8/10/2019 Tutorial_13 - Wave Reflection and Transmission at Normal Incidence

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Exercise 13 - Wave reflection and transmission at normal incidenceEMT 238 - Electromagnetic Theory

Lecture Review

When a plane wave propagating in medium 1, with permittivity 1 and permeability 1encounters an interface with a different medium 2, with permittivity 2 and permeability2, a portion of the wave is reflected from the interface while the remainder of the wave is

transmitted. The electric field amplitude of each of the waves takes the form:

Incident wave: Ei =Ei0

n cos(k1z t)

Reflected wave: Er =Er0

n cos(k1z+t)

Transmited wave: Et =Et0

n cos(k2z t)

Notice that the wavenumbers ofEi and Er are the same because both waves are in the samemedium, whereas the wavenumber of Et is different since it is in a different medium. Theangular frequencies of all the waves are of course the same as frequency does not dependon medium. You should also be caution with the sign of the t-term which indicates the

propagation direction of the respective wave.

The amplitudes of the reflected wave, Er0

, and the transmitted wave, Et0, are related to the

amplitude of the incident wave, Ei0

via the relations

=Er

0

Ei0

=2 12+1

(reflection coefficient)

=Et

0

Ei0

= 222+1

(trasmission coefficient)

where is the intrinsic impedance of the medium

=

The standing wave ration in medium 1 is defined as

S=1 + ||

1 ||

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8/10/2019 Tutorial_13 - Wave Reflection and Transmission at Normal Incidence

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Exercises

1. A plane wave in air with an electric field amplitude of 5 V/m is incident normally uponthe surface of a lossless, nonmagnetic medium with r = 1.69. Determine the following

(a) The reflection and transmission coefficients.

(b) The standing-wave ratio in the air medium.

(c) The average power densities of the incident, reflected, and transmitted waves.

2. A plane wave travelling in a medium with r1 = 9 is normally incident upon a secondmedium withr2= 4. Both media are made of nonmagnetic, non-conducting materials.If the magnetic field of the incident plane wave is given by

Hi = 2 azcos(2 109t ky) (A/m)

(a) Obtain time-domain expressions for the electric and magnetic fields in each of the

two media.(b) Determine the average power densities of the incident, reflected, and transmitted

waves.

3. A 400 MHz, with an electric field amplitude of 20 V/m, travelling in the +z-directionis normally incident in air upon a dielectric medium with r = 9. The electric field ispolarized along the x-direction.

(a) Obtain the reflection and transmission coefficients.

(b) Calculate the standing-wave ratio in the air medium.

(c) Obtain time-domain expressions for the electric and magnetic fields in each of the

two media.

(d) Determine the percentages of the incident average power reflected by the boundaryand transmitted into the second medium.

4. Repeat Problem 3, but replace the dielectric medium with a medium withr = 1.44 andr = 1.

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