A MATHCAD ELECTRONIC BOOK

Visual Electromagnetics for Mathcad: Propagating Electromagnetic Waves in Lossless Media

From Visual Electromagnetics for Mathcad by Keith W. Whites. Copyright © 1998 by The McGraw-Hill Companies. Used with permission of The McGraw-Hill Companies.

A uniform plane wave will be assumed propagating in the +z or -z directions as shown in Fig. 6.2 of the text and in the figure below. The electric field is directed only along the x axis while the corresponding magnetic field is directed along the y axis.

Choose the parameters for the wave and the space in which it propagates:

Frequency (Hz).

Relative permittivity and permeability of the space.

Electric field amplitude of waves propagating in the +z and -z directions (V/m).

Compute the radian frequency and wavenumber of the wave:

Period of one time cycle (s).

Wavenumber (rad/m) from (37) in Chap. 6.

In this worksheet, we will generate a number of plots for uniform plane waves that propagate both in the +z and in the -z directions. The first two plots we will construct are the variation of the electric field amplitudes of these two waves as a function of z at three different times.

Choose the number of points to plot in z:

Number of points to plot in z.

z ending point for the plots (m).

Construct a list of zi points at which to plot the Ex fields:

Define the functions Ex_plus and Ex_minus which compute the x components of the electric field propagating in the +z and -z directions, respectively, for a uniform plane wave. Refer to Equations (40a) and (41a) of Chap. 6 in the text:

+z propagating wave.

-z propagating wave.

Now plot the electric field of the +z propagating wave at three different instances of time:

For a wave with amplitude (V/m)

In this plot, Ex_plus is shown as a function of z at three different time values: t = 0, t = T/4 and t = T/2 where T is the period of one time cycle. At the chosen frequency

(Hz) then (s).

The primary purpose of this plot is to illustrate that the function defined above for E x_plus (and in Equation (40a) of the text) represents an electric field that varies in space and time as a wave . If we examine the variation of E x_plus as a function of z but at a fixed time (say, for example, t = 0), we can observe in this plot that Ex_plus varies sinusoidally with z. That is, for increasing z this x-directed electric field has an increasing then decreasing positive value, then goes to zero at some point, then points in the -x direction with increasing then decreasing value, goes to zero again, etc. This sinusoidal variation of the electric field in space at a fixed time is like that of a wave.

We can observe more wavelike behavior of this electric field if we now fix our attention to one specific value for Ex_plus at some z and at t = 0. Now if we look for this point on the plot of Ex_plus at the next time step (t = T/4 ) we can see that this point has "moved" in space. Similarly, for the next time step (t = T/2 ) this point on the waveform has moved again in space. In fact, this entire waveform is moving through space as a function of time, again just like that of a wave.

Although not shown in this worksheet, Hy_plus has a similar variation in space and time which we can deduce from the similarity of the cosine terms in Equations (40a) and (40b) in Chap. 6 of the text.

Now plot the electric field of the -z propagating wave (Ex_minus ) as a function of z at three different times, t:

For a wave with amplitude (V/m)

This plot shows the variation of the Ex field as a function of z for the -z propagating wave and at the same time values, t, as in the previous plot. We can observe a similar behavior here as in the previous plot except that if we fix our attention to one point on the wave at t = 0, we can see that as time increases, the wave moves in the -z direction.

This wave propagation behavior of the electric field is probably best observed through animation. We will now generate animation clips that very clearly illustrate the combined space and time variation of these electric field waves that propagate in the +z and -z direction.

Choose the number of time periods for which to plot the electric field wave:

Number of time periods to plot.

Number of points to plot per period.

Time to start and end plot(s).

Frame

Define the variable time in terms of the variable FRAME.

Animate

Now generate the animation clip for the +z propagating electric field wave. For best results, in the Animate dialog box, choose

Time (in periods, T)

Animate

Next, generate the animation clip for the -z propagating electric field wave. For best results, in the Animate dialog box, choose

Time (in periods, T)

The first animation clip is for the E x wave that propagates in the +z direction (labeled the “forward” wave) and the second clip is for the Ex wave that propagates in the -z direction (labeled the “reverse” wave).

After viewing these animation clips i t should be very apparent why these E x functions of space and time in Equations (40a) and (41a) in Chap. 6 of the text (which are equivalent to the functions Ex_plus and Ex_minus defined earlier in this worksheet) are called “waves.” We observe in the animation clips that these Ex fields “move” through space in a wavelike fashion.

The final property of these electric field waves that we will examine is the effect of different dielectric materials on their behavior. We will also discuss the concept of wavelength, l , and the physical significance of the wavenumber, b.

First, c hoose two different relative permittivities of the space for a comparison of the waves in these two materials:

Now define the wavenumber as a function of the relative permittivity of the space:

Construct the plot of the electric field waves which propagate in the +z direction in both of these materials:

Construct a list of zj values at which to plot Ex :

We need to redefine the functions for the x component of the electric fields that propagate in the plus and minus z directions. The only difference between the following definitions and those earlier in this worksheet is that we are allowing for different relative permittivities of the space in which the waves propagate:

Compute the wavelengths of the waves that propagate in the two materials using (38) in Chap. 6 of the text:

Now, plot Ex as a function of z for the +z propagating plane wave at t = 0:

For a wave with amplitude (V/m)

The (solid) red line and the (dashed) blue line indicate the propagating Ex fields for waves in materials 1 and 2, respectively, which have relative permittivities

and .

Both Ex fields are shown at the time t = 0.

A striking feature we observe in this plot is that there is more variation of Ex per unit distance for the wave that propagates in the material with a larger e r than in the material with a smaller er. The wavelength in the material with a larger er is smaller than the wavelength in the other material as you can see in the above plot. The vertical markers for l1 and l2 in this plot indicate a distance of one wavelength of the two waves as measured from z = 0 to the marker for the respective wavelength.

One way that we can develop a better appreciation of these new concepts of wavelength and wavenumber is through an analogy between the spatial variation of this wave and its time variation. For example, the +z propagating Ex wave is given from (40a) in Chap. 6 as:

We can observe in the argument of the cosine function that wt and bz have the same units, radians, which is actually a dimensionless unit. At a fixed time, the spatial variation of Ex is dictated by b z while at a fixed z, the time variation is dictated by w t. There is a “dual” relationship between the wt and b z terms in this cosine function.

The cos(w t) variation is quite common to all students of electrical circuit analysis. The factor w is the radian frequency of the waveform indicating how many radians the argument will change per unit of time. The period T of this time waveform is, of course, given by

The period indicates the length of time in seconds for which the time waveform will repeat its sinusoidal behavior.

Because of the “dual” relationship between the wt and b z terms in the wave function for Ex just mentioned, we can develop a physical interpretation of b as the spatial analog to w. That is, b indicates how many radians the argument will change per unit of distance (rather than time). Similarly, the period of the spatial waveform – which is called the wavelength, l – is given by

Relating this to the last plot, we can see that increasing the relative permittivity (and, similarly, the permeability) has the effect of increasing the wavenumber, b . This is analogous to the effects of increasing the frequency of oscillation, w, to the time variation. Since b has increased we would expect that the spatial period of oscillation, l , would decrease as does T when w is increased.