Simulation-based optimization tolerance analysis

In order to find an optimal geometry for a 90° hybrid coupler, a minimal model containing only the critical coupling region is designed in CST Microwave Studio, which is shown in Fig. 1. The cross-section including the set of parameters, which are used for parameter sweeps, is provided in Fig. 2, where the two cylindrical shapes that are inserted into the coupling region from both sides exhibit both the same radius r _inset. The conductor material is modeled as a rough surface exhibiting a bulk conductivity σ_DC = 10 MS/m and an RMS surface roughness of R _q = 800 nm based on prior work.

Fig. 1.

Minimal CAD model for optimization of coupling region.

Fig. 2.

Cross-section of the optimized simulation model.

Some of the parameters, such as t _edge are bound within the manufacturing limitations of the utilized 3D printer, hence, it should not fall below 200 μm. The optimal parameters acquired by simulation sweeps are shown in Table 1.

Table 1.

Overview of geometrical parameters as optimized by simulation

However, deviations from the optimum values are expected in the 3D-printed specimen due to tolerances of the 3D printer. Although the final precision depends on many aspects of the manufacturing process including the choice of resin, cleaning, and curing settings as well as the printer resolution itself, it has been shown [Reference Cook, Joye and Calame11] that a relative error $\leq 3.3\%$ and – depending on the orientation – even $\leq 1.8\%$ can be achieved with a printer of nominal 30 μm resolution in xy-plane.

Measurements in [Reference Lomakin, Klein, Sippel, Helmreich and Gold10] suggest deviations in the order of ±50 μm along the critical geometry of the coupling region in Fig. 2 which resembles a relative deviation of approximately 1–$5\%$ and is in line with the utilized machine resolution of 50 μm in xy-plane and 10 μm in z-direction. Hence, parameter sweeps with the model in Fig. 2 for l _c, d _inset, t, and t _edge varying each within ±50 μm are evaluated to visualize the worst-case impact of imperfections in the coupling region without taking the waveguide connections into account. Therefore, Fig. 3 shows the simulated range for through (S ₃₁) and couple (S ₄₁) paths as well as the maximum imbalance in magnitude (‖S ₃₁| − |S ₄₁‖) among all combinations. Within the range from 132 to 165 GHz, the worst-case imbalance remains below 5 dB. The corresponding impact of such variations in geometry on the phase difference at the output ports between S ₃₁ and S ₄₁ is summarized in Fig. 4, where the phase deviates by only 7° within the frequency range from 130 to 165 GHz in worst-case combination of geometry variation.

Fig. 3.

Maximum range of simulated Through and Couple paths for variation of ± 50 μm within critical geometry of coupling region (l _c, d _inset, t, and t _edge) in Fig. 1.

Fig. 4.

Maximum range of simulated phase difference at output between S ₃₁ and S ₄₁ for variation of ± 50 μm within critical geometry of coupling region (l _c, d _inset, t, and t _edge) in Fig. 1.

Since the minimal model cannot be directly measured due to the size of the measurement equipment, test fixtures need to be designed in order to attach the coupling region to the measurement interface – which in turn should finally be realized as a standard WR06 flange connection. Therefore, Fig. 5 shows the extended simulation model, where continuous wall waveguide paths are added to account for the necessary test fixture. Since the final specimens will be fabricated following the slotted waveguide approach [Reference Lomakin, Herold, Simon, Sippel, Sion, Vossiek, Helmreich and Gold2, Reference Lomakin, Herold, Ringel, Ringel, Simon, Sippel, Sion, Vossiek, Helmreich and Gold6, Reference Lomakin, Klein, Ringel, Ringel, Sippel, Helmreich and Gold7], where the waveguide is implemented similar to an SIW with regular posts miming a continuous wall, another rather complex simulation model is proposed in Fig. 6. Therein, the previously continuous narrow walls are resembled by posts with a thickness of d = 350 μm and a periodicity s = 500 μm. The same material properties are deployed in all three cases.

Fig. 5.

CAD model for simulation including waveguide test fixture.

Fig. 6.

CAD model for simulation including slotted waveguide test fixture.

Figure 7 shows the simulated insertion losses along the through (|S ₃₁|) and couple (|S ₄₁|) paths for the three presented simulation models. While the minimal model achieves about 0.3 dB less than the ideal −3 dB due to conductor loss and surface roughness along the short coupling region, both complex models drop down toward −4.3 dB due to the losses along the test fixtures. However, there is no significant difference between the continuous wall and the slotted waveguide model, and in all cases, the imbalance in output magnitude at through and couple paths remains within the same order.

Fig. 7.

Comparison of different simulation setups with: continuous line ⇔ S ₃₁ and dashed line ⇔ S ₄₁.

