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How to Extract Distributed Circuit Parameters from the Scattering Parameters of a Transmission Line Nathan D How to extract distributed circuit parameters from the scattering parameters of a transmission line Nathan D. Orloff1, Nina Popovic2, Jasper A. Drisko1, Angela Stelson1, Charles A. E. Little1,2, James C. Booth1, Jordi Mateu3, and Christian J. Long1 1National Institute of Standards and Technology, Boulder, CO, 2University of Colorado, Boulder, CO, 3Universitat Politecnica de Catalunya, Catalunya, Spain Abstract — Distributed circuit parameters parameterize the as a function of frequency. At a high enough frequency, these transmission and reflection off a given transmission line in terms phases produce a branch cut in the argument used to compute of a distributed resistance, inductance, capacitance, and conductance, which are per-unit-length, frequency-dependent the propagation constant, which results in a large discontinuity quantities. While there are analytical models for extracting the in the extracted distributed circuit parameters. distributed circuit parameters, these models are discontinuous as Inspired by previous work, we have developed a nonlinear a function of frequency when the argument approaches a branch least-squares algorithm to mitigate this problem, and permit the cut. Here, we develop a nonlinear least-square regression extraction of the distributed circuit parameters over the full algorithm that accurately extracts the distributed circuit parameters. Compared to existing approaches and finite element frequency regime. Our approach starts with corrected S- models, our algorithm successfully extracts the distributed circuit parameters of a transmission line. We then computed estimates parameters as a function of frequency, all while being less sensitive for the distributed circuit parameters, and scale them such that to these phase conditions. Such an algorithm is useful for they all varied with the same frequency dependence and had an understanding how to deembed transmission lines, and how to order of magnitude close to one. We then passed the estimates extract electrical properties of the materials used in a circuit. Index Terms — transmission line, fitting, distributed circuit to the nonlinear least-squares optimization algorithm, parameters, S-parameters. computed the difference between the model and the data, and minimized the difference. The key idea was to scale the distributed circuit parameters, which improved how we used I. INTRODUCTION trust-region-reflective [6] and Levenberg-Marquardt [7], [8] Understanding how voltages and currents propagate in a optimization. There are some potential trade-offs to our transmission line has been important to electrical engineers algorithm. If the distributed circuit parameters are negative as a since the invention of the first working telegraph in 1816 by function of frequency, then our algorithm may need to be Francis Rolands [1]. Indeed, some of the greatest pioneers in modified to work. electricity and magnetism got their start on this very topic. In this paper, we demonstrate that the distributed circuit Chief among them is Oliver Heaviside, who was the first to parameters can be fit with a nonlinear least-squares algorithm, write down what we commonly refer to as the telegrapher’s which considers the relative scaling of the fit parameters to equations [2]. The Telegrapher’s equations describe how a minimize the sensitivity of the fit to phase of the S-parameters. voltage and current change as a function of time as they We describe our algorithm step-by-step and explain the propagate down a transmission line. At their conception, these reasoning behind each step. We validate our algorithm with equations allowed electrical engineers to predict how far their coplanar waveguide transmission lines on fused silica substrate, signals could propagate and account for distortion. which were corrected with an optimized multiline thru-reflect- Since then, Telegrapher’s equations have become an line (TRL) calibration algorithm. After applying our algorithm, essential part of every electrical engineer’s toolbox. We use we compare the distributed circuit parameters extracted with them to model the magnitude and phase of a voltage and current analytical expressions and our algorithm, which we contrast to at a given position and time, whether that be in an integrated finite element simulations of the same transmission lines. circuit or coaxial cable. In fact, there are very good analytical Taken together, this paper provides a detailed description of an models of the distributed circuit parameters in the literature, algorithm to extract the distributed circuit parameters of a with several outstanding examples for coplanar waveguide transmission line as a function of frequency that is more robust (CPW) transmission lines [3], [4]. More generally, we can against the phase uncertainties than analytical models. measure