Critical Wavelengths in Elliptical Waveguides
Abstract Generated abstract
This paper examines how ellipticity affects the critical wavelengths of electric and magnetic modes in elliptical waveguides, a problem relevant to deformed circular waveguides where nonsymmetric mode degeneracy is removed. The analysis formulates the field problem in elliptical coordinates, uses separated Mathieu functions and boundary conditions for E and H waves to compute dimensionless critical wavelengths as functions of eccentricity. Calculations for several low-frequency modes show that the normalized critical wavelength decreases with increasing eccentricity, with odd waves having smaller values than corresponding even waves. Measurements of resonant frequencies in three elliptical resonators confirmed the calculated values within a reported error not exceeding 2.4 percent.
Full Text
PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
AWARDED THE ORDER OF THE OCTOBER REVOLUTION AND THE ORDER
OF THE RED BANNER OF LABOR, named after S. M. KIROV
Vol. 232 1975
CRITICAL WAVELENGTHS IN ELLIPTICAL WAVEGUIDES
S. A. NOVIKOV
(Presented by the seminar of the Microwave and Technical Devices Sector, Research Institute of Nuclear Physics)
Resonators of various types and waveguides of rectangular and circular cross sections have found the broadest application in microwave technology. The principal parameter determining their dispersion characteristics is the critical wavelength \(\lambda\) (or the critical frequency \(f_{\mathrm{cr}}\)), which in each particular case determines the choice of geometrical dimensions or the frequency of the feeding generator.
Considerable interest attaches to the study of critical wavelengths in elliptical waveguides. This is explained by the fact that any deformation of a circular waveguide in the course of manufacture or operation makes it elliptical. This leads to the removal of the degeneracy of nonsymmetric modes of oscillation (even and odd waves [1]), i.e., each nonsymmetric mode of oscillation splits into two components with unequal phase velocities.
In this work the influence of ellipticity on the critical wavelengths of \(E\)- and \(H\)-types is considered, and measurements have been made of the eigenfrequencies of resonators consisting of sections of elliptical waveguides with various values of the eccentricity.
Let the field vary in time according to the law \(e^{-i(\omega t-k_z z)}\). Then, in the elliptical system, the equation for \(\Pi_z\) takes the form
\[ \frac{\partial^2 \Pi_z}{\partial \xi^2} + \frac{\partial^2 \Pi_z}{\partial \eta^2} + \frac{\sqrt{S}}{2}(\operatorname{ch} 2\xi-\cos 2\eta)\Pi_z=0, \tag{1} \]
where
\[ \sqrt{S}=k_\perp h, \]
\(h\) is the semifocal distance of the ellipse,
\(\xi,\eta\) are elliptical coordinates.
Using the method of separation of variables, equation (1) can be transformed into the following system of equations:
\[ \frac{d^2\Phi}{d\eta^2} + \left( a-\frac{\sqrt{S}}{2}\cos 2\eta \right)\Phi=0, \tag{2} \]
\[ \frac{d^2\psi}{d\xi^2} - \left( a-\frac{\sqrt{S}}{2}\operatorname{ch} 2\xi \right)\psi=0. \tag{3} \]
where \(a\) is an arbitrary constant.
The solutions of these equations are Mathieu functions describing the field distribution in the azimuthal and radial directions. A suitable solution will be
\[ \Pi_z=\sum_{m=0}^{\infty} C_m Ce_m(\xi,\sqrt{S})ce_m(\eta,\sqrt{S})+ \sum_{m=0}^{\infty} S_m Se_m(\xi,\sqrt{S})Se_m(\eta,\sqrt{S}), \tag{4} \]
where \(C_m, S_m\) are arbitrary constants determined in the usual way; \(ce_m(\eta,\sqrt{S})\), \(se_m(\eta,\sqrt{S})\) are, respectively, the cosine and sine of the elliptic cylinder, which, when the ellipse becomes a circle, become the ordinary trigonometric functions \(\cos m\eta\), \(\sin m\eta\); \(ce_m\) is an even function, \(se_m\) an odd one.
\(Ce_m(\xi,\sqrt{S})\), \(Se_m(\xi,\sqrt{S})\) are radial functions, which for \(h\to 0\) become the Bessel functions \(I_m(k_1 r)\). Nothing can be said about the evenness and oddness of these functions, but since these functions are obtained by replacing \(\eta\) in \(ce_m(\eta,\sqrt{S})\) and \(se_m(\eta,\sqrt{S})\) by \(i\xi\), in general \(\Pi_{zmc}\) and \(\Pi_{zms}\) may be characterized as even and odd solutions.
The solution of the ordinary and modified Mathieu equation may be sought in the form of series of Bessel functions [2]
\[ \Pi_{xmc}=C_m\left[\sum_{r=0}^{\infty}(-1)^r De_{2r+p}^{\,l} I_{2r+p}(\sqrt{S}\cdot \operatorname{ch}\xi)\right]\cdot \]
\[ \cdot\left[\sum_{r=0}^{\infty}(-1)^r De_{2r+p}^{\,l} I_{2r+p}(\sqrt{S}\cdot \cos\eta)\right], \tag{5} \]
\[ \Pi_{zms}=S_m\left[\sum_{r=0}^{\infty}(-1)^r(2r+p)Do_{2r+p}^{\,l} I_{2r+p}(\sqrt{S}\operatorname{ch}\xi)\right]\cdot \]
\[ \left[\sum_{r=0}^{\infty}(-1)^r(2r+p)Do_{2r+p}^{\,l} I_{2r+p}(\sqrt{S}\cos\eta)\right], \tag{6} \]
where \(De\) and \(Do\) are tabulated expansion coefficients. For even \(m\) it is necessary to take \(p=0\), and for odd \(m\), \(p=1\).
