Interdisciplinary Lively Applications Project

Interdisciplinary Lively Applications Project (ILAP) for Students

Title: Designing an Electric Car

Authors: Peter LoPresti, Department of Electrical Engineering Doug Jussaume, Department of Electrical Engineering Shirley Pomeranz, Department of Mathematical & Computer Sciences, The University of Tulsa, Tulsa, OK

Mathematics Classifications: Calculus I - MATH 2014 (Introduction to differential equations)

Disciplinary Classification: Electrical Engineering - EE 2003 (Electric circuit analysis)

Prerequisite Skills: 1. Graphing and analyzing graphs 2. Differentiation

Physical Concepts Examined: 1. RC circuits (resistors and capacitors)

Materials Available: 1. Problem statement and discussion; Student 2. Supplemental background material; Student 3. Sample solution; Instructor 4. Notes for Instructors

Computing Requirements: Basic graphing capability using Excel, Mathematica, MATLAB, or another graphing package/tool

Class Requirements: One class period for problem assignment; One class period for laboratory verification (requires RC circuit components, etc.)

Partial support for this work was provided by the National Science Foundation's Course, Curriculum, and Laboratory Improvement program under grant #0410653

1 Problem statement and discussion - Charge and current in an RC circuit:

Normally, an electric car is powered with a battery. The battery stores charge at a given potential (voltage), and when a motor is attached, the battery delivers charge over a period of time (current) to the motor to make the car move. You can, however, achieve the same effect if you charge up a capacitor and then connect it to the motor. The capacitor stores charge at a given voltage and delivers it to the motor, just like a battery. This system of capacitor and motor is a first-order resistor-capacitor (RC) circuit. The capacitor discharges through the resistor (motor) and drives the car for as long as there is enough current to drive the motor. The presence of the resistor prevents the capacitor from discharging its voltage (stored potential energy) all at once (instantaneously). Therefore, the resistor permits the discharge over a period of time, during which the motor runs. Whether or not there is enough current to drive the motor is dependent on the rate of change (derivative) of the capacitor voltage drop, Vc, and the charge capacity of the capacitor (capacitance) (see Eq (3)).

The voltage drop across the capacitor is given by the following function:

t  RC (1) VC (t)  VC (0) e , t  0,

where t is the time (in seconds), Vc(t) is the voltage drop across the capacitor (in volts) as a function of time; Vc(0) is the initial capacitor voltage drop (in volts), at time t=0; R is the resistance of the resistor (in ohms); and C is the capacitance of the capacitor (in Farads = coulombs/volt).

The relationship between q, the charge on the capacitor (in coulombs), and the voltage drop across the capacitor is

q(t)  C VC (t), t  0. (2)

The motor/capacitor current, i (in amps), is related to the capacitor voltage drop as described in the following equation:

d q(t) d V (t) i(t)   C C , t  0 . (3) d t d t

The capacitor of capacitance C is initially charged to some electric potential, Vc(0). The capacitor is then discharged by suddenly connecting it across a resistor of resistance R by closing a switch at time, t=0. As current flows, the capacitor gradually loses its charge, the voltage drop across the capacitor will diminish, as will the rate of change of the capacitor voltage, and this in turn will decrease the flow of current.

For a graphical representation of some of the relationships involved in an RC circuit, go to the web site (for example) http://lectureonline.cl.msu.edu/~mmp/kap23/RC/app.htm .

2 You are to “design” a car that will move for a specified amount of time, tmax. Use the formulas d V (t) given above (Eqs (1)-(3)). You will need to compute the derivative, C . You are given the d t information that the car will move provided that the current has at least a specified magnitude, imin . Note: This means that| i(t) |  imin . The parameters that you can vary are the capacitance, C, and the initial voltage drop across the capacitor, Vc(0). (i) For a fixed value of C, find the minimum value of Vc(0) that “works” within the constraints given above. (ii) Then repeat the process, but this time fix the value of Vc(0) and find the minimum acceptable value of C (i.e., reverse the roles of Vc(0) and C). The value of parameter R is fixed. Note that q, Vc , and i are functions of time.

