week. The course text is now Electronics
with Discrete Components by Galvez,20 whereas Principles
of Electronic Instrumentation by Diefenderfer and Holton21
was used during the first two years of the investigation.
Students typically have one midterm exam and one cumulative final exam.
The goal of the investigation was to characterize student
thinking in sufficient detail to help guide instructional interventions; for this reason, the specific difficulties empirical
framework of the University of Washington Physics
Education Group was adopted.14–16 The participants in the
study were undergraduates enrolled in upper-division
physics courses on analog electronics at three different
institutions, as well as undergraduates in introductory and
upper-division electrical engineering courses at one of the
institutions.
We begin this article with a brief overview of our research
context and methodology (Sec. II). In Secs. III and IV, we
present two tasks used to probe student understanding along
with the data collected and difficulties identified. We then
describe additional data collected in electrical engineering
courses (Sec. V). Finally, in Secs. VI and VII, we discuss
implications for instruction and summarize our findings.
In all three courses, students learn that an operational amplifier (see Fig. 1) is a high-gain differential amplifier, with
non-inverting (þ) and inverting (–) inputs, and is typically
powered by connections to positive and negative rails (e.g.,
615 V). Both inputs are characterized by extremely large
input impedances (modeled as 1 in an ideal op-amp) and
therefore negligible currents. When the op-amp is in a circuit
with negative feedback (e.g., all circuits in Figs. 2 and 3),
Horowitz and Hill use two Golden Rules to describe op-amp
behavior: “I. The output attempts to do whatever is necessary
to make the voltage difference between the inputs zero… II.
The inputs draw no current.”17 At UW and UM, these
Golden Rules are covered explicitly; at UA, the same ideas
are motivated and discussed in instruction, but they are not
formulated as rules.
II. CONTEXT FOR INVESTIGATION
E. Research methods
This study was primarily conducted in upper-division
physics courses on analog electronics at the University of
Washington (UW), the University of Athens (UA) in Greece,
and the University of Maine (UM). Brief course overviews
are provided below.
In this investigation, we wished to examine and document
the level of student understanding of op-amp circuits after
relevant instruction in lecture and laboratory. Free-response
questions requiring explanations of reasoning as well as
closely related task-centered clinical interviews were used in
order to identify persistent and prevalent conceptual and
reasoning difficulties. Free-response probes were included
on course exams and also administered as short ungraded
“quizzes” after relevant instruction. The interviews
employed a think-aloud protocol and were conducted in the
quarter or semester after students had completed their electronics courses.
A. Upper-division laboratory course at the University of
Washington
The junior-level analog electronics course at UW is one
quarter in length (10 weeks), and consists of two 50-min
lectures and one 3-h laboratory session each week. The text
is The Art of Electronics by Horowitz and Hill.17 The laboratories are drawn or adapted from the Student Manual for the
Art of Electronics by Hayes and Horowitz.18 There are
weekly homework assignments and two 50-min exams.
B. Upper-division laboratory course at the University of
Athens
At UA, the analog electronics course is one semester in
length (10 weeks in practice). There are two lectures each
week (for a weekly total of 3.75 h), and one 2-h laboratory
session every other week. The course is typically taken by
juniors. The course text is Introduction to Electronics by
Tombras.19 The laboratories were developed by the textbook
author and are used exclusively at UA. The lecture and laboratory components of the course each have final exams.
D. Brief overview of op-amp coverage
III. PERTURBATIONS OF THE NON-INVERTING
AMPLIFIER
The work of Mazzolini et al. indicated that students
encountered difficulties when they were asked to analyze
standard op-amp circuits drawn in non-traditional ways.13
This suggests that memorization of specific circuits, gain formulas, and key results may play a substantive role in student
ability to solve canonical op-amp circuits successfully. For
this reason, we were interested in exploring how well
students could predict the behavior of circuits that were
C. Upper-division laboratory course at the University of
Maine
At UM, the physical electronics course is one semester in
length (15 weeks), and is the first half of a full-year juniorlaboratory experience. The course originally consisted of one
50-min lecture and one 3-h laboratory session each week,
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Am. J. Phys., Vol. 83, No. 12, December 2015
Fig. 1. Standard schematic of an operational amplifier or op-amp. The
op-amp has two input terminals (the non-inverting input indicated by a “þ”
and the inverting input indicated by a “–”) and one output terminal.
Papanikolaou et al.
1040
Fig. 2. Three amplifiers task in which students are asked to compare the
absolute values of the output voltages from three non-inverting amplifier circuits with identical positive input voltages Vin. Note that the text has been
slightly abridged/paraphrased and that explanations were required.
slight perturbations of standard op-amp circuits. It was hoped
that student responses to such tasks would provide insight
into the extent to which students were simply applying memorized results rather than reasoning through the behavior of
the “perturbed” circuit from fundamental principles and
device rules.
A. Three amplifiers task
In this section, we first describe the three amplifiers task
and then discuss the elements of a correct response to the
task.
1. Overview of task
In the three amplifiers task (Fig. 2), students are shown
three circuits that are all non-inverting amplifiers. Circuit B
is the canonical non-inverting amplifier. In circuit A, a single
10-kX resistor is inserted between Vin and the non-inverting
input of the op-amp. In circuit C, the same resistor is inserted
between the output of the op-amp and the output of the circuit (the point at which VC is measured). All op-amps are
identical and ideal, and all three circuits have identical and
unchanging positive input voltages Vin (from ideal voltage
sources). Students are told to assume that no loads are connected to the outputs of the circuits. Students are asked (a) to
compare the absolute value of the output voltage VB to that
of VA, and (b) to compare the absolute value of the output
voltage VC to that of VB. In an earlier version, students were
asked to rank, from largest to smallest, all three output voltages (VA, VB, and VC) according to absolute value. Since
student performance was similar on both versions, the results
are presented together for simplicity.
2. Correct response
A correct response to the task does not necessarily require
explicit determination of all three output voltages, but
instead relies on a careful analysis of whether or not each
modification to the canonical inverting amplifier (circuit B)
will impact the output voltage. There are many approaches
that students can use to determine VB. For example, students
may simply apply the gain formula for the non-inverting amplifier (1þR2/R1, where R1 corresponds to the 5-kX resistor
and R2 corresponds to the 20-kX resistor) and correctly
determine that VB ¼ 5Vin. Alternatively, students may apply
Golden Rule I to conclude that the voltage at the inverting
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Am. J. Phys., Vol. 83, No. 12, December 2015
input must be Vin. Thus, there is a voltage drop of Vin across
the 5-kX resistor. Since the current through the 5-kX resistor
is equal to that through the 20-kX resistor (due to Golden
Rule II and Kirchhoff’s junction rule), there must be a drop
of 4Vin across the 20-kX resistor and VB ¼ 5Vin. In circuit A,
since there can be no current through and no voltage drop
across the 10-kX input resistor (due to Golden Rule II),
VA ¼ VB ¼ 5Vin. In circuit C, the voltage at the inverting
input is again equal to Vin (from Golden Rule I), and the voltage across and the current through the 5-kX resistor are necessarily the same as in circuit B. The subsequent analysis is
therefore identical, so VC ¼ VB ¼ 5Vin. (Note that the output
voltage of the op-amp in circuit C, 7Vin in this case, must be
larger than in circuit B because there is a single current
through all three resistors and there will be a voltage drop
across the 10-kX output resistor.) All three output voltages
are thus equal in absolute value and non-zero.