Having determined the vectors \(\Pi_{zm}^{H}\) and \(\Pi_{zm}^{E}\) by means of the relations that connect them with the field components, one may write expressions for the latter [1]. By imposing boundary conditions for the tangential components of the field, one can determine the phase velocities and the critical waves of each of the \(E\)- or \(H\)-type waves.
For \(E\)-waves, the boundary condition, as usual, reduces to \(E_z=0\) on the inner surface of the waveguide \(\xi=\xi_0\)
\[ Ce_m(\xi_0,\sqrt{S})=0,\qquad Se_m(\xi_0,\sqrt{S})=0. \tag{7} \]
Let \(S_{mn}\) and \(\bar S_{mn}\) be, respectively, the \(n\)-th roots of these equations. With the corresponding substitution of \(S_{mn}\) and \(\bar S_{mn}\) into the solution of equation (1), we obtain various modes of \(E\)-waves, which we shall denote by the indices \(E_{mn}\). Here \(m\) is the number of variations of the field with respect to the angular coordinate \(\eta\).
For \(H\)-waves the boundary conditions have the following form:
\[ Ce'_m(\xi_0,\sqrt{S})=0,\qquad Se'_m(\xi_0,\sqrt{S})=0. \tag{8} \]
The types of oscillations of the \(H_{mn}\)-waves are determined analogously.
If, in a circular waveguide, \(\lambda_\perp\) is determined by the value of the radius (or of the perimeter \(2\pi R\)), then for an elliptical waveguide one must take into account the shape of the ellipse. We shall characterize the ellipticity by the eccentricity \(e=\dfrac{h}{a}\), where \(a\) is the major radius of the ellipse. For waveguides of this type the critical wavelength is a function of the variables
\[ \lambda_\perp=f(a,e). \]
The calculation is complicated by the fact that the expansion coefficients \(C_m, S_m\) depend on the parameter \(\sqrt{S}\).
\[ \text{If } \sqrt{S}=k_\perp h,\qquad \text{then } \lambda_\perp(m,n)=\frac{2\pi h}{\sqrt{S_{mn}}}. \]
Since we are interested in the influence of eccentricity on the critical wavelength, it is convenient to express the value of the function in dimensionless quantities.
\[ \frac{\lambda_\perp}{a}=\frac{2\pi e}{\sqrt{S}}. \tag{9} \]
Of greatest interest in the UHF range are the lowest-frequency types of oscillations \(H_{01}\), \(E_{01}\), \(E_{\mathrm{II}}^{o}\), \(H_{\mathrm{II}}^{eo}\), where the index \(o\) denotes odd types of oscillations, and \(e\) denotes even types of oscillations.
Let us consider the \(E_{01}\)-wave. To determine the critical wavelength of this wave one must solve the equation
\[ \sum_{r=0}^{\infty}(-1)^r De_{2r} I_{2r}\bigl(\sqrt{S_1}\cdot \operatorname{ch}\xi_0\bigr)=0. \]
Owing to the rapid convergence of the series, taking into account only four terms gives sufficient accuracy. The coefficients \(De_{2r}(S_1)\) are found by interpolation of tabulated values at three points.
Similarly, one can determine the critical wavelengths of other types of oscillations. Graphs of dependences of the form (9) are presented in Fig. 1.
Fig. 1. Dependence of \(\lambda_\perp\) on eccentricity: solid line — calculated curve; dashed line with squares — experiment for \(E_{01}\); squares — experiment for \(E_{\mathrm{II}}^{e}, E_{\mathrm{II}}^{o}\).
In order to verify the calculated data, the resonant frequencies of three elliptical resonators with eccentricities \(e_1=0.616\), \(e_2=0.845\), \(e_3=0.903\) were determined.
The wavelength in the waveguide was measured for the entire set of resonant frequencies within the operating range of the generator. The critical wavelength was determined from the relation [3]
\[ \lambda_0= \frac{1}{ \sqrt{\dfrac{1}{\lambda_\perp^2}+\dfrac{P}{2L^2}} }, \tag{10} \]
which is applicable to all waveguide systems and types of oscillations; \(p\) is the number of half-waves fitting along the resonator length \(L\). The values of \(\dfrac{\lambda_\perp}{a}\) determined in this way are plotted in Fig. 1.
Calculations showed that, as the eccentricity increases, \(\dfrac{\lambda_\perp}{a}\) decreases for all waves. As follows from Fig. 1, the values of the parameter \(\dfrac{\lambda_\perp}{a}\) for odd waves are smaller than the values of \(\dfrac{\lambda_\perp}{a}\) for the corresponding even waves.
The experimentally determined values of \(\lambda_\perp\) confirmed the correctness of the calculation; the error did not exceed \(2.4\%\). It is evident that taking into account a larger number of terms in the expansion of the Mathieu functions and interpolating with a larger number of points will increase the accuracy of agreement between the experimental and calculated data.
REFERENCES
- I. V. Mack-Lachlan. Theory and Application of Mathieu Functions. IL, Moscow, 1953.
- Tables for the Computation of Mathieu Functions. Computing Center of the Academy of Sciences of the USSR. Moscow, 1967.
- I. V. Lebedev. Microwave Engineering and Instruments. Vol. 1. “Vysshaya shkola,” 1970.