1. You are to write a complete report in which you explain how you solved the problem mathematically. Use the notation provided in this discussion. Work with the general parameters, and only substitute the numerical data at the end of your calculations. Numerical data will be given in the laboratory class, about two weeks after this project is assigned. Include relevant graphs; in particular, include clearly labeled graphs of current magnitude versus time for your mathematical models. Use the guidelines that you have been given for writing a science/engineering/mathematics report. (See the Technical Report Format and Writing Guide at the TU ILAPs web site: http://www.ilaps.utulsa.edu/). You can use your calculus book to get examples of professional writing styles for mathematics and text (your instructors may have additional handouts or references). Cite and/or acknowledge all sources, as appropriate.

2. After you write your basic report, you will then investigate/verify your results experimentally in the electrical engineering laboratory. The instructors will provide a listing of the available capacitor and/or voltage ratings, as needed. In your choices for these parameters, you are constrained by (a) the limited values of the capacitance that are commercially available and (b) the maximum voltage which a given capacitor can hold without failing. We assume in solving this problem that the weight of the car, tire grip of the “road,” and other parameters are the same as when the tests were performed to determine imin . Assumptions that have been made in the mathematical modeling of the problem may account for discrepancies between what you have obtained graphically, numerically and/or analytically versus what you observe experimentally. Include in your report a brief discussion as to how well your experiment verified what you predicted mathematically (and possible effects of experimental error and/or idealized assumptions in the mathematical model).

The following are some web sites with mathematics modules that you may find helpful with supplementary information about separable ordinary differential equations (you may find other web sites with useful information about ordinary differential equations that describe exponential decay): http://www.math.eku.edu/jones/sect1_5.pdf http://www.physics.uoguelph.ca/tutorials/exp/decay.html

3 Supplemental background material: An analogy between mechanical, torsional, and electrical systems

One of the most important and interesting subjects of applied mathematics is the theory of small oscillations of mechanical systems in the neighborhood of an equilibrium position or a state of uniform motion. There is an analogy between the oscillations of a “vibrating system” consisting of an electrical circuit and the oscillations of a vibrating system consisting of a mechanical system or torsional (twisting) system. The same types of differential equations and boundary/initial conditions apply to each. It is only the physical interpretations (and related notation, terminology, etc.) that differ. By this analogy, the mathematical methods used to analyze electrical circuits can be used in the analysis of mechanical systems (torsional systems; and vice-versa). In the mechanical system, one is usually concerned with the determination of the natural frequencies and modes of oscillation rather than the complete solution for the amplitudes, subject to the initial conditions of the system.

We can consider oscillating or vibrating systems with one degree of freedom (the external applied force happens to be an oscillating force in these examples). Consider the vibrating systems depicted below in Figure 1. System (a) represents a mass that is constrained to move in a linear path. It is attached to a spring of spring constant K and is acted upon by a dashpot mechanism, i.e., a mechanism that introduces a frictional constraint proportional to the velocity of the mass. The mass has exerted upon it an external applied force, P0 sin t .

Figure 1.

By Newton’s second law, we have

d 2 x d x M  K x  R  P sin t, (1) dt 2 d t 0 where K is the spring constant and R is the friction coefficient of the dashpot. System (b) represents a system undergoing torsional oscillations. It consists of a massive disk of moment of inertia J attached to a shaft of torsional stiffness K. The disk undergoes torsional

4 damping proportional to its angular velocity. The disk has exerted upon it an external applied

oscillatory torque, T0 sin t . By Newton's second law, we have

d 2 d J  K   R  T sin t. (2) dt 2 d t 0

System (c) is a series electrical circuit having inductance (L), resistance (R), and capacitance (C). By Kirchhoff's law, the equation satisfied by the charge (q) is

d 2 q 1 d q L   q  R  E sin t. (3) dt 2 C d t 0

By comparing these three equations, (1)–(3), we obtain the following table of analogs:

Mechanical system Torsional system Electrical system Mass M Moment of inertia J Inductance L Stiffness K Torsional stiffness K Elastance = 1/Capacitance 1 S  C Damping R Torsional damping R Resistance R Applied force Applied torque Applied potential

P0 sin t T0 sin t E0 sin t Displacement x Angular displacement  Capacitor charge q d x d d q Velocity v  Angular velocity   Current i  d t d t d t Figure 2.

We see from Figure 2, the table of analogs, that it is necessary only for us to (mathematically) analyze one system and then (by means of the relationships given in table) we may obtain the corresponding solution for the others.