(a) Propagation Constant in parallel planes

γ_g  = √ K²_g – w²με

γ_g  = √(mπ /α)² – w²με

Where                      K_g = mπ /α

At very high frequency, so that

w²με >> (mπ /α)²

Thus   γ_g  = √-[-( mπ /α)² + w²με ]

γ_g  = √-[ w²με – ( mπ /α)² ]

This shows that quantity under the radical will be negative and then γ_g  will be pure imaginary that is

γ_g  = j √ w²με – (mπ /α)²

Also               γ_g  = α_g + jb_g

Where α_g is attenuation constant and b_g is phase constant.

Definition of  attenuation constant α_g  in parallel planes: α_g is defined as a constant which indicates the rate at which the wave amplitude reduces as it propagates from one point to another.

It is real part of propagation constant. It has units of  dB/m or Neper/m.

Definition of  phase shift constant  in parallel planes b_g . b_g is defined as a measure of the phase shift in radians per unit length.

It is imaginary part of propagation  constant, g_g  with units      radians/m.

Comparing the imaginary parts of above two equations, we get

Phase Constant

b_g = √w²με – ( mπ /α)² ]

Under  these conditions, the fields will progress in the +z direction as waves and the attenuation of such waves will be zero for perfectly conducting planes

that is Attenuation Constant α_g = 0

(b) Cut-Off Frequency

As the frequency is decreased, there will be a stage at critical frequency,

f_c = w_c/2π, at which

w²με = ( mπ /α)²

or                                     w_c = ( mπ /α) 1/√ με

or                                     2πf_c = ( mπ /α) 1/√με

or                                     f_c = (m/2 α) (1/√ με) = (m/2 α) ( 1/√ μ₀ε₀)  (if μ = μ₀, ε = ε₀)

or             f_c = m/2 α υ₀

Here f_c is the cut – off frequency.

For all frequencies less than f_c , the quantity under the radical of equation will be positive and γ_g  will be a real number, that is γ_g  = α_g + j 0 = α_g, as b_g = 0. This implies that fields will be attenuated exponentially in the +z direction and there will be no wave motions as b_g = 0.

Definition of Cut-off Frequency (f_c ). The frequency at which wave motion cases is known as the cut-off frequency of the guide.

Another Definition. It is defined as a frequency below which there exists only attenuation constant, α_g and phase shift constant, b_g = 0 and above which α_g = 0 and γ_g exists.

As                        f_c = m/2 α υ₀

Thus for each value of m, there is a corresponding cut-off frequency below which wave propagation cannot occur. Above the f_c , the wave propagation does occur and there will be no attenuation (α_g = 0) of the wave for perfectly conducting planes.

I will discuss more characteristics or properties of TE and TM waves in parallel planes in next article.

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