Comparison of SIS receiver performance.
)In millimeter and submillimeter radio astronomy, waveguide heterodyne receivers are often used in signal mixing. Wave guiding structures such as circular and rectangular waveguides are widely used in such receiver systems to direct and couple extraterrestrial signals at millimeter and submillimeter wavelengths to a mixer circuit (Carter et al., 2004; Boifot et al., 1990; Withington et al., 2003).
To illustrate in detail the applications of waveguides in receiver systems, a functional block diagram of a typical heterodyne receiver in radio telescopes is shown in Fig. 1 (Chattopadhyay et al., 2002). The electromagnetic signal (RF signal) from the antenna is directed down to the front end of the receiver system via mirrors and beam waveguides (Paine et al., 1994). At the front end of the receiver system, such as the sideband separating receiver designed for the ALMA band 7 cartridge (Vassilev and Belitsky, 2001a; Vassilev and Belitsky, 2001b; Vassilev et al., 2004), the RF signal is channelled from the aperture of the horn through a circular and subsequently a rectangular waveguide, before being coupled to the mixer. In the mixer circuit, a local oscillator (LO) signal which is generally of lower frequency is then mixed with the RF signal, to down convert the RF signal to a lower intermediate frequency (IF) signal. Here, a superconductor-insulator-superconductor (SIS) heterodyne mixer is commonly implemented for the process of down conversion. At the back end of the system, the IF signal goes through multiple stages of amplification and is, eventually, fed to a data analysis system such as an acousto-optic spectrometer. The data analysis system will then be able to perform Fourier transformation and record spectral information about the input signal.
The front-end receiver noise temperature
|Center Frequency (GHz)||230||345||492|
Since the input power level of the weak millimeter and submillimeter signals is quite small – i.e. of the order of 10–18 to 10–20 W (Shankar, 1986), it is therefore of primary importance to minimize the conversion loss
2. Related work
Analysis of the propagation of wave in circular cylindrical waveguides has already been widely performed (Glaser, 1969; Yassin et al., 2003; Claricoats, 1960a; Claricoats, 1960b; Chou and Lee, 1988). The analyses by these authors are all based on the rigorous method formulated by Stratton (1941). In Stratton’s formulation, the fields at the wall surface are made continuous into the wall material. Assumption made on the field decaying inside the wall material yields relations which allow the propagation constant to be determined. Due to the difficulty in matching the boundary conditions in Cartesian coordinates, this approach, however fails to be implemented in the case of rectangular waveguides. A similar rigorous technique to study the attenuation of rectangular waveguides is not available hitherto.
The perturbation power-loss method has been commonly used in analyzing wave attenuation in lossy (Stratton, 1941; Seida, 2003; Collin 1991; Cheng, 1989) and superconducting (Winters and Rose, 1991; Ma, 1998; Wang et al., 1994; Yalamanchili et al., 1995) rectangular waveguides; respectively. This is partly due to its ability to produce simple analytical solution, and also partly because it gives reasonably accurate result at frequencies
It can be seen that the assumption of lossless fields fail to give an insight or deeper understanding on the mechanism of the propagation of wave in practical 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. The workhorse of this chapter is, therefore, to develop a novel and accurate formulation – i.e. one that does not assume lossless boundary conditions – to investigate the loss of waves in rectangular waveguides. In particular, the new method shall be found more accurate and useful for waveguides operating at very high frequencies, such as those in the millimeter and submillimeter wavelengths. In Yeap et al. (2009), a simple method to compute the loss of waves in rectangular waveguides has been developed. However, the drawback of the method is that different sets of characteristic equation are required to solve for the propagation constants of different modes. Here, the method proposed in Yeap et al. (2009) shall be developed further so that the loss of different modes can be conveniently computed using only a single set of equation. For convenience purpose, the new method shall be referred to as the boundary-matching method in the subsequent sections.
3. General wave behaviours along uniform guiding structures
As depicted in Fig. 2, a time harmonic field propagating in the z direction of a uniform guiding structure with arbitrary geometry can be expressed as a combination of elementary waves having a general functional form (Cheng, 1989)
Hence, in using phasor representation in equations relating field quantities, the partial derivatives with respect to
The field intensities in a charge-free dielectric region (such as free-space), satisfy the following homogeneous vector Helmholtz’s equation
It is convenient to classify propagating waves into three types, in correspond to the existence of the longitudinal electric field
Transverse electromagnetic (TEM) waves. A TEM wave consists of neither electric fields nor magnetic fields in the longitudinal direction.
Transverse magnetic (TM) waves. A TM wave consists of a nonzero electric field but zero magnetic field in the longitudinal direction.
Transverse electric (TE) waves. A TE wave consists of a zero electric field but nonzero magnetic field in the longitudinal direction.
Single-conductor waveguides, such as a hollow (or dielectric-filled) circular and rectangular waveguide, cannot support TEM waves. This is because according to Ampere’s circuital law, the line integral of a magnetic field around any closed loop in a transverse plane must equal the sum of the longitudinal conduction and displacement currents through the loop. However, since a single-conductor waveguide does not have an inner conductor and that the longitudinal electric field is zero, there are no longitudinal conduction and displacement current. Hence, transverse magnetic field of a TEM mode cannot propagate in the waveguide (Cheng, 1989).
