MATHEMATICAL PHYSICS

P. E. KRASNOUSHKIN

A METHOD FOR CALCULATING AN INHOMOGENEOUS DIAPHRAGMED RADIO WAVEGUIDE OF FINITE LENGTH

(Presented by Academician I. M. Vinogradov, 31 VIII 1964)

Any inhomogeneous cellular waveguide may be regarded as a chain of nonidentical functional cells (j=1,2,\ldots,R), transforming the distribution functions of the amplitudes of forced oscillations of the tangential fields (\mathbf{E}{\tau_j}^{(1)}(q)) and (\mathbf{H}}^{(1)}(q)), (q\in S^{(1)}(j)), at the input apertures (S^{(1)}(j)) of the cells into the functions (\mathbf{E{\tau_j}^{(2)}(q)) and (\mathbf{H}(q)}}^{(2)}(q)), (q\in S^{(2)}(j)), at the output apertures (S^{(2)}(j)) ((^1)). To construct the (N)-th matrix approximation, let us expand these fields in the basis functions ({e_{k,j}^{(1,2){k=1}^{\infty}) and ({h(q)}}^{(1,2){k=1}^{\infty}), (q\in S^{(1,2)}(j)), and, taking the first (N) Fourier coefficients of these expansions ({V_k(j)}) as the amplitudes specified at the strips (k=1,2,\ldots,N) of the multiport with number (j), we obtain, in place of the waveguide, a chain of (4N)-ports. The forced oscillations in it, in source representation, will be:}^{N}) and ({I_k(j)}_{k=1}^{N

[
\left|\begin{matrix}\mathbf{V}(j)\ \mathbf{I}(j)\end{matrix}\right| e^{i\omega t}
=
e^{i\omega t}
\left{
\sum_{j'=0}^{j-1} G'(j,j') Z_1(j')
+
\sum_{j'=j}^{R-1} G''(j,j') Z_2(j')
\right}
\left|\begin{matrix}\mathbf{V}^0(j')\ \mathbf{I}^0(j')\end{matrix}\right|,
\tag{1}
]

where (\mathbf{V}(j)) and (\mathbf{I}(j)) are (N)-dimensional vectors with components ({V_k}1^N) and ({I_k}_1^N). If the ports of adjacent cells are matched, i.e. (e), then}^{(1)}=e_{k,j}^{(2)}), (h_{k,j+1}^{(1)}=h_{k,j}^{(2)

[
\begin{aligned}
G'(j,j') &= A(j-1)\ldots A(j'+1)\,\mathcal{G}(j');\
G''(j,j') &= A^{-1}(j)\ldots A^{-1}(j'-1)A^{-1}(j'),
\end{aligned}
\tag{2}
]

where (A(j)) is the (2N\times 2N) Brillouin matrix of the (j)-th cell. The Green matrix operator (2) acts on the vectors of external voltages (\mathbf{V}^0(j')) and currents (\mathbf{I}^0(j')) through the matrices (Z_1) and (Z_2):

[
Z_1(j')=
\left(
\begin{matrix}
\widetilde{Z}(R+1,j'+1)\,\Delta Z^{-1}
&
-\widetilde{Z}(R+1,j'+1)\,\Delta Z^{-1}\widetilde{Z}(0,j'+1)
\
\Delta Z^{-1}
&
-\Delta Z^{-1}\widetilde{Z}(0,j'+1)
\end{matrix}
\right),
\tag{3}
]

where (Z(0,j'+1)) and (Z(R+1,j'+1)) are the input-impedance matrices (Z(0)) and (Z(R+1)) of the end cells (j=0) and (j=R+1), recalculated from the ends of the chain to the section (S^{(1)}(j'+1)), and (\Delta Z=\widetilde{Z}(0,j'+1)+\widetilde{Z}(R+1,j'+1)). (Z_2(j')) is obtained from (3) by interchanging (\widetilde{Z}(0,j'+1)) and (\widetilde{Z}(R+1,j'+1)). The introduction of (Z_1) and (Z_2) removes the pathological properties of the Brillouin matrix as (N\to\infty) ((^1)), and also ensures in (1) the automatic fulfillment of the boundary conditions for (j=0) and (j=R+1). Equation (1) is valid for arbitrary cells with matrices (A(j)) possessing inverses (A^{-1}(j)), and

for any frequencies (\omega), except for the resonant (\omega_s), determined from the equation (|\Delta Z|=0), whose roots do not depend on (j'). For arbitrary cells, (1) is an expansion in generalized normal waves of rank (2N). The appearance in (1) of waves of lower rank is possible for cells possessing identity elements (2). When all (A(j)) are identical (identical cells) and (Z_1) and (Z_2) commute, then from (1) we obtain an expansion in ordinary normal waves of rank 1.

