SHORT NOTES

tion. Among these are the interactive aspect and the relativeease of material development.The author's use of the method has grown "like Topsy" and

although student evaluations have been very favorable no for-mal evaluation has been made to date. However an effort isbeing made to evaluate the method in conjunction with theelectromagnetic fields course.A final observation is in order. It has been the author's

experience that a detailed analysis is required to produceworthwhile programmed learning material. This detail forcesthe instructor to think more about what objectives the studentis expected to achieve and the order in which they should betreated.

ACKNOWLEDGMENTThe author wishes to thank J. I. Young of the Instructional

Systems Center, University of Maine, Orono for his many help-ful comments and criticisms.

REFERENCES1. L. L. Kazmerski, J. F. Vetelino, J. C. Field, C. F. Hannon, and

J. Peter Wiberg, "A University-Industry Approach to ContinuingEducation for Engineers," this issue, pp. 155-158.

2. Markle, S. M., Good Frames and Bad, John Wiley & Sons, NewYork 1969, p. v.

Introducing Nonlinear Problems to UndergraduateEngineering Students

MULUKUTLA S. SARMA, SENIOR MEMBER, IEEE,KARL E. LONNGREN, SENIOR MEMBER, IEEE, AND

GLEN L. JACKSON

Abstract-It is argued that preparing engineers for the challengingproblems of present-ay industry should involve the extension ofhomework problems to include realistic complications of nonlinearityand extensive use of the available computing facilities. This point ofview has been illustrated through a simple Faraday's law problem of anelectromechanical system that exhibits nonlinear effects if the resis-tance of the rais is taken into account. The procedure that is employedin this short note to solve the nonlinear equation is simple enough to beintrduced at an undergraduate level for the engineering students.

INTRODUCTIONIt is argued that preparing engineers and applied physicists

for the challenging problems of present-day industry should in-volve the extension of homework problems to include realisticcomplications of nonlinearity and extensive use of the avail-able computing facilities. Wherever possible, simple systemsfor which differential equations to be solved are linear may bepresented to the undergraduate engineering student at first,and these may then be extended to become nonlinear as a real-istic and easily visualized complication. While the solution ofnonlinear differential equations is in general very difficult andsuch methods of solution as the method of perturbations (11and numerical procedures [21 are usually introduced only at agraduate leveL simpler solution techniques should be explored

Manuscript received February 14, 1975. This work was supported inpart by NSF Grant ENG 74-00704 at the University of Iowa.M. S. Sarma is with the Department of Electrical Engineering, North-

eastem University, Boston, Mass. 02115.K. E. Lonngren and G. L. Jackson are with the Department of Elec-

trical Engineering, University of Iowa, Iowa City, Iowa 52242.

wherever possible. This point of view has been illustrated herethrough a simple Faraday's law problem of an electromechani-cal system that exhibits nonlinear effects.

THE ELECTROMECHANICAL SYSTEMUNDER CONSIDERATION

The motion of a crossbar of Mass Mo and resistance RO,which accelerates without friction on a pair of parallel rails isto be analyzed. A battery having potential difference .,, thetwo rails, and the crossbar constitute a closed electrical circuit,as shown in Figure 1. When a constant external magnetic in-duction field Bo, directed perpendicular to the circuit loop ispresent and a current is flowing in the circuit, the crossbar isaccelerated by the Lorentz force. The motion of the crossbarchanges the magnetic flux 0 linked by the circuit, and a backelectromotive-force is induced according to Faraday's law.For simplicity and convenience of analysis, all other inductive,capacitive, radiative, and frictional effects are neglected, ex-cept for the ohmic resistance within the circuit. The solutionfor the time dependence of the motion of the crossbar and thecurrent in the circuit when the switch S is closed is a wellknown undergraduate physics problem [31.

LINEAR ANALYSISAs simple as this problem may appear, the differential equa-

tion of motion to be solved is linear only if the resistance ofthe rails may be neglected. Practical design may often justifylinearization to obtain an approximate solution. In such acase, the application of Kirchoffs second law gives for thecircuit loop:

Vo= IRo +dO14I?,+dtwhere

0 =BoYox.

