Title of the Paper
A ONE-SEMESTER COURSE IN ELECTRIC CIRCUIT ANALYSIS
Clayton R. Paul[1]
1 Abstract
A one-semester course in electric circuit analysis is described. This course has been class tested for the past five years and is taught to electrical, computer, biomedical, mechanical, environmental and industrial engineers in one class as a part of a new, integrated engineering curriculum. Because of the need to prepare students from all of these engineering disciplines, a substantial revision of the course content and the sequence of topics was required. The new content and organization is discussed and a rationale for each is given.
I. Introduction
Electrical engineering (EE) as well as computer engineering (CE) courses, like all other engineering courses, need a continuing reassessment of not only their content but also the sequencing of the material in them. This reassessment is needed in order to effectively and efficiently address how each course interfaces with and serves the remaining courses in the curriculum. These remaining courses are also under constant reassessment due to advances in technology. A considerable amount of new technology has been inserted into the EE/CE curriculum in the last twenty years. Perhaps the largest segment of this new technology is the increased use of integrated circuits and computers. There is constant pressure to insert new courses into an already crowded curriculum in order to include this new technology. The length of all undergraduate engineering programs have remained four years since the inception of the discipline of engineering, and there is little indication that the degree process will be lengthened to more than four years. With the insertion of new courses such as those demanded by the increased use of digital technology, many of the traditional courses have either been relegated to elective courses or compressed in their topic coverage. For example, traditional topics such as automatic controls and feedback, as well as communications are no longer required courses in most curricula. An electrical engineer can graduate without knowledge of AM and FM communication techniques unless he/she took the elective course covering those aspects. In order to make way for the increasing new technologies, several other traditional courses have been compressed from two semesters to one semester. For example, electromagnetics, arguably the most fundamental course for EE’s as well as CE’s, has been compressed from a two-semester sequence to a one-semester sequence in a majority of schools. This has resulted in rather severe “course compression”. The four-year engineering curriculum is essentially a zero-sum game; insertion of a new course requires the elimination or compression of an existing course. This compression of the curriculum requires that we become more efficient and effective in how we cover the material in these courses.
Perhaps the most important skill set possessed by an EE or CE is the ability to analyze lumped electric circuits. All other EE/CE courses except the digital ones require that the student has mastered this skill. For example, in electronics courses lumped-circuit models are constructed from the physical laws that govern the electronic elements. In order to understand electronic circuits, the student must be able to analyze the resulting circuit model. All EE/CE students take, as their first discipline-specific course, an electric circuit analysis course in their sophomore year. This course has remained one of the few, if not the only, two-semester required courses in the curriculum. (Many technical electives in the senior year are two-semester sequences.) Due to future technological developments, there will no doubt be ever-increasing pressure to reduce this course to one semester since the four-year curriculum is unlikely to be increased in length. This argues for a re-examination of the content and delivery of that course.
Prior to World War II, textbooks on electric circuit analysis concentrated on electric power and machinery [1]. In the early 50’s and 60’s, the dramatic increase in electronics technology required a radical revision of such textbooks. One of the earliest such revisions is the text by Hayt and Kemmerly which bears the copyright of 1962 [2]. Subjects such as controlled sources and two-port networks which were absent from the earlier texts were included in these post WWII texts to accommodate the dramatic change in technology. Virtually all of these later texts contained additional topics such as Fourier and Laplace transform methods. In addition, certain topics such as magnetic circuits and extensive analysis of transformers that were included in the pre WWII texts were retained in those texts. Hence the traditional circuit analysis course grew from one semester in length to two semesters to accommodate this additional material. In the 70’s and 80’s the EE curriculum was re-examined and significant changes were made. Additional courses were inserted. It is common to follow the circuit analysis courses with a course on Signals and Systems. This course covers the Fourier and Laplace methods as well as more detailed aspects such as filters, constructing Bode plots, etc. In spite of this, the circuit analysis texts still contained those topics.