B. Overview of student performance on three amplifiers
task
The three amplifiers task has been administered at UW
(N ¼ 160), UA (N ¼ 181), and UM (N ¼ 49) after all relevant
instruction. Results are summarized in Table I and discussed
in detail below.
Approximately one-quarter to one-third of UW and UM
students correctly ranked the absolute values of all three circuits (jVAj ¼ jVBj ¼ jVCj). At UA, roughly 10% of students
gave a correct ranking.22 The percentages of students who
supported a correct ranking with correct reasoning ranged
from 2% to nearly 30% at the different institutions.
At all three institutions, approximately 40%–50% of
students correctly recognized that jVBj ¼ jVAj. A similar percentage (40%–55%) of students indicated that jVBj > jVAj.
Approximately 35%–45% of all students justified this incorrect comparison by explicitly focusing on a voltage drop
across the input resistor. For example, one student wrote:
“Because Vin is the same for all circuits, since A
has a 10 k resistor before the noninverting input,
VþA < VþB, thus VA < VB. Since VA < VB,
more current in case B flows through the 5 k. By V
¼ IR, if R is the same, but I ", V ", so VB > VA.”
Table I. Overview of student performance on the three amplifiers task in
physics courses on analog electronics at three different institutions. The
question is shown in Fig. 2.
Percentage of total responses
UW
UA
UM
(N ¼ 160) (N ¼ 181) (N ¼ 49)
jVAj 5 jVBj 5 jVCj (correct ranking) (%)
Correct reasoning (%)
jVBj 5 jVAj (correct) (%)
Correct reasoning (%)
jVBj > jVAj (%)
Voltage drop due to input resistor (%)
jVCj 5 jVBj (correct) (%)
Correct reasoning (%)
jVCj < jVBj (%)
Voltage drop due to output resistor (%)
jVCj > jVBj (%)
Circuit vs. op-amp output confusion (%)
23
8
40
29
54
44
39
10
46
31
15
5
9
2
40
11
48
39
18
2
42
16
23
8
Papanikolaou et al.
33
29
49
45
41
35
61
51
31
27
8
4
1041
Note that there is no mention of any current through the
input 10-kX resistor. Upon examining all responses given by
UW and UM students to the three amplifiers task and to a
very similar pilot task (administered once only at UW), there
were 121 responses in support of this incorrect comparison;
however, only two out of those responses explicitly attributed the voltage drop to an input current, suggesting an
almost automatic mapping of a voltage drop to the input
resistor. At UA, on the other hand, of the 86 written
responses indicating that jVBj > jVAj, 26 explicitly argued
that there was a voltage drop due to an input current, whereas
43 solely spoke of a voltage drop. The source of this discrepancy between the UA responses and those from UW and UM
students is not clear; the same ideas are stressed at UA,
although they are not formulated as Golden Rule II.23
Performance on the comparison of circuits C and B was
somewhat more varied. The percentage of correct comparisons ranged from 18% at UA to 61% at UM. At all three
institutions, however, approximately 30%–45% of students
stated that jVCj < jVBj. Responses in support of jVCj < jVBj
tended to draw on productive elements of reasoning about
voltage dividers. One student explicitly mentioned the
divider chain, writing:
“In C, the voltage divider now has a voltage drop
across 10 k as well as 20 k þ 5 k so less voltage is
dropped across 20 k þ 5 k and Vc is less.”
This student argued that the addition of a third resistor to the
divider chain meant that less voltage was dropped across the
original two resistors, which is consistent with an incorrect
assumption that the voltage across the entire chain remains
constant. Other students were more explicit about this
assumption. For example, one student wrote:
“Circuit C is similar to B, but the input resistor
from A has taken up residence between the op-amp
output and Vout C, thus creating a voltage
divider….”
This student correctly drew the divider chain for the circuit,
but noted that “jVopamp outj ¼ jVBj.” Indeed, roughly
15%–30% of all students focused on the voltage drop due to
the output resistor and appeared to be implicitly assuming
that the outputs of the op-amps in circuits B and C were
identical. Relatively few of the written responses offered
insight into the thinking behind this assumption. In one
response, however, a student wrote:
“… there is a potential drop across the 10 k
resistor in circuit C between the output of the opamp and the output VC, but circuit B is identical
otherwise, so VB > VC.”
This student appeared to be arguing that since most of the
circuit (i.e., that to the left of and below the op-amp) is
“identical” in both cases, both op-amps should have the
same output. Such responses seem to draw on a combination
of localized and sequential reasoning,24 arguing that any
change after the op-amp shouldn’t impact its output. Of
course, this line of reasoning is inconsistent with the notion
of negative feedback, which is critical for many op-amp
circuits (including the non-inverting amplifier).
Approximately 10%–25% of all students incorrectly
claimed that jVCj > jVBj. The most prevalent line of incorrect
reasoning supporting this comparison (given by roughly
5%–10% of all three populations) involved the erroneous
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Am. J. Phys., Vol. 83, No. 12, December 2015
claim that the additional output resistor increased the gain of
the circuit. For example, one student wrote:
“In circuit C the 10 kX resistor is being added in
series to the others. So 1/AVo ¼ R1/(R1 þ R2 þ R3)
…. So VC ¼ AVoVin ¼ 7Vin > 5Vin….”
If the output of the op-amp were identical to the output of
the circuit (as in circuit B), this reasoning would be correct.
For the given circuit, however, such responses suggest a failure to differentiate between the output of the circuit and the
output of the op-amp.
C. Related interview task
As part of this investigation, think-aloud interviews were
conducted with 31 students: 29 at UA and 2 at UW. Most
were undergraduates who had just recently completed an
electronics course, but three participants from UA were firstyear physics graduate students.25 Rather than presenting
quantitative results about student performance on the threeamplifier interview task, which are generally consistent with
those from the written task, we limit our discussion to the
additional insight into student reasoning provided by the
interviews.
In the interviews, students often invoked a voltage drop
across the input resistor in circuit A without any consideration of whether or not there is current through that resistor,
as illustrated by the following transcript excerpt (in which
we use S to indicate the student being interviewed and I to
indicate the interviewer):
S: I do have a voltage drop here, of course.
I: How do you know that?
S: Because of the resistor! This voltage will be Vin – VR.
I: And how do we get that voltage drop? That is my
question.
S: Because of the resistor.
Even the interviewer’s subtle prompting did not elicit a statement about current through the resistor. The interviewer then
let the student proceed through the rest of the task, and only
revisited this issue more explicitly near the end of the task.