Fig. 1. An inhomogeneous cellular waveguide and its equivalent multiconductor network. Unshaded boxes ((j=0,1,2,3,\ldots,R+1)) are the principal cells with Breizig matrices (A(j)) (formula (9)); for (j=1,2,\ldots,R), these are terminal cells, and (j=0) and (j=R+1) are cells with impedance matrices (Z(0)) and (Z(R+1)); shaded boxes are matching cells at the junctions of the principal cells ((j=1,2,\ldots,R+1)) with matrices (A_{\mathrm{match}}(j)) (formula (11)).

Construct (1) for axially symmetric forced oscillations of the (TH) type in an inhomogeneous iris-loaded waveguide of circular cross section (Fig. 1). In the case when all (b_j=b), it can be represented as a chain of cells of two kinds: symmetric cells with diaphragms, characterized by the parameters (a_j, t_j, d_j=D_j-t_j), and cells without diaphragms ((a_j=b,\ t=0)). The latter are sections of a cylindrical tube of radius (b) and length (D_j=d_j). Taking for them as basis functions ({e_k}k) and ({h_k}_k) the forms of the normal waves (TH), i.e.
[
e_k(r/b)=J_1[\rho_{0k}(r/b)]/b\sqrt{\pi}J_1(\rho_{0k}),
]
where (J_1) is the Bessel function of order 1, and (\rho_{0k}) is the (k)-th root of (J_0(\rho)), we write in this basis the Breizig matrices in the form
[
A(j)=
\left(
\begin{array}{cc}
\left|\operatorname{ch}\gamma_k d_j \delta_{kk'}\right|^{NN} &
\left|-(\gamma_k/i\omega\varepsilon)\operatorname{sh}\gamma_k d_j \delta_{kk'}\right|^{NN} \[2mm]
\left|-(i\omega\varepsilon/\gamma_k)\operatorname{sh}\gamma_k d_j \delta_{kk'}\right|^{NN} &
\left|\operatorname{ch}\gamma_k d_j \delta_{kk'}\right|^{NN}
\end{array}
\right),
\tag{4}
]
where
[
\gamma_k=\sqrt{(\rho_{0k}/b)^2-\omega^2/c^2}
]
is the wave number of wave number (k); (\delta_{kk'}) is the Kronecker symbol. For cells with diaphragms we introduce the basis system of functions (\mathscr{E}_s=\mathscr{E}'_s(r/a)), (r<a); (\mathscr{E}_s=0), (a<r<b), (s=1,2,\ldots,N), in sections (\langle 2\rangle) and (\langle 4\rangle) (Fig. 1), where the sharp edges of the diaphragms are located, and take into account in (\mathscr{E}'_s) for (r=a) the electrostatic singularities of the field (E_r).

Expand the fields (E_r) in sections (\langle 2\rangle) and (\langle 4\rangle) in the functions (\mathscr{E}s):
[
E_r \simeq \sum_s(r).}^{N} K_s^{(1,2)}\mathscr{E
]
The vectors (\mathbf K_1) and (\mathbf K_2) with components ({K_s^{(1)}}^{N}) and ({K_s^{(2)}}^{N}), according to the first boundary-value problem of electrodynamics, uniquely determine the fields of the cell. However, the junction of cells in the chain occurs in the aperture sections (S^{(1)}) and (S^{(2)}) ((\langle 1\rangle) and (\langle 5\rangle) in Fig. 1); therefore the basis system ({\mathscr{E}_s}_1^N) in (\langle 2\rangle) and (\langle 4\rangle) must be transformed to sections (\langle 1\rangle) and (\langle 5\rangle). As a result of solving the first problem of electrodynamics (1), we obtain the following relations of the vectors (\mathbf K^{(1,2)})

with the vectors V and I in sections (\langle 1\rangle) (output) and (\langle 5\rangle) (output)

[
\begin{pmatrix}
v_{11} & -v_{12}\
v_{12} & -v_{11}
\end{pmatrix}
\left|
\begin{matrix}
{K_s^{(1)}}1^N\
{K_s^{(2)}}_1^N
\end{matrix}
\right|
=
\left|
\begin{matrix}
{V_s^{\mathrm{in}}}_1^N\
{V_s^{\mathrm{out}}}_1^N
\end{matrix}
\right|;
\qquad
\begin{pmatrix}
i\} & -i_{12
i_{12} & -i_{11}
\end{pmatrix}
\left|
\begin{matrix}
{K_s^{(1)}}_1^N\
{K_s^{(2)}}_1^N
\end{matrix}
\right|
=
\left|
\begin{matrix}
{I_s^{\mathrm{in}}}_1^N\
{I_s^{\mathrm{out}}}_1^N
\end{matrix}
\right|.
\tag{5}
]