This paper describes a one-semester course in electric circuit analysis that focuses on the efficient presentation of the fundamental circuit analysis skills in order to fit into a severely compacted curriculum. It has been documented in the form of a textbook [3]. The course is a 4 credit hour course consisting of three 50 minute lectures each week along with a 2 hour “recitation” in which students work problems under the guidance of the instructor. We have found that this recitation is a valuable part of the course. At this stage in their academic career, students need considerable guidance in showing them how to conceptualize the solution of electric circuits. The recitation provides this important guidance. The course has been taught in the Mercer University School of Engineering as a part of an novel integrated engineering curriculum. All engineering students - electrical, computer, mechanical, bio-medical, environmental and industrial - are required to complete this course. This integrated curriculum is intended to address the emerging industry requirement that all engineers are increasingly being required to work in teams and therefore must be cognizant of the disciplines of the other team members. In the past non-EE’s were required to take a circuit analysis course in their senior year. Those students tended to view that course as an unnecessary requirement. Students from other disciplines who take the course described in this paper in their sophomore year understand its importance through the repeated emphasis on design and team work throughout the Mercer engineering program.
The basic outline of the course consists of four major sections. These are
1) basic concepts and laws (the physical concepts of voltage and current as well as the basic laws of Kirchhoff’s voltage and current laws),
2) resistive circuit analysis,
3) frequency-domain or phasor analysis, and
4) time-domain or transient analysis.
Although this overall organization seems to be quite traditional, there are unique aspects of reorganization and emphasis that we now discuss.
II. Basic Concepts and Laws
Students today are less likely than those of the past to simply accept a teacher’s assertion that “they will need to know this”. Students require that a link between the abstract symbols and concepts and the real world be shown. Hence it is important to explain the physical meaning of the important items of voltage and current in terms of the movement of charge and the energy required to move that charge.
Mastery of the important laws of Kirchhoff, Kirchhoff’s voltage law (KVL) and Kirchhoff’s current law (KCL), are, in this author’s view, the most important skills to be mastered in the course. Without mastery of these seemingly simple skills, none of the later methods can be accomplished. Students have little difficulty understanding, in general terms, how to apply these laws. The principal difficulty they have in successfully implementing the laws is in making sign errors. Sign errors in applying KCL are fairly simple to avoid if the student is taught to equate current entering a node to current leaving that node and in determining these to focus on the direction of the current arrow and not its numerical value. Once KCL has been written at a node with the determination of whether the current is entering or leaving that node being determined solely by the direction of the current arrow, then the actual numerical value (which may be negative) is substituted.
Kirchhoff’s voltage law seems to be more prone to sign errors than KCL. The author has developed a seemingly innocuous but very effective way of writing KVL that helps to avoid sign errors. This is illustrated in Fig. 1. The method is to simply draw a bisector separating the loop (around which KVL is to be written) into two parts: a part on the left and a part on the right. Then label one end of the bisector with a + sign and the other end with a - sign. Once this is done, simply equate the voltages on the left to those on the right by focusing on the polarity of each voltage and not the numerical value. If the polarity of a voltage agrees with the polarity assigned to the bisector then it is entered as a positive quantity (again irrespective of its numerical value) otherwise it is entered as a negative value. Once this has been done the actual numerical value (which may be negative) is substituted. For example, consider the loop of Fig. 1(a). Writing KVL in this fashion gives
[pic]
giving V=16V. Observe that the values are contained in parentheses in order to obtain the proper sign with regard to the [pic] signs assigned to that voltage. Perhaps the most common use of KVL is to determine an unknown voltage in a loop where all the other voltages are known. In this case the bisector is drawn such that one side has only the unknown on it as illustrated in Fig. 1(b). In addition, the + of the bisector is chosen to coincide with the + of the unknown. For this application we obtain
[pic]
which agrees with the previous result.
The remaining variable to be computed is the power delivered to or by an element. Students easily master the fact that the power is the product of voltage and current. The problem that must be corrected is in determining the proper sign. Again, it helps to instruct them to focus on the + sign of the voltage polarity and the direction of the current arrow and disregard (for the moment) their numerical values. If the current arrow points into the positive voltage terminal then the power delivered to the element is the product VI. It is at this point that the actual numerical values can be substituted and not before.
