A Concept Diagram for Transient Circuit Analysis Instruction

Transient circuit analysis is a particularly challenging topic in introductory circuit analysis courses. In illustrating the solution of transient analysis problems, many textbooks emphasize the procedure of problem solving. While such focus provides valuable training for students, we believe an in-depth understanding of critical concepts in transient process is crucial and needs to be emphasized in classroom instruction. In this paper, a concept diagram is presented that focuses on three critical time instants for transient analysis. Important concepts for each time instant are clearly presented by this simple diagram, which integrates all relevant concepts in an organized manner. The diagram is an effective tool for outlining and reviewing transient analysis concepts. Background and Motivation This paper addresses the instruction of an important subject in introductory electrical engineering (EE) courses, transient circuit analysis. It is well known that most engineering students, including both EE and non-EE majors, find transient analysis challenging and hard to grasp. In fact, ba'ied on our teaching experience, non-EE students consider transient analysis the most difficult topic in introductory circuit analysis courses. When there is a switch in the circuit, the change in the switch position results in a sudden change in the circuit structure. If there is a capacitor or an inductor in the circuit, a transient process will begin as the circuit rebalances itself toward a new steady state. Since capacitor voltage and inductor current cannot change suddenly to achieve a new steady state value, transient analysis is very important when we want to study how the circuit parameters, such as voltages and currents, evolve with time. Practical examples include charging a camera flashlight, shutting off a motor while avoiding a sudden voltage change to destroy the motor, etc. The major challenge students face when learning transient analysis is to establish fundamental connections between key concepts in differential equations and in transient process. This subject not only requires solid foundation in first and second order ordinary differential equations, but also the ability to describe and analyze a physical process with mathematical equations, a critical skill to be developed for engineering students. Many textbooks provide a comprehensive coverage of materials necessary for transient analysis, and illustrate the problem solving process through various example problems (e.g., Hambley, 2007; Kerns & Irwin, 2004; Rizzoni, 2007 & 2009). The presentation often focuses on the procedure of problem solving, such as identifying the initial and final conditions, establishing the differential equations, and determining the parameters in the general-form solution of differential equations. While such presentation provides valuable training for students to follow certain standard solution procedure·, a coherent insight into critical concepts in transient process is crucial and needs to be emphasized in classroom instruction. Otherwise, students can easily lose the view of "big picture" when trying to solve problems using the standard procedure. It is our belief that a unified framework is needed to present essential transient process concepts in a succinct and logical manner. This will help students acquire an organic system of knowledge on this subject. For such a purpose, a concept diagram is proposed in this paper for effective instruction of transient analysis. The diagram concisely presents key concepts in transient analysis and illustrates their logical connections. These include: steadystate equivalent models for capacitors and inductors, continuity of Ve (voltage across capacitor) and h (current across inductor), initial and final conditions identification, and differential equations describing transient processes using KCL (Kirchhoff's Current Law) or KVL (Kirchhoff's Voltage Law). The Concept Diagram Figure 1 is the concept diagram developed to facilitate the instruction of transient analysis. The focus is on three critical time instants essential to the solution of transient circuits: t = 0-, the instant before the switch changes position; t = O+, the instant immediately after the switch changes position; and t -? ro, after the switch has been changed for a long period of time. Steady Initial State l · . Switch Changes Condition t = 0-----t=O+ C: open Vc(O-) = Vc(O+) L: short IL (0-) =IL (0+) Vc(O+) IL (O+) Figure J. Concept diagram for transient analysis Steady Transient Process State 2 t ~00 Diff. Equation by C: open KCL or KVL L: short As can be seen, key information at each time instant is provided in the diagram. Also, the transitions between time instants are clearly described. Explanation of the Concept Diagram 1) t = 0-, the instant before the switch changes position This time instant is important because it is useful in determining the initial condition needed to solve a transient circuit problem. At this instant, circuit is in a steady state, Le., voltages and currents are not varying with time (assuming DC sources). Therefore, there is no current across capacitors and no voltage across inductors. As a result, capacitors are considered open circuits, and inductors are considered short circuits. The two equivalent models allow us to solve for V c(O-) or h (0-) conveniently. 2) t = O+, the instant immediately after the switch changes position This time instant is important because it is the starting point of the transient process, i.e., the initial condition for the differential equation corresponding to the transient process. The value of Vc(O+) or h (O+) needs to be obtained for the solution of the differential equation. In the case of second order transient circuits, dV ddt(O+) or dh ldt(O+) is also needed. 