Newton's second law yields

Bo Yol9 = Mox

(1)

(2)

(3)for the motion of the crossbar, where the left-hand side of Eq.(3) is the Lorentz force on the bar. Substitution of Eqs. (2)and (3) into Eq. (1) leads to the linear differential equation

x +(B2Y2/MoRo)i = VOBO Yo/MoRo.Assuming the initial conditions

x(O) =x0

x(0) =0

and

I(O+) = Vo IR,the solution of Eq. (4) for t >0 is given by

x = x0 + Vft - Vfr [ I - exp (- tlr)]xX +Vf [1 - exp (-et/T)]x = (VfI/') exp (-tir)

and

I= (VO/RO) exp (-t/T)

where

=MORo /B Y4

is the time constant for I and x to decay to Zero and

Vf = Vo /Bo Yo

(4)

(5a)

(Sb)

(Sc)

(6a)

(6b)(6c)

(7)

(8)

(9)

159

IEEE TRANSACTIONS ON EDUCATION, AUGUST 1975

x x x

0 X X X I X

x x x x

- xX X X B

0

(14b)

Using the new variables as defined by Eqs. (13) and (14), onemay rewrite Eq. (10) as follows:

y0

x

Fig. 1. Circuit diagram of a simple Faraday's law problem.

is the final velocity of the crossbar. The direction of motionof the crossbar is determined by the relative signs of VO andBo, as per Eq. (9). Plots of the time dependence of the posi-tion, velocity and acceleration of the crossbar, based on Eq.(6), can easily be obtained [3]. It can also be shown that theenergy supplied by the battery is equal to the sum of the ki-netic energy of the crossbar and the heat dissipated in thecrossbar.

NONLINEAR PROBLEM

Equation (4) becomes a nonlinear differential equation, ifany of the parameters Bo, YO, MO, or Ro becomes explicitlyor implicitly time dependent. For undergraduates the exten-sion of the model to account for the resistance of the rails isboth a realistic and easily visualized complication. In such acase, Ro in Eq. (1) is to be replaced by (Ro + 2rx), where r isthe resistance per unit length of the rails. Equation (4) maythen be rewritten as

x +P(X -X,)X + (l/r')x - (VfIT')= 0

where

?==MO(Ro + 2rxo)/B2Y2and

P = 2r/(Ro + 2rxO).

Ti7I /t - + - f =0

P d~ PT TIwhich may be rearranged as

7T dT d,

T T

Integrating both the sides of Eq. (1 5b), one obtains

-, Ti- T'PVf ln( Vf In) K

(1Sa)

(lSb)

(16)

where K is a constant to be evaluated from the initial condi-tions given by

att=O,x =xo and x =0

t0 = 1 and 7Ti =0.

The constant K may be found to be

PVfK= -r'PVfln f.

(17a)

(17b)

(18)

On substituting the value of K and rearranging terms, oneobtains

TI71 PVf In (1 - p7 = In t. (19)(10)

Using the definitions given by Eqs. (13) and (14), replacing tand Ti in terms of the original variables x and t, the following

(11) equation can be written:

(12)PTr'.i +(PT'Vf(1- + + In (1 +P(x - x,)] =0. (20)

The nonlinear equation (10) has been solved [3] by the usualperturbation procedure [2]. However, it can be much moreeasily solved by integrating once to obtain a first-order differ-ential equation and then developing a simple Fortran programto obtain the solution on a digital computer. Such an ap-proach can be better appreciated by an undergraduate student,as it will be more consistent with his academic background.

METHOD OF SOLUTION TO THE NONLINEAR EQUATIONThe second-order nonlinear differential equation (10) will

now be transformed into a first-order differential equation.For convenience one may define new variables:

= 1 +P(x -xO)from which it follows that

t = Pxand

t =Px.Also let

n=Tifrom which it follows that

(13a)

Another change of variables will be made for convenience asfollows:

r =PvrT'x

I/f TI

x- p0

T =t

Using the above, Eq. (20) may be rewritten as

(13b) dT n( dT)

(21a)

(21 b)

(21c)

(21 d)

(22)

A simple digital-computer program in Fortran language iswritten to yield an iterative solution of the Eq. (22), while

(1 3c) (x - xo) goes from 0 to 10 in 100 equal steps, and the corre-sponding time step is estimated as

(23a)(Iw4a) Tn + i=aTnp+t(yT)nwhere A\T is approximately given by

160

d?? di?i .= d. - t = 77 dt-

SHORT NOTES

10.

