A.S2 Modal Analysis of an H-Plane Step in Rectangular Waveguide
The field analysis of most discontinuity problems is very difficult, and beyond the scope of this book. The technique of modal analysis, however, is relatively straightforward and similar in principle to the reflection/transmission problems which were discussed in Chapters 2 and 3. In addition, modal analysis is a rigorous and versatile technique that can be applied to many coax, waveguide, and planar transmission line discontinuity problems, and lends itself well to computer implementation. We will present the technique of modal analysis by applying it to the problem of finding the equivalent circuit of an H-plane step (change in width) in rectangular waveguide.
The geometry of the H -plane step is shown in Figure A.S2.1. It is assumed that only the dominant TE10 mode is propagating in guide 1 (z < 0), 1lnd that such a mode is incident on the junction from z < 0. It is also assumed that no modes are propagating in guide 2, although the analysis to follow is still valid if propagation can occur in guide 2. From Section 4.3, the transverse components of the incident TE10 mode can then be written, for z < 0,
(A.S2.1)
(A.S2.2)
where
(A.S2.3)
is the propagation constant of the TEn0 mode in guide 1 (of width a ), and
(A.S2.4)
is the wave impedance of the TEn0 mode in guide 1. Because of the discontinuity at z = 0 there will be reflected and transmitted waves in both guides, consisting of infinite sets of TEn0 modes in guides 1 and 2 . Only the TE10 mode will propagate in guide 1, but the higher-order modes are also important in this problem because they account for stored energy, localized near z = 0. Because there is no y variation introduced by this discontinuity, TEnm modes for m 0 are not excited, nor are any TM modes. A more general discontinuity, however may excite such modes.
Figure A.S2.1 Geometry of a H –plane step (change in width) in rectangular wave guide .
The reflected modes in guide 1 may then be written for z < 0 as
(A.S2.5.a)
(A.S2.5.b)
where An is the known amplitude coefficient of the reflected TEn0 mode in guide 1. The reflection coefficient of the incident TE10 mode in guide 1. The reflected coefficient of the incident TE10 mode is then A1. Similarly , the transmitted modes into guide 2 can be written for z>0, as
(A.S2.6.a)
(A.S2.6.b)
where the propagation constant in guide 2 is
(A.S2.7)
and the wave impedance in guide 2 is
(A.S2.8)
Now at z = 0, the transverse fields (Ey, Hx) must be continuous for 0 < x < c; in addition, Ey must be zero for c < x < a because of the step. Enforcing these boundary conditions leads to the following equations:
(A.S2.9.a)
(A.S2.9.b)
Equations (A.S2.9.a) and (A.S2.9.b) constitute a doubly infinite set of linear equations for the modal coefficients An and Bn. We will first eliminate the Bns, and then truncate the resulting equation to a finite number of terms and solve for the Ans.
Multiplying (A.S2.9.a) by sin (nx/a), integrating from x = 0 to a, and using the orthogonality relations from Appendix D yields
(A.S2.10)
where
(A.S2.11)
is an integral that can be easily evaluated, and
(A.S2.12)
is the Kronecker delta symbol. Now solve (A.S2.9.b) for Bk by multiplying (A.S2.9.b) by
sin (kx/c) and integrating from x = 0 to c. After using the orthogonality relations, we
obtain
(A.S2.13)
Substituting Bk from (A.S2.13) into (A.S2.10) gives an infinite set of linear equations for the Ans, where m = 1, 2, ...,
(A.S2.14)
For numerical calculation we can truncate the above summations to N terms, which will result in N linear equations for the first N coefficients, An. For example, let N = 1.
Then (A.S2.14) reduces to
(A.S2.15)
Solving for A1 (the reflection coefficient of the incident TE10 mode) gives
(A.S2.16)
where , which looks like an effective load impedance to guide 1. Accuracy is improved by using larger values of N, and leads to a set of equations which can be written in matrix form as
[Q][A] = [P], (A.S2.18)
where [Q] is a square N N matrix of coefficients,
Q(A.S2.19)
[P] is an N 1 column vector of coefficients given by
P(A.S2.20)
and [A] is an N 1 column vector of the coefficients An. After the Ans are found,the Bns can be calculated from (A.S2.13), if desired. Equations (A.S2.18)- (A.S2.20) lend themselves well to computer implementation.
Figure A.S2.2 shows the results of such a calculation. If the width, c, of guide 2 is such that all modes are cutoff (evanescent), then no real power can be transmitted into guide 2, and all the incident power is reflected back into guide I. The evanescent fields on both sides of the discontinuity store reactive power, however, which implies that the step discontinuity and guide 2 beyond the discontinuity look like a reactance (in this case an inductive reactance) to an incident TE10 mode in guide I. Thus the equivalent circuit of the H -plane step looks like an inductor at the z = 0 plane of guide I, as shown in Figure A.III.1.1.e. The equivalent reactance can be found from the reflection coefficient Al as
X=-jZ(A.S2.21)
Figure A.S2.2 shows the normalized equivalent inductance versus the ratio of the guide widths, c/a, for a free-space wavelength . = 1.4a and for N = 1, 2, and 10 equations. The modal analysis results are compared to calculated data from reference [7]. Note that the solution converges very quickly (because of the fast exponential decay of the higher-order evanescent modes), and that the result using just two modes is very close to the data of reference [7] .
The fact that the equivalent circuit of the H -plane step looks inductive is a result of the actual value of the reflection coefficient, A1, but we can verify this result by
Figure A.S2.2 Equivalent inductance of H-plane asymmetric step.
computing the complex power flow into the evanescent modes on either side of the discontinuity. For example, the complex power flow into guide 2 can be found as
P=
=
=
=
=(A.S2.22)
where the orthogonality property of the sine functions was used, as well as (A.S2.6) to (A.S2.8). Equation (A.S2.22) shows that the complex power flow into guide 2 is purely inductive. A similar result can be derived for the evanescent modes in guide 1.