Characteristic Impedance, Power, and Causality

Dylan F. Williams, Senior Member, IEEE, and Bradley K. Alpert, Member, IEEE 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- A new causal power-normalized and the current i by waveguide equivalent-circuit theory determines z z  i(z) uniquely both the magnitude and phase of the Ht(r,z) [ce c e ]ht(r) ht(r), (2) i0 characteristic impedance of a waveguide.

where r = (x,y) is the transverse coordinate, Et and Ht INTRODUCTION are the total transverse electric and magnetic fields in

the guide, et and ht are the transverse modal electric and We summarize the causal waveguide circuit theory magnetic fields of the single propagating mode, is the of [1] and show that it determines uniquely the modal propagation constant, and c+ and c- are the characteristic impedance of a single-mode waveguide. forward and reverse amplitudes of the mode. The time The theory marries the power normalization of [2] with dependence ej7t in (1) and (2) have been suppressed, additional constraints that enforce simultaneity of the and all of the parameters are functions of 7. The two theory’s voltages and currents and the actual fields in factors v0 and i0 define v and i in terms of the fields and the circuit. These additional constraints guarantee that can be thought of as voltage and current normalization the network parameters of passive devices in this theory factors. are causal. We will show that the new theory requires The total time-averaged power p in the waveguide Z0 be real and constant in a lossless coaxial waveguide. is found by integrating the Poynting vector over the We will also show that in lossless rectangular guide’s cross section S: waveguide, must be proportional to the wave Z0 1 1 vi impedance: the choice |Z0| = 1 is not allowed. p E ×H #zdS e ×h #zdS. 2 P t t 2 P t t (3) S v0i0 S THE CAUSAL CIRCUIT THEORY The power normalization of [2] is achieved by The equivalent-circuit theory of [1] begins with a imposing the constraint waveguide that is uniform in the axial direction and v i p  e ×h #zdS, 0 0 0 P t t (4) supports only a single mode of propagation at the S reference plane where v and i are defined. The voltage v is defined by which ensures that the time-averaged power is p = ½vi*. z z  v(z) Et(r,z) [ce c e ]et(r) et(r) (1) v0

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1 CHARACTERISTIC IMPEDANCE DOMINANT TE10 MODE OF LOSSLESS RECTANGULAR WAVEGUIDE

The characteristic impedance Z0 of a waveguide is defined by the ratio of v to i when only the forward The power flow p0 and therefore Z0 of the TE10 mode is present [2]: mode of a lossless rectangular waveguide are real

2 above its cutoff frequency 7 and imaginary below 7 . v v v p c c Z  0 0 0 . % 7 0 2 (5) So arg(Z0) is equal to ± /2 below c, and 0 above. The i c 0 i p i 0 0 0 Hilbert transform of

Equation (5) shows that the power normalization 1 72 ln  . (6) 2  72 7 2  requires arg(Z0) = arg(p0), which from (4) is a fixed  c  property of the guide uniquely determined by the modal % 7 7 % 7 7 field solutions et and ht. is equal to - /2 for - c < < 0, /2 for 0 < < c, The causal circuit theory of [1] requires that both and 0 elsewhere [4]. So the causality constraints of [1] 7 7  7 Z0( ) and Y0( ) 1/Z0( ) be causal. That is, the require that ˆ ˆ theory requires that Z0(t) Y0(t) 0 for t < 0, where Zˆ (t) and Yˆ (t) are the inverse Fourier transforms of 72 0 0 Z    , (7) Z (7) and Y (7). 0  72 7 2  0 0  c  These conditions ensure that the waveguide responds to input signals after, not before, their onset.  where indicates proportionality. That is, Z0 must be 7 They also imply that Z0( ) is minimum phase [3]. proportional to the wave impedance of the guide: the The minimum phase constraint is a strong one. It choice |Z0| = 1 is not admissible in the causal theory. implies that the real and imaginary parts of the complex logarithm of a minimum phase function are a Hilbert MIS TRANSMISSION LINE transform pair: that is, arg(Z0) must be equal to the Hilbert transform of ln|Z | [3]. As a result, we can 0 Different choices of voltage and current paths in determine ln|Z |, where  is a constant, from arg(Z ) 0 0 conventional waveguide circuit theories result in [1]. different characteristic impedances in metal-insulator- semiconductor (MIS) transmission lines. Not all of LOSSLESS COAXIAL TRANSMISSION LINE these choices are consistent with causality. Figure 1 compares three characteristic impedances The power flow p0 is real in a lossless coaxial for the TM01 mode of the infinitely wide MIS line transmission line, so the phase of Z0 is 0. The set of investigated in [5]. This MIS line consists of a 1.0 µm constant functions form the null space of the Hilbert thick metal signal plane with a conductivity of 3×107 transform, so in the causal circuit theory of [1] Z0 must S/m separated from the 100 µm thick 100 6-cm silicon be real and constant. supporting substrate by a 1.0 µm thick oxide with conductivity of 10-3 S/m. The ground conductor on the back of the silicon substrate is infinitely thin and perfectly conducting.

