Scattering Parameters Representing Imperfections in Precision Coaxial Air Lines

Scattering parameter expressions are developed for the principal mode of a coaxial air line. The model allows for skin-effect loss and dimensional variations in the inner and outer conductors. Small deviations from conductor circular cross sections are conformally mapped by the Bergman kernel technique. Numerical results are illustrated for a 7 mm air line. An error analysis reveals that the accuracy of the scattering parameters is limited primarily by the conductor radii measurement precision.


Introduction
To acctirately characterize imperfections of precision coaxial air lines, skin effect and surface roughness need to be considered. Skin effect is now well documented [1] and conductor surface finish has been studied in detail by Rice [2] and Ament [3] through the use of Fourier series methods. While Karbowiak [4] points out that Fourier analysis reveals useful knowledge of the spectral components which principally affect scattering parameters, it is also appropriate to examine local pointwise influence along the axial (z) coordinate. In this connection, Hill [5] developed perturbation expressions for the scattering parameters for a lossless circular air line. When the conductor surface exhibits transverse angular variation, Roimeliotes, Houssain and Fikioris report the effects of ellipticity and eccentricity on cutoff wave numbers [6].
The purpose of this paper is to develop numerically accurate pointwise coaxial air-line scattering parameters that account for skin effect loss and conductor surface variations in the transverse an-gular and axial directions. Following Schelkunoff [7], Reiter [8], Solymar [9] and Gallawa [10], generalized telegraphist equations for the principal mode are derived in section 2 for a circular air line. Transformation to forward and backward wave differential equations enables general solutions for the scattering parameters in section 3. To allow for conductor surface measurements along the z-axis, cubic spline polynomials provide a starting point for establishing pointwise recursion formulas of forward and backward waves in section 4. In section 5, the Bergman's kernel technique is used to establish a conformal mapping for transforming noncircular conductors into equivalent circular conductors in correspondence to the principal mode. Computational results illustrating | S^ \ versus air-line length are given in section 6. An error analysis of the computational algorithms for the accuracy resolution of the measurement system is developed in section 7.

Generalized Telegraphist Equations for the Principal Mode
Consider the coaxial air line in figure 1. With inner radius a{z) and outer radius b(z) the field components of primary interest are radial electric field (Er), angular magnetic field (H^, and axial electric field {E^. We assume the fields E^ He, and E^ are composed of TEM and TM modes, and coupling of the modes is caused by skin effect with variations of the conductor surfaces. Appropriate Maxwell equations for determining transverse fields E^ and Hg in the air dielectric region of the air line are [11] iEr . rr , ^Ez dHg az = -jote Er. The parameters (a, jn, and e are defined as radian frequency, permeability and permittivity, respectively, In addition the fields are assumed to vary with time according to the complex exponential function e'"'. To find the generalized telegraphist equations it is convenient to assume the fields possess orthogonal expansions in r and 6. In view of TEM and TM modes together with impedance boundary condi- tions a set of orthogonal basis functions needs to be constructed from the Gram-Schmidt process. Assmning E^, He, and E^ possess continuous first and second derivatives implies their expansions are absolutely and uniformly convergent [12]. In Appendix B these properties are used to rearrange the expansions into the form E,(ir,e,z)= 'x' C (^) e^iU'-.^.^) (n,p)=(0,0) where the superscripts (1) and (2)  where ~ stands for the complex conjugate. In particular for the TEM mode e^(r,z)= 1 1 A^oo(z) r ' (1.10) iV.(z)=J2.1n^)".
Higher order modes are usual linear combinations of the first derivative Bessel functions J'" and Y' Following Reiter [8] by taking the inner product of eq (1.3) with the basis function e% yields J ^ e:%dS = -jo^x J afle ■ {ae^^'^, XaJdS

\'-f^-^%^
(1-11) where S{z) denotes the cross sectional air dielectric region between the conductors. The left side calls for differentiation of a variable surface integral and the second member of the right side integrates by parts.' Hence, eq (1.11) evolves into the form Now substituting basis function definitions eq (1.7) and calling for the principal mode yields Examining eqs (1.16) and (1.21) reveals that continuous mode coupling occurs through the voltage and current transfer coefficients (left side), respectively, a phenomenon observed by Schelkunoff [7]. Skin effect coupling on the conductor surfaces was also reported by Schelkunoff and Gallawa [10]. When the air line is operated at frequencies appropriate to the principal mode, all TM modes attenuate rapidly below their cutoff frequencies.
Consequently, dominant coupling occurs between the forward and backward waves of the principal mode.^ In this regard, eq (1.16) assumes the form where the superscript, (1), has been dropped since only one mode is involved. Rearranging terms produces the expression In a(z) The equation for current proceeds similarly. Equation (1.20) yields ^=_ya,eroo+mz)/oo (1.26) ^ Higher order mode influence on the TEM mode will be reported in a later issue of this journal.
where the current transfer coefficient is defined as ^6(z)ln^ a(z)ln^J

Conversion of Generalized Telegraphist Equations to Forward and Backward Wave Equations
Following Solymar [9] we define the amplitudes of the forward and backward waves A ^ and A ^ from the relations

Too(z)A^,
In view of eqs (1.25) and (1.27) the last expression possesses the form
Returning to eqs (2.2) and (2.3) and retaining Solymar's assumption above leaves the terms Tooiz) A «). Since coupling in this sense is meaningless, we drop the terms Tooiz) A ^ and obtain' To incorporate appropriate boundary conditions, let the incident wave be A oo(P)=Ao with perfect termination at z=L, that is AMiL)=0.
At this point the forward wave solution yields T;-'^^\O<Z<L (2.10) and, at z=0, the reflected wave expression +IoMy2i!imAo^ (2.11) show general forms which remain to be useful for using conductor radii measurements.