4. Fields in cartesian coordinates
By applying the method of separation of variables,
The transverse field components can be derived by substituting the longitudinal field components into Maxwell’s source free curl equations
5. Review of the power-loss method
In the subsequent sections, analysis and comparison between the perturbation power-loss method and the new boundary matching method shall be performed. Hence, in order to present a complete scheme, the derivation of the conventional power-loss method is briefly outlined in this section.
The attenuation of electromagnetic waves in waveguides can be caused by two factors – i.e. the attenuation due to the lossy dielectric material
For a conducting waveguide, the inner core is usually filled with low-loss dielectric material, such as air. Hence,
The approximate power-loss method assumes that the fields’ expression in a highly but imperfectly conducting waveguide, to be the same as those of a lossless waveguide. Hence,
Conduction loss is assumed to occur due to small fields’ penetration into the conductor surfaces. According to the law of conservation of energy, the attenuation constant due to conduction loss can be derived as (Cheng, 1989)
6. The new boundary-matching method
It is apparent that, in order to derive the approximate power-loss equations illustrated in section 5, the field equations must be assumed to be lossless. In a lossless waveguide, the boundary condition requires that the resultant tangential electric field
In order to model the field expressions closer to those in a lossy waveguide and to account for the presence of fields inside the walls, two phase parameters have been introduced in the new method. The phase parameters – i.e x and y, are referred to as the field’s penetration factors in the x and y directions, respectively. It is worthwhile noting that, with the introduction of the penetration factors, Et and do not necessarily decay to zero at the boundary, therefore allowing the effect of not being a perfect conductor at the waveguide wall.
6.1. Fields in a lossy rectangular waveguide
For waves propagating in a lossy hollow rectangular waveguide, as shown in Figure 3, a superposition of TM and TE waves is necessary to satisfy the boundary condition at the wall (Stratton, 1941; Yassin et al., 2003). The longitudinal electric and magnetic field components Ez and Hz, respectively, can be derived by solving Helmholtz’s homogeneous equation in Cartesian coordinate. Using the method of separation of variables (Cheng, 1989), the following set of field equations is obtained
where E0 and H0 are constant amplitudes of the fields.
The propagation constant kz for each mode will be found by solving for kx and ky and substituting the results into the dispersion relation
Equations (21)and (22) must also apply to a perfectly conducting waveguide. In that case Ez and are either at their maximum magnitude or zero at both x = a/2 and y = b/2 – i.e. the centre of the waveguide, therefore
Solving (24), the penetration factors are obtained as,
For waveguides with perfectly conducting wall, kx = mπ/a and ky = nπ/b, (25) and (26) result in zero penetration and Ez and Hz in (21) and (22) are reduced to the fields of a lossless waveguide. To take the finite conductivity into account, kx and ky are allowed to take complex values yielding non-zero penetration of the fields into the waveguide material
where βx and βy are the phase constants and αx and αy are the attenuation constants in the x and y directions, respectively. Substituting the transverse wavenumbers in (27) and (28) into (23), the propagation constant of the waveguide kz results in a complex value, therefore, yielding loss in wave propagation.
where μ0 and ε0 are the permeability and permittivity of free space, respectively.
At the wall, the tangential fields must satisfy the relationship defined by the constitutive properties μc and εc of the material. The ratio of the tangential component of the electric field to the surface current density at the conductor surface is represented by (Yeap et al., 2009b; Yeap et al., 2010)
where μc and εc are the permeability and permittivity of the wall material, respectively, and is the intrinsic impedance of the wall material. The dielectric constant is complex and εc may be written as
where σc is the conductivity of the wall.
In order to estimate the loss of waves in millimeter and submillimeter wavelengths more accurately, a more evolved model than the conventional constant conductivity model used at microwave frequencies is necessary. Here, Drude’s model is applied for the frequency dependent conductivity σc (Booker, 1982)
where σ is the conventional constant conductivity of the wall material and τ the mean free time. For most conductors, such as Copper, the mean free time τ is in the range of 10–13 to 10–14 s (Kittel, 1986).
In order to obtain nontrivial solutions for (36) and (37), the determinant of the equations must be zero (Yeap et al., 2009a). By letting the determinant of the coefficients of E0 and H0 in (36) and (37) vanish the following transcendental equations are obtained
Since the dominant TE10 mode has the lowest cutoff frequency among all modes and is the only possible mode propagating alone, it is of engineering importance. In the subsequent sections, comparison and detail analysis of the TE10 mode shall be performed. For the TE10 mode, m and n are set to 1 and 0, respectively. Substituting m = 1 and n = 0 into the penetration factors in (25) and (26), the transcendental equations in (38) for TE10 mode can be simplified to
In the above equations, kx and ky are the unknowns and kz can then be obtained from the dispersion relation in (23). A multi root searching algorithm, such as the Powell Hybrid root searching algorithm in a NAG routine, can be used to find the roots of kx and ky. The routine requires initial guesses of kx and ky for the search. For good conductors, suitable guess values are clearly those close to the perfect conductor values i.e. βx = mπ/a, βy = nπ/b, αx = αy = 0. Hence, for the TE10 mode, the initial guesses for kx and ky are π/a and 0 respectively.)
