TM Modes in Rectangular Wave Guide
المؤلف:
Sidney B. Cahn Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 1 , p 75
11-8-2016
1618
TM Modes in Rectangular Wave Guide
A rectangular wave guide of sides a = 7.21 cm and b = 3.40 cm is used in the transverse magnetic (TM) mode (see Figure 1.1). TM modes are modes in which the magnetic field is perpendicular to the direction of propagation, here z. Assume that the walls are perfect conductors.
Figure 1.1
a) By calculating the lowest cutoff frequency, determine whether TM radiation of angular frequency ω = 6.1 × 1010 s-1 will propagate in the wave guide.
b) What is the dispersion relation (i.e., the relationship between ω and the wave vector k) for this guide.
c) Find the attenuation length, i.e., the distance over which the power drops to e-1 of its starting value, for a frequency ω that is half the cutoff frequency.
SOLUTION
a) We can express all the fields in terms of a single longitudinal component. In this problem, we are considering TM waves so Bz = 0, and we use Ez instead. We find for the field components
(1)
where again
The wave equation for the Ez component is
(2)
where
The solution to (2) with the boundary condition E × n|s = 0 is given by
(3)
where n and m are integers. So
(4)
The frequency ω is given by
(5)
The cutoff frequency corresponds to k = 0, so
(6)
For TM waves, we cannot take any of the n or m = 0 modes because that would make Ez = 0 So the lowest cutoff frequency corresponds to m = n = 1
So the TM radiation with frequency 6.1 × 1010 s-1 will propagate in the guide.
b) The dispersion relation was given in (a):
c) The wave number as a function of the cutoff frequency ωλ can be written in the form
The wave of frequency ω < ωλ cannot propagate (k becomes imaginary), and in fact the attenuation of the field will be given by kλ. In our case,
We may write Ez in the form
The power dissipation will be proportional to |Ez|2:
We wish to find the point where P(z)/P(0) = 1/e. Hence,
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