Cubic Spline Fitting of Conductor Radius Measurements
Underlying an accurate solution to A^ and A ^ are two critical items: (a) fitting conductor radii measurements with acceptable error bounds and (b) expansion of all known functions in a systematic manner to sufficient powers of r.
To handle (a) consider cubic spline polynomials [13] for the inner (or outer) conductor measurements such that where &jt_i(z) approximates a(z) or b(z), zic-i<z<,Zk such that k = l,-• ;N and Zi^=L. It is desirable to transform the cubic spline eq (3.1) over the interval [z*^_i,Z;t] into the representation" in such a way that the condition, holds at z=zic-i and z=Zk where l^=Zk-zic_i. In addition we require Q_i(0)=&o,A-i, and such that Co,*_i represents the measurement of a(z) or b(z) at Z=ZA_I. To implement (b), recall that E" Hg, and the second derivatives of Er and Hg are assumed to be analytic functions in r, 9, and z. Hence, the expansions of Ibiz) In^)-' and L(z) In^l"' can be rearranged in powers of z. In Appendix C the following expressions are derived over the interval  where 0<^<z-z"_i.
To establish A^ and ^ob at each point z" it is convenient to employ recursion relations. Inserting eq (3.3) into eq (2.6) and examining the interval Zi<z<Z2 yields is the forward wave emerging at z=Zi.
For the interval z^_i<z<z eq (3.5) immediately generalizes to the recursion relation Proceeding to the backward wave ^4 M by using eqs (3.3) and (3.4) in eq (2.7) for the interval Zi<z<Z2 produces the relation •e-^t^oi"*>^"''«>d7, (3.8) where the transformation r}=z-Zi introduces the term e"^"^'. Now eq (3.8) also generalizes to the recursion relation From Appendix C eq (3.9) assumes the solution

Conductor Surface Variations in the Transverse Coordinates
When the outer conductor is bored, circular cross sections are the exception rather than the rule. Most likely, an elliptical cross section evolves with some degree of rotation. Consequently, it is desirable to perform mechanical measurements of conductor radii in the transverse plane to characterize the deviation from circular cross sections. Since the principal mode is TEM in the transverse plane a direct conformal mapping of the measvurement contour into an equivalent circular contour eliminates any difficulty of solving Laplace's equation for an irregular boundary. If an equivalent circular contour is found for each transverse measurement plane on the air line, a corresponding set of scattering parameters represents the original air line of measurement contours.
The solution of Laplace's equation for a TEM mode with the inner conductor potential held to VQ and the outer conductor potential set at 0 is ■ " ln{r/6(z)} 'P-*^»ln{a(z)/6(z)} (4.1) We initially state that Riemann's mapping theorem assures a mapping from the contour L to the unit circle and a particular expression for mapping evolves from the Bergman kernel expansion [14]. Thus, for a contour L centered at ^o=8 e'*=0 the Bergman kernel is defined as 5(0,0= X^v"(0) Pv(0.  Carrying out the above procedures yields the conductor radii accurate to third order, where Hence, any asymmetry in the contour L is expected to be noticeable through the off-diagonal elements To find the equivalent circular conductor radius, integrals in eqs (4.5) and (4.6) need to be determined from measurements of p and 6 on the contour L. Let the following cubic spline be defined p,_i(e)=2p,k-ie' (4.11a) such that Pk-iiQ) approximates a(d,z) or b(6,z) over the interval 6k-i<0<,dic for k = l,-, N, i.e., 0^^ = 277.
Following the same procedures as in the transformation from z to { in eqs (3.2a) to (3.2c) enables the cubic spline.
Equations (4.5) and (4.11) yield an expression for length using the binomial expansion:

Computation Results
The amplitude of Sn has been computed from eqs (2.12) and (3.9) for a 7 mm air line approximately 15.6 cm in length using the frequencies 6, 12, and 18 GHz. In addition the number of conductor dimensional measurements in three sections of air line with variable spacing is shown in figure Figure 2 illustrates that conductor radius measurements near either end are more volatile-particularly the outer conductor. Figures 3, 4, and 5 reveal that changes in conductor radii in the z-axis provide the dominant contribution to |5n| while skin effect loss amplifies the in-phase and out-ofphase behavior of the lossless air line (as shown in fig. 2). In addition, skin effect loss affects the most significant digit of \Su\ even for short lengths of line. On comparison of figures 3, 4, and 5 with Hill's results [5], the most noticeable feature is the overall difference in magnitudes of Sn, which evolves from a uniform inner conductor model and lossless boundary conditions in Hill's work. Conformal mapping effects from elliptical measurement contours do not affect Su and ^21 unless the eccentricity is greater than 5x10"' meters. However, if the inner conductor has an eccentric position with respect to the outer conductor, conformal mapping by Bergman's kernel reveals scattering parameters Sn and ^21 are noticeably af-