I: Let us go back to something you said before. You said
that there is a voltage drop here before the þ input. I asked
you why and you replied that it happens because of the
resistor. Right?
S: Yes.
I: Remind me: What is the current there, before the þ input?
S: It must be zero… Wait a minute… Now that I think of it,
there can be no voltage drop! There is no current, no voltage drop. I guess I assumed there was a current before.
Only after explicit questioning about current into the noninverting input did the student consider the current (or lack
thereof) through the input resistor, recognize the inconsistency, and revise his response. Even students who arrived at
a correct response without interviewer assistance often struggled to reconcile their initial claims about the voltage at the
non-inverting input with an analysis of the current through
the input resistor:
S: There must be a voltage drop… but I remember that the
op-amp has a high input resistance… so it must be zero current there! But then you have a potential difference across
the 10 k without a current, but that cannot be done!
Papanikolaou et al.
1042
Thus, even in the interviews, we observed a tendency (at
least initially) to associate a voltage drop with the input
resistor.
The interviews were also extremely helpful in clarifying
student thinking about how the behavior of circuit B is
impacted by the addition of an output resistor in circuit C.
Indeed, all four students who concluded that jVCj is greater
than jVBj explicitly confused the circuit output and the
op-amp output (e.g., “The current goes to the 10 k, the 20 k,
then the 5 k… So they are in series… Then it must be more
than [VB]—it must be 7Vin.”). All but two of the students
who concluded that jVCj is less than jVBj talked about the
voltage drop associated with the 10-kX output resistor, as
illustrated below:
S: Basically it will be the 5Vin minus the voltage drop across
this resistor.
I: Why do you decide to say 5Vin?
S: We have the same circuit up to here. So it must be it…
This student focused on the fact that the circuit to the left
and below the op-amp is the same, and incorrectly concluded
that the op-amp output is the same (5Vin), thereby demonstrating the local and sequential reasoning observed in the
written responses.
D. Specific difficulties identified
Student performance on the three amplifiers task suggests
that all students struggled with the application of basic
circuits concepts and op-amp rules to circuits that differ only
slightly from canonical op-amp circuits. Several specific difficulties were identified.
Lack of a functional understanding of Golden Rule II.
Roughly half of the students at all three institutions provided
reasoning when comparing circuits B and A that would only
be appropriate if there were a current into the non-inverting
input of the op-amp. The reasoning given by all of these
students is inconsistent with Golden Rule II, and calls into
question the extent to which students have developed a truly
functional understanding of Golden Rule II (i.e., there is no
current into the inverting and non-inverting inputs due to
their high input impedances). At the very least, many
students are not drawing on the Golden Rules to check the
feasibility of their responses, possibly reflecting a lack of
familiarity and practice with such consistency checking strategies. It is worth noting, however, that some students did
change their interview responses without any prompting after
they considered the high input impedance of the noninverting input.
Tendency to ascribe a voltage drop to a resistor through
which there is no current. From Golden Rule II, there cannot be a voltage drop across the input resistor since there is
no current into the non-inverting input. When examining all
explanations given in support of the jVBj > jVAj responses
(N ¼ 207) on both this task and a related pilot task, only 28
students explicitly mentioned a current through the input
resistor in their written responses. While this doesn’t
preclude the possibility that many of the other students
thought there was a current through the resistor (and into the
non-inverting terminal), it suggests that some students may
in fact be automatically (and possibly subconsciously)
ascribing a voltage drop to the resistor without analyzing the
situation through the more formal lens of Ohm’s law.26
Moreover, given the interview results, it is likely that a
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Am. J. Phys., Vol. 83, No. 12, December 2015
significant percentage of students simply attributed a voltage
drop to the resistor without even considering the presence or
absence of current.
Lack of a functional understanding of Golden Rule I.
Approximately 30%–45% of all students incorrectly claimed
that jVCj < jVBj. Roughly 15%–30% of the students at both
institutions indicated either implicitly or explicitly through
their reasoning that the output voltages of the op-amps in
both circuits are the same. If this were the case, however, the
potential at the inverting input (V–) would necessarily be less
than Vþ ¼ Vin since the “same” output voltage is now
divided over three resistors. Thus, the reasoning given by
nearly one-third of students and the answers given by up to
45% of students are inconsistent with Golden Rule I. Again,
students did not appear to draw on the Golden Rules in order
to test the viability of their responses.
Tendency to reason locally and sequentially about the
behavior of op-amp circuits. Up to one-third of students
assumed that the addition of a resistor after an op-amp
would not change the output of that op-amp. Reasoning that
a change “downstream” will not affect the “upstream”
behavior of the circuit is typically referred to as local or
sequential reasoning, and it is well documented in the literature on introductory circuits.24 Although this particular
instantiation is a relatively clear-cut example of local reasoning, it is somewhat more surprising given the emphasis
on negative feedback and feedback loops during op-amp
instruction. In these less familiar situations, however,
students appear to be relying on reasoning that they have
largely abandoned but may have used when first studying
simple dc circuits.27
E. Discussion
Our findings suggest that students lack a robust conceptual
understanding of the standard non-inverting amplifier circuit
after instruction. Small perturbations (e.g., the addition of a
resistor immediately before or after the op-amp) typically
resulted in up to 50% of students making incorrect predictions about the behavior of the modified circuits. These predictions were often inconsistent with the Golden Rules and
fundamental circuits concepts, and sometimes employed
local (or sequential) circuit reasoning.
IV. BEHAVIOR OF THE INVERTING AMPLIFIER
The findings from the three amplifiers task suggested that
many students likely did not possess a robust understanding
of the behavior of the non-inverting amplifier circuit itself,
even after all instruction on basic op-amp circuits. For this
reason, we were interested in developing a task in which students would be forced to think deeply about the currents and
voltages in another canonical op-amp circuit: the inverting
amplifier. In essence, we wished to document the extent to
which students possessed the level of understanding required
to derive the inverting amplifier’s gain formula.
A. Inverting amplifier task
In this section, we describe the standard inverting amplifier task developed for this investigation and then discuss
some modified versions of the task.
Papanikolaou et al.
1043
op-amp through point D must split into the current down
through point E to the negative rail and the current to the
right through point C. Thus, jIDj > jIAj ¼ jIBj ¼ jICj > 0 and
jIDj > jIEj > 0.
3. Modified versions of task
Fig. 3. Standard version of the inverting amplifier task in which students are
asked to determine the output voltage of the amplifier and to characterize
various currents in the circuit. Note that the text has been slightly abridged/
paraphrased and that explanations were required.
1. Overview of task
In all versions of the task, students are shown the inverting
amplifier circuit in Fig. 3. There are seven points labeled
A–G on the diagram. Students are told that the op-amp is
ideal and that there is no load connected to the output of the
circuit. The input voltage Vin is constant and is equal to
–5 V. In part 1, students are asked to find the value of the
circuit’s output voltage Vout. In part 2, students are asked
to indicate the direction of the current through point A or to
state explicitly if there is no current through that point. In
part 3, students are asked to compare the absolute values of
the currents through points F and G (i.e., into the two inputs)
and to indicate explicitly if any current is equal to zero.
