IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 26, NO. 2, MAY 2003 165 Causal Characteristic Impedance of Planar Transmission Lines Dylan F. Williams, Fellow, IEEE, Bradley K. Alpert, Uwe Arz, Member, IEEE, David K. Walker, Member, IEEE, and Hartmut Grabinski Abstract—We compute power-voltage, power-current, and quickly with the spectral-domain algorithm they employed to causal definitions of the characteristic impedance of microstrip calculate it. and coplanar-waveguide transmission lines on insulating and In 1978, Jansen [7] observed similar differences between conducting silicon substrates, and compare to measurement. characteristic impedances defined with different definitions. Index Terms—Causal, characteristic impedance, circuit theory, Jansen finally chose a voltage-current definition for reasons of coplanar waveguide, microstrip. “numerical efficiency.” In 1978, Bianco, et al. [8] performed a careful study of the issue, examining the dependence of I. INTRODUCTION power-voltage, power-current, and voltage-current definitions of characteristic impedance on the path choices, and obtained E COMPUTE the traditional power-voltage and power- very different frequency behaviors. current definitions of the characteristic impedance [1], W In 1979, Getsinger [9] argued that the characteristic [2] of planar transmission lines on insulating and conductive impedance of a microstrip should be set equal to the wave silicon substrates with the full-wave method of [3] and com- impedance of a simplified longitudinal-section electric (LSE) pare them to the causal minimum-phase definition proposed in model of the microstrip line. Bianco, et al. [10] criticized [4]. Where possible we compare computed values of the causal Getsinger’s conclusions, arguing that there was no sound impedance to measurement. In all cases we find good agree- basis for choosing Getsinger’s LSE-mode wave-impedance ment. The microstrip simulations have been reported in confer- definition over any other definition. ence [5]. In 1982, Jansen and Koster [11] argued that the definition Classical waveguide circuit theories define characteristic of characteristic impedance with the weakest frequency depen- impedance within the context of the circuit theory itself. That dence is best. On that basis, Jansen and Koster recommended is, they develop expressions for the voltage and current in terms using the power-current definition. of the fields in the guide, and then define the characteristic In 1991, Rautio [12] proposed a “three-dimensional defini- impedance of the guide as the ratio of that voltage and current tion” of characteristic impedance. This characteristic impedance when only the forward mode is present. This results in a unique is defined as that which best models the electrical behavior of a definition of characteristic impedance in terms of the wave microstrip line embedded in an idealized coaxial test fixture. Re- impedance. cently Zhu and Wu [13] attempted to refine Rautio’s approach. Classical waveguide circuit theories cannot be applied in What the early attempts at defining the characteristic planar transmission lines because they do not have a unique impedance of a microstrip transmission line lacked was a wave impedance. This has led to an animated debate in the suitable equivalent-circuit theory with well-defined properties literature over the relative merits of various definitions of to provide the context for the choice. The causal waveguide the characteristic impedance of microstrip and other planar circuit theory of [4], which avoids the TEM, TE, and TM transmission lines. restrictions of classical waveguide circuit theories, provides In 1975, Knorr and Tufekcioglu [6] observed that the just this context. power-current and power-voltage definitions of characteristic The causal waveguide circuit theory of [4] marries the power impedance did not agree well in microstrip lines they studied, normalization of [1] and [2] with additional constraints that en- fueling a debate over the appropriate choice of characteristic force simultaneity of the theory’s voltages and currents and the impedance in microstrip. They concluded that the power-cur- actual fields in the circuit. These additional constraints not only rent definition was preferable because it converged more guarantee that the network parameters of passive devices in this theory are causal, but a minimum-phase condition determines Manuscript received August 29, 2002; revised April 22, 2003. This work was the characteristic impedance of a single-mode waveguide supported in part by an ARFTG Student Fellowship. uniquely within a real positive frequency-independent multi- D. F. Williams, B. K. Alpert, and D. K. Walker are with the National plier. Institute of Standards and Technology, Boulder, CO 80303 USA (e-mail: [email protected]). This approach to defining characteristic impedance differs U. Arz was with the Universität Hannover, Hannover D-30167, Germany, and fundamentally from previous approaches. It replaces the often is now with the Physikalisch-Technische Bundesanstalt, Braunschweig, Ger- vague criteria employed in the past with a unique definition many. H. Grabinski is with the Universität Hannover, Hannover D-30167, Germany. based on the temporal properties and power normalization of Digital Object Identifier 10.1109/TADVP.2003.817339 the microwave circuit theory. 