6.3. Experimental setup
To validate the results, experimental measurements had been carried out. The loss as a function of frequency for a rectangular waveguide was measured using a Vector Network Analyzer (VNA). A 20 cm copper rectangular waveguide with dimensions of a = 1.30 cm and b = 0.64 cm such as that shown in Fig. 4 were used in the measurement.
In order to allow the waveguide to be connected to the adapters which are of different sizes, a pair of taper transitions had also been used as shown in Fig. 7. Fig. 8 depicts the complete setup of the experiment where the rectangular waveguide was connected to the VNA via tapers, chokes, coaxial cables, and adapters. Before measurement was carried out, the coaxial cables and waveguide adapters were calibrated to eliminate noise from the two devices. The loss in the waveguide was then observed from the S21 or S12 parameter of the scattering matrix. The measurement was performed in the frequencies at the vicinity of cutoff.
6.4. Results and discussion
As shown in Fig. 9, a comparison among the attenuation of the TE10 mode near cutoff as computed by the new method, the conventional power-loss method, and the measured S21 result was performed. Clearly, the attenuation constant αz computed from the power-loss method diverges sharply to infinity, as the frequency approaches fc, and is very different to the simulated results, which show clearly that the loss at frequencies below fc is high but finite. The attenuation computed using the new boundary-matching method, on the other hand, matches very closely with the S21 curve, measured using from the VNA. As shown in Table 2, the loss between 11.47025 GHz and 11.49950 GHz computed by the boundary-matching method agrees with measurement to within 5% which is comparable to the error in the measurement. The inaccuracy in the power-loss method is due to the fact that the fields expressions are assumed to be lossless – i.e. kx and ky are taken as real variables. Analyzing the dispersion relation in (23), it could be seen that, in order to obtain αz, kx and/or ky must be complex, given that the wavenumber in free space is purely real. Although the initial guesses for kx and ky applied in the new boundary-matching method are assumed to be identical with the lossless case, the final results actually converge to complex values when the characteristic equations are solved numerically.
Fig. 10 shows the attenuation curve when the frequency is extended to higher values. Here, the loss due to TE10 alone could no longer be measured alone, since higher-order modes, such as TE11 and TM11, etc., start to propagate. Close inspection shows that the loss predicted by the two methods at higher frequencies is in very close agreement. It is, therefore, sufficed to say that, although the power-loss method fails to predict the attenuation near fc accurately, it is still considered adequate in computing the attenuation of TE10 in lossy waveguides, provided that the frequency f is reasonably above the cutoff fc.
As depicted in Fig. 11, at frequencies beyond millimeter wavelengths, however, the loss computed by the boundary-matching method appears to be much higher than those by the power-loss method. The differences can be attributed to the fact that at extremely high frequencies, the loss tangent of the wall material decreases and the field in a lossy waveguide can no longer be approximated to those derived from a perfectly conducting waveguide. At such high frequencies, the wave propagating in the waveguide is a hybrid mode and the presence of the longitudinal electric field Ez can no longer be neglected.
|Frequency GHz||Experiment||Boundary-matching method||%∆|
Unlike the power-loss method which only gives the value of the attenuation constant, one other advantage of the boundary-matching method is that it is able to account for the phase
constant of the wave as well. A comparison between the attenuation constant and phase constant of a TE10 mode is shown in Fig. 12. As can be observed, as the attenuation in the waveguide gradually decreases, the phase constant increases. Fig. 12 illustrates the change in the mode – i.e. from evanescent below cutoff to propagating mode above cutoff.
A fundamental and accurate technique to compute the propagation constant of waves in a lossy rectangular waveguide is proposed. The formulation is based on matching the fields to the constitutive properties of the material at the boundary. The electromagnetic fields are used in conjunction of the concept of surface impedance to derive transcendental equations, whose roots give values for the wavenumbers in the x and y directions for different TE or TM modes. The wave propagation constant kz could then be obtained from kx, ky, and k0 using the dispersion relation.
The new boundary-matching method has been validated by comparing the attenuation of the dominant mode with the S21 measurement, as well as, that obtained from the power-loss method. The attenuation curve plotted using the new method matches with the power-loss method at a reasonable range of frequencies above the cutoff. There are however two regions where both curves are found to differ significantly. At frequencies below the cutoff fc, the power-loss method diverges to infinity with a singularity at frequency f = fc. The new method, however, shows that the signal increases to a highly attenuating mode as the frequencies drop below fc. Indeed, such result agrees very closely with the measurement result, therefore, verifying the validity of the new method. At frequencies above 100 GHz, the attenuation obtained using the new method increases beyond that predicted by the power-loss method. At f above the millimeter wavelengths, the field in a lossy waveguide can no longer be approximated to those of the lossless case. The additional loss predicted by the new boundary-matching method is attributed to the presence of the longitudinal Ez component in hybrid modes.