Finally, in part 4, students are asked to rank, from largest to
smallest, the absolute values of the currents through points
A–C. If any of the currents are equal in absolute value or are
equal to zero, students are prompted to indicate that explicitly. For all parts of the task, students are required to explain
their reasoning.
Over the course of this investigation, specific parts of the
task have been modified or added in order to examine student
understanding of the rail currents.
Rail currents: Directions and magnitudes. In some
versions, students are asked to indicate the directions of the
currents through points D and E (i.e., the rail currents). In
other versions, students are asked to compare the absolute
values of the currents through points D and E, and to indicate
explicitly if any current is equal to zero.
Current ranking for points A, B, C, and D. In versions
administered at UW and UM, students are asked to rank,
according to absolute value, the currents through points A, B,
C, and D from largest to smallest. This prompt replaced the
original, simpler part 4.
B. Overview of student performance on basic inverting
amplifier task
Versions of the inverting amplifier task have been administered at UW (N ¼ 183), UA (N ¼ 242), and UM (N ¼ 45)
after all relevant instruction. In this section, we describe student performance on the standard version shown in Fig. 3
(see Table II).
On part 1, between 55% and 70% of students at all three
institutions gave correct values or expressions for Vout. An
Table II. Overview of student performance on the inverting amplifier task in
physics courses on analog electronics physics at three different institutions.
The question is shown in Fig. 3.
Percentage of total responses
2. Correct response
In order to most clearly outline the reasoning required for
all parts of this task (including additional prompts in the
modified versions), we present an analysis of the entire circuit. From Golden Rule II, the currents through points F and
G are both equal to zero due to the high input impedances of
both inputs (part 3). From Golden Rule I, the electric potential at point F is 0 V, so the current through point A is to the
left because the potential at point F is higher than Vin ¼ –5 V
(part 2). From Kirchhoff’s junction rule, the current through
the 20-kX resistor is equal to that through the 10-kX resistor,
so the current through point B is up the page. Since there is a
single current through both resistors, a voltage drop of 5 V
across the 20-kX resistor implies that there is a voltage drop
of 2.5 V across the 10-kX resistor (from Ohm’s law). Thus,
Vout ¼ þ2.5 V (part 1). Because no load is attached to the
output of the circuit, the current through B must equal that
through C via the junction rule. Thus, jIAj ¼ jIBj ¼ jICj (part
4). Since the direction of current is from high to low potential, the currents through points D and E are both oriented
down the page. By recognizing that the total current into the
op-amp must equal the total current out of the op-amp (from
the junction rule) and that the currents through points F and
G are both zero (from Golden Rule II), the current into the
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Am. J. Phys., Vol. 83, No. 12, December 2015
UW
UA
UM
(N ¼ 183) (N ¼ 242) (N ¼ 45)
1. Vout 5 12.5 V (correct) (%)
Sign error (%)
2. Left (correct) (%)
Correct reasoning (%)
Right (%)
Current from Vin or Vin to Vout (%)
Correct reasoning (%)
Zero (%)
Golden Rule II (%)
3. jIFj 5 jIGj 5 0 (correct) (%)
Correct reasoning (%)
VF ¼ VG ¼ 0 (%)
4. jIAj 5 jIBj 5 jICj > 0 (correct) (%)
Correct reasoning (%)
jIAj ¼ jIBj > jICj ¼ 0 (%)
jIAj ¼ jIBj ¼ jICj ¼ 0 (%)
Reasoning for jICj ¼ 0:
Overgeneralization of Golden Rule II (%)
Junction rule with rail difficulties (%)
jICj > jIAj ¼ jIBj > 0 (%)
jIAj ¼ jIBj > jICj > 0 (%)
55
17
63
50
27
5
5
10
4
79
55
5
37
27
16
1
…
6
5
11
8
57
5
21
14
59
18
2
12
4
50
27
5
7
4
16
8
…
5
2
8
2
Papanikolaou et al.
69
13
69
60
29
9
7
2
2
84
73
2
53
36
11
0
…
9
0
16
0
1044
additional 5%–15% of students made a sign error, indicating
that the output voltage would be negative. On what is arguably the most standard question one can pose about an
op-amp circuit, approximately 20% or more of all populations gave fundamentally incorrect responses.
On part 2, approximately two-thirds of students at UW
and UM correctly recognized that current is to the left
through point A, with roughly half of all students in these
populations giving correct reasoning. Only 21% of UA students indicated the correct direction, with 14% supporting
their correct answers with correct reasoning. Roughly 30%
of students at UW and UM and 60% of students at UA incorrectly indicated that the current through point A is to the
right. Of these incorrect responses, approximately 20%–30%
were supported by statements indicating that current either
comes from Vin into the circuit or from Vin to Vout. For example, one student wrote, “Current flows from the power supply
through the 20 k R then through the 10 k R.” Another simply
stated that current “flows from in toward out.” This idea that
current comes from the voltage source seemed to be the most
prevalent incorrect explanation offered for a current to the
right through point A. An additional 20%–25% of these
incorrect responses at UW and UM were supported by correct reasoning (e.g., the direction of the current is from high
to low potential), suggesting that some students may have
been treating Vin as a positive voltage. It is also conceivable,
however, that some of these students were trying unsuccessfully to reconcile correct formal reasoning with a perhaps
more intuitive sense that current should come from the
voltage source. Between 2% and 12% of students at all three
institutions claimed that there was no current through point
A. A large portion of these responses (40%–100%) were
incorrectly justified on the basis of Golden Rule II, suggesting that many students either failed to recognize that point A
is located to the left of the junction or did not realize that it
is possible to have current through the feedback loop. For
example, one student wrote, “Zero. The op-amp draws no
current.”
On part 3, the majority of students (80%–85%) at UW and
UM and half of the UA students correctly indicated that the
currents through points F and G were both zero. Roughly
55%–75% of students at UW and UM gave correct answers
with correct reasoning; just over 25% of students gave
correct reasoning at UA. Approximately 5% of all students
incorrectly argued that both currents are zero because both
points are grounded. For example, one student wrote:
“The current through F & G are both equal to
zero because F & G’s potentials are equal
(Golden rule of op amps) & the value of VG is
zero, because it’s grounded.”
On part 4, between 35%–55% of all UW and UM students
and only 7% of UA students correctly determined that, for
the currents through points A–C, jIAj ¼ jIBj ¼ jICj > 0. The
most prevalent incorrect ranking, given by about 10%–15%
of all students, was jIAj ¼ jIBj > jICj ¼ 0. (A small percentage
of students gave a similar ranking in which jICj was not equal
to zero.) An additional 10%–15% of all students indicated
that jICj > jIAj ¼ jIBj > 0. Nearly 10% of the students at UA
stated that jIAj ¼ jIBj ¼ jICj ¼ 0, whereas almost no one at
UW and UM gave this response.