2 0.012 the power constraint of (4) and the voltage across the Causal Z oxide, which is labeled “Power/oxide-voltage,” differs 0.010 0 Power/total-voltage significantly from the characteristic impedance required Power/oxide-voltage 0.008 by the causal theory presented here. Further

-m) ˆ Ω investigation shows that Z0(t) defined from the oxide ( 0.006  0

Z voltage is indeed nonzero for t < 0.  0.004

0.002 CONCLUSION

0 0.01 0.1 1 10 We have studied some of the implications of the Frequency (GHz) causal power-normalized waveguide circuit theory of

Fig. 1. |Z0| for the metal-insulator-semiconductor [1]. The examples illustrate an important contribution transmission line of [5]. The two solid curves are so of the causal theory: it replaces the subjective and close as to be indistinguishable. sometimes misleading “common-sense” criteria for The two solid curves in Fig. 1, which are labeled defining Z0 with a clear and unambiguous procedure “Causal Z0” and “Power/total-voltage,” agree so that guarantees causal responses. This new approach closely as to be indistinguishable on the graph. The should be especially useful in complex transmission curve “Causal Z0” is the magnitude of the characteristic structures where the choice of voltage and current paths impedance determined from the phase of p0 and the are not intuitively obvious. minimum phase properties of Z0, in accordance with the causal theory of [1]. The curve “Power/total-voltage” REFERENCES is the magnitude of the characteristic impedance defined with a power-voltage definition. Here the power 1. D.F. Williams and B.K. Alpert, “Causality and normalization is based on (4) (the integral of the waveguide circuit theory,” submitted to IEEE Trans. Poynting vector over the guide cross section) and the Microwave Theory Tech. Advance copies posted on voltage normalization on http://www.boulder.nist.gov/div813/dylan/ and http://math.nist.gov/mcsd/Staff/BAlpert/. v e #dl, 0 P t (8) path 2. R. B. Marks and D. F. Williams, “A general waveguide circuit theory,” J. Res. Natl. Inst. Stand. where the path begins at the ground on the back of the Technol., vol. 97, no. 5, pp. 533-562, Sept.-Oct., 1992. silicon substrate and terminates on the conductor metal on top of the oxide. This normalization implies that 3. E.A. Guillemin, Synthesis of Passive Networks. John Wiley and Sons: New York, 1957. v E #dl. P t (9) path 4. Bateman Manuscript Project, Tables of Integral Transforms. McGraw-Hill: New York, 1954, Table Conventional circuit theories do not specify voltage 15.2 (42). path uniquely, and a voltage path in the MIS line from 5. D.F. Williams, “Metal-insulator-silicon the silicon surface through the oxide to the signal line transmission line model,” 51st ARFTG Conf. Dig., is equally consistent with those theories. However, Fig. pp. 65-71, June 12, 1998. 1 shows that the characteristic impedance defined from

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