CHAPTER 2 CHAPTER 2
2.1 Introduction
Propagation of electromagnetic waves in circular waveguides has been widely investigated, for waveguides with lossy (Glaser, 1969) and superconducting (Yassin et al., 2001; Yassin et al., 2003) walls, unbounded dielectric rod (Claricoats, 1960a), bounded dielectric rod in a waveguide (Claricoats, 1960b), and multilayered coated circular waveguide (Chou and Lee, 1988). The computation given by these authors were based on a method suggested by Stratton (1941). The circular symmetry of the waveguide allows the boundary matching equations to be expressed in a single variable which is the propagation constant kz. The eigenmodes could therefore be obtained from a single transcendental equation. This approach, however, cannot be
implemented in the case of rectangular symmetry where a 2D Cartesian coordinate system must be used (Krammer, 1976). A similar rigorous technique to study the attenuation of rectangular waveguides is not available hitherto. It is to be noted, however, that in practice, rectangular waveguides are more widely used than circular waveguides. This is especially true in receivers of radio telescopes (Carter et al., 2004; Boifot et al., 1990;
Withington et al., 2003) where rectangular waveguide-to-microstrip transition is commonly used to couple the field to the detector circuit. Indeed, rectangular waveguides are much easier to manipulate than circular waveguides (bend, twist, etc.) and also offer significantly lower cross polarization component.
The approximate power-loss method has been widely used in analyzing wave attenuation in lossy rectangular waveguides as a result of its simplicity and because it gives reasonably accurate result, when the frequency of the signal is well above cutoff (Stratton, 1941; Seida, 2003; Collin, 1991;
Cheng, 1989). In this method, the fields’ expression are derived assuming perfectly conducting walls, allowing the solution to be separated into pure TE and TM modes. For a practical waveguide with finite conductivity, however, a superposition of both TE and TM modes is necessary to satisfy the boundary conditions (Stratton, 1941; Yassin et al., 2003). To calculate the attenuation using the power-loss method, ohmic losses are assumed to exist due to small field penetration into the conductor walls. Results however show that this method fails near cutoff, as the attenuation obtained diverges to infinity when the signal frequency f approaches the cutoff frequency fc. Clearly, it is more
realistic to expect losses to be high but finite rather than diverging to infinity.
The inaccuracy in the power-loss method at cutoff is due to the fact that the fields’ equation is assumed to be the same as those of a lossless waveguide. A lossless waveguide behaves exactly like an ideal high pass filter where signal ceases to propagate at frequency f below the cutoff fc. Since waveguides are commonly used as filters, an accurate calculation of the power loss at frequencies at the vicinity of cutoff would hence be substantial.
Robson (1963) and Bladel (1971) discussed degenerate modes propagation in lossy rectangular waveguides, but neither of them was able to compute the attenuation values accurately near cutoff. Like the power-loss method, their theories predict infinite attenuation at cutoff. An expression valid at all frequencies is given by Kohler and Bayer (1964) and reiterated by Somlo and Hunter (1996). This expression however is only applicable to the TE10 dominant mode. The perturbation solution developed by Papadopoulos (1954) shows that the propagation of a mode does not merely stop at fc. Rather, as the frequency approaches fc, transition from a propagating mode to a highly attenuated mode takes place. The propagation of waves will only cease when f = 0. Papadopoulos’ perturbation method (PPM) shows that the attenuation at frequencies well above fc remains in close agreement with that computed using the power loss method for non-degenerate modes. Because of this reason, PPM is perceived as a more accurate technique in computing the loss of waves travelling in waveguides. A similar solution has been derived by Gustincic using the variational approach (Collin, 1991; Gustincic, 1963).
Nevertheless, the PPM is merely an approximate solution based on the
perturbation from the lossless case. Therefore, it is not an accurate derivation from fundamental principles. Although this method takes into account the co- existence of TE and TM modes, the boundary conditions are still assumed to be the same as those of the perfectly conducting waveguide.
It can be seen that almost all analysis techniques are based on certain approximations and assumptions. The most commonly used assumption is that based on the boundary conditions of lossless waveguides. Due to such assumption, most methods fail to give an insight or deeper understanding on the mechanism of the propagation of waves in lossy waveguides. Moreover, at very high frequency – especially that approaches the millimeter and submillimeter wavelengths – the loss tangent of the conducting wall decreases.
Therefore, such assumption turns out to be inaccurate at very high frequency.
Although Stratton (1941) has developed a truly fundamental approach to analyze waveguides, his approach is only restricted to the case of circular waveguides and could not be applied to rectangular waveguides. Because of these reasons, a more accurate approach – one that does not assume lossless boundary condition, is essential to accurately compute the loss of waves in waveguides – in particular, at frequencies operating in the millimeter and submillimeter wavelengths.
In this chapter, a novel and fundamental technique to compute the attenuation of waves in rectangular waveguides with imperfectly conducting walls is introduced. The method is derived from fundamental principles without assumptions made in its formulation. In this method, the solution for
the attenuation constant is found by solving two transcendental equations derived from matching the tangential components of the electromagnetic field at the waveguide walls with the constitutive properties of the wall material, expressed as surface impedance. The attenuation constants for the dominant non-degenerate TE10 mode and the degenerate TE11 and TM11 modes are computed and compared with the power-loss method and the PPM. As demonstrated in the subsequent sections, the new method gives more realistic values for the degenerate modes since the formulation allows co-existence and exchange of power between these modes while other methods treat each one independently.