Error Sources
Error sources that contribute to scattering parameters evolve from (a) spline interpolation with respect to z, (b) spline interpolation with respect to 6, (c) conforfflal mapping using the Bergman kernel, and (d) expressions for backward and forward waves.
To examine (a) and (b) consider the error bound from cubic spline interpolation theory [13], where SAt(x) defines a cubic spline, ju, signifies the modulus of continuity,* /" denotes the second derivative of the function /, and A^ stands for the mesh size between arguments x^. An approxima-tion to the modulus of continuity is where ff ] denotes the first order divided difference of/. For a 7 mm air line with a mesh size (Ajt) equal to 1 mm, a value of ju, from observations of conductor radii measurements as functions of z indicate ;LI=0.1 is reasonable, and in the angular direction /A < 0.01.' Therefore, the total error from spline interpolation is (considering the errors as additive) Error,otai=Error2+Error«<2.8 X lO"'' m. (  The error source (c) from conformal mapping an elliptical measurement contour of eccentricity equal to 38.1x10"' m (150 ju. in) is illustrated below.
First term of outer conductor equivalent (mapped radius) 3.4989 mm.
Second term 2.32 X10"'^ mm. Third term -3.35x10-' mm. Keeping in mind that ellipses are symmetrical with respect to the origin reveals that even terms are effectively zero (on the order of 10"'^ in view of machine precision). Since the convergence above is very strong, the fifth term is likely on the order of 10"' mm. The erroe from source (d) depends on the number of expansion terms representing the functions l/a(z), \/b{z), and ln{6(z)/fl(z)}.
At this point accuracy considerations of the forward and backward wave eqs (3.7) and (3.10) in correspondence to the measurement system are in order. For instance, to examine A^a consider the measurements of a(z) and b(z) at z=0 and z=Zi. where Oi and bi are the coefficients from differentiating the cubic spline representations of a{z) and b(z), respectively. In computing AAQO and AA^ for a 7 mm air line let the measurement precision be Az=Aro=2.8xlO~^ m for a frequency range of 1-18 GHz. Using the measurements of a(z) and b(z) in figure 3, we select the maximvmi divided to a conductor surface measurement resolution of 2.8X10-' m over frequencies appropriate to the principal mode for 7 mm air lines. (6.8) 9^ Acknowledgments (6.9) To examine the total uncertainty in ^ ^ over N measurements of a(zi) and b(zk) for k = l,-,N let AAffyhe represented as N A^ oo(0;zjv)= 2 f{a(zk),b(Z")}AA ^(z,_"z"), (6.10) where/" is on the order of A^(z"^i-^"). Since the computation of AAa)(Zn-il2") proceeds in the same way as L4^(0;zi), in eq (6.5), the total imcertainty over all measurement positions is found when the individual uncertainties in z and /"o at each measurement position are known.

Efficiency Improvements in Cubic Spline Approximation
While fitting the surfaces of coaxial air line geometries with products of cubic splines over the variables 6 and z successfully meets error bounds consistent with measurement precision, significant reductions in the number of measurements yields equivalent error bounds with Gordon's successive decomposition spline [18]. The number of measurements required for successive decomposition splines in comparison to usual spline products is generally less than 50 percent.

Summary
Generalized telegraphist equations for the coaxial air line have been derived under two assumptions: (a) skin effect losses are present, and (b) conductor surface variations occur in the axial and transverse coordinates. Product cubic spline expressions to accurately fit conductor surface measurements were employed to arrive at pointwise scattering parameter expressions. Error bounds from eqs (6.8) and (6.9) reveal at least four significant figures can be obtained to characterize the scattering parameters 52i and 5ii in correspondence The author expresses sincere appreciation to the following staff members of NIST: (a) B. C. Yates for many helpful and critical comments over the project duration (especially examining the numerical results of |Sii|); (b) R. L. Jesch for general leadership and advice during the manuscript preparation, and (c) E. G. Johnson for stimulating theoretical discussions on various aspects of air line modeling.

About the author: Donald R. Holt is a mathematician in the Electromagnetic Fields Division of the National Institute of Standards and Technology, Boulder, CO.
In reference to figure 6 the boundary condition is where the relation E^ tan<|>(z)=.E, has been employed.
The inner conductor boundary condition relation is similarly derived as follows: We have

Appendix B. Transformation of Basis Functions for the Field Components
The transverse fields Er and He are represented as where v(n)=2'n-if « =0

='jTiin^Q
The expansion for H^ is obtained in the same way, i.e.,