Design and performance of millimeter-wave thermocouple sensors - the HP Q8486A and HP R8486A Power Sensors - technical

Hewlett-Packard Journal, April, 1988 by Lee H. Colby

The input impedance Z.sub.i at a radius r.sub.b for a radial transmission line shorted at its outer radius r.sub.a is given by: Z.sub.i = jZ.sub.o.sub.i.sin([theta].sub.kr.sub.b - [theta].sub.kr.sub.a)/cos([psi].sub.kr.sub.b - [theta].sub.kr.sub.a.)

where Z.sub.o.sub.i is the characteristic wave impedance for radial transmission lines: Z.sub.o.sub.i = 377G.sub.o.(kr.sub.b)/G.sub.1.(kr.sub.b) and [theta].sub.x = tan.sup.-1.[N.sub.o.(X)/J.sub.o.(X)] [psi].sub.x = tan.sup.-1.[J.sub.1(X)/-N.sub.1.(X)] G.sub.n.(X) = [square root of](J.sup.2.sub.n.(X) N.sup.2.sub.n.(X)) k = 2[pi]/[lambda]

J.sub.n.(X) is a Bessel function of the first kind and N.sub.n.(X) is a Bessel function of the second kind.

Solving for Z.sub.i resonant at the center of the frequency band, or Z.sub.i equal to infinity by varying r.sub.a., obtains: cos([psi].sub.kr.sub.b - [theta].sub.kr.sub.a.) = 0

The total impedance at any frequency can be found by taking k at the frequency of interest and solving for the radial choke's input impedance. Then: Z.sub.total = dZ.sub.i./2[pi]r.sub.b

where d is the width of the choke (see Fig. 3).

Other methods of trying to place an electrical high-frequency short at the waveguide were tried, such as a low-high-low-impedance dumbell choke filter, but the radial choke filter reflects an impedance closer to zero ohms at zero degrees at the waveguide entrance. For calibration purposes, a 50-MHz signal can be fed through the choke assembly, across the waveguide with minimal effect, through the coaxial line, and into the thermocouple.

Constant-impedance Taper

The thermocouple is mounted on a suspended substrate transmission line whose inner and outer conductors are tapered from the wide center conductor at the coaxial end of the transmission line down to a narrow center conductor at the thermocouple end of the transmission line. Usually this is done by using straight tapers for the inner and outer conductors. Straight tapers are a compromise and a better method that maintains a constant impedance as the conductors move closer together is desired.

The suspended substrate is mounted in a circular housing, and for wide outer conductor spacings approaching the outer housing, closed-form equations do not exist for calculating the spacing of the outer-conductor-to-inner-conductor width versus impedance. The required spacing for the wide center conductor cases was determined using a finite-difference program called FCAP.sup.3 For the narrower center conductor region where the effect of the outer housing is negligible, analysis equations.sup.4 were used to determine the correct spacing because they are faster and more accurate than the finite-difference approach. The results were then curve-fitted into a fourth-order polynomial describing the outer conductor width versus the center conductor width for a constant impedance of 50 ohms.

An arbitrary shape called an ogee curve was selected for the center conductor. An ogee curve is just two opposing arcs that meet at a point where their slopes are identical, giving a smooth continuous curve from the wide to the narrow end of the center conductor length. The beginning and ending center conductor widths and the transition length for the desired center conductor were selected. Then an outer conductor position was located that gave an impedance of 50 ohms for many different closely spaced positions down the length of the center conductor (see Fig. 4).