The characteristics or properties of transverse electric (TE) and transverse magnetic (TM) in parallel planes or plates waves can be studied with the help of propagation constant gg for these waves.

(a) Propagation Constant in parallel planes

γg  = √ K2gw2με

γg  = √(mπ /α)2 – w2με

Where                      Kg = mπ /α

At very high frequency, so that

w2με >> (mπ /α)2

Thus   γg  = √-[-( mπ /α)2 + w2με ]

γg  = √-[ w2με – ( mπ /α)2 ]

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

γg  = j √ w2με – (mπ /α)2

Also               γg  = αg + jbg

Where αg is attenuation constant and bg is phase constant.

Definition of  attenuation constant αg  in parallel planes: αg 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 bg . bg is defined as a measure of the phase shift in radians per unit length.

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

Comparing the imaginary parts of above two equations, we get

Phase Constant

bg = √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 αg = 0

(b) Cut-Off Frequency

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

fc = wc/2π, at which

w2με = ( mπ /α)2

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

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

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

or             fc = m/2 α υ0

Here fc is the cut – off frequency.

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

Definition of Cut-off Frequency (fc ). 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, αg and phase shift constant, bg = 0 and above which αg = 0 and γg exists.

As                        fc = 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 fc , the wave propagation does occur and there will be no attenuation (αg = 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.