1521-3323/03$17.00 © 2003 IEEE 166 IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 26, NO. 2, MAY 2003 In [14] we examined some of the implications of the cir- where is the unit vector tangential to the integration path. cuit theory described in [4], determining the characteristic The phase angles of the characteristic impedances and impedance required by the minimum-phase constraint of that are equal to the phase angle of , which is a fixed property theory in a lossless coaxial waveguide, a lossless rectangular of the guide. This condition on the phase of the characteristic waveguide, and an infinitely wide metal-insulator-semicon- impedance is a consequence of the power-normalization of the ductor transmission line. In [5] we investigated microstrip circuit theory; it is required to ensure that the time-averaged lines of finite width on silicon substrates, using full-wave power in the guide is equal to the product of the voltage and calculations to compute the power-voltage and power-current the conjugate of the current [1]. definitions of the characteristic impedance of the microstrip The magnitude of the characteristic impedance is determined lines, and using a Hilbert-transform relationship to deter- by the choice of voltage or current path, and is therefore not mine the causal minimum-phase characteristic impedance. A defined uniquely by the traditional circuit theories of [1] and comparison showed that the minimum-phase characteristic [2]. impedance agrees well with some, but not all, of the conven- The causal circuit theory of [4] also imposes the power nor- tional definitions in microstrip lines. malization of [1], so the phase angle of the causal characteristic In this paper we expand on [5], treating microstrips and impedance is also equal to the phase angle of . However, coplanar waveguides on both lossless and lossy substrates, and the causal theory requires in addition that be minimum extending the study to include common measurement methods. phase, which implies that In each case studied, we compare the causal power-normalized definition to common conventional definitions considered (6) earlier by previous workers. We use these studies to show where is the Hilbert transform. This condition ensures that explicitly how the theory of [4] resolves the earlier debates cen- voltage (or current) excitations in the guide do not give rise to a tered around the best definition of the characteristic impedance current (or voltage) response before the excitation begins. of a planar transmission line. Finally, we demonstrate that the Once is determined by the power condition (3), the constant-capacitance method [15] and the calibration-compar- space of solutions for is defined by ison method [16] measure causal minimum-phase characteristic impedances. (7) II. CHARACTERISTIC IMPEDANCE where is a real positive frequency-independent constant that determines the overall impedance normalization [4]. Equation Following [1] and [5] we define the power-voltage character- (7) results from two facts: the Hilbert transform has a null space istic impedance from consisting of the constant functions, and elsewhere the inverse of the Hilbert transform is its negative. (1) We fixed in (7) by matching and at a single frequency, as discussed in [4] and [17]. We used the lowest fre- and the power-current characteristic impedance from quency at which we had performed calculations to match to , because we believed our calculation errors to be smallest (2) there. where the complex power of the forward mode is III. COMPARISON OF DEFINITIONS We used the full-wave method of [3] to calculate the charac- (3) teristic impedance of the 5 wide microstrip line of Fig. 1. Fig. 2 compares this microstrip’s power-voltage, power-cur- is the angular frequency, is the unit vector in the direction rent, and causal minimum-phase definitions of characteristic of propagation, is the transverse coordinate, and impedance. are the transverse electric and magnetic fields of the forward The curve in Fig. 2 labeled “ ” is the magnitude of the char- mode, and the integral of Poynting’s vector in (3) is performed acteristic impedance determined from the phase of , which