3) The transition from t = 0to t = O+ The common problem for many textbooks is the lack of emphasis on distinguishing between t = 0and t = O+. It is true that from the perspective of problem solving, Vc(O+) equals to Vc(O-), and h (0+) equals to h (0-). However, understanding the difference between the two time instants and their connection is essential to transient analysis. The sudden change in the switch position causes a change in the circuit structure, which forces the circuit to rebalance itself and establish a new steady state through the transient process, during which voltages and currents evolve towards their new stabilized values. Since the quantities that cannot change suddenly are Ve and h, we can easily get the value of the initial condition V c(O+) or h (O+ ), which is identical to V c(O-) or h (0-). For this very reason, we always establish differential equations in terms of V c or h. The continuity of these two quantities makes it possible for us to determine the initial conditions. In the case of second-order circuits, dV ddt(O+) or dh /dt(O+) also needs to be obtained from lc(O+) or VL (O+ ), which can be solved through KCL or KVL equations at t = O+. 4) t ~ ro, after the switch has been changed for a very long period of time This is the new steady state called "final condition." Like the first steady state, all voltages and currents are again stabilized (assuming DC sources), so capacitors are considered open circuits, and inductors are considered short circuits. The value of V c or h at t ~ ro can be easily obtained by simple circuit analysis. It is worth mentioning that the final condition is embedded in the differential equation also. In fact, by letting the derivative terms go to zero, the final condition can be determined conveniently from the differential equation. 5) The transient process: from t = O+ to t -7 oo The evolution of the current and voltage quantities during the transient process is described by differential equations. Since KCL and KVL are always valid, they are often used to establish these differential equations. For parallel circuits, KCL is used; and for series circuits, KVL is used. In establishing the differential equations, it is important to focus on the desired variable (either Ve or h) and express other voltages and currents appearing in KCL or KVL in terms of Ve or h. Instruction Methodology of Transient Analysis Here we present our suggested methodology for the instruction of transient analysis: 1) It is highly beneficial to first review key concepts in first and second order differential equations and the general forms of their solutions. In particular, the importance of initial and final conditions, and how they are used to determine constants in general-form solutions need to be emphasized. This review refreshes students' memory about differential equations they have previously learned about. 2) The concept diagram is presented, as explained in the previous section. 3) Example transient problems are solved to illustrate the determination of constants in general-form solutions for first and second order transient circuits. For example, in first order networks, the general-form solution of Ve or his: x(t) = K1 + K2e-clr, where: x(t) is either Ve or h; K1 is a constant that is equal to the final condition; K1 is another constant equal to the difference between the initial and final conditions, i.e., x(O )-x( oo); Tis the time constant. There is no need to list the differential equation to determine T, only the equivalent resistance Rrn (with Lor C considered as the load) is needed. For RL network, T=URrn; for RC first order network, T=RrnC. For second order networks, the standard procedure to solve a second order differential equation needs to be followed. The most challenging part here is the identification of the second initial condition, i.e., dV ddt(O+) or dh /dt(O+ ). This involves applying KCL or KVL at t = O+ to solve for I c(O+) or VL (O+ ). 4) Transient responses for sample circuits are plotted in MATLAB windows. It is important for students to see dynamic evolution of transient responses, and see the solution as a combination of forced and natural responses. 5) During the final review of transient analysis at the end of the semester, the concept diagram is again presented to outline key concepts and ensure students' grasp of the "whole picture." Assessment questions can be designed at each stage to evaluate the effectiveness of this teaching practice. In particular, drawing the concept diagram as a quiz or one-minute paper is a good assessment tool. Also, asking students to explain the concept diagram to each other as a think-pair-share activity is also a valuable exercise. Based on our experience, students respond very positively to the introduction of the concept diagram. After learning the concept diagram, many students indicated that they had a clear mental picture of transient process and expressed appreciation to us. We are currently planning to conduct a formal survey for assessment purpose in the near future. The survey questions are under preparation. Conclusions In this paper, a concept diagram is introduced to assist the instruction of transient analysis, a challenging topic in introductory electric circuit courses. The concept diagram clearly illustrates three important instants that are essential to the solution of transient circuit problems, and helps students establish a clear picture of the dynamic evolution of voltages and currents during the transient process. Combined with relevant example problems and assessment activities, the concept diagram is a highly effective tool for teaching andreviewing critical concepts in transient analysis. ReferencesHambley, A. R. (2007). Electrical Engineering: Principles and Applications, 4/e, Prentice Hall.Kerns, D. V., & Irwin, J. D. (2004). Essentials of Electrical and Computer Engineering,