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Nanoelectronic COupled Problems Solutions: uncertainty quantification for analysis and optimization of an RFIC interference problem
 Piotr Putek^{1, 2}Email authorView ORCID ID profile,
 Rick Janssen^{3},
 Jan Niehof^{3},
 E. Jan W. ter Maten^{1},
 Roland Pulch^{2},
 Bratislav Tasić^{3} and
 Michael Günther^{1}
https://doi.org/10.1186/s1336201800543
© The Author(s) 2018
 Received: 28 March 2018
 Accepted: 12 November 2018
 Published: 20 November 2018
Abstract
The FP7 project nanoCOPS (the 7th Framework Programme project Nanoelectronic COupled Problems Solutions) derived new methods for simulation during development of designs of integrated products. It has covered advanced simulation techniques for electromagnetics with feedback couplings to electronic circuits, heat and stress. It was inspired by interest from semiconductor industry and by a simulation tool vendor in electronic design automation.
Due to the application of higher frequencies and the continuous downscaling process, there is a higher probability of unforeseen interactions between different domains of a Radio Frequency Integrated Circuit (RFIC), which can lead to the variability of the output performance functions. Since these undesired phenomena ought to be investigated in the early phases of the integrated circuit (IC) design, in this work we formulate the robust optimization problem in terms of the expectation and the standard deviation values under the uncertainties of material parameters.
Therein, the statistical information included in the multiobjective functional can be provided by a response surface model. For this purpose the Stochastic Collocation Method (SCM) combined with Polynomial Chaos Expansion (PCE) has been used. The reason for analyzing the variability of the Electromagnetic Interference (EMI) is, on the one hand, to quantify the uncertainty in an integrated RadioFrequency Complementary MetalOxide SemiConductor (RFCMOS) transceiver design, and, on the other hand, to improve this design in a robust sense. We have illustrated our methodology for an integrated RadioFrequency Complementary MetalOxide SemiConductor (RFCMOS) transceiver design.
Keywords
 Floorplan modeling
 Isolation grounding
 Polynomial chaos expansion
 Stochastic collocation method
 Uncertainty quantification
 Robust design optimization
 Sobol decomposition
 Variancebased and local sensitivity analysis
1 Introduction
Nowadays the computational electromagnetic (CEM) modeling methods are the integral part of the advanced tools for electromagnetic compatibility (EMC) problems of integrated circuits (ICs), encountered in engineering practice [2]. In contrast to the traditional EMC product testing, which are both timeconsuming and expensive, computational modeling and simulation offer more flexibility in design modification and are potentially faster and cheaper [39]. They can be applied to resolve a broad range of real life problems, including interference issues and coupling effects in modern mixedsignal and radio frequency integrated circuits. The latter has become more and more important with the continuous integration process of RF, mixed signal and digital blocks on a single die, which is additionally combined with observable trends on semiconductor market to apply higher frequencies for accommodating higher data rates [20]. These trends towards miniaturization of RFIC allows primarily for easier implementation of multiple functions in a compact unit, which implies also substantial reductions in product cost. In fact, the complexity poses many challenges in the integration process of various subsystems, such as the socalled aggressors (the noisy blocks), the victims (the sensitive part, affected by noise) and other intellectual property (IP) blocks to ensure their proper and interferencefree operation [15]. Additionally, provisions should be made at the influence of the miniaturization on the failure probability associated with yield loss due to defects, faults, process variations and design issues [37]. In fact, the statistical variations in input parameters, originating from manufacturing tolerances of industrial processes, may result in a thermal destruction of devices due to thermal runaway [30, 32]. In this respect, taking the input statistical variations into account in modeling allows for providing the predictable and reliable RFIC simulations. Moreover, unintended RF coupling, caused by industrial imperfections and miniaturization due to the scalingdown process, could significantly downgrade the quality of products and their performance or even be dangerous for safety of both environment and the endusers [8]. Thus, to meet the stringent design specification requirements for electromagnetic compatibility standards [4] such as speeds, bandwidth, noise margin crosstalk, etc., the ICs designers have to be aware of interference issues caused by highfrequency phenomena and randomness involved in the manufacturing process. Considering these phenomena at early design stages allows for avoiding expensive respins and to reduce iterations in the timetomarket cycle.