Here (v_{11}) and (v_{12}) are matrices with elements

[
v_{ss'}^{11}
=
\sum_{k=1}^{\infty}
\frac{\alpha_{ks}\alpha_{ks'}\,\operatorname{cth}\alpha_k t}{\alpha_k}
+
\sum_{k=1}^{\infty}
\frac{\beta_{ks}\beta_{ks'}\,\operatorname{cth}\gamma_k d/2}{\gamma_k};
\qquad
\alpha_k
=
\sqrt{\left(\frac{\rho_{0k}}{a}\right)^2-\frac{\omega^2}{c^2}};
]

[
v_{ss'}^{12}
=
\sum_{k=1}^{\infty}
\frac{\alpha_{ks}\alpha_{ks'}}{\alpha_k\,\operatorname{sh}\alpha_k t};
\qquad
\alpha_{ks}
=
\left(\mathscr E_s^{3},\, e_k\left(\frac r a\right)\right);
\qquad
\beta_{ks}
=
\left(\mathscr E_s,\, e_k\left(\frac r b\right)\right),
]

and the matrices (i_{11}) and (i_{12}) have elements

[
i_{ss'}^{11}
=
v_{ss'}^{11}
-
\sum_{k=1}^{\infty}
\frac{\beta_{ks}\beta_{ks'}}
{\gamma_k\,\operatorname{ch}\gamma_k \frac d2\, \operatorname{sh}\gamma_k \frac d2};
\qquad
i_{ss'}^{12}
=
v_{ss'}^{12}.
]

The index (j) is omitted. The coefficients (\alpha_{ks}) and (\beta_{ks}) for (\mathscr E_s'=(r/a)^{2s-1}/\sqrt{1-(r/a)^2}) are given in (3).

The transformation of the components ({V_s}1^N) and ({I_s}_1^N) of the vectors V and I, specified in the basis ({\mathscr E_s}_1^N) recalculated into (\langle 1\rangle) or (\langle 5\rangle), into components of these same vectors specified in the natural basis ({e_k(r/b)}), i.e., into the components (V_k=(E_r,e_k(r/b))) and (I_k=(H\varphi,e_k(r/b))), is expressed by the (N\times\infty) matrices (|\beta_{ks}^{V}|) and (|\beta_{ks}^{I}|):

[
{V_s}1^N=|\beta{V_k}}^{V}|^{N,\infty1^\infty;
\qquad
{I_s}_1^N=|\beta{I_k}_1^\infty,}^{I}|^{N,\infty
\tag{6}
]

[
\beta_{ks}^{V}=\beta_{ks}/\gamma_k\,\operatorname{sh}\gamma_k d/2
\quad\text{and}\quad
\beta_{ks}^{I}=\beta_{ks}/i\omega\varepsilon\gamma_k\,\operatorname{ch}\gamma_k d/2.
]

From (5) we obtain the admittance matrix of a cell with a diaphragm in the basis of functions ({\mathscr E_s}_1^N), recalculated to the output and input sections of the cell:

[
Y=
\begin{pmatrix}
Y_{11} & Y_{12}\
Y_{21} & Y_{22}
\end{pmatrix}
=
\begin{pmatrix}
i_{11} & -i_{12}\
i_{12} & -i_{11}
\end{pmatrix}
\begin{pmatrix}
v_{11} & -v_{12}\
v_{12} & -v_{11}
\end{pmatrix}^{-1}.
\tag{7}
]

Here

[
Y_{11}=i_{11}\Delta-v_{12}\Delta v_{12}v_{11}^{-1};
\qquad
Y_{12}=i_{11}\Delta v_{12}v_{11}^{-1}-v_{12}\Delta,
\tag{8}
]

[
\Delta=(v_{11}-v_{12}v_{11}^{-1}v_{12})^{-1};
]

(Y_{22}=Y_{11}) by symmetry of the cell, and (Y_{21}=Y_{12}) owing to the satisfaction of the reciprocity principle. The Bracewell matrix of the cell in this same basis has the form

[
A(j)=
\begin{pmatrix}
A_{11} & A_{12}\
A_{21} & A_{22}
\end{pmatrix}
=
\begin{pmatrix}
-Y_{12}^{-1}Y_{11} & Y_{12}^{-1}\
Y_{11}Y_{12}^{-1}Y_{11}-Y_{12} & -Y_{11}Y_{12}^{-1}
\end{pmatrix},
\tag{9}
]

where (A_{11}A_{21}^{-1}A_{22}A_{21}-A_{12}A_{21}=\mathscr E).

