To: Lynn Baker

From: Jon Hagen

Date: 4-22-03

Subject: Design of thermal transition choke joint

Lynn - I guess I'd never written this down, so I'll redo it. (You'd probably better check my work). The choke arrangement consists of a radial waveguide and then a coaxial waveguide (whose inner and outer diameters are nearly the same). The radial waveguide presents a short at the wall of the main circular waveguide. This means that the Bessel function for the field in the radial guide must have a zero at the radius of the wall. Moving out in radius from this zero, the Bessel function passes through a maximum. At this radius the current is a minimum. which is ensured by putting the radial waveguide in series with a shorted quarter wave section of coaxial guide.

1. Coaxial Waveguide

The figure below shows how a thin rectangular waveguide can be rolled up to become the coaxial waveguide.

In the rolled up waveguide, the x axis becomes the theta axis. But since the height of the guide is small compared to the radius, we can analyze it as if it were not rolled up at all. Except for one thing: the boundary conditions that the field be zero at the left and right ends of the guide are replaced by the condition that the field be periodic, i.e., when x changes by an amount equal to the circumference of the guide C, the field be unchanged. The E field will be in the Y direction.

A couple of steps removed from Maxwell's curl equations, we have

where k0 = 2 / 0 and 0 is the free space wavelength. Now the boundary condition requires that the x dependence of E be cos(2 nx/C). Also, since we have a TE11 mode in the circular waveguide, we have that n=1. Thus d2E/dx2 = -(2 /C)2E and we have

or

where C = 2 Rcoax and Rcoa x is the mean radius of the coaxial guide. Next we must determine Rcoax.

2. The radial waveguide

In the radial wave guide, E is in the z direction and is a function of of only r and . (If it were a function of z, its divergence wouldn't be zero). For this waveguide, we have

The angular dependence of E must again be proportional to cos(n ) where, for the same reason as before n = 1. In polar coordinates the equation then becomes

The solutions to this equation are J1(k0 r) and Y1(k0 r), Bessel functions of the first and second kind (Mathcad notation). These are plotted below.

You can use Mathcad (or whatever) to find a value a so that the linear combination J1(x) + a Y1(x) has a zero at x = k0 Rguide where Rguide is the radius of the circular waveguide. Using that value of a, increase x until the linear combination reaches a relative maximum (or minimum). That value of x = k0 Rcoax. Now that you have found Rcoax, you can calculate the guide wavelength in the coaxial guide and determine the length of a quarter wave section.

3. Frequency Response

At the design frequency, the transmission of the thermal transistion should be unity, even if the gap and the coaxial waveguide section are not particularly small. But, by making these gaps small, impedances remain low (relative to Z0 of the circular waveguide) as the frequency departs from the design frequency, and the short at the gap in the main circular waveguide remains fairly good. A rigourous analysis is complicated by the need to deal with the Bessel functions at every frequency (unlike the simpler analysis of choke joints in rectangular waveguide). Note also, that away from the design frequency, the dimension labeled on the drawing as "This dimension not important" does play a role and thata wide-band device could use several coaxial waveguides, rather than just one.