This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 1 Model Order Reduction of Nonlinear Transmission Lines Using Interpolatory Proper Orthogonal Decomposition Behzad Nouri , Member, IEEE, and Michel S. Nakhla , Life Fellow, IEEE Abstract—A new method is presented for simulation of nonlinear transmission line circuits based on proper orthogonal decomposition reduction techniques coupled with an efficient interpolatory algorithm. Evaluation of the nonlinear function and corresponding Jacobian is performed in the reduced domain. A key criterion is developed for a priori determination of the dimension of the interpolation space leading to a substantial reduction in the computational cost. The proposed algorithm is applicable to general nonlinear circuits and does not impose any constraints on the topology of the pertinent circuit or type of the nonlinear components. Index Terms— Interpolation space, model order reduc- tion (MOR), nonlinear transmission line (NLTL), orthogonal basis, proper orthogonal decomposition (POD), singular-value decomposition (SVD). I. I NTRODUCTION W ITH the rapid advances in both optical and millimeter- wave semiconductor devices and systems, the demand for incorporation of nonlinear transmission lines (NLTLs) in high-frequency designs is rapidly increasing [1]–[5]. NLTLs have shown significant promise in the applications such as high-power RF pulse generation [6], [7], large-bandwidth fre- quency multipliers [2], electrical soliton oscillators [8]–[11], picosecond pulse generators [12], low-power ultra-wideband (UWB) impulse radio transmitters [13], [14], passive inter- modulation of analog and digital signals [1], [15], pulse shaping [16], and pulse compressor in both UWB and collision avoidance radar systems [17]. However, simulation and optimization of circuits containing NLTLs are a challenging task due to the large computational complexity associated with the traditional high-frequency large-signal transmission line models. On the other hand, model order reduction (MOR) has proven to be an effective tool in reducing the computational cost associated with large Manuscript received May 2, 2018; revised August 18, 2018 and October 26, 2018; accepted October 30, 2018. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). This paper is an expanded version from the IEEE MTT-S International Microwave Symposium (IMS2018), Philadelphia, PA, USA, June 10–15, 2018. (Corresponding author: Behzad Nouri.) The authors are with the Department of Electronics, Carleton Uni- versity, Ottawa, ON K1S 5B6, Canada (e-mail: sbnouri@doe.carleton.ca; msn@doe.carleton.ca). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2018.2880759 linear and nonlinear circuits and enabling design tasks that are not possible otherwise. MOR is based on developing systematic algorithms for replacing complex models with much simpler ones that still accurately capture the essential features of the original large system. This is achieved by projecting the equations describing the large system into a lower dimensional subspace. Several methods have been proposed in [18] and [19] for the reduction of nonlinear circuits. Due to its reliability, proper orthogonal decomposition (POD) [20]–[23] is the most suitable reduction method for large strongly nonlinear circuit applications. While a straight- forward implementation of the nonlinear reduction algorithms, in general, and POD-Galerkin, in particular, often yields a very low-order reduced model, the computational complexity for the evaluation of the resulting reduced-order model (ROM) is often the same as the complexity of the full (unreduced) circuit. This major computational drawback is due to the fact that the ROM still requires the evaluation of the unreduced nonlinear function and its Jacobian in the original domain instead of the reduced domain. This limitation can undermine any advantage that could be gained from reducing the size of the original model. To avoid this computational drawback, the trajectory piece- wise linear approach [24] was proposed based on the approxi- mation of nonlinear dynamics using a weighted sum of a set of linear models derived from the linearization of the nonlinear circuit at several operating points. However, the accuracy and efficiency of this approach are not consistently reliable particu- larly for circuits with highly nonlinear functions which cannot be approximated well using low-degree piecewise polynomi- als. In order to achieve a sufficient accuracy, a large number of linearization points and higher degree polynomials are gen- erally required leading to increased computational cost [25]. This fact eventually renders the trajectory-piecewise-based method intractable. Several alternative approaches have been proposed to tackle the problem of reducing the complexity of evaluating the nonlinear function and associated Jacobian. The main variants of these approaches can be categorized as gappy POD [26], [27], missing point estimation [28], [29], the empirical interpolation method (EIM) [30] and its dis- crete variant so-called as discrete EIM (DEIM) [31]. These approaches are closely related in the sense that they all use some selected set of spatial points (the subset of components of the vector-valued nonlinear function that are to be computed). 0018-9480 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.