Finite Element Analysis

Typically, accurate scale model measurements are difficult, time-consuming and prone to uncertainties for the following reasons: (1) the wide range of sizes (> 1000:1) in an SIS mount are difficult or impossible to achieve in a model (2) The iterative process of modifying the scale model to study the effect of changes involves re-machining and/or re-layout of probe circuits and (3) the problem of providing small coaxial probes to the location of the SIS junction involves uncertainties in the scale model measurements due to calibration errors and the fact that the probe itself is disturbing the field at the measurement point.

We have performed numerical electromagnetic simulations of the waveguide mount. The advantages of numerical analysis are that one may study the effects of the dielectric, optimize the SIS tuning circuit using the embedding impedance, and modify the structure easily. Other advantages include the ability to study the small size scales of the junction mount with the antenna probe, and the ability to reduce the complexity of the problem by exploiting symmetry considerations. The finite element analysis was done using Ansoft's HFSS [4]. The accuracy of HFSS in predicting embedding impedances has already been demonstrated in the design of multipliers [16].

 

Figure 6: Comparison of the embedding impedance of the scale model with HFSS analysis. The equivalent back-short distance of the scale-model at the frequency of operation is 0.2 mm, and is the same for the HFSS model. The HFSS results are shown after renormalizing to $50 ~\Omega$. The two plots are for an equivalent frequency range of 300 to 400 GHz.

 

Figure 7: View of the HFSS model and the defined ports of the 4-port model. The zoomed-in view to the right shows the definition of the SIS junction as a square gap, with a TEM port and a capped feed.

 

Figure 8: Equivalent MDS circuit model. The backshort is treated as a short-circuited transmission line.

 

Figure 9: Optimized input RF match into a tuned SIS junction of impedance $40+j20~\Omega$ using embedding impedance model from HFSS and optimization in MDS. The equivalent backshort distance is 0.25 mm.

We tried several different approaches in the numerical analysis. Initially, voltage sources were used at the location of the gap between the two antenna probes in the center of the waveguide. The field-calculator in the post-processor of HFSS was then used to determine impedance by calculating the Poynting power flow through the gap and using the Z_PV definition to determine the impedance. This technique gave reasonable results in measuring the embedding impedance, but was slow and laborious. It also suffers from the fact that these impedances are time-dependent, and hence care must be taken to set the phase of the excitation right. The next method was to ``subtract'' the RF choke and antenna structure from the waveguide structure. Subtraction in HFSS is an ``exclusive-or'' operation, and results in a new structure that contains one of the two objects but not both. This has the effect of bringing the buried gap in the center of the waveguide to the outside world, thereby allowing us to define a port and excite the gap with a TEM-type transmission line. In Figure 6, we show the comparison of the scale model from the previous section to an HFSS model configured using this technique. The backshort distance in the HFSS model was set to 0.2 mm, and is the same as the scale model. It can be seen that although their location is in the same general vicinity in the Smith Chart, the agreement between the scale models and finite element analysis is not very good. We modeled the antenna structure as a perfect conductor, and this could account for some of the discrepancy. However, since the HFSS model is performed at the frequency of operation, much of the discrepancy is attributed to uncertainties in the scaling and calibration of the scaled model measurements.

A third approach to the numerical analysis is to reproduce as faithfully as possible, the actual layout of the junction with relation to the antenna probe. Figure 7 shows the view of the HFSS model used. The model is a 4-port network with port 1 being the SIS junction, port 2 the IF port and ports 3 and 4 the input and the output waveguide ports respectively. The output port is deembedded at a later stage and a backshort attached to it using analysis outside HFSS. The zoomed-in view of the antenna-probe to junction transition shows the junction defined as a square area. The insulator gap of the junction for this problem is a virtual object to help with creating an adequate mesh for the problem. The actual port for the TEM transmission line is capped off at the end with a perfect conductor. The capped feed ensures that the field propagates only into the junction area, which is the area of interest. Once the problem is analyzed in HFSS, the four-port S-parameter is exported (after renormalization to $50 ~\Omega$ and deembedding) to a linear circuit simulator (MDS) and optimized there. Figure 8 shows the MDS equivalent circuit model. The backshort is modeled with a shorted half-height rectangular waveguide transmission line in the circuit simulator. For the purpose of this analysis, the tuned SIS junction was replaced with an equivalent impedance of $40+j20~\Omega$. The circuit was then optimized for the best backshort distance, resulting in the best case input RF match shown in Figure 9. In practice, the SIS junction and tuning circuit as well as the backshort can be optimized together with the embedding impedance S-parameter set derived from HFSS for the best optimization. It can be seen from Figure 9 that the input match is much better than that shown in Figure 5 using the scale-model measurements, and that one backshort setting covers the entire band of interest. Another advantage of this analysis is that the coupling of RF to the IF port and the effectiveness of the RF choke is easily calculated, and any transverse resonance modes in the substrate channel can be studied carefully.