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Correspondence

Letter to the Editor

Yale Jay Lubkin

For many years, there has been unease about using a periodic function, such as a sinusoid, as an input to determine the response of a network or of a structure excited by an electromagnetic field. It has been argued that periodic functions have infinite energy and violate causality because there is no way that one cycle of a periodic function differs from any other. The following argument shows that the traditional method of using sinusoids or other periodic functions to design equipment is asymptotically correct.

Consider the behavior of a network (or such) to a periodic function existing between t , and t,, such that

--oo < t , c t c t , c 00. ( 1 ) Let f ( t ) be a periodic function of t . The response of a network

(or such) to an input f ( t ) will be F ( t ) . Consider f ( t ) as the sum of three functions:

f ( t ) = f , ( t ) +f,(t) + f 3 ( f ) (2)

f l ( t ) = f ( t ) [ l - U ( t - t l ) ]

f2(f) = f ( t ) [ W - t i ) - U ( t - f,)]

f 3 ( t ) = f ( t ) [ W - f 2 ) I .

F( t ) = F,( t ) + F,( t ) + F3( t ) .

F ( t ) = F , ( t ) + F2( t ) t < t , (4)

where

Maxwell’s equations and the networks are assumed linear. Then

(3) We are interested in F,(t) in the interval defined in (1). Consider f 3 ( t ) . This is zero for t < t ,; therefore

since the network has no way of knowing the future. Now, consider F,(t) . For t < t , , F ( t ) = F,(t) because for

t < t , , the network does not know of the existence of f 2 ( t ) . For t > t ,

F l ( t ) = R ( t ) t > t , ( 5 ) where R ( t ) is some function of t . If the network is dissipative, as are all real networks, the amplitude of R ( t ) then will decline towards zero, and R ( t ) is a transient. It represents the network response to the energy stored in the network at the time of turnoff of fi. The energy may be considered to be an initial boundary condi- tion for the network.

Substituting in (4), we get

F ( t ) = R ( t ) + F , ( t ) t , < t < t , ( 6 )

(7)

so that F, ( t ) = F ( t ) - R ( t ) .

Hence, the desired time-limited response for the time of interest F,(t) is just the periodic response minus a transient caused by the turnoff of the periodic input. The turn-on transient is just the negative of the turn-off transient.

Manuscript received August 20, 1990. The author is with Ben Franklin Industries, Owings, MD 20736. IEEE Log Number 9041042.

For real signals and real networks, R ( t ) will be limited in both amplitude and duration for practical purposes, except in pathological cases like superconductive networks.

The restriction to linear networks may be removed by noting that R ( t ) is simply the network response to an initial state. From the value of R ( t , ) , we can determine the energy stored in each compo- nent at t , and, thus, the response to f,.

I hope that this answers an annoying philosophical question.

Comments on “ Riemann-Green Function Solution of Transient Electromagnetic

Plane Waves in Lossy Media”

Henning F. Harmuth, Member. IEEE

Asfar’s paper [ l ] shows that the Green’s function method can be used successfully for the calculation of the electric field strength caused by an electric excitation force in a lossy medium. This is the third method, after Fourier’s method of standing waves and the Laplace transform, that has been applied successfully to the prob- lem. What has not been obtained correctly by anyone without the addition of a magnetic current density that can be made to approach zero at the end of the calculation, or some other less clearly stated assumption, is the associated magnetic field strength [2], [3]. One might think that the associated magnetic field strength should follow readily from the electric field strength. This is correct as far as the formalism goes, but one obtains an undefined result due to a singularity.

It is important to understand that one does not get a wrong result but merely an undefined one. This means one can write infinitely many solutions for the associated magnetic field strength that will all satisfy Maxwell’s equations, but Maxwell’s equations cannot tell us which one of these infinitely many solutions is the right one. It can be verified by substitution that the unique solution for the associated magnetic field strength obtained with a magnetic current density [2], [3] indeed satisfies Maxwell’s equations even though it cannot be derived from them.

These mathematical subtleties are now mainly of interest for demonstrating that one should never ignore fine points encountered in the course of a mathematical investigation. The physical signifi- cance of the magnetic current density term used became clear when transients in lossy media were investigated with Lorentz’s equations of electron theory, which allow for the fact that electric charges are always connected with particles having a mass, whereas Maxwell’s original equations do not contain the concept of mass. One finds the following simple physical explanation.

Currents can come in the form of monopole currents carried by charges or monopoles; examples are the current in a resistor carried by electrons or the current in a gas discharge carried by ions and electrons. However, currents can also come in the form of dipole currents or polarization currents carried by dipoles; an example is the current flowing in the dielectric of a capacitor. Generally

Manuscript received July 27, 1990. The author is with the Department of Electrical Engineering, Catholic

IEEE Log Number 9041041. University of America, Washington, DC 20064.

0018-9375/91/0200-00~2$01.00 0 1991 IEEE