Page 434 - Engineering Electromagnetics, 8th Edition

416 ENGINEERING ELECTROMAGNETICS

√
9
Solution. We calculate ω = 6π × 10 rad/s, β 1 = ω µ 1 1 = 40π rad/m, and
√
β 2 = ω µ 2 2 = 60π rad/m. Although the wavelength would be 10 cm in air,
we find here that λ 1 = 2π/β 1 = 5 cm, λ 2 = 2π/β 2 = 3.33 cm, η 1 = 60π
,
η 2 = 40π
, and = (η 2 − η 1 )/(η 2 + η 1 ) =−0.2. Because is real and negative
(η 2 <η 1 ), there will be a minimum of the electric field at the boundary, and it will be
repeated at half-wavelength (2.5 cm) intervals in dielectric l. From (23), we see that
|E x1T | min = 80 V/m.
Maxima of E are found at distances of 1.25, 3.75, 6.25,... cm from z = 0.
These maxima all have amplitudes of 120 V/m, as predicted by (20).
There are no maxima or minima in region 2 because there is no reflected wave
there.

The ratio of the maximum to minimum amplitudes is the standing wave ratio:

|E x1T | max 1 +| |
s = = (27)
|E x1T | min 1 −| |
Because | | < 1, s is always positive and greater than or equal to unity. For the
preceding example,

1 +|−0.2| 1.2
s = = = 1.5
1 −|−0.2| 0.8
If | | = 1, the reflected and incident amplitudes are equal, all the incident energy
is reflected, and s is infinite. Planes separated by multiples of λ 1 /2 can be found on
which E x1 is zero at all times. Midway between these planes, E x1 has a maximum
amplitude twice that of the incident wave.
If η 2 = η 1 , then = 0, no energy is reflected, and s = 1; the maximum and
minimum amplitudes are equal.
2
If one-half the incident power is reflected, | | = 0.5, | |= 0.707, and s = 5.83.

D12.2. What value of s results when =±1/2?
Ans. 3

Because the standing wave ratio is a ratio of amplitudes, the relative amplitudes,
as measured by a probe, permit its use to determine s experimentally.

EXAMPLE 12.3
A uniform plane wave in air partially reflects from the surface of a material whose
properties are unknown. Measurements of the electric field in the region in front of the
interface yield a 1.5-m spacing between maxima, with the first maximum occurring
0.75 m from the interface. A standing wave ratio of 5 is measured. Determine the
intrinsic impedance, η u ,of the unknown material.