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 = √ K2g – w2με
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
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.