Waveguide tapers are often necessary when making a transition between two different types of waveguide transmission lines, such as singleridge waveguide and double-ridge waveguide. While three-dimensional electromagnetic (EM) analysis programs can help achieve good performance for such waveguide tapers, they also depend on a fairly close starting point in terms of the geometric dimensions for the transition. Fortunately, the author has developed a useful software program based on a mathematical model that provides geometric dimensions for tapered waveguide transformers.

Designing waveguide transformers usually starts with an initial geometric structure that can be perfected by means of EM analysis, using a commercial EM computer-aidedengineering (CAE) software tool such as the High-Frequency Structure Simulator (HFSS) software from Ansoft Corp. The latest versions of such EM simulator include powerful optimization routines but, depending upon the starting point, can require a long time to make meaningful refinements to a waveguide transition design. But by the use of mathematical models, it is possible to optimize a geometric design for a required frequency range. These results can then be used with an EM simulator for fine-tuning the results.

Those “old” mathematical models, combined with a powerful algorithm, are the basis for a program that provides a geometric starting point for the design of waveguide tapers and transitions. The program will be used to design various examples of waveguide transitions, and geometric representations of those transitions will be created from the dimensions provided by the program. The program can be used to calculate dimensions for three different types of waveguide: rectangular, single-ridge, and double-ridge waveguide (Fig. 1). It can also be used for calculations when designing horn antennas radiating into free space.^1-5

The three ways to calculate waveguide impedance6 are shown in Eqs. 1a, 1b, and 1c in the box. The impedance for waveguide, which does not support transverse electromagnetic (TEM) mode propagation, is not unique and can be readily calculated for a given structure. However, the useful relationship shown in Eq. 2 makes it possible to find the intrinsic impedance, Z_PV = Z_int, for a given waveguide profile according to whether it is rectangular or ridged waveguide. For rectangular waveguide², the relationship of Eq. 3 can be applied, where the impedance of free air can be found by

Z_AIR = (µ₀/e₀)^0.5

For ridged waveguide,^1,3-6 the relationship of Eq. 4 can be applied, where m = 2 for a single-ridge waveguide structure and m = 1 for a double-ridge waveguide, and

θ₁ = [ 1 - (S/A)](A/λC)

and

θ₂ = [ 1 - (S/A)](A/λC)

where λ_C is the wavelength at the cutoff frequency.

For given input and output impedances and a required voltage standing wave ratio (VSWR), any tapered geometric profile that is long enough will give good results for the different types of waveguide. The software program presented here is not meant to deal with particularly long waveguide structures, but is designed to calculate the impedance taper with the minimum length necessary to ensure a required reflection coefficient, ρ. The reflection coefficient is a parameter of the phase in the impedance equation of the waveguide taper. The method for solving for it is to use the simplified Riccati equation, with ρ² << 1, as shown in Eq. 5.

where

F(z) = 0.5{d/dz[lnZ(z)]}

and

ß = 2/π = 2f/c

and

ß = the wave coefficient at frequency f;
c = the speed of light in a vacuum;
z = the position on the taper; and
Z(z) = the impedance at the taper location indicated by z.

The software program provides the option of calculating one of three types of electrical impedance tapers. These three types are near optimum taper, cosine square taper, and exponential taper. For the near optimum taper,⁷ the impedance function can be found by applying the relationship of Eq. 6, where Z_in = the impedance at the transformer’s input port;
Z_out = the impedance at the transformer’s output port, with the value of parameter y₀ = 0.5ln(Z_out/Z_in);
b = a calculated parameter that is dependent upon the required maximum acceptable VSWR;
L = the taper length derived also from the required VSWR value;
and I₀ = the modified Bessel function of the first type and zero order.

For the cosine square taper,⁸ the impedance function can be found by applying the relationship of Eq. 7, while for the exponential taper,⁸ the impedance function can be found by applying the relationship of Eq. 8.

For finding the key parameters for a given waveguide taper, the first step is to calculate the input and output impedances from the given dimensions. Then, it is a matter of

calculating the impedances for any number of points along a taper profile for a desired taper type. The last step involved converting the impedance values to wavguide geometric dimensions. This last step, of course, is not trivial. In order to create the proper taper for a given waveguide, the software program must certify that the predicted geometric profile is continuous and that there are no drastic dimensional changes along the taper—i.e., that the taper is smooth and continuous. The right algorithm was needed to perform this certification.

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