9-

8.

x< 7

6.<cCV 5m

J 4.

Clt 3X::C)

2.

0

r-o IX

-jLLJ

CD

M:CYC)Z:

0 1 2 3 4 5 6 7 8 9 10NORMALIZED ELAPSED TIME T

Fig. 2. Plot of the time dependence of the position of the crossbar.

Ax _ 0.01(dxT (ddx\\dTr \dt/

(23b)

Plots of time dependence of the position of the crossbar(x - xo VS T) and plots of time dependence of the velocity ofthe crossbar (dx/dT VS T for different values of F (0, 0. 1, 1.o0,and 10.0) are shown in Figures 2 and 3.

It may be finally recalled that

r=PVf'TB,Y2MO Vfr (24a)

M0(R0+2rx0)O(Ro Y 2) VfX (24b)(Bo~Y0)2

r =0

0 1 2 3 4 5 6 7 8 9

NORMALIZED ELAPSED TIME T

Fig. 3. Plot of the time dependence of the velocity of the crossbar.

REFERENCES[11 Marion, J. B., "Classical Dynamics ofParticles and Systems," Text

Book, Academic Press, New York, 1970. (2nd Edition, Chap. 5,and Cited References).

12] Ames, W. F., "Nonlinear Differential Equations in Engineering,"Text Book, Academic Press, New York, 1965.

[3] White, John J., "Solution of a Faraday's Law Problem Includinga Nonlinear Term,"A merican Journal ofPhysics, Vol. 41, pp.644-47, May 1973.

Faulty Analysis

R. J. DISTLER, SENIOR MEMBER, IEEE

and

xV= f dx.

Abstract-A way to encourage a student to examine critically what heis reading is to give him a system analysis which contains some fault

(24c) which leads to a false conclusion. The fault should be rather subtle andcan be either a logical error or a bad assumption. Three examples arepresented:

DISCUSSION OF THE RESULTS AND CONCLUSIONSThe normalized velocity dx/dT increased monotonically

from 0 to 1, as expected, as can be seen from Fig. (3). As thevalue of r is made to increase, the velocity that is reached in agiven time decreases. This behaviour can be predicted by thedependence of F on the resistance r per unit length of therails given by Eq. (24a).A particular feature of the computer program may be men-

tioned here: As (x - xo) and dx/dT appear in Eq. (22) to besolved, steps were chosen for the normalized distance (x - xo)conveniently instead of for the normalized time T.The nonlinear problem described in Fig. 1 has been solved

through a change of variables, a single integration, and a simpledigital-computer-program in Fortran language. While the non-linear equation (10) can be solved by the method of perturba-tions, or by numerical methods, or by some other sophisti-cated procedure normally used to solve nonlinear equations, ithas been successfully solved here much more easily, by a pro-cedure that is simple enough to be introduced at the under-graduate level.

(1) The magnetic rectifier, a coil which produces a dc component inthe current when an ac voltage is applied.

(2) The signal-to-noise ratio amplifier, a circuit for a communicationreceiver which improves the signal-to-noise ratio by 3 db.

(3) A single-phase-to-three-phase converter, a device which producesa three-phase output with a single-phase input voltage.

INTRODUCTION

The engineer is often required to study a report or paper anddetermine if it is correct and if it applies to his current prob-lem. The reading of technical material which is known or sus-pected of being faulty requires a more critical examinationthan the reading of a textbook which is assumed to be correct.At some point in his education, then, for pedagogical reasons,the engineering student should be presented with an analysis

Manuscript received November 5, 1974.The author is with the Department of Electrical Engineering, Uni-

versity of Kentucky, Lexington, Ky. 40506.

161