The input-impedance matrices of the terminal cells are determined by the formulas (Z(0)=Y_{11}^{-1}(0)); (Z(R+1)=Y_{11}^{-1}(R+1)), if the input aperture of the 0-th cell and the output aperture of the ((R+1))-th cell are closed by perfectly conducting partitions. If the cells are nonidentical, then the matrices (9) cannot be used directly to construct the Green’s function (2), since they are written in different bases. Let us reduce them all to the basis ({e_n(r/b)}_k), assuming that the field forms in the apertures of the cells are approximated with sufficient accuracy by (N) basis functions ({e_k}_1^N). Then, truncating the matrices in (6) to the (N)-th column and inverting these relations, we obtain

[
{V_k}1^N \simeq |\bar\beta{V_s}}^{V}|^{-11^N;
\qquad
{I_k}_1^N \simeq |\bar\beta{I_s}_1^N.}^{I}|^{-1
\tag{10}
]

Here it is assumed that the matrices (|\beta_{ks}^{V}|) and (|\bar{\beta}{ks}^{I}|) are nonsingular. Using the abbreviated forms (5) and (10), we obtain that the matching (4N)-terminal network, which must be inserted between the (j)-th and ((j+1))-st cells with diaphragms, has the quasi-diagonal Breisig matrix (A)}

[
\left(
\begin{array}{cc}
|\bar{\beta}{ks}^{V}(j+1)|\cdot|\bar{\beta} & 0 \[2mm]}^{J}(j)|^{-1
0 & |\bar{\beta}{ks}^{V}(j+1)|\cdot|\bar{\beta}}^{J}(j)|^{-1
\end{array}
\right).
\tag{11}
]

If adjacent cells turn out to be a cell with a diaphragm and a cell without one, then between them one should insert a matching (4N)-terminal network with a quasi-diagonal matrix (A_{\mathrm{match}}) with blocks (|\beta_{ks}^{V}(j)|) and (|\beta_{ks}^{J}(j)|), or with blocks (|\bar{\beta}{ks}^{V}(j)|^{-1}) and (|\beta), depending on the order in which the cells are arranged. The general circuit diagram with matching cells is given in the lower half of Fig. 1. In constructing the Green operator (2), the matrices (4), (9), (11) should be successively multiplied in the order of their arrangement in the diagram of Fig. 1. We note that, in order to obtain the operator expression (1), it is also necessary to compute (Z_1) and (Z_2) by formula (3). The transformed impedances entering into it are determined by the propagation method with the aid of the formula}^{J}(j)|^{-1

[
Z(j_0,j' + 1)=[A_{11}Z(j_0)-A_{12}][-A_{21}Z(j_0)+A_{22}]^{-1},
]

where (A_{ik}) are the blocks of the matrix (G'(j' + 1,1)) for (j_0=0), and of the matrix (G''(j' + 1,R)) for (j_0=R+1) (see in more detail ((^1))). The vectors (V^0(j')) and (I^0(j')) in (1) are determined by expanding the fields (E_r(r)) and (H_\varphi(r)), specified at the input cross sections of the cells of the inhomogeneous waveguide, in the corresponding systems of basis functions of the input cross sections.

If the cellular waveguide is infinite and consists of identical cells (A_j=A), then the matrices (G'(j,j')) and (G''(j,j'')) in (1) can be simultaneously diagonalized by the same linear change of basis. In this case there are no matching cells, and the basic cells form a countable set of mutually unconnected chains of two-terminal networks (l=1,2,3,\ldots). In each such chain there exists one normal wave with wave number (\psi_l,\ l=1,2,\ldots), determined through the eigenvalue (\lambda_l=\exp(-\psi_l)) of the matrix (A). By virtue of the symmetry of the cells and the fulfillment of the reciprocity principle, the eigenvalue equation for (\lambda_l) reduces to the form

[
\det\left(Y_{11}-\cos\psi\,Y_{12}\right)=0.
\tag{12}
]

It is one of the modifications of the dispersion equation for a homogeneous cellular waveguide considered in ((^3)) (formula (16)).

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
31 VIII 1964

REFERENCES

(^1) P. E. Krasnushkin, Radio Engineering and Electronics, 10 (1965).
(^2) P. E. Krasnushkin, DAN, 155, No. 5, 1042 (1964).
(^3) P. E. Krasnushkin, S. P. Lomnev, A. G. Tragos, DAN, 159, No. 3, 528 (1964).
(^4) Ya. N. Fel’d, Fundamentals of the Theory of Slot Antennas, 1948.