Uniform Plane Waves Infinite Current Sheets.
Over the past weeks, we’ve seen that voltage and current waves can propagate along transmission lines. Examples of TLs, as we saw, included coaxial cables, twin lead, and lands on a circuit board. These voltage and current waves are also examples of electromagnetic (EM) waves.
There are many more examples of EM waves:
• Radio
• Radar
• Satellite communications
• Light
• Microwave ovens
The next major topic in this course is the simplest examples of EM waves propagating without a supporting structure. These simplest examples are called uniform plane waves (UPWs). Uniform plane waves are produced by infinite planar sheets of surface current density, ` J_{s} . In our work here, we will assume only sinusoidal steady state so that ` J_{s} is a phasor.
There is no supporting structure here. We are simply assuming that the current exists as an impressed (or source) current. The task before us is to solve for the `E and `H produced everywhere in space by this electric current sheet.
Wave Equation
We will begin this solution process with the phasor forms of Maxwell’s curl equations
Ñ ´ `E = - jw`B = - Jwm`H
Ñ ´ `H = jw`D = - Jwe`E
Since there is no variation in the source (or space) in the x and y directions, we also would not expect any variation of `E and `H in x and y. Consequently, ¶ / ¶x¶0 and ¶ / ¶y¶0.
Therefore, (1) becomes
Or
Similarly for (2) we find
We wish to determine separate differential equations for E_{x} alone and E_{y} alone from (3) and (4). To do this, we take d / dz of (3) and substitute (4) giving
We can recognize (5) and (6) as phasor wave equations for E_{x} and E_{y} . They are identical in form to the voltage and current phasor wave equations for TLs we derived in Lecture 16. Defining the phase constant (also called the wavenumber) b as
then the solutions to (5) and (6) are
E_{y}(z) = C_{1e}^{-jbz} + C_{2e}^{+jbz}
E_{x}(z) = C_{3e}^{-jbz} + C_{4e}^{+jbz}
C_{i} , =1,…,4 are (complex) constants that are evaluated by applying the boundary conditions presented by a particular problem. As we’ve seen before with phasor voltages and currents on TLs, the first terms in (8) and (9) are the phasor representations of electric field waves propagating in the +z direction. Conversely, the second terms represent waves propagating in the –z direction.
Consequently, this current sheet is producing electromagnetic waves! These EM waves propagate in space without any supporting structure, such as conductors in a coaxial cable, for example.
EM Waves Produced by the Current Sheet
Returning to the current sheet problem:
In general, E_{x} and E_{y} in each region 1 and 2 will have the form (8) and (9). It can be shown that a uniform ` J_{s} directed only in a_{x} produces
`E only in the same a_{x} direction. Therefore, there will only exist an E_{x} in both regions 1 and 2 according to (9) as
Region 1: E_{x1} = Ae^{-jbz} + Be^{+jbz}
Region 2: E_{x1} = Ce^{-jbz} + De^{+jbz}
In both of these regions, we would expect only “outgoing” waves; that is, waves that travel away from the current sheet. This means that
Therefore, B = C = 0 in (10) and (11).
Also associated with these electric fields are magnetic fields ` H that we can determine from these electric fields using Maxwell’s
Using (10) and (11) in (12) with B = C = 0, we can determine the magnetic fields as
To determine the remaining constants A and D, we apply the boundary conditions for the electric current sheet at z = 0:
a_{z}´(`E_{1}-`E_{2})=0|_{z=0}
a_{z}´(`E_{1}-`E_{2})=`J|_{z=0}
Substituting (with B = C = 0) we find
A = D
while substituting we find
To finish this solution, we substitute and giving
We have successfully solved for the EM fields produced everywhere in space by this current sheet. Equations (19)–(22) are the complete solution for the EM waves produced by the uniform electric current sheet ` J_{s} = - a_{x} J _{s0} A/m at z = 0.
Discussion
• From
. ` E ` H .
. Both ` E and ` H are perpendicular to z, which is the direction of propagation.
.The direction of the cross product ` E ×` H is also the direction of propagation.
• The ratio of E_{x} to H_{y} in region 1 is
while in region 2
This is similar to TLs where
.
• In planes perpendicular to the directions of propagation (±z ) the `E and `H fields are constant – in both magnitude and phase. That is, the fields are uniform. Because of this property, these types of EM fields are called uniform plane waves (UPWs).
• As with TLs
• How fast does an EM wave travel in vacuum
which is the speed of light in a vacuum!
Example N25.1: A UPW is propagating in a material with e_{r} = 3 and m_{r} = 4. This UPW has a magnetic field given as
`H (z,t) = a_{x}0.1 cos(6p´10^{8}t-21.780z)A/m
Determine (a) the direction of propagation, (b) the frequency, (c) the wavenumber, (d) the wavelength, and (e) write the complete time domain expression for E(z,t ). From the given magnetic field:
(a) Because of the ‘-’ sign, wave propagation is in the +z direction.
(b) = 2 f = 6 ×10^{8} rad/s f = 3×10^{8} = 300 MHz.
(c) Directly from the phase term, = 21.780 rad/m.
(d) The wavelength can be computed from as
(e) To determine the direction of `E, it is very helpful to draw a sketch:
Because
With
There fore
`E(z,t)=-a_{y} 43.501 cos (6p ´10^{8}t-21.780z)V/m