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Causal Characteristic Impedance of Planar Transmission Lines Causal Characteristic Impedance of Planar Transmission Lines Dylan F. Williams, Fellow, IEEE, and Bradley K. Alpert, Member, IEEE, Uwe Arz1, David K. Walker, and Hartmut Grabinski1 National Institute of Standards and Technology, 325 Broadway, Boulder, CO 80303 Ph: [+1] (303)497-3138 Fax: [+1] (303)497-3122 E-mail: [email protected] Abstract- We compute power-voltage, power- not have a unique wave impedance. This has led to an current, and causal definitions of the characteristic animated debate in the literature over the relative impedance of microstrip and coplanar-waveguide merits of various definitions of the characteristic transmission lines on insulating and conducting impedance of microstrip and other planar transmission silicon substrates, and compare to measurement. lines. In 1975, Knorr and Tufekcioglu [6] observed that INTRODUCTION the power-current and power-voltage definitions of characteristic impedance did not agree well in We compute the traditional power-voltage and microstrip lines they studied, fueling a debate over the power-current definitions of the characteristic appropriate choice of characteristic impedance in impedance [1], [2] of planar transmission lines on microstrip. They concluded that the power-current insulating and conductive silicon substrates with the definition was preferable because it converged more full-wave method of [3] and compare them to the quickly with the spectral-domain algorithm they causal minimum-phase definition proposed in [4]. employed to calculate it. Where possible we compare computed values of the In 1978, Jansen [7] observed similar differences causal impedance to measurement. In all cases we find between characteristic impedances defined with good agreement. The microstrip simulations havebeen different definitions. Jansen finally chose a voltage- reported in conference [5]. current definition for reasons of “numerical Classical waveguide circuit theories define efficiency.” In 1978, Bianco, et. al [8] performed a characteristic impedance within the context of the careful study of the issue, examining the dependence circuit theory itself. That is, they develop expressions of power-voltage, power-current, and voltage-current for the voltage and current in terms of the fields in the definitions of characteristic impedance on the path guide, and then define the characteristic impedance of choices, and obtained very different frequency the guide as the ratio of that voltage and current when behaviors. only the forward mode is present. This results in a In 1979, Getsinger [9] argued that the unique definition of characteristic impedance in terms characteristic impedance of a microstrip should be set of the wave impedance. equal to the wave impedance of a simplified LSE Classic waveguide circuit theories cannot be model of the microstrip line. Bianco, et. al [10] applied in planar transmission lines because they do criticized Getsinger’s conclusions, arguing that there This work was partially supported by an ARFTG Student Fellowship. Publication of the National Institute of Standards and Technology, not subject to copyright. 1Universität Hannover, D-30167 Hannover, Germany. 1 was no sound basis for choosing Getsinger’s LSE- phase constraint of that theory in a lossless coaxial mode wave-impedance definition over any other waveguide, a lossless rectangular waveguide, and an definition. infinitely wide metal-insulator-semiconductor In 1982, Jansen and Koster [11] argued that the transmission line. In [5] we investigated microstrip definition of characteristic impedance with the lines of finite width on silicon substrates, using full- weakest frequency dependence is best. On that basis, wave calculations to compute the power-voltage and Jansen and Koster recommended using the power- power-current definitions of the characteristic current definition. impedance of the microstrip lines, and using a Hilbert- In 1991, Rautio [12] proposed a “three- transform relationship to determine the causal dimensional definition” of characteristic impedance. minimum-phase characteristic impedance. A This characteristic impedance is defined as that which comparison showed that the minimum-phase best models the electrical behavior of a microstrip line characteristic impedance agrees well with some, but embedded in an idealized coaxial test fixture.Recently not all, of the conventional definitions in microstrip Zhu and Wu [13] attempted to refine Rautio’s lines. approach. In this paper we expand on [5], treating What the early attempts at defining the microstrips and coplanar waveguides on both lossless characteristic impedance of a microstrip transmission and lossy substrates, and extending the study to line lacked was a suitable equivalent-circuit