Designing high frequency devices requires a thorough understanding of theoretical precision and how prototyping or manufacturing tolerances affect performance. Real-world factors, such as variations in fabrication processes and material properties, result in discrepancies between ideal simulations and actual performance that must be addressed. By predicting how tolerances influence performance, uncertainty quantification (UQ) enables engineers to optimize designs for more reliable, consistent outcomes.
The COMSOL Multiphysics® software offers functionality for UQ studies and electromagnetics modeling. The latest release, version 6.3, introduces many updates for electromagnetics and important additions to UQ functionality for microwave components. This update was introduced in the RF Module, an add-on to the software that allows a user to examine how different input parameters impact device S-parameter performance.
THE CHALLENGE: FROM SIMULATION TO REAL-WORLD PERFORMANCE
Figure 1 Microstrip patch antenna mockup models.
Figure 2 MOAT screening results.
Figure 3 Sobol indices for a sensitivity analysis.
Figure 1 shows a batch of 100 microstrip patch antennas fabricated from a simulation model that estimates an ideal S11 of -20 dB. Considering manufacturing and material variations, how many of these antennas will meet a design target of S11 ≤ -10 dB?
In real-world manufacturing, variations in critical parameters, like the patch dimensions, substrate thickness and dielectric constant, are inevitable. These variations create antenna performance deviations. For example, small patch length changes may shift the resonant frequency, while inconsistent thickness or dielectric properties can cause impedance mismatching. Without accommodating these factors, even well-designed antennas may not achieve the intended performance.
UQ systematically analyzes how design parameter variations affect system performance and estimates how many antennas will meet the specified performance criteria. The standard process involves a screening study, a sensitivity analysis, uncertainty propagation and a reliability analysis. COMSOL Multiphysics® offers an intuitive workflow for performing these analyses.
MORRIS ONE-AT-A-TIME (MOAT) SCREENING
The goal of a screening study is to perform a qualitative analysis to identify input parameters with the greatest influence on performance indicators or quantity of interest (QoI), S11 in this case. The MOAT method computes the mean and standard deviation. They are the average QoI effect of varying a parameter and assessing the variability in that parameter’s effect, indicating potential interactions with other parameters.
Typical parameters and ranges in printed antenna simulations include:
- Substrate thickness, d, can vary by approximately ±7 percent from 0.060 in. (~1.524 mm), based on manufacturer data. Setting σ to 3.5 percent of d captures about 95 percent of the variations within ±2σ.
- Patch length (l_patch) has a nominal value of 52 mm and the tolerance depends on the fabrication method. Non-high-precision PCB etching can reduce the tolerance to 0.127 mm, whereas milling with loose anchoring can yield a tolerance of ±2σ ≈ 0.520 mm.
- Dielectric constant, dk, is often 3.38 ± 0.05. For stricter coverage (e.g., ±3σ), σ = 0.005 × dk ≈0.0169, resulting in ±0.0507 around the nominal value.
Filtering out less important parameters can make subsequent UQ analyses more efficient. Figure 2 shows results with l_patch, dk and d. The screening analysis indicates that substrate thickness is less influential than the other parameters.
SENSITIVITY ANALYSIS WITH SOBOL INDICES
After identifying the most influential parameters, a sensitivity analysis quantifies how each parameter, individually and in combination, affects the QoI. Sobol indices include:
- First-order: reflects the direct contribution of each parameter to the QoI
- Total: captures direct effects and interaction effects with other parameters.
Figure 3 shows first-order and total Sobol indices for an analysis that evaluates parameter sensitivity and interactions. The analysis shows that the length of the patch is more influential than the substrate’s dielectric constant.
UNCERTAINTY PROPAGATION WITH KERNEL DENSITY ESTIMATION (KDE)
Figure 4 KDE plot.
Uncertainty propagation evaluates how variations in input parameters affect the S11 distribution. A KDE plot shows a smoothed probability density function estimate of the QoI, illustrating how input parameter uncertainties influence the distribution. By depicting likely QoI values in a continuous manner, KDE provides clearer insight into probabilistic behavior and variability. This approach helps determine the probability of different outcomes, revealing the range of QoI values under inherent uncertainties. The KDE in Figure 4 illustrates the probability density function of the QoI.
RELIABILITY ANALYSIS
A reliability analysis calculates the probability of meeting a predefined performance criterion. It provides a probability value for conditions, indicating the fraction of scenarios falling below the threshold.
Example scenarios:
- Milling with loose anchoring: large tolerances on l_patch result in a 55 percent probability of S11 < -10 dB
- Non-high-precision PCB etching: reduced tolerances increase the probability of antennas passing to 86 percent.
Such findings highlight how tightening tolerances or employing more consistent manufacturing processes significantly raises the proportion of devices that meet the target specification.
RESULTS AND DISCUSSION
Key takeaways from the UQ studies include:
- MOAT screening frequently identifies l_patch and dk as primary drivers of S11 variability
- Sobol indices confirm direct effects and the importance of interactions among parameters
- KDE plots display S11 distribution, revealing whether most samples cluster around or deviate from the -10 dB threshold
- Reliability analysis quantifies the fraction of cases ≤ -10 dB, informing decisions on tolerance tightening and design adjustments.
Narrowband devices, particularly those operating at higher frequencies, are especially sensitive to slight geometric or material deviations. Even a minor 5 to 10 MHz shift in the resonant frequency can result in unacceptable return loss. By employing UQ, engineers can anticipate these shifts, optimize designs and determine when to tighten certain tolerances.
Integrating UQ into the design process helps engineers bridge the gap between simulation and reality, ensuring reliable performance across a range of manufacturing variations. For the 100 antennas, predicting that most will meet the -10 dB S11 target provides a solid production foundation while offering refinement opportunities for the fraction not meeting the target. This continuous improvement loop elevates both design and manufacturing processes.
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