Therefore, the structured methodology is needed to model the adverse effects caused by the manufacturing process using polynomial chaos expansion combined with the stochastic collocation method [43–46] for analysis of the EMC problem of integrated circuits. Here, we focus more on the new application of the methodology mainly developed in [32, 33, 35]. Taking this into account, the special attention is paid to the analysis of both the means of the gradient of the output characteristics with respect to parameter variations and on the variancebased sensitivity, which allows for quantifying impact of particular parameters to the variance of output functions. For this purpose, the Sobol decomposition [38] has been applied. The proposed approach is innovative, since statistics based on uncertainty quantification (UQ) allows for assessing the most influential input random parameter, which can be further used for the physicallybased design of RFIC model. Additionally, the UQ can be incorporated in the GaussNewton optimization algorithm [27], which yields the powerful tools for the automatic design of RFIC system, while taking both noise and uncertain input parameters into account. More specifically, based on results achieved in [28], the socalled mean gradient sensitivity analysis has been explored to approximate the derivative of the robust functional, which furthermore, has been applied in the optimization process. Hence, the current paper is an extended version of [29] providing detailed insights on the floorplanning and grounding strategy as well as new results for the variancebased and the local sensitivity analysis.
The paper is organized as follows. Section 2 overviews the methodology developed and used for modeling and simulation of an integrated RFCMOS transceiver for automotive applications, that is the socalled floorplan modeling and isolation strategy. Section 3, in turn, considers the benefits of the PCEbased stochastic collocation methods applied for solving the randomdependent DifferentialAlgebraic Equations (DAEs) and additionally provides the results for the variancebased and the local sensitivity analysis. Section 4 deals with the robust optimization problem, which definition includes the statistical moments, used as cost functional. Section 5 gives some achievements obtained during the project with respect to reliable RFIC isolation. And finally, in Sect. 6 we briefly summarize our paper including remarks and future work on this research.
2 Methodology for deterministic modeling

Onchip: domainregions, padring, sealring, splitter cells, substrate effects (including deepNwell and Pwell regions).

Package: ground and power pins, bondwires/downbonds, exposed diepad connection.

Printed Circuit Board (PCB): ground plane and exposed diepad connections.
2.1 Mathematical modeling for an RFCMOS automotive test case
In our work, to avoid computationally expensive simulations of the fully threedimensional model of RFIC [39, 48], the Equivalent Circuit Model (ECM) with the appropriate floorplan modeling and isolation strategy has been used. More specifically, the ECM has been created based on the simulation of a chip architecture using the Advanced Design System (ADS)/Momentum software from Keysight Technologies [18]. Hence, for the solution of systems of partial differential equations the Methods of Moments (MoM) [6] has been employed, where the concept of Green functions is explored to model the proper behavior of the substrate [12].
2.2 Testbench model of the integrated circuit
As the first step, an initial floorplan model, describing the interaction between individual IP blocks/substrate domains via the substrate and well structures, is created. If present, also the connection of the backside of the die is included by taking the vertical substrate noise propagation path into account. Each IP block is represented by a single port, representing substrate connections. Such a firstorder model can be created with a number of commercially available EM simulators. Initially, a rough estimate of the technology parameters can be sufficient (e.g., substrate conductivity). The output (either Sparameters or lumped circuit components) is then included in the simulation testbench shown later. In addition in finding the optimal relative placement of the IP blocks and the spacing between them, this model can also be used to study the impact of, e.g., type of substrate (bulk vs. silicononinsulator), doping levels (lowlydoped vs. highlydoped), backside connection (soldered vs. glued), etc. Figure 3 shows an example floorplan model indicating the typical complexity (number of used substrate/ground domains) of the analyzed design.
The aggressor current, representing the switching activity of, e.g., the digital block, produces a bounce on the supply and ground nodes due to the finite impedance of the supply/ground network. This noise then couples to the victim through the substrate represented by the floorplan model. The values of the well junction capacitances representing the well structures are determined by the technology, the aggressor area and the design style (e.g., presence of Deep NWell). In addition, currents through the bondwires can also inductively couple to the bondwires of the victim circuit. Onchip voltage regulators and decoupling capacitances have an impact on the coupling and need to be included in the model, either as Sparameters or by a simplified circuit model. Typical aggressors include: digital logic, digital IOs, clock generation circuits, a digitaltoanalog converter (DAC) and ADC.
When more details about the victim IP architecture become available, a simple victim circuit can be created. For example, a lownoise amplifier can be represented by a voltagecontrolled current source, or an ideal operational amplifier with a simplified feedback circuit. In this case, in addition to the substrate noise rejection properties of the circuit, the noise coupling to the amplifier input can be included in the model. Finally, when the design of the victim block is complete, an Sparameter file, containing the relevant terminals (e.g. substrate and sensitive input), should be generated by the IP designer for inclusion in the interference testbench. Typical victims are: sensitive RF tuners, analog IOs, ADC, voltage controlled oscillators (VCO), etc.
