Wave Propagation

Prof. Dr. Jabir S. AZIZ Computer Communications Department Al-Rafidain University College Propagation in a Lossless Medium

• Consider the uniform plane wave case in which there is no variation with respect to y or z, the wave equation in phasor form may be expressed as

or

where

+ −푗훽푥 − 푗훽푥 Consider 퐸푦 component : 퐸푦 = 퐸 푒 + 퐸 푒 ,

where 퐸+and 퐸−are arbitrarily real constants Propagation in a Lossless Medium • The corresponding time-varying field is :

The above equation becomes Propagation in a Lossless Medium • By identifying some point in the waveform and observing its velocity for a wave travelling in the +ve x-axis, the point is given by:

Differentiation yields:

훽 is the phase shift constant (rad/unit length) Propagation in a Lossless Medium

• Another important quantity is the wavelength 휆, defined as that distance over which the sinusoidal waveform passes over the full cycle of 2휋 radians. Since the fields vary in space as of 훽푥, then

The phase velocity for the wave given above: Propagation in a Lossless Medium

• A complete specification of the plane wave electromagnetic field should include the magnetic field. • In general, whenever 푬 or 푯 is known, the other field vector can be readily found by using one of Maxwell’s curl equations. Thus, applying to the electric field of 휕푩 퐸 = 퐸+푒−푗훽푥 + 퐸−푒푗훽푥, 훻 × 푬 = − 푦 휕푡 푗 휕퐸 1 gives 퐻 = 퐻 = 0, and 퐻 = 푦 = (퐸+푒−푗훽푥 − 퐸−푒푗훽푥), 푥 푦 푧 휔휇 휕푥 휂

• where 휂 = 휔휇/β = 휇/휀 is known as the intrinsic impedance of the medium. • The ratio of the 푬 and 푯 field components is seen to have units of impedance, known as the wave impedance; for planes waves the wave impedance is equal to the intrinsic impedance of the medium.

• In free-space the intrinsic impedance is 휂0 = 휇0/휀 = 377. • Note that the 푬 and 푯 vectors are orthogonal to each other and orthogonal to the direction of propagation (±푥); this is a characteristic of transverse electromagnetic (TEM) waves. Propagation in a Lossless Medium

Example:- A plane wave propagating in a lossless dielectric medium has an electric field given as 퐸푥 = 퐸0 cos(휔푡 − 훽푧) with a frequency of 5.0 GHz and a wavelength in the material of 3.0 cm. Determine the propagation constant, the phase velocity, the relative permittivity of the medium, and the wave impedance. Solution:-

The relative permittivity can be calculated as from (νo = c): Conductors and Dielectrics

• In Electromagnetics, materials are divided roughly in two characteristic

1. Conductors, and 2. Dielectric ( or insulators).

• The dividing line between the two classes is not sharp and some media (the earth for ex.) are considered as conductors in one part of the frequency range, but as dielectric (with loss) in another part of the range.

In Maxwell equation:

• The ratio 휎 휔휀 is the ratio of conduction current density to displacement current density in the medium. Hence (휎 휔휀 ) can be considered as the line witch divide conductors and dielectric. Conductors and Dielectrics

• For good conductors (휎 휔휀 ≫ 1) over the entire radiating spectrum.

ex. for copper at 푓 = 30 GHz

• For good dielectric (휎 휔휀 ≪ 1)

ex. for mica at audio frequency

• For good conductors 휎 and 휀 are nearly independent of frequency, but for most materials classed as dielectric 휎 and 휀 are function of frequency.

• Dissipation factor D of the dielectric is (휎 휔휀)

• Power factor Wave Propagation in a Conducting Medium

• The wave equation may be written in the form of a Helmholtz as:

where:

• 훾 is taken as that of the square root which has a positive real part (훼) and if is positive, then 훽 should also be positive. Wave Propagation in a Conducting Medium

We obtain 훼 and 훽 from previous equations as and

From the above two equations, we obtain Wave Propagation in a Conducting Medium

• Consider a uniform wave travelling in the x-direction 푬 has the form

Which has one possible solution

In time varying form:

−훼푥 푬 푥, 푡 = 퐸0푒 cos(휔푡 − 훽푥)풂푦

A sketch of |E| at times t = 0 and t = ∆t is portrayed in Figure, where it is evident that E has only an y-component and it is traveling along the +x- direction. Wave Propagation in a Conducting Medium

Having obtained 푬 (푥, 푡), we obtain 푯 (푥, 푡)

−훼푥 푯 푥, 푡 = 퐻0푒 cos(휔푡 − 훽푥 − 휃휂)풂푧

퐸 where 퐻 = 0 0 휂 and 휂 is a complex quantity known as the intrinsic impedance (in ohms) of the medium. It is given by Wave Propagation in good Dielectric

• For the case (휎 휔휀 ≪ 1) So that

The attenuation equation becomes:

and the phase constant equation becomes: Wave Propagation in good Dielectric

• The velocity of wave in the dielectric is:

where

• The intrinsic or characteristic impedance of a medium which has a finite conductivity is:

Using the same approximation as above: Wave Propagation in a good Conductor

For this case (휎 휔휀 ≫ 1) , so that

The velocity of the wave in the conductor will be: Wave Propagation in a good Conductor

and

• It is seem that: Depth of Penetration

• In a medium with a conductivity, the wave is attenuated as it progresses owing to the losses which occur. • In a good conductor at radio frequencies the rate of attenuation is very great and the wave may penetrate only a very short distance before being reduced to a negligibly small percentage of its original strength. Therefore, the depth of penetration 훿, is defined as:

That depth in which the wave has been attenuated to ퟏ 풆 or approximately ퟑퟕ% of its original value. Since the amplitude decreases by the factor 푒−훼푥, then:

The general expression: For a good conductor: Depth of Penetration

7 Example: 훿 in metals, 푓 = 1 MHz into copper (휎 = 5.8 × 10 S and 휇 = 휇0)

⇒

• At 100 MHz 훿 = 0.00667 mm

• At 60 MHz 훿 = 8.67 mm

• At 1 MHz (in to sea water) 훿 = 25 cm

• At 1 MHz (in to fresh water) 훿 = 7.1 m Summary of Results for Plane Wave Propagation in Various Media Surface Impedance

• It has been seen that at high frequencies the current is confined almost entirely to a very thin sheet at the surface of the conductor. It’s therefore, conventional to make use of surface impedance defined a

where 퐸푡푎푛 is the EF strength parallel to 푧-axis, and at the surface of the conductor and 푱풔 is the linear current density that flows as a result of this 퐸푡푎푛. The linear current density 푱풔 represents the total conduction current per meter width flowing in the thin sheet. • If it is assumed that the conductor is a flat plate with its surface at the y=0 plane (Figure below), the current distribution in the y-direction will be given by:

Current distribution in a thick flat-plate conductor Surface Impedance

• It is assumed that the thickness of the conductor is ≫ 훿 so that there is no reflection from the back surface of the conductor.

• The total conduction current per meter width is:

⇒

But 퐽0, the current density at the surface is ⇒

• In a conductor medium is Surface Impedance

This gives for a thick conductor,