[pic]
III. Resistive Circuit Analysis
It is important to realize and to point out to the student that the resistive circuit analysis skills of this segment are critical to mastery of the remaining two important segments: frequency-domain (phasor) and time-domain (transient) analysis. Circuits in these two last major categories will be transformed into “resistive-like” circuits wherein the resistive circuit analysis skills will, once again, be called on. Deficiencies in mastering the resistive circuit analysis skills will therefore prevent the student from mastering the skills of these remaining two segments.
It is at this point that we introduce the independent voltage and current sources and the resistor. In the previous section where KVL and KCL were introduced, the elements are represented by boxes. This was intended to demonstrate that KVL and KCL are independent of the types of circuit elements. Students seem to readily grasp Ohm’s law but routinely make errors in getting the correct sign in that law. As in the power calculation we must emphasize the passive sign convention and write Ohm’s law based on the polarity of the assumed voltage and the direction of the assumed current arrow. Once this is done, the numerical values of the voltage or current (which may be negative) are substituted.
The sequence of resistive analysis skills is as follows and in this order:
1) single-loop and single-node-pair circuits,
2) voltage and current division,
3) single-source circuit reduction,
4) the direct solution method,
5) source transformation,
6) superposition,
7) Thevenin and Norton equivalent circuits, and
8) Node-voltage and mesh-current equations.
The first actual analysis skills we introduce are for the single-loop and single-node-pair circuits. These are perhaps the most fundamental circuits of the course since many of the remaining analysis methods seek to reduce a circuit to one of these forms. Hence the student must be able to rapidly and flawlessly analyze such circuits.
At this point the important time saving techniques of voltage and current division are introduced. The students seem to master this skill easily except that they have the usual propensity to make a sign error. It helps to instruct them to observe the polarity of the source that is being divided (the + of the voltage source or the direction of the arrow in a current source) in order to determine the correct sign.
The next analysis skill we introduce is that of circuit reduction. This method works only for circuits that contain only one independent source. It requires the continuous use of series and parallel resistor reductions. The students adapt well to the general concept. The main problem they encounter in this technique is when they fail to observe when a voltage or current has been lost in a reduction and they mistakenly assign that variable in the reduced circuit. For example, in reducing two resistors that are in parallel, the currents through each of them are lost but the common voltage across them is retained. Similarly, in reducing two resistors that are in series, the voltages across each of them are lost but the common current through them is retained. Also they must develop a willingness to assign additional, as yet unknown, voltages and currents at each stage of the reduction to assist them in the solution.
A powerful analysis method that works well for simple circuits is the direct application of KCL, KVL and Ohm’s law which is referred to as the direct method. The key to this method is to select an unknown voltage or current to be solved for and write all other voltages and currents in terms of this variable using KCL, KVL, and Ohm’s law. It is imperative that the student resist the temptation to define anymore unknowns. We wish to avoid obtaining simultaneous equations in two or more variables. For example, consider the circuit shown in Fig. 2(a). Although there are two unknowns to be solved for, [pic] and [pic], we select the controlling variable of the controlled source, [pic], to first solve for. Once [pic] has been solved for the solution for [pic] is easily determined in terms of [pic]. Draw three checkoff boxes beside the circuit for each of the three laws that will and must be used. Then proceed through this list to see which can be applied. In Fig. 2(b) we apply KCL at the upper node and check it off. Next we apply Ohm’s law to the two resistors and check that off. This leaves KVL around the loop giving
[pic]
which is readily solved for [pic]. Then [pic] can be determined as
[pic]
[pic]
The remaining fundamental solution method is that of source transformations. Students adapt to this method very well. However, there are two mistakes that are commonly made. The first mistake is to get the polarity of transformed source wrong. This can be easily remedied by instructing them in the memory aid that “the voltage source is pushing current out of its + terminal”. Another very common mistake is illustrated in Fig. 3 where a voltage or current associated with the resistor is incorrectly assumed to remain with that resistor in the transformed circuit.