When examining responses in support of jIAj ¼ jIBj > jICj
¼ 0, the explanations tended to focus on why the current
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Am. J. Phys., Vol. 83, No. 12, December 2015
through point C must be zero. After a careful analysis of all
justifications for IC ¼ 0, regardless of ranking, two distinct
categories emerged.
Tendency to generalize Golden Rule II inappropriately
(i.e., assumption that there is no current into or out of any
terminal of the op-amp). Many students cited the rules/properties of the op-amp as justification for IC ¼ 0. For example,
one student wrote:
“We know because of the axioms of the op-amp,
that there is no current flowing in or out of the
op-amp at the inputs/outputs.”
This student appears to have incorrectly generalized Golden
Rule II to the op-amp output in addition to its inverting and
non-inverting inputs. Some student explanations were considerably more specific, with one student noting, “Op amp
output gives no current because it has infinite output
impedance.” Although students are typically taught that the
op-amp’s extremely low output impedance is an important
and useful characteristic, this student appears to have applied
the idea of infinite input impedance to the output of the opamp. Approximately 5%–10% of all explanations fell into
this category.
Failure to account for the correct behavior of the rails
when applying Kirchhoff’s junction rule to the op-amp (e.g.,
incorrectly stating IF 1 IG 5 IC or treating ID 5 IE 5 0).
Several students’ explanations for IC ¼ 0, however, differed
substantively from those described above. In particular, some
students gave responses similar to the following:
“No current at Vin6 because op amp doesn’t
intake current. The op-amp has 0 current flow
through it so all outputs and inputs of op-amp
[are] 0 current.”
This student emphasized that because no current enters the
inverting (–) or non-inverting (þ) inputs, there is no current
through the op-amp, and therefore, there is no current through
the output of the op-amp. It is important to stress that this student recognized the importance of applying the junction rule
to the terminals of the op-amp and tried to ensure that the
junction rule was satisfied; however, the student did not correctly account for those currents entering and exiting the opamp via the power rails. Given that these rails are omitted
from many diagrams of op-amp circuits (including, for example, the inverting amplifier diagrams that appear in the texts
of all three courses),28 it is perhaps not surprising that some
students struggle to apply the junction rule to the op-amp.
Indeed, if a student solely focuses on the two inputs and the
output, the op-amp’s behavior appears to violate the junction
rule. Up to 5% of all students failed to account for the power
rails correctly when applying the junction rule to the op-amp.
Student understanding of the rail currents is discussed in more
detail with the results from the modified versions of this task.
While the jIAj ¼ jIBj > jICj ¼ 0 response was the most common incorrect ranking, roughly 10%–15% of all students
indicated that jICj > jIAj ¼ jIBj > 0. Despite the prevalence of
this ranking, our analysis suggests that many different lines
of reasoning (all at very low percentages) were used to justify this response. Some students appeared to treat the circuit
as if there were a load, whereas others focused on the low
output impedance of the op-amp or the absence of an output
resistor. A few students emphasized the notion of feedback
and the idea that only a part of the output is fed back to the
Papanikolaou et al.
1045
op-amp, but they incorrectly applied these arguments to
currents rather than voltages.
Although not common at UW or UM, 8% of students at
UA stated that jIAj ¼ jIBj ¼ jICj ¼ 0. Most of these students
argued that there is no current anywhere in the circuit due to
the absence of a load. Interview data (discussed in Sec.
IV D) provided considerably more insight into the reasoning
behind these responses.
C. Overview of student performance on questions
involving rail currents
In this section, we discuss student performance on additional prompts designed to explore student understanding of
currents to and from the power rails.
When asked about the rail current directions (through
points D and E), only about 20% of UA students (N ¼ 242)
and 55% of UW (N ¼ 56) students correctly indicated that
both currents were down the page. Roughly 20% of students indicated that the currents through both points were
zero. Approximately 15% of UW students and nearly 30%
of UA students indicated that the currents through both
points were directed toward the op-amp (i.e., down through
point D and up through point E). While explanations in
support of this answer were often unclear, up to 5% of all
students focused on the idea that the rails supply power to
the op-amp (e.g., “E and D are going into the op amp to
power it.”).
When asked to compare the relative magnitudes of the rail
currents, only 1% of UA students (N ¼ 242) and 13% of UW
students (N ¼ 76) correctly recognized that the absolute
value of the current through D is larger than that through E.
Approximately 45% of all students indicated that the absolute values of the currents through D and E are equal and
non-zero. Roughly 10% of UW and UA students supported
this comparison by noting that the rail voltages are identical
in absolute value (e.g., “Both of the absolute values of voltages are the same, and they have the same resistance so
they must have the same current.”). In addition, approximately 10% of UW students simply focused on the role of the
rails in providing power when arguing that the two current are
equal (e.g., “The abs value of current through D & E are the
same & the function of this power source is to power the opamp, so the currents will be the same.”). Roughly 15% of all
students said that both currents are equal to zero. Many of
these responses were consistent with an overgeneralization of
Golden Rule II. For example, one student wrote:
“Under ideal operation, op amps sink & source no current.
Since power dissipation ¼ I2R, the op amp receives no
power at inputs and sources no power. This leads me to
conclude that jIDj ¼ jIEj ¼ 0 assuming no heat losses etc.”
An additional 15% of the jIDj ¼ jIEj ¼ 0 responses given by
UA students were supported by the argument that there is no
current anywhere in the circuit in the absence of a load.
In general, however, the explanations offered by students
responding to the two questions on rail currents were rather
unclear and hard to categorize; many students did not justify
their answers at all. The results suggest that most of the students had not had previous opportunities to think carefully
and deeply about the role of the power rails.
It is therefore perhaps not surprising that only about 15%
of students at UW (N ¼ 56) and UM (N ¼ 45) gave correct
rankings of the currents through points A–D. Moreover, only
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Am. J. Phys., Vol. 83, No. 12, December 2015
3 UW students and one UM student gave correct explanations for why the current through D must be larger than that
through points A, B, and C.
D. Related interview task
The inverting amplifier task was also used for interviews at
UA (N ¼ 29) and UW (N ¼ 2). Performance was similar to that
reported in Sec. IV B, so we limit our discussion to student reasoning about the op-amp as a junction and the power rails.
The interviews provided considerably more insight into
how students are applying Kirchhoff’s junction rule to the
op-amp. For example, in the excerpt below, the interview
participant had just stated that there was no current through
point C. (As before, I is the interviewer and S is a student
interview participant.)
I: Why did you think that current is not coming out of the
op-amp?
S: Since no current is coming in! We said that the potential at
F and G is zero, so there is no current coming in the inputs.
I: Ok. Are you thinking of some op-amp rule or maybe a
Kirchhoff’s knot?