The test fixture waveguides are in turn prone to tolerances within printing and handling, which may have significant impact on the performance due to the relatively long waveguide sections. A dimensional mismatch between the different waveguide paths – which may be related to an asymmetric distortion of the model due to non-homogeneous shrinking – causes a deviation of the output phase $\triangle \varphi_{\text{TC}}$ between the through and coupling path from the ideal 90°. This impact is simulated in CST in order to provide an estimation on the overall precision that can be realized with desktop grade printing systems in the D-band frequency range. Therefore, port 3 of the minimal model is extended by a short variable waveguide section, which is swept by simulation within a range from 0 to 300 μm. This setup is identical to an asymmetry within the test fixture paths of the complex models and can be spread along the entire model. The deviation of $\triangle \varphi _{\text{TC}}$ from the ideal 90° is shown in Fig. 8. Obviously, an asymmetry as little as 100 μm can already lead to a phase error of > 10°, therefore, imposing a tough challenge for the manufacturing process even for conventional systems already. Furthermore, simulation suggests that asymmetry within the manufactured model has a significantly higher impact on the phase deviation than symmetrical tolerances within the coupling region itself, while symmetric ones have, in turn, a more significant impact on the magnitude imbalance.

Fig. 8.

Simulated deviation of phase difference from the ideal of 90° between through and couple ports due to asymmetry within path lengths.

Design for manufacturing and manufactured specimens

As previously indicated, the minimal model of the coupling region is extended by test fixtures and flanges for measurement purpose. Since the model should be kept as small as possible to avoid losing too much precision on the attenuation within the waveguide sections, the flanges are attached in an angle of 45° to the model to provide space for the frequency converters in the measurement assembly. Figure 9 shows the CAD model of the proposed coupler that includes the particularly intersecting flanges and the slotted waveguide sections. Furthermore, Fig. 10 provides a cross-section view on the manufacturing model, where especially the posts of the narrow waveguide walls can be seen.

Fig. 9.

Adjusted manufacturing model of the proposed hybrid coupler.

Fig. 10.

Cross-section of the coupling region.

Although the test fixtures are simulated including conductor loss and surface roughness, the final impact of printing and metal plating is rather difficult to simulate in advance. Therefore, a simple two-port model is designed as shown in Fig. 11 as a CAD-model and in Fig. 12 as manufactured part, where the test fixture sections are pointed straight onto each other – similar to a through-calibration structure for the case of measuring the coupling-path (e.g. S ₄₁). This model is manufactured by the same process as the actual specimens, but measured separately in order to get an estimation of the attenuation within the test fixture paths, which can afterwards be used to correct the coupler measurements.

Fig. 11.

Model for estimating attenuation of test fixture.

Fig. 12.

Manufactured through-type structure for estimating attenuation of test fixture.

Finally, the actual coupler model is printed in three different orientations (A, B, and C) as shown in Fig. 13. In case of A, all four ports are placed on the xy-plane of the printer, consequently, being parallel to the build platform. With orientation B, the model is rotated by 90° along the long side and placed on ports 1 and 3. Lastly, in case of C, the model is placed under a 45° angle against the xy-plane.

Fig. 13.

Build orientations of specimens A, B, and C as considered in this work.

The utilized printer is a desktop grade DLP printer with a nominal xy-resolution of roughly 50 μm and a z-axis accuracy of up to 10 μm. After printing the plastic body, it is cleaned using isopropanol alcohol and post-processed in a UV-curing chamber followed by an electroless silver plating process, which in turn is repeated three times. One of the manufactured specimens is shown in Fig. 14.

Fig. 14.

One of the manufactured specimens (orientation A).

Estimation of test fixture attenuation and operational tolerances

Since the simulation of phase deviation suggests a high sensitivity toward small dimensional mismatch, high accuracy is required within the measurement. Hence, also the tolerances that are imposed by operation, especially the flange attachment, are analyzed. Therefore, the manufactured specimens are measured four times with different torques manually applied to the flanges, which is shown in Fig. 15, where the spread within the magnitude of the through and coupling path among the four different settings is plotted over frequency. It shows that the manual handling of screws at the flanges can lead to an error of up to 1 dB in magnitude since the utilized material has a certain elasticity and a relatively simple flange design is used, which relies on a plane face contact. Figure 16 shows the impact of flange attachment on the deviation of $\triangle \varphi _{\text{TC}}$, which can reach 15° when too little torque is applied or the flange alignment is not correct. Consequently, comparing these measurements to Fig. 8, handling the measurement setup can lead to a phase error which is similar to a geometry deviation of ≈ 100 μm and moreover, even surpass the impact of geometry deviations within the coupling region as simulated in Fig. 4. However, the impact on the magnitude imbalance should remain within < 2 dB and is therefore less critical as compared to the deviations within the coupling region as stated in Fig. 3.