the scattering (S-) parameters of a transmission line, correct them, and then use analytical models to extract the II. THEORY distributed circuit parameters [5]. This analytical approach is useful when the exact geometry of the transmission line is For a lossy transmission line, we parameterize the distributed unknown, or when there is no analytical model that describes circuit in terms of a resistance (ܴ), inductance (ܮ), capacitance this specific type of transmission line. The problem with this (ܥ), and conductance (ܩ) per unit length. approach lies in the phase of the reflection and the transmission 978-1-5386-4356-3/17/$31.00 ©2017 IEEE ሺߛκሻ݄݊݅ݏڄ ሺߛκሻ ܼ௥݄ݏ݋ܿ ࡹൌ ቌͳ ቍ (6) ሺߛκሻ݄ݏሺߛκሻ ܿ݋݄݊݅ݏڄ ܼ௥ We take (6), combining terms and geometrical identities, to derive an argument involving the propagation constant ሺߛሻ and length of the transmission line (κ), ݁ିఊκ ൌܿ݋ݏ݄ሺߛκሻ ൅ݏ݄݅݊ሺߛκሻ Fig. 1. The distributed circuit for a transmission line modeled by a ܣ൅ܦ (7) ܥڄܤൌ ൅ ξ resistance (ࡾ), inductance (ࡸ), capacitance (࡯) and conductance (ࡳ) ʹ per unit length. Eisenstadt and Eo [5] took (7) one step further and solved for ߛ In the most general sense, ܴǡ ܮǡ ܥ and ܩ depend on frequency and ܼ in terms of the corrected S-parameters. We use this analytical model for comparison here. For completeness, we (Fig. 1). Applying Kirchhoff’s point and loop rules to Fig. 1, reproduce these expressions here, we obtain the Telegrapher’s equations, ିଵ ͳ ͳെܵଶ ൅ܵଶ ߲ ߲ ߛൌെ ൥ቊ ଵଵ ଶଵ േܭቋ ൩ ܵʹ ሚሺݔሻ κܫڄሚሺݔሻ െܴܫ ܮ෨ሺݔሻ ൌെܸ ߲ݔ ߲ݐ (1) ଶଵ డ డ భ (8) ෨ሺݔሻ. ଶ ଶ ଶ ଶ మܸڄܩ෨ሺݔሻ െܸ ܥሚሺݔሻ ൌെܫ డ௫ డ௧ ሺܵଵଵ െܵଶଵ ൅ͳሻ െ ሺʹܵଵଵሻ ܭൌ ቊ ଶ ቋ ሺʹܵଶଵሻ Next, we assume a sinusoidal time dependence, and the characteristic impedance, ሺݔሻ݁௜ఠ௧ ሺͳ൅ܵ ሻଶ െܵଶܫሚሺݔሻ ൌܫሺݔǡݐሻ ൌܫ (2) ܼଶ ൌܼଶ ଵଵ ଶଵǤ (9) ௜ఠ௧ ௥ ଶ ଶ ሺݔǡ ݐሻ ൌܸ෨ሺݔሻ ൌܸሺݔሻ݁ . ሺͳെܵଵଵሻ െܵଶଵܸ The problem with (8) and (9) are the potential sign ambiguities that arise when the phase of the argument in (8) go through ʌ or Then, we insert (2) into (1), and solve the wave equation to -ʌ. As in Ref. [5], we compute ܴǡ ܮǡ ܥ and ܩ, (ሻ (10ܼڄdefine the propagation constant, ܴൌԸሺߛ (ሻȀ߱ (11ܼڄൌԱሺߛܮ (ሻ, (3ܥ൅݅߱ܩሻඥሺܮߛൌඥሺܴ൅݅߱ ఊ ܥൌԱቀቁȀ߱, and (12) and the characteristic impedance, ௓ ఊ ඥሺܴ൅݅߱ܮሻ ܩൌԸቀቁ. (13) ܼൌ Ǥ (4) ௓ ඥሺܩ൅݅߱ܥሻ A. Eiensentadt’s analytical model for transmission lines B. Our analytical model for transmission lines Now that we understand where the distributed circuit Instead of the Eisenstadt and Eo [5] model, we used the T- parameters come from, we can use them to develop analytical matrix form of a transmission line, because the T-matrix models of a transmission line. In [5], Eisenstadt and Eo formalism does not require a defined reference impedance. In developed their analytical expressions for ߛ and ܼ from ABCD this case, the model for the T-matrix of a transmission line in a matrices. In general, ABCD-parameters are convenient for 50 ȳ reference impedance is, modeling both lumped element and distributed circuits, as well ିఊκ ࢀൌ ܳହ଴ ൬݁ Ͳ ൰ܳ௓ ǡ (14) as for cascading elements together [9]. A general ABCD matrix ௓ Ͳ݁ఊκ ହ଴ is given by, where the impedance transformer is given by, ܣܤ ࡹൌቀ ቁǤ (5) ͳ ܥܦ ࢆ࢔ ܼ௠ ൅ܼ௡ ܼ௠ െܼ௡ ࡽࢆ ൌ ൬ ൰Ǥ (15) ௠ െܼ௡ ܼ௠ ൅ܼ௡ܼ ܼڄ ࢓ ʹඥܼ ABCD-parameters are also intuitive, because they relate ௡ ௠ physical currents and voltages in a two-port network. We can We have used this model with great success in the past; see, for convert S-parameters to ABCD-parameters following the example, Refs. [10]–[14]. expressions in Refs. [5] and [9]. For a uniform two-port transmission line of length κ III. FABRICATION corrected to a characteristic impedance ܼ௥, the ABCD- parameters are given by, We fabricated the CPW devices on a 500 ȝm thick, 7.62 cm diameter fused silica (quartz) wafer. We chose quartz, because it has low electrical conductivity and isotropic permittivity [15]. The electrode layer consisted of a 5 nm Ti adhesion layer related to Maxwell’s equations through the material properties followed by 500 nm of Au that we deposited by electron beam used to make the transmission line. For simplicity, the evaporation. The CPWs on the calibration chip had a nominal distributed circuit parameters are related to physical material ,characteristic impedance of ׽50 ȍ with 20 ȝm-wide center properties and associated integrals through (ξ߱ǡ (16 ڄ ߱ሺߤ௜ሻȀɐןܴ conductors, 2 ȝm-wide gaps, and 200 ȝm-wide ground planes (Fig. 2). The devices on the calibration chip had contact pads (ߤ௥, (17ןܮ for on-wafer probe measurements. The pads were reinforced (߳௥, and (18ןܥ with ׽115 nm of palladium, which was used for the contact (߱߳. (19ןܩ pads due to its low evaporation temperature, minimal oxidation, ௜ Here, the complex permittivity of the transmission line is given and hardness [14]. by ߳ൌ߳௥ െ݅߳௜, the complex permeability is ߤൌߤ௥ െ݅ߤ௜, and the conductivity of the conductors is ߪ. From these relationships, (14)-(15), we know that ܩȀ߱ has the same units as ܥ, and that (14)-(15) are proportional to the complex permittivity.
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