Plane Wave Propagation in Lossy Media Asst Prof

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Plane Wave Propagation in Lossy Media Asst. Prof. Dr. Özgür Tamer

Plane Wave Propagation in Lossy Media Asst. Prof. Dr. Özgür Tamer

Lossy Media A lossy dielectric is a medium in which an EM wave loses

Lossy Media A lossy dielectric is a medium in which an EM wave loses power as it propagates due to poor conduction. A lossy dielectric is a partially conducting medium

Lossy Media In a lossless dielectric medium, the relative permittivity εr is real, For

Lossy Media In a lossless dielectric medium, the relative permittivity εr is real, For non-magnetic materials, the relative permeability μr =1 For lossy media permittivity is a complex number

Lossy Media In a lossy medium

Lossy Media In a lossy medium

Loss Tangent is called the loss tangent › The complex permittivity can be written

Loss Tangent is called the loss tangent › The complex permittivity can be written as

Loss Tangent Current Terms

Loss Tangent Current Terms

Helmholtz’s equation By replacing k with kc, all the previous results derived for lossless

Helmholtz’s equation By replacing k with kc, all the previous results derived for lossless media are applicable to lossy media. But since kc is a complex number, the plane wave will experience loss when it propagates.

Helmholtz’s equation γ is known as the complex propagation constant › solutions with the

Helmholtz’s equation γ is known as the complex propagation constant › solutions with the +α constant have been discarded as they imply waves with increasing amplitudes which is impossible

Complex Propagation

Complex Propagation

Complex Propagation The general form of a plane wave in a lossy medium is

Complex Propagation The general form of a plane wave in a lossy medium is

Complex Propagation

Complex Propagation

Complex Propagation

Complex Propagation

Good Conductor Loss tangent for a good conductor is;

Good Conductor Loss tangent for a good conductor is;

Good Conductor In a good conductor, the intrinsic impedance ηc is a complex number,

Good Conductor In a good conductor, the intrinsic impedance ηc is a complex number, meaning that the electric and magnetic fields are not in phase as in the case of a lossless medium.

Good Conductor Phase velocity and wavelength

Good Conductor Phase velocity and wavelength

Good Conductor H and E inside a good conductor

Good Conductor H and E inside a good conductor

Good Conductor Skin Depth › Because of the attenuation constant α, the wave amplitude

Good Conductor Skin Depth › Because of the attenuation constant α, the wave amplitude becomes smaller when it propagates through a good conductor. › amplitude of a travelling plane wave decreases by a factor of e-1 = 0. 368 = 36. 8% is called the skin depth

Skin Depth

Skin Depth

Skin Depth

Skin Depth

Skin Depth For copper σ = 5. 8 × 107 S/m and μr =1

Skin Depth For copper σ = 5. 8 × 107 S/m and μr =1

Skin Depth

Skin Depth

Low Loss Dielectrics For a low loss dielectric loss tangent satisfies

Low Loss Dielectrics For a low loss dielectric loss tangent satisfies

Low Loss Dielectrics

Low Loss Dielectrics

Low Loss Dielectrics A skin depth δ can be similarly defined as in the

Low Loss Dielectrics A skin depth δ can be similarly defined as in the good conductor case:

Example 1 The electric field intensity of a linearly polarised uniform plane wave propagating

Example 1 The electric field intensity of a linearly polarised uniform plane wave propagating in the +z direction in seawater is E = x 100 cos(107 πt) at z=0. The constitutive parameters of seawater are εr = 72, μr = 1, and σ = 4 S/m.

 Determine the attenuation constant, intrinsic impedance, phase velocity, wavelength, and skin depth.

Determine the attenuation constant, intrinsic impedance, phase velocity, wavelength, and skin depth.

 Write expressions for H(z, t) and E(z, t).

Write expressions for H(z, t) and E(z, t).

 Find the distance z 1 at which the amplitude of the electric field

Find the distance z 1 at which the amplitude of the electric field is 1% of its value at z = 0.

 Compute the skin depth at a frequency of 1 GHz.

Compute the skin depth at a frequency of 1 GHz.

 find the power densities at distances of skin depth z = δ and

find the power densities at distances of skin depth z = δ and z = 0.