Fig. 15.

Measured spread of through and couple path for variation of screw torque.

Fig. 16.

Measured maximum spread of phase difference between through and couple path for variation of screw torque.

Finally, Fig. 17 shows the measured attenuation within the test fixture waveguide sections on both sides, which sum up to a value between −1 and −1.5 dB in the interesting frequency region which is in line with the simulation results in Fig. 7, where an additional attenuation of ≈ 1 dB is introduced between minimal and complex model. Consequently, an overall drop of around −1 and −1.5 dB below the ideal −3 dB attenuation in through and couple path is expected in the measurement, while the magnitude imbalance between through and couple paths should not be affected due to symmetry in the design.

Fig. 17.

Measured test fixture attenuation.

Measurement and characterization

The three different coupler orientations A, B, and C are all measured at a VNA with D-band frequency converters that are calibrated to the flange interface by a TOSM algorithm with waveguide calibration standards. Since the four-port coupler is measured by a two-port measurement setup, the complete S-parameters are assembled from the respective permutations, while the other two ports are therefore terminated by waveguide matches.

Figure 18 shows the return- (|S ₁₁|) and isolation (|S ₂₁|) loss of the three specimen. Isolation and matching are in the same order of magnitude within each orientation and generally remain below −10 dB within 120–160 GHz. Specimen C even achieves a −15 dB-bandwidth from 116 to 165 GHz.

Fig. 18.

Measured matching |S ₁₁| (solid) and isolation |S ₂₁| (dashed) of the manufactured specimen for all orientation types A, B, and C.

The measured magnitude of through and couple paths at each specimen (A, B, and C) for source at port 1 (|S ₄₁| and |S ₃₁|) and source at port 2 (|S ₄₂| and |S ₃₂|) is shown in Figs 19–21, respectively. As suggested by simulation in Fig. 7, the overall magnitude drops significantly toward −4… − 6 dB in case of A and C and even reaches −8 dB in case of sample B. However, the general functionality of the hybrid coupler is still given in all three samples. Sample C is even almost identical to simulated responses with a deviation of < 0.5 dB as shown in the closeup comparison in Fig. 22. The magnitude imbalance between through and couple output is summarized in Fig. 23 where sample C even remains within a narrow margin of ±0.7dB from 115 to 155 GHz, therefore, suggesting very high degree of accuracy and quality, while sample A also shows acceptable behavior. Moreover, considering the measured attenuation within the test fixture sections of 1–1.5 dB, the actual coupling region of coupler C even reaches almost ideal −3 dB for through and coupling paths.

Fig. 19.

Measured through (solid) and couple (dashed) path for source at ports 1 and 2 of specimen orientation type A.

Fig. 20.

Measured through (solid) and couple (dashed) path for source at ports 1 and 2 of specimen orientation type B.

Fig. 21.

Measured through (solid) and couple (dashed) path for source at ports 1 and 2 of specimen orientation type C.

Fig. 22.

Closeup on measured through (solid) and couple (dashed) path for source at ports 1 and 2 of specimen orientation type C compared with simulation results for slotted WG from Fig. 7.

Fig. 23.

Measured magnitude imbalance between coupling and through paths with source at port 1 and 2 of specimens A, B, and C.

Similar results can be observed within the measured phase difference $\triangle \varphi _{\text{TC}}$ between through and couple output paths for source at port 1 and 2 respectively as shown in Figs 24–26 for all three specimens. While sample B defines a range around the desired 90° from 50° to 120° – hence, spanning 70° – sample C achieves a very narrow range from 80° to 94° from 120 to 160 GHz, leaving sample A in between. Consequently, build orientation C allows for a realized precision that is in the order of the phase tolerances imposed by manual handling of flange interconnections and resembles a geometrical asymmetry within the narrow margin of ≈ 100 μm as indicated by the simulated range of deviations by 0–100 μm at the complex model in Fig. 5 corresponding to a phase variation of Δφ = βΔl ≈ 13° at 140 GHz. The solid green line indicates an asymmetry of Δl = 50 μm between the waveguides connecting port 3 and port 4.

Fig. 24.

Measured phase difference between through and coupling paths for source at ports 1, 2, 3, and 4 of specimen A.

Fig. 25.

Measured phase difference between through and coupling paths for source at ports 1, 2, 3, and 4 of specimen B.

Fig. 26.

Measured phase difference between through and coupling paths for source at ports 1, 2, 3, and 4 of specimen C as well as simulated range for asymmetry in waveguide feed paths of 0–100 μm with center at 50 μm length difference (green solid).