S: No, just Kirchhoff.
This student claimed that since no current is coming into the opamp through points F and G (argued via incorrect reasoning),
there is no output current. In this and other interviews, students
frequently applied the junction rule to the op-amp without
accounting for the correct behavior of the power rails.
Students at UA who concluded that there was no current
through points A–C due to the absence of a load also drew
upon the junction rule in the same manner.
S: Sure, A ¼ B. But there is no load. Kirchhoff must hold,
the op-amp is a knot, so no current anywhere. Because
nothing comes into F and G, nothing comes out at C… so A
¼ B ¼ 0. There is no way we can have a current.
This student applied Kirchhoff’s junction rule at multiple
points in the circuit, including the op-amp itself, and incorrectly concluded that all three currents must be zero since
jIAj ¼ jIBj ¼ jICj and there can be no current through points F
and G. By failing to recognize that the op-amp output can
serve as a source and sink of current due to the power rails,
this student argued that a load was required for non-zero
current. The interviews suggest that the responses from UA
students claiming that jIAj ¼ jIBj ¼ jICj ¼ 0 likely stem from
difficulties with the analysis of the op-amp itself (described
earlier) combined with a more thorough application of the
junction rule to the circuit.
In the interviews, students struggled to compare the rail
currents. Only 5 of 31 students correctly concluded that
jIDj > jIEj. Twelve students argued that the currents are equal
and non-zero for a variety of reasons, including the claim
that they are the “only” currents, the fact that the rail voltages are equal in absolute value, etc. Two students said that
both currents were equal to zero, focusing on the related
ideas of high input impedances and no input currents. In general, there was a sense that the students had not really
thought about the rail currents before. One student even
noted, “I have not found that anywhere, nobody explained
them to us….” Indeed, the op-amp power rails are typically
introduced in instruction (primarily due to the constraints
they impose on Vout), but rarely analyzed in detail when considering the behavior of basic op-amp circuits.
Papanikolaou et al.
1046
E. Additional difficulties identified
In addition to the difficulties highlighted in Sec. IV B, several other difficulties were identified through the use of the
inverting amplifier task.
Tendency to apply Kirchhoff’s junction rule inconsistently in op-amp circuits. A significant percentage of
students gave rankings in which the currents through points
A–C were not equal. The junction rule was often applied to
certain junctions but not others. A focus on student rules
about op-amps (e.g., an overgeneralized Golden Rule II) often seemed to preclude the application of the junction rule to
the B-C junction. While most students have a basic understanding of the junction rule, the salience of specific features
in these advanced circuits appeared to trigger alternative
lines of reasoning, making it more difficult for students to
recognize the need to apply the junction rule in such cases.
Kautz reported similar phenomena in the context of ac
circuits.9
Tendency to assume current always comes from Vin or
always goes from Vin to Vout. A significant percentage of
students (5%–15%) expressed the idea that current always
comes from the power supply, apparently ignoring the sign
of Vin and treating the supply as though it is only able to output current. This is reminiscent of and may be related to the
tendency of introductory students to think of the battery as a
constant current source, a prevalent difficulty that has been
documented in the literature.24 Moreover, for some students,
the voltage input and output of an op-amp circuit seemed to
correspond to the input and output of current, respectively.
As a result, these students struggled to analyze the circuit’s
currents in a productive manner and typically failed to draw
on relevant fundamental circuits concepts.
Tendency to argue that I 5 0 if V 5 0 at a point. Some
students claimed that there is no current through points at
electrical ground. While students rarely articulated this line
of reasoning in detail, we suspect it stems from either confusion between voltage at a point (an electric potential) and
voltage across an element (an electric potential difference)
or an incorrect generalization of Ohm’s law for a single
point. Approximately 5% of all responses to the inverting
amplifier question included such arguments.
V. EXTENSION TO ELECTRICAL ENGINEERING
COURSES
At UM, we are currently investigating the learning and
teaching of thermodynamics and electronics in both physics
and engineering courses. As part of this project, we have
been examining student understanding of basic op-amp circuits in electrical engineering courses. Data were collected
in three courses: an introductory circuits course required for
all electrical and computer engineering (ECE) majors (typically taken in the sophomore year), an introductory circuits
courses taken by non-ECE engineering majors (in the junior
or senior year), and a junior-level analog electronics course
required for all ECE majors. Neither introductory course has
a formal lab component, but students in the introductory
course for ECE majors purchase a circuits kit and assemble
basic op-amp circuits.
A. Overview of student performance in engineering
courses
The percentage of correct rankings on the three amplifiers
task ranged from 14% to 24% in the engineering courses.
(See Table III, which also includes data from the UM
physics course for comparison.) Many engineering students
argued that jVBj > jVAj due to the voltage drop across the
input resistor, but essentially none mentioned an input current. Students also often stated that jVCj < jVBj for reasons
similar to those of physics students.
On the inverting amplifier task, many engineering students incorrectly concluded that the current through point A
is to the right (see Table IV). Approximately 5%–25% of
engineering students explicitly claimed that current comes
from the power supply. Only 9%–33% of students in the engineering courses correctly stated that jIAj ¼ jIBj ¼ jICj > 0,
and a large percentage of rankings (60%–90%) were
inconsistent with the junction rule. Roughly 15%–25% of
engineering students claimed there was no current through
point C, with up to 10% overgeneralizing Golden Rule II. In
addition, many engineering students did not demonstrate an
understanding of the role of the power rails in their
responses to the modified current ranking task (e.g.,
20%–40% indicated that ID was zero, 15%–25% indicated that jICj > jIDj > 0, and 10%–20% indicated that
jICj ¼ jIDj > 0).
B. Discussion
While there was considerable variation in the exact percentages of students giving particular incorrect responses,
results suggest that engineering students encounter essentially the same conceptual and reasoning difficulties as
physics students. It is thus clear that the difficulties identified
Table III. Overview of student performance on the three amplifiers task in electrical engineering courses at UM. Data from the UM physics course from Table I
are included for comparison. The question is shown in Fig. 2.
Percentage of total responses
Engineering
Physics
Circuits, non-ECE majors (N ¼ 63) Circuits, ECE majors (N ¼ 97) Electronics (N ¼ 59) Electronics (N ¼ 49)
jVAj 5 jVBj 5 jVCj (Correct ranking) (%)
jVBj 5 jVAj (Correct) (%)
jVBj > jVAj (%)
jVCj 5 jVBj (Correct) (%)
jVCj < jVBj (%)
jVCj > jVBj (%)
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Am. J. Phys., Vol. 83, No. 12, December 2015
24
44
43
40
38
16
19
46
48
29
57
11
14
53
42
22
61
8
Papanikolaou et al.
33
49
41
61
31
8
1047
Table IV. Overview of student performance on the inverting amplifier task in electrical engineering courses at UM. Data from the UM physics course from
Table II are included for comparison. The question is shown in Fig. 3.