The above methods of (1) solution of single-loop and single-node-pair circuits, (2) voltage and current division, (3) circuit reduction, (4) the direct method, and (5) source transformations form the basic solution techniques. Every simple circuit which one is willing to solve by hand (as opposed to computer simulation) can be easily solved with one or more of these methods. Furthermore, the remaining methods, superposition and the Thevenin and Norton equivalent circuits, rely on the use of the above simple methods because they reduce the circuit to single-loop or single-node-pair circuits or simple circuits where voltage and current division or circuit reduction or the direct method or source transformations can be used to complete the solution.
[pic]
Once the above five basic solutions methods are mastered we introduce the three remaining solution methods - superposition, Thevenin and Norton reductions, and node-voltage and mesh-current methods. Superposition is readily understood and applied with skill by the students so long as they have mastered the above five basic solution methods to analyze the resulting single-source circuits. Thevenin and Norton reduction methods are also mastered rather easily by the students because they have mastered the above five basic methods that are used to determine the open-circuit voltage, short-circuit current, and the Thevenin resistance.
The last resistive circuit analysis methods to be introduced are the node-voltage and mesh-current methods. Traditionally these methods have been introduced before the above fundamental resistive circuit analysis methods are introduced. We have chosen to introduce them as the last methods in our list for the following reasons. The law of primacy exerts a strong influence on the student’s selection of a solution technique in that first things learned are often used to the exclusion of later things learned. If the node-voltage or mesh-current methods are introduced first, one routinely sees students analyzing relatively simple circuits with these methods and generating simultaneous equations to be solved. The same circuit can be analyzed much more rapidly and with more insight obtained using the simpler resistive analysis techniques previously introduced. Another important reason for not introducing the node-voltage or mesh-current methods before the simpler analysis methods are mastered is that solution of the node-voltage or mesh-current equations simply give numerical answers and provide little insight into the factors that determine this result. It is important to remember that the reason we teach students the circuit analysis skills is not so much for the purposes of analysis but more importantly to prepare them for design. The simpler resistive analysis techniques of single-loop and single-node-pair circuits, circuit reduction, voltage and current division, the direct method, source transformations, and superposition, show how various parts of the circuit affect the other parts and act to determine the result, whereas the node-voltage and mesh-current methods do not.
IV. Frequency-Domain or Phasor Methods
The next most fundamental and frequently-used skill of an EE is the ability to analyze a circuit which has an independent source whose time variation is sinusoidal. EEs call upon this skill in electronics, signals and systems, energy conversion and electromagnetics courses. The sinusoidal excitation of a circuit is a basic problem that permeates all of electrical engineering.
Charles Steinmetz developed the phasor method for analyzing these circuits in 1893. The simplicity of the method is beautiful, and its logic should be made clear to beginning students. The heart of the method is Euler’s identity
[pic]
where [pic]. (Other disciplines such as Physics use i for this quantity.) The phasor method relies on superposition. Hence the circuit must be linear for it to apply. It is a simple and logical method for the student to understand if its development is approached in the following manner. Replace the desired source, [pic] or [pic] with a source whose value is
[pic]
By Euler’s identity this result can be written as
[pic]
Show the student the replacement of the actual source with this source and that the new source consists of two sources (voltages sources in series and current sources in parallel). Once the student is reminded that superposition holds, he/she can readily see that the solution for this new source has a real part that is due to the cosine part and an imaginary part that is due to the sine part. In addition, the inductor and capacitor terminal relations can be easily shown to be “resistor-like” and hence all the resistive circuit analysis methods previously discussed can be applied to solve this resulting phasor circuit with only the added burden of complex arithmetic.
