tems. As noted in the preceding section, the usual precision waveguide and connector requirement is often imposed to permit the specification and evaluation in terms of these parameters, rather than as a result of more fundamental considerations.

The new concept, which has been outlined in this dissertation, proposes an alternative description in which power, maximum efficiency, and mismatch factor (M, N, R) are the fundamental parameters. The advantages of this concept are basically twofold. First, the inevitable departures from waveguide uniformity in any physically realizable system no longer impose a limitation on its experimental evaluation, and precision measurements are now possible in coaxial systems which are fitted with the Type N connector, for example. The second advantage of this concept is in the more easily understood physical model which it provides, where instead of taking account of the traveling wave amplitudes at the several ports, the attention is directed to the power transfer characteristics. (The terms Mga and Nat in eq (19), for example, take the place of a rather complicated algebraic expression involving the six complex reflection and scattering coefficients.) As a consequence of this simplification, it should now be practical to instruct field personnel in the use and application of mismatch corrections.

It should perhaps be noted that many of the basic parameters, upon which the new concept is built, are not new. The quantities which are represented by Pg, na, Mgt, for example, are well known in the prior art. The contribution of this dissertation, is in the recognition of the invariant features of these parameters and the introduction of others such that independence of connector discontinuity is achieved. In addition, practical techniques, for the measurement of these terms, have been developed and described.

For completeness, it should be observed that while the analysis of nonuniform transmission lines and waveguides is a subject of current interest, the emphasis in this dissertation has been on the development of a measurement theory and practice rather than a detailed description of the fields associated with nonuniform nonuniform waveguide. For convenience, the measurement procedures have been developed in terms of the existing circuit representation and then examined for their dependence upon uniformity requirements. In particular these procedures are characterized by their dependence only upon a phasable short, and methods for recognizing the equality of two "impedances." It is possible that these results could be obtained more directly from an alternative formulation.

Finally, the discussion has been limited primarily to the power measurement problem, with attenu

"This may be demonstrated in several ways. From physical considerations it is evident that T1 when = 1. In terms of eqs (26) and (27) this calls for R = 1, R0, and which, with the appropriate changes in notation, leads to eqs (A-3) and (A-4).

where C and D are two additional constants of the system. Taking the ratio of eqs (A-19) and (A-22) yields eq (15), which is the starting point in the usual reflectometer development [3].

7.4. Appendix 4

to

It will be shown that the adjustment of Tx, achieve the C+DT=0 condition, is invariant to the adjustment of T.

This could be done in a completely formal manner by finding the appropriate expressions for C and D in terms of the parameters of the two couplers and tuning transformers; this approach, however, is tedious. Instead, in figure 6, let an auxiliary terminal plane be temporarily inserted between the tuner Ty and the second coupler (the one on the right). Within this second coupler (including Tr) the electromagnetic fields are completely determined by a and b2 and, in particular, the field in arm 4 can be written

where C' and D' are functions of the parameters of the second coupler and Tr (but not of Ty). If b4 is eliminated between eqs (15) and (A-23) it becomes apparent that a nontrivial linear relationship exists between b3, b2, and a2 if, and only if, C/D C'/D'. Since the ratio C'/D' is invariant to the adjustment of Ty, the same is true of C/D. Thus, if this ratio is such that eq (14) is satisfied for one particular adjustment of Ty, the same is true for all possible adjustments.

7.5. Appendix 5

It is the purpose of this appendix to investigate, analytically, the wave propagation in a perturbed waveguide.

Beginning with figure 16 it will prove useful to consider the perturbed waveguide and a circumscribed uniform waveguide. Within the volume V, which is bounded by the uniform waveguide and the

Uniform Waveguide

FIGURE 16. System used to analyze wave propagation in `perturbed waveguide.

4 cf reference 16, pages 170-223.

47 See for example, reference 15, p. 250.

where the integration is over the transverse plane and n is in the positive z direction.

It is next necessary to introduce the vector analog of Green's second identity 47 which states that given a closed region of space, bounded by a regular sur

After making these

(A-35) substitutions, eq (A-32)

=

Ss1 241a(-ea ha) · n ds + Ss2 2A 2a (— eå × hå) · n ds Sv(ea-Eaz) ea2 · J dv. (A-41)

Finally, observing that the outward normal from the volume V in figure 16 coincides with the positive z direction at S2, but is opposite to it at S1, and using eq (A-29) gives:

A2a=A1a-So(eå - Eaz) · Jea2 dv. (A-42)

Equation (A-42) is interpreted as follows. At terminal 2 the coefficients A2a, A2b, A2c, . . . represent a collection of waves whose direction of propagation is in the positive z direction. From eq (A-42) each of these coefficients is comprised of two parts. . . the complex amplitude which would obtain in the absence of the perturbation, plus a contribution due to the perturbation. Note that whereas the coefficients of the field expansion at terminals 1 and 2 are equal in the absence of the perturbation, the expression for the transverse field (cf. eq (A-37)) also includes the propagation constant eyaz such that the usual attenuation is obtained for those components of the field which represent evanescent waves.

With regard to the second term on the right of eq (A-42) it is convenient to choose the origin of the coordinate system such that z=0 on S2. Then z is negative throughout the volume of integration. If in eq (A-42), A2a represents the amplitude of one of the nonpropagating modes, the exponential factor indicates that the contribution to A2ɑ from the induced current density, J, is small for those current elements which are remote from S2. Indeed, it is both possible and convenient to visualize the induced current J as a secondary source of waves whose propagation is then governed by the dimensions of the unperturbed guide.

In the foregoing treatment, the only type of perturbation explicitly considered has been that of a deformed boundary, and this is certainly the problem of greatest practical interest. It is possible to obtain the same general conclusion if an arbitrary distribution of dielectric and magnetic media is also included in the perturbed guide.

This discussion provides the basis for the description given in section 3c.

Equation (A-48) may be rearranged in two different ways to yield:

and

Substitution of these expressions in the denominator and numerator respectively of eq (A-53) leads to:

| S21|2 (1 − | Tm |2)

(A-43)