theory include common measurement methods. In each case with well-defined properties to provide the context for studied, we compare the causal power-normalized the choice. The causal waveguide circuit theory of [4], definition to common conventional definitions which avoids the TEM, TE, and TM restrictions of considered earlier by previous workers. We use these classical waveguide circuit theories, provides just this studies to show explicitly how the theory of [4] context. resolves the earlier debates centered around the best The causal waveguide circuit theory of [4] marries definition of the characteristic impedance of a planar the power normalization of [1] and [2] with additional transmission line. Finally, we demonstrate that the constraints that enforce simultaneity of the theory’s constant-capacitance method [15] and the calibration- voltages and currents and the actual fields in the comparison method [16] measure causal minimum- circuit. These additional constraints not only phase characteristic impedances. guarantee that the network parameters of passive devices in this theory are causal, but a minimum-phase CHARACTERISTIC IMPEDANCE condition determines the characteristic impedance Z0 of a single-mode waveguide uniquely within a real Following[1] and [5] we define the power-voltage positive frequency-independent multiplier. characteristic impedance ZPV from This approach to defining characteristic impedance differs fundamentally from previous (1) approaches. It replaces the often vague criteria employed in the past with a unique definition based on the temporal properties and power normalization of and the power-current characteristic impedance ZPI the microwave circuit theory. from In [14] we examined some of the implications of the circuit theory described in [4], determining the characteristic impedance required by the minimum- 2 (5) where l is the unit vector tangential to the integration path. The phase angles of the characteristic impedances ZPV and ZPI are equal to the phase angle of p0,whichis a fixed property of the guide. This condition on the phase of the characteristic impedance is a consequence of the power-normalization of the circuit Fig. 1. Microstrip line geometry. The signal conductor had theory; it is required to ensure that the time-averaged a conductivity of 3x107 S/m, a width of 5 :m, and a power in the guide is equal to the product of the thickness of 1 :m. The oxide layer was 1 :m thick and had voltage and the conjugate of the current [1]. a relative dielectric constant of 3.9. The silicon substrate was 1 :m thick and had a relative dielectric constant of The magnitude of the characteristic impedance is 11.7. The silicon substrate may be conductive. We call out determined by the choice of voltage or current path, the conductivity explicitly in the text and other figures. and is therefore not defined uniquely by the traditional (From [5].) circuit theories of [1] and [2]. The causal circuit theory of [4] also imposes the (2) power normalization of [1], so the phase angle of the causal characteristic impedance ZC is equal to the phase angle of p0. However, the causal theory requires where the complex power p0 of the forward mode is in addition that ZC(T) be minimum phase, which (3) implies that (6) T is the angular frequency, z is the unit vector in the direction of propagation, r =(x,y) is the transverse where , is the Hilbert transform. This condition coordinate, et and ht are the transverse electric and ensures that voltage (or current) excitations in the magnetic fields of the forward mode, and the integral guide do not give rise to a current (or voltage) of Poynting’s vector in (3) is performed over the entire response before the excitation begins. cross section of the guide. The voltage v0 of the Once arg[p0] is determined by the power condition forward mode is found by integrating the electric field (3), the space of solutions for |ZC| is defined by over a path with (7) (4) where 8 is a real positive frequency-independent constant that determines the overall impedance and current i0 is found by integrating the magnetic normalization [4]. Eqn. (7) results from two facts: the filed over a closed path with Hilbert transform has a null space consisting of the constant functions, and elsewhere the inverse of the Hilbert transform is its negative. 3 50 40 Power/oxide-voltage Causal 30 20 (arbitrary units) 0 Z 10 0 -20 0 20 40 Time (ns) Fig. 2. Comparison of definitions of characteristic Fig. 3. Comparison of the inverse Fourier transforms of the impedance of a microstrip line on a 100 SAcm substrate. We causal and power/oxide-voltage characteristic impedances. matched |Z | with |Z | at 5 MHz by adjusting 8 in (7).(From C PV The power/oxide-voltage definition predicts a response to [17].) current excitations
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