The student easily sees that, by superposition, if the original source is a sine then the result is a sine with the same frequency but different magnitude and phase angle. Similarly, if the original source is a cosine then the result is a cosine with the same frequency but different magnitude and phase angle. It is at this point that we often make an unnecessary and confusing requirement of the student: the student is required to convert all sine sources to an equivalent cosine form or all sine sources to an equivalent cosine form. (The particular requirement depends on the author of the text although the majority seem to require converting to a cosine.) This conversion is an unnecessary and, moreover, an error-prone requirement of the student. The student knows that the conversion from sine to cosine or cosine to sine is made with the addition or subtraction of [pic] but frequently adds [pic] when [pic] should have been subtracted and vice-versa. Granted, it is a relatively straightforward matter to instruct the student in the correct conversion from sine to cosine and vice-versa, but this is unnecessary and introduces a potential pitfall for the student. An additional hindrance to understanding what is going on is the propensity to simply give the student the phasor circuit already labeled with element impedances. Simply giving the student the phasor circuit may be the reason for the common requirement to convert all time-domain sources to either cosine or sine form. If the student is simply given the phasor circuit, then the student and the teacher must agree on a common time form for the source. This author believes that the proper sequence is to give the time-domain circuit and ask the student to determine the time-domain solution for an element voltage or current in that circuit. Thus the student must (a) replace the sine or cosine source with [pic], suppress the [pic] (after explaining that it is common to all the circuit variables and need not be carried along) (b) compute the element impedances, [pic] and [pic], (c) solve for the result using resistive circuit analysis skills, and (d) finally convert the complex result back to the time domain as a sine if the original source was a sine and a cosine if the original source was a cosine. This logical process is illustrated in Fig. 4. For circuits that contain more than one sinusoidal source, it is imperative to stress the fact that we still have the use of superposition in the time domain. In other words, regardless of the frequency or form of the sinusoidal sources, they can be put in individual phasor circuits which are individually solved and each complex result returned to the time domain with the result being the sum of these time-domain responses.
[pic]
V. Time-Domain (Transient) Analysis Methods
At this point it should be stressed that the student’s body of accumulated knowledge and skills allow the solution of a wide range of circuits: those that have sinusoidal and dc ([pic]) sources. Hence the student obtains a large sense of accomplishment and satisfaction. For circuits whose time variation is not sinusoidal or dc, or for circuits that have some disturbance such as a switching operation, the usual method is to (a) write the differential equation relating the desired response to the source, and (b) solve that differential equation by classical methods. While this method is traditional and somewhat straightforward in principle, it too is fraught with some practical difficulties for the student. First the student is required to derive the differential equation relating the desired response to the source. The second problem that arises with the classical method is the determination of the required initial conditions on the variable of interest for second-order circuits: its value at [pic] and its derivative at [pic]. Students have great difficulty determining the derivative of the desired variable at [pic].
All of these difficulties with the classical solution method can be avoided by introducing the Laplace transform solution method. It has been said that students at this stage do not comprehend the complex variable implications of what they are doing. An equal argument can be made that students at this stage don’t understand the programming details in the background of a spreadsheet program like Excel but they are able to use it to effectively solve problems. Why should an early introduction of the Laplace transform be any different? The key to an effective teaching of the Laplace transform at this early stage is to avoid introducing any unnecessary detail. For example, it is not necessary to show the use of the Laplace transform in solving ordinary differential equations. All that is needed to be discussed are (a) linearity, (b) time shift, (c) s shift, and (d) the transform of the first derivative. With the transform of the first derivative, the impedances of the R, L, and C elements, including the initial conditions, can be readily understood by the student. Once again, the Laplace-transformed circuit becomes a “resistive-like” circuit having impedances of R, sL, and [pic], and all the previously-discussed resistive circuit analysis techniques can be brought to bear to determine the transform of the desired variable. The final skill that needs to be introduced is the partial fraction expansion which the students seem to readily master.
What have we omitted by going directly to the Laplace transform rather than first introducing the classical solution? Any of the insights gained with the classical solution can be easily duplicated with the Laplace transform method. For example, for first-order circuits, the important concepts are those of the transient or natural response, the forced or steady-state response, and the time constants [pic] and RC. Consider a first-order circuit having a switching operation as shown in Fig. 5(a). We first draw the [pic] circuit as shown in Fig. 5(b) by replacing the inductor with a short circuit since the only source in this circuit is dc. If the source were sinusoidal, the result could be obtained with the phasor method and evaluated at [pic]. This is another argument for the introduction of the phasor method prior to the time-domain methods. Next we draw the transformed t>0 circuit as shown in Fig. 5(c). Again resistive circuit analysis skills are called upon to determine the transform of this result as
[pic]
This result can be obtained by a variety of resistive circuit analysis methods such as superposition, source transformation or the direct application of KVL, KCL, and Ohm’s law. Superposition was used to obtain the above result and the first part was obtained using current division. The partial fraction expansion of this result easily yields, using linearity,
[pic]
The inverse transform of this result is easily obtained as
[pic]
Each of these terms can be discussed in the usual fashion as with the classical solution. The classical solution does not provide any more insight than the Laplace transform method since in both methods we look at the final mathematical relation for interpretation rather than the intermediate steps leading to that solution. The Laplace transform method provides a significant aspect of insight over the classical solution. The contribution of the initial condition, the initial inductor current [pic], becomes apparent through its source in the transformed circuit of Fig. 5(c). This makes it crystal clear to the student that initial conditions should be treated no differently than actual independent sources.