Similar to victim IP models, different levels of abstraction are used to incorporate the package coupling effects in the analysis. Again, at early stages, a very simple model could be useful. Lumped inductances with estimated values based on expected lengths of bondwires [3] and leads are added to the testbench. Later, it can be extended with estimates of mutual inductances and capacitances, based on previous designs or simple models. At this stage, a concurrent optimization of the floorplan and the IO ring takes place. Finally, when the IO ring and pin assignment is fixed and the package choice is finalized, the fullpackage model can be included in the test bench for a more accurate prediction of inductive and capacitive effects. The models of individual IP blocks, the floorplan and the package are combined in a single testbench, which is used to investigate the total effect of all implemented isolation measures (see Fig. 3).
2.3 Verification of the applied methodology
Validation of the methodology has been reported in [20]. The same relative effect of 8–9 [dB] by adding the downbonds for the ground connections has been observed in both simulated transfer values, as well as in spur measurements of packaged IC samples.
3 Stochastic modeling and uncertainty quantification
In the model (2), we replace physical parameters \(\textbf {p} \in\varPi\subseteq\mathbb{R}^{q}\) by independent random variables \(\textbf{p}: \varOmega_{\mathrm{pr}} \rightarrow\varPi\) on some probability space \((\varOmega_{\mathrm{pr}},\mathcal{A},\mathbb{P})\) to quantify uncertainties. Let \(p_{i}\) for \(i=1,\ldots,q\) be a random variable with a traditional probability distribution like uniform distribution or Gaussian distribution, for example. The solution of the system (2) becomes a random field, i.e., each component of the solution represents a function depending on the random variables p for fixed frequency \(f \in\mathcal {F}\) in some frequency range \(\mathcal{F} \subseteq\mathbb{R}\). The chosen physical parameters belong to the four different parts (a)–(d) of the functional chip specified in Sect. 2.1.
3.1 Polynomial chaos expansion
3.2 Pseudospectral approach
3.3 Sobol decomposition and variancebased sensitivity analysis
Given a function \(z : \varPi\rightarrow\mathbb{R}\) depending on parameters p, we are interested in sensitivities. Local sensitivities are given by the partial derivatives of the deterministic parameterdependent function z provided that the function is smooth.
4 Optimization under noise emission and uncertainties
The problem of the empirical design of the crossdomain coupling on RFIC under uncertainties based on the Sobol decomposition [38] has been has carried out in paper [48]. Within the nanoCOPS project (http://fp7nanocops.eu/), the variancebased and the local sensitivity analysis based on the PCE surface response model and the adjoint variable method has been explored in many engineering fields. For instance, in [29] the forward stochastic problem of the RFIC isolation under uncertainties has been studied in [28]. In works [30–32], in turn, the PCEbased stochastic collocation methods has been applied for the UQ analysis and the robust shape/topology optimization of the power transistor device under material and geometric uncertainties. Additionally, in the paper [23] the stochastic inverse problem has been investigated. Here, we focus mainly on presenting the robust framework, tailored for the application of the electromagnetic interference variability on the RFIC, which is partially based on the preliminary results achieved in [29].
4.1 Objectives
4.2 Robust formulation
In the conventional optimization, it is assumed, for simplicity, that the parameters of interest are treated as deterministic variables. This, in turn, can lead to an optimal configuration, which is very sensitive to the variation of the nominal parameters [47]. From this perspective, there is a need to take into account uncertainties of material and/or geometric parameters, including excitation term that comes from manufacture imperfection, during the optimization process. To solve this problem, we propose to apply the robust framework [40], invented by the Japanese engineer Taguchi. This allows for providing an optimal design of the RFIC, which is insensitive to variations of random parameters as well as robust versus the noise generated by the aggressors.
4.3 Sensitivity analysis for gradient calculation
The sensitivity analysis allows for investigating the influence of inputs perturbation into the variation of output performance functions, i.e., the tolerance analysis. Its application can be further extended to the problems of network synthesis, which is based on the optimization of the assumed network performance function, e.g., expressed by the robust functional (15a).
Moreover, in case of the robust formulation, the abovedescribed sensitivity techniques can be combined with the PCE (6), (7) and (8) in order to calculate the derivative of mean and of standard deviation [1, 31, 32]. Within this context, in the nanoCOPS project, the variancebased sensitivity analysis (10) has been used in [34] for the solution of a shape optimization problem. In the current work, based on results achieved in [28], the socalled mean gradient sensitivity analysis (12) has been applied to approximate the derivative of the robust functional.