Percentage of total responses
Engineering
1. Vout 5 12.5 V (correct) (%)
2. Left (correct) (%)
Right (%)
Zero (%)
3. jIFj 5 jIGj 5 0 (correct) (%)
4. jIAj 5 jIBj 5 jICj > 0 (correct) (%)
jIAj ¼ jIBj > jICj ¼ 0 (%)
jIAj ¼ jIBj ¼ jICj ¼ 0 (%)
jICj > jIAj ¼ jIBj > 0 (%)
jIAj ¼ jIBj > jICj > 0 (%)
Circuits, non-ECE majors (N ¼ 76)
Circuits, ECE majors (N ¼ 101)
Electronics (N ¼ 68)
Electronics (N ¼ 45)
54
29
53
13
67
11
21
0
4
22
68
46
44
10
70
9
7
1
13
10
82
62
28
7
79
33
12
0
3
33
69
69
29
2
84
53
11
0
16
0
in the UW, UA, and UM physics populations are not simply
artifacts of approaches to op-amp instruction solely
employed in physics courses.
VI. IMPLICATIONS FOR INSTRUCTION
In all populations studied, we have found evidence of substantive difficulties associated with the analysis of canonical
op-amp circuits after relevant instruction on op-amps. We
are actively working to develop and refine research-based
laboratory activities and in-class tutorials similar to those in
Tutorials in Introductory Physics29 that draw upon our findings. Here, however, we provide some general recommendations for instruction on op-amp circuits based on the results
of our investigation.
Emphasize the role of the op-amp’s power rails. The
majority of students struggled with the role of the power
rails in the behavior of a simple op-amp circuit. Even those
students who attempted to apply the junction rule to the device were frequently unable to do so productively because
they did not realize that the rails enable the op-amp output
to source and sink current. Without a robust understanding
of the power rails, the op-amp may seem to violate the
junction rule. Simple measurements of rail currents in the
laboratory for amplifiers with values of Vin that are positive, negative, and zero can be particularly valuable for
helping students recognize that the relationship between
the rail currents depends on whether or not the op-amp is
sinking or sourcing current. Questions similar to the inverting amplifier task can provide students with additional
practice.
Use “perturbed” circuits in instruction, even if they
aren’t touchstone circuits. There is a tendency to focus on a
small number of touchstone circuits that may serve as building blocks in more complex designs. Our research, however,
has indicated that slight modifications of standard circuits,
even if those modifications are pointless or non-ideal from a
design perspective, can be productive in that they force students to set aside memorized gain formulas and reason from
fundamental concepts and device rules.
Examine currents and voltages in canonical circuits.
While students also need to think about a basic op-amp
circuit as one “chunk” in a larger circuit design, our findings
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Am. J. Phys., Vol. 83, No. 12, December 2015
suggest that students are often unable to describe what is
actually going on in some of these very basic op-amp circuits. Providing students with an opportunity to think about
currents and voltages at various locations in a circuit for one
or more values of Vin can help students reason through the
behavior of the circuit on their own.
Provide additional support for students as they apply
fundamental circuits concepts in more advanced contexts.
The kinds of circuits covered in analog electronics provide
additional challenges to the successful application of
Kirchhoff’s rules, Ohm’s law, etc. Indeed, students are
encountering new devices with new rules, new conventions
(e.g., voltage at a point), and new and increasingly abstract
representations (e.g., the omission of rails in many diagrams). It is perhaps not surprising that students applied the
junction rule to certain junctions in the inverting amplifier
but not to others; in some cases, overgeneralized op-amp
rules appeared to be more salient for students and somehow
hindered their ability to apply the junction rule to the B-C
junction (Fig. 3). By providing students with opportunities to
apply these fundamental concepts in challenging circuits
throughout the course, students can receive productive feedback and be guided to recognize the importance of, and
subtleties associated with, applying fundamental circuits
concepts in all circuits.
Provide explicit prompts for students to check that their
predictions are consistent with the Golden Rules and
fundamental circuits concepts. We have found that students
often make predictions about the behavior of basic op-amp
circuits that are inconsistent with the Golden Rules and/or
fundamental circuits concepts. We believe that, after students have analyzed (individually or collaboratively) a circuit that elicits such inconsistencies, it may be beneficial to
prompt students explicitly to verify that their predictions are
consistent with the Golden Rules and fundamental concepts.
This may help students explore productive lines of reasoning
that they did not originally consider.
VII. CONCLUSIONS
In this multi-year study, we investigated student conceptual understanding of basic operational-amplifier circuits in
the context of upper-division physics courses on analog
Papanikolaou et al.
1048
electronics as well as electrical engineering courses on introductory circuits and analog electronics. We found that students in all populations struggled to analyze basic op-amp
circuits after relevant instruction; in particular, tasks requiring predictions of the behavior of “perturbed” op-amp circuits or detailed examinations of the currents and voltages in
a canonical circuit allowed for the identification of prevalent
difficulties. Students often gave reasoning and drew conclusions that were inconsistent with the Golden Rules. In
addition, students largely failed to demonstrate a basic
understanding of the role of the op-amp’s power rails, and
many students did not apply fundamental circuits concepts
consistently and systematically. Our findings suggest a
need for increased emphasis on certain relevant topics (e.g.,
the power rails) and for research-based and researchvalidated instructional materials that address the difficulties
identified.
ACKNOWLEDGMENTS
The authors would like to acknowledge Leslie J.
Rosenberg for his critical role in catalyzing this multi-year
investigation of student understanding of analog electronics.
The authors appreciate the substantive contributions made
by colleagues in the University of Washington Physics
Education Group, especially Lillian C. McDermott and Peter
S. Shaffer. The authors also acknowledge the efforts of
Benjamin Pratt, Matiah Shaman, Mark Onufer, David P.
Smith, and Jason Alferness. In addition, the authors would
like to thank many faculty members who graciously allowed
us to collect project data in their classrooms: Ali Abedi, Nuri
Emanetoglu, Duane Hanselman, David Kotecki, Miguel
Morales, Donald Mountcastle, Hector E. Nistazakis, and
Leslie Rosenberg. This material is based upon work
supported by the National Science Foundation under Grant
Nos. DUE-0618185, DUE-0962805, DUE-1022449, and
DUE-1323426.
1
B. M. Zwickl, N. Finkelstein, and H. J. Lewandowski, “The process of
transforming an advanced lab course: Goals, curriculum, and assessments,” Am. J. Phys. 81, 63–70 (2013).
2
L. C. McDermott and P. S. Shaffer, “Research as a guide for curriculum
development: An example from introductory electricity. Part I:
Investigation of student understanding,” Am. J. Phys. 60(11), 994–1003
(1992); printer’s erratum to Part I, 61(11), 81 (1993).
3
P. S. Shaffer and L. C. McDermott, “Research as a guide for curriculum development: An example from introductory electricity. Part II:
Design of instructional strategies,” Am. J. Phys. 60(11), 1003–1013
(1992).