[pic]
Introducing the Laplace transform method early also provides the student with an early introduction and development of fundamental skills that will be reinforced and investigated more deeply in the later Signals and Systems courses. This author has found that this process provides a more rapid assimilation of the Laplace method in these later courses. Furthermore, there seems to be nothing lost in going directly to it rather than starting with the classical solution method. All important aspects of the solution obtained through the classical solution process can be obtained with the Laplace transform method and more simply. In addition there are a number of insights that can be obtained from the Laplace transform solution such as the role of initial conditions that are obscured in the classical solution.
VI. Summary
This paper describes a logical and pedagogically effective sequence of topics and methods for teaching electric circuit analysis in a one-semester course. A cursory glance might indicate that there is nothing new here. However, this author believes that a student’s learning of a method can be hindered by unnecessary detail and failure to focus on basic techniques that relate only to learning the skill. It is important to keep a logical and consistent pattern that the student can perceive and continue to follow. For example, by continually forming the methods to relate back to resistive circuit analysis skills such as in the phasor method and the Laplace transform method, the student can begin to “distinguish the trees from the forest”. Minimizing the things a student has to memorize and placing the material in a form that he/she can build on with previously learned methods is crucial to providing the student with the ability to rapidly and flawless analyze a given circuit and to understand why he/she is doing it. A student who has not mastered circuit skills (in particular, the resistive circuit analysis skills) will have little hope of mastering the later material since virtually all of that material relies on developing a circuit model from which an understanding of the physical behavior is obtained. Hence mastery of these circuit analysis skills is fundamental to the success of EE/CE students. It is also important to point out the common mistakes students make so that they can see why they are incorrect and avoid duplicating them in future solutions.
References
[1] R.M. Kerchner and G.F. Corcoran, Alternating-Current Circuits, second edition, John Wiley & Sons, NY, 1943.
[2] W. H. Hayt and J.E. Kemmerly, Engineering Circuit Analysis, McGraw-Hill, NY, 1962.
[3] C.R. Paul, Fundamentals of Electric Circuit Analysis, John Wiley, NY, 2001.
.
1 Clayton R. Paul
Clayton R. Paul was born in Macon, GA on September 6, 1941. He received the B.S. degree, from The Citadel, Charleston, SC, in 1963, the M.S. degree, from Georgia Institute of Technology, Atlanta, GA, in 1964, and the Ph.D. degree, from Purdue Univesity, Lafayette, IN, in 1970, all in electrical engineering. He is emeritus professor of electrical engineering at the University of Kentucky where he was a member of the faculty in the department of electrical engineering for 27 years. He is currently the Sam Nunn Eminent Professor of Aerospace Systems Engineering and Professor of Electrical and Computer Engineering in the department of electrical and computer engineering at Mercer University in Macon, GA. He is the author of 12 textbooks on electrical engineering subjects, and has published numerous technical papers, the majority of which are in his primary research area of electromagnetic compatibility (EMC) of electronic systems. From 1970 to 1984, he conducted extensive research for the US Air Force in modeling crosstalk in multiconductor transmission lines and printed circuit boards. From 1984 to 1990 he served as a consultant to the IBM corporation, in the area of product EMC design. Dr. Paul is a Fellow of the Institute of Electrical and Electronics Engineers (IEEE) and is an Honorary Life Member of the IEEE EMC Society. He is also a member of Tau Beta Pi and Eta Kappa Nu.
-----------------------
[1] Department of Electrical and Computer Engineering, Mercer University, Macon, GA.
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