5 Results for numerical experiment and discussion
Chosen values for the deterministic elements of the EMC model
\({R}_{\mathrm{db}}\)  \({L}_{\mathrm{db}}\)  \({R}_{\mathrm {bw}}\)  \({L}_{\mathrm{bw}}\)  \(C_{\mathrm{d}}\)  \(C_{1}\)  \(C_{2}\)  

Elems. values  100.0 [mΩ]  0.1 [nH]  100.0 [mΩ]  2.0 [nH]  2.3 [nF]  0.4547 [nF]  0.2412 [nF] 
The mean values for the initial configuration
\({\overline{R}}_{\mathrm{db}\_\mathrm{rxpa}}^{0}\)  \({\overline{L}}_{\mathrm {db}\_\mathrm{rxpa}}^{0}\)  \({\overline{R}}_{\mathrm{via}\_\mathrm{exp}}^{0}\)  \({\overline{L}}_{\mathrm{via}\_\mathrm{exp}}^{0}\)  \({\overline{R}}_{\mathrm {lb}\_\mathrm{xolo}}^{0}\)  \({\overline{L}}_{\mathrm{lb}\_\mathrm{xolo}}^{0}\)  \({\overline{R}}_{\mathrm{lb}\_\mathrm{rxpa}}^{0}\)  \({\overline{L}}_{\mathrm {lb}\_\mathrm{rxpa}}^{0}\)  

Mean values  10.0 [mΩ]  0.02 [nH]  0.1 [mΩ]  0.01 [nH]  20.0 [mΩ]  0.4 [nH]  16.7 [mΩ]  0.33 [nH] 
The mean values for the optimized configuration [28]
\({\overline{R}}_{\mathrm{db}\_\mathrm{rxpa}}\)  \({\overline{L}}_{\mathrm{db}\_\mathrm{rxpa}}\)  \({\overline{R}}_{\mathrm{via}\_\mathrm{exp}}\)  \({\overline{L}}_{\mathrm{via}\_\mathrm{exp}}\)  \({ \overline{R}}_{\mathrm{lb}\_\mathrm{xolo}}\)  \({\overline{L}}_{\mathrm{lb}\_\mathrm{xolo}}\)  \({\overline{R}}_{\mathrm{lb}\_\mathrm{rxpa}}\)  \({\overline{L}}_{\mathrm{lb}\_\mathrm{rxpa}}\)  

Mean values  9.37 [mΩ]  0.0187 [nH]  0.13 [mΩ]  0.0138 [nH]  25.0 [mΩ]  0.5 [nH]  0.36 [mΩ]  7.22 [nH] 
6 Conclusion
In our work, the PCEbased stochastic collocation method has been applied to study uncertainty propagation throughout the ECM of a reallife RFIC device. In order to reduce the computationally expansive simulation of the EMC, we have applied the equivalent lumped model developed by the NXP Semiconductor. It allows for conducting each simulation within the realistic time of about several seconds. Furthermore, based on the response surface model by truncated PCE, it has been possible to provide both the variancebased and local sensitivity analysis of the crossdomain coupling. Thanks to this analysis we could identify the most influential input parameters in a very efficient way. This information can be further used for the physicsbased design of an RFIC.
Another effective solution relies in incorporating the UQ into the regularized GaussNewton procedure in order to find automatically impedances z. This approach applied in our work allows to reduce the coupling effect of \(y_{3}(f)\) in the mean sense approximately by 27 [dB] across the considered range of frequency (Fig. 9). Also the standard deviation of \(y_{2}(f)\) and \(y_{3}(f)\) have been considerably reduced by 90% in average sense.
However, in our opinion, the application of the Pareto front method to solve a multiobjective problem can further improve the provided results for the RFIC interference problem due to competing objective functions \(y_{2}\) and \(y_{3}\). In such a case, the AWM might not approximate properly the Pareto front. However, this issue is considered as a further direction of our research.
In Fig. 6 \(\mathrm{Digital}_{\mathrm{vdd}} = \mathrm{vdd}_{\mathrm {dig}}\); \(\mathrm{Digital}_{\mathrm{gnd}}=\mathrm{gnd}_{\mathrm{dig}}\), \(\mathrm{RxPA}_{\mathrm{gnd}}=\mathrm{gnd}_{\mathrm{RxPA}}\); not found in Fig. 6: \(\mathrm{gnd}_{\mathrm{adc}}\), \(\mathrm{XoLO}_{\mathrm{gnd}}\) are mentioned in Fig. 2.
Declarations
Acknowledgements
The authors are grateful to the financial support from the European Union in the FP7ICT201311 Programme.
Availability of data and materials
The data that support the findings of this study are available from NXP, but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available.
Funding
This FP7 Collaborative Project nanoCOPS is supported by the European Union in the FP7ICT201311 Programme under Grant Agreement Number 619166 (Project nanoCOPS—nanoelectronic COupled Problems Solutions). For further details see http://www.fp7nanocops.eu/.
Authors’ contributions
All authors contributed to the writing of the final version of this paper. However, special merits go to RJ, JN and TB for sharing their experiences from industry, which yielded the development of the methodology for UQ modeling and optimizing of the reallife device; to RP for his work on the UQ; to PP, JtM and MG for their contribution to the robust optimization. All authors read and approved the final version of manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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