4
P. V. Engelhardt and R. J. Beichner, “Students’ understanding of direct
current resistive electrical circuits,” Am. J. Phys. 72, 98–115 (2004).
5
M. R. Stetzer, P. van Kampen, P. S. Shaffer, and L. C. McDermott, “New
insights into student understanding of complete circuits and the conservation of current,” Am. J. Phys. 81, 134–143 (2013).
6
J. G. Getty, “Assessing inquiry learning in a circuits/electronics course,”
Proceedings of the 39th IEEE International Conference on Frontiers in
Education Conference (IEEE Press, Piscataway, NJ, 2009), pp. 817–822.
7
See, for example: E. C. Sayre, M. C. Wittmann, and J. R. Thompson,
“Resource selection in nearly-novel situations,” AIP Conf. Proc. 720,
101–104 (2004). In this proceedings paper, the authors reported some data
on student understanding of diodes and simple diode circuits.
8
See, for example: A. O’Dwyer, “Prior understanding of basic electrical
circuit concepts by first year engineering students,” All-Ireland Society
for Higher Education (AISHE) Conference, NUI Maynooth, 2009; T.
Ogunfunmi and M. Rahman, “A concept inventory for an electric circuits course: Rationale and fundamental topics,” in Proceedings of the
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Am. J. Phys., Vol. 83, No. 12, December 2015
2010 IEEE International Symposium on Circuits and Systems (ISCAS),
2010, pp. 2804–2807; N. H. Hussain, L. A. Latiff, and N. Yahaya,
“Alternative conceptions about open and short circuit concepts,”
Procedia—Soc. Behav. Sci. 56, 466–473 (2012).
9
C. H. Kautz, “Development of instructional materials to address student
difficulties in introductory electrical engineering,” in Proceedings of the
40th SEFI Annual Conference 2012, Lisbon, Portugal, 2011, pp.
228–235.
10
P. Coppens, M. de Cock, and C. H. Kautz, “Student understanding of filters in analog electronics lab courses,” in Proceedings of the 40th SEFI
Annual Conference, Thessaloniki, Greece, 2012.
11
See, for example: A. S. Andreatos and G. S. Kliros, “Identifying transistor roles in teaching microelectronic circuits,” in Proceedings of 2006
IEEE Mediterranean Electrotechnical Conference, Bernalmadena,
Spain, 2006, pp. 1221–1224; A. Andreatos and G. Michalareas,
“Engineering education e-assessment with Matlab; Case study in electronic design,” in Proceedings of the 5th WSEAS/IASME International
Conference on Engineering Education, Heraklion, Greece, 2008, pp.
172–177.
12
See, for example: M. F. Simoni, M. E. Herniter, and B. A. Ferguson,
“Concepts to questions: Creating an electronics concept inventory
exam,” in Proceedings of the 2004 American Society for Engineering
Education Annual Conference & Exposition, 2004; T. A. Hudson, M.
Goldman, and S. M. Sexton, “Using behavioral analysis to improve student confidence with analog circuits,” IEEE Trans. Educ. 51(3),
370–377 (2008).
13
A. Mazzolini, T. Edwards, W. Rachinger, S. Nopparatjamjomras, and O.
Shepherd, “The use of interactive lecture demonstrations to improve students’ understanding of operational amplifiers in a tertiary introductory electronics course,” Lat. Am. J. Phys. Educ. 5(1), 147–153 (2011); available at
http://www.lajpe.org/march11/LAJPE_456_Alexander_Mazzolini_preprint_
corr_f.pdf.
14
L. C. McDermott, “Millikan lecture 1990: What we teach and what is
learned—Closing the gap,” Am. J. Phys. 59(4), 301–315 (1991).
15
L. C. McDermott, “Oersted medal lecture 2001: Physics education
research—The key to student learning,” Am. J. Phys. 69(11), 1127–1137
(2001).
16
P. R. L. Heron, “Empirical investigations of learning and teaching, Part I:
Examining and interpreting student thinking,” in Enrico Fermi Summer
School on Physics Education Research, edited by E. F. Redish and M.
Vicentini (Italian Physical Society, Varenna, Italy, 2003), pp. 341–350.
17
P. Horowitz and W. Hill, The Art of Electronics, 2nd ed. (Cambridge U.
P., NY, 1991).
18
T. C. Hayes and P. Horowitz, Student Manual for the Art of Electronics
(Cambridge U.P., NY, 1992).
19
G. S. Tombras, Introduction to Electronics, 2nd ed. (Diavlos Books,
Athens, 2006).
20
E. J. Galvez, Electronics with Discrete Components (John Wiley & Sons,
Hoboken, NJ, 2013).
21
A. J. Diefenderfer and B. E. Holton, Principles of Electronic
Instrumentation, 3rd ed. (Brooks/Cole, Belmont, CA, 1994).
22
On the tasks in this study, we have often noted significant performance differences between UA and the other two institutions. We speculate that
they may be related to systemic factors at UA including larger class sizes
(200þ), less laboratory time (labs every other week), and lower course
attendance.
23
It is possible that some of the UA students were more attentive to the relationship between the voltage across and the current through an ohmic
resistor.
24
See, for example, Ref. 2.
25
These graduate students were either working as TAs in the electronics
course or just beginning research in electronics.
26
Such behavior is consistent with a “knowledge in pieces” or resources model of student thinking (in which, for example, a student
might draw upon a more informal notion that “increased resistance
leads to less result”) and dual process theories of reasoning. See, for
example: D. Hammer, “Student resources for learning introductory
physics,” Phys. Educ. Res., Am. J. Phys. Suppl. 68(7), S52–S59
(2000); “Misconceptions or p-prims: How may alternative perspectives of cognitive structure influence instructional perceptions and
intentions?,” J. Learn. Sci. 5(2), 97–127 (1996); A. diSessa,
“Towards an epistemology of physics,” Cognit. Instruct. 10(2–3),
105–225 (1993); M. Kryjevskaia, M. R. Stetzer, and N. Grosz,
“Answer first: Applying the heuristic-analytic theory of reasoning to
Papanikolaou et al.
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examine student intuitive thinking in the context of physics,” Phys.
Rev. ST Phys. Educ. Res. 10, 020109 (2014).
27
A similar phenomenon has been reported in Ref. 5.
28
Many electronics texts show and discuss the rails when first introducing the
op-amp, but subsequently omit them from diagrams and discussions when
covering the canonical op-amp circuits. This was generally true for the texts
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Am. J. Phys., Vol. 83, No. 12, December 2015
and materials used in this study, although both Galvez and Tombras explicitly discuss rail currents at least once when covering the canonical circuits.
29
L. C. McDermott, P. S. Shaffer, and the Physics Education Group at the
University of Washington, Tutorials in Introductory Physics, 1st ed.
(Prentice Hall, Upper Saddle River, NJ, 2002); Instructor’s Guide
(Prentice Hall, Upper Saddle River, NJ, 2003).
Papanikolaou et al.
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