Exact principal mode field for a lossy coaxial line

Exact field equations for a lossy coaxial transmission line with an infinite outer conductor are presented. The corresponding determinantal equation is solved to obtain an exact propagation constant from which errors in the usual microwave approximation and an alternative full frequency range approximation are calculated. The calculations show that the microwave approximation, although containing a large relative error at the lower frequencies, is still useful in practical applications. >


I. INTRODUCTION
N exact field solution to Maxwell's equations for a A coaxial transmission line has been in existence since at least 1941 [l], but has gone unrecognized or at least unexploited since that time. This is probably because in 1941 the means of calculation available today were nonexistent so the exact solution was inadvertently buried in the need to approximate. In any case, this solution has been recently developed and is presented in this paper.
As six-port and automatic network analyzers have become more accurate and sensitive and are being utilized at lower and lower frequencies, increasing emphasis is placed upon accuracy of the transmission line standards used to characterize or calibrate these systems. In particular, accurate calculation of the line's propagation constant acquires added importance. The approximation used at present [l], [2] to calculate this constant, however, needs to be reexamined below the microwave frequency region for which it was derived since its accuracy could only be estimated in the past from Russell's equations 131, which themselves are approximate. The work described here was pursued in part to correct this situation and represents the first real check on the microwave approximation; it also suggests an alternative approximation to the propagation constant that is more accurate and that is applicable over the full frequency range of the line.
Russell's 1909 paper [3], which was used [2] to estimate the accuracy of the microwave approximation to the propagation constant and the distributed line parameters, is interesting in its own right. It discusses early theoretical Manuscript received January 15, 1991; revised April 9, 1991. The author is with the National Institute of Standards and Technol-IEEE Log Number 9101008. ogy, 325 Broadway, Boulder, CO 80303. developments concerning the transmission of highfrequency currents down "concentric mains" (involving the efforts of Maxwell, Heaviside, and Lord Rayleigh) and contributes significantly to that development in addition to deriving approximations to Kelvin's "ber," "bei," "ker," and "kei" functions that are standard today. The approximations in Russell's solution appear in his equations (5914611, where Faraday's, Ohm's, and an integral form of Ampere's law (not the exact version containing the displacement current) are used in addition to the approximation that the axial electric field is constant across the conductors.
The establishment of impedance standards for integrated circuit applications in stripline and microstrip transmission lines is considerably more difficult than similar endeavors for coaxial lines such as the one described here. Indeed, even an accurate description of the wave propagation on these miniature lines is complex, and it is useful to have at least one exact transmission line solution against which to compare results. The fields and results to be described below constitute one of these exact solutions.
The exact coaxial line solution is presented in the next section, preceding which is a short review of the pertinent theory that is included to collect the equations in one place and because of the importance of air line standards to precision network analyzer calibrations. A slight change has been made in the derivation to avoid divergences present in the usual treatment [l] as the frequency or the conductor resistivity approaches zero.
Boundary conditions requiring continuity of the fields tangential to the conducting surfaces are applied in Section I11 to derive the determinantal equation, and a simple iteration scheme to quickly find the root (which is used to calculate the propagation constant) is presented. Approximation$ are developed in Section IV, where an expression for the root of the determinantal equation is presented that is more accurate and applies over a wider frequency range than the microwave version [ll, [2l presently in use. Full range, first-order fields are also presented in that section.
A number of results are discussed in Section V, including field graphs within and without the conductors; a comparison of the conductor skin depths with the planar approximation; graphs showing the accuracy of the approximations, in particular the accuracy of the propaga-U.S. Government work not protected by U.S. copyright tion constant calculations; a comparison of the inner and outer conductor currents; and, finally, graphs of the wave impedance and phase velocity versus frequency. Conclusions drawn from the results are presented in Section VI.
11. EXACT FIELDS A derivation of the exact electric and magnetic fields propagating along the axis of a coaxial line is briefly reviewed in this section. The line (Fig. 1) is assumed to be homogeneous and isotropic with a uniform cross section consisting of a center conductor (region 11, a dielectric region between the conductors (region 2), and an outer conductor (region 3). The magnetic permeabilities, electric permittivities, and conductivities are denoted by p,, E , , and at in the various regions ( i = 1,2,3) and a, = 0 in the dielectric region 2. The wavenumbers are denoted by k , , and the h, are parameters to be described later. The cootdinate system is right-handed with the directions (F, 4 , i ) shown in the figure, the z direction coming out of the paper toward the reader. The conductor resistivities are denoted by p, ( = l/a,) and will be used instead of the conductivities when convenient.
The three regions are assumed to be source free so that Maxwell's equations in SI units take the form

V X H = ( a i + j w c i ) E V * E = O ( 2 )
where the harmonic variation e"' is assumed. Performing the standard vector operations [1] on (1) and (2) leads to the Helmholtz equation nates. Furthermore, (3) implies that each cylindrical component of both E and H must satisfy this Helmholtz equation and have the form [4] ?

E, =C, h y -' Z , ( h r ) , a < r < b
(10) The quantity h is obtained from the determinantal equation (28) discussed in the next section. The Z, ( n = 0,l) functions in (10) are defined by the equation where G is a constant determined by the boundary conditions. This last definition was chosen in place of the usual definition, Z , = J, + GN, [l], to avoid divergences in both G and N, as h approaches 0 owing to the frequency or the conductor losses (resistivities) approaching 0. Both terms in (16) are well behaved in these limits, the first term approaching 0 and the second term approaching 0 or -2 b /~r depending on whether n = 0 or n = 1.
Each set of fields given by (9), (lo), and (11) satisfies Maxwell's equations identically, as can be seen by substitution into (1) and (2), using the Bessel function recursion relations to reduce the resulting terms containing n = 2 to combinations of terms containing only n = 0 or n = 1.
The boundary conditions require the continuity of E, and H4 at r = a , b, and yield the following field expressions which are obtained from the fields in (9)-(11) by solving the boundary conditions for C , and C, in terms of C, and using (14) for the wave admittances: Region 1: Magnetic losses within the conductors and dielectric losses between the conductors can be included by using a com- The field expressions (17)-(25) formally satisfy both Maxwell's equations and the boundary conditions exactly, but the value of h must still be determined in order to calculate the propagation constant y and complex fields. This is accomplished in the next section.
plex p l y p 3 , and k 2 .

DETERMINANTAL EQUATION
Calculation of the constant h is necessary to utilize the field equations of the last section. It is obtained, like the C, constants, by applying boundary conditions at r = a and r = b. The resulting equation from which h is determined and a simple means of solving that equation are described in this section.
Two expressions containing the constants C, and C, are obtained by requiring continuity in E, and H+ at the boundary r = a (see (91-411)). Eliminating these constants between the two expressions yields where The last equality in (28) is the determinantal equation from which the root h is extracted. There are actually an infinity of roots: the root with magnitude close to 0 which is the desired h of the principal mode; and an infinite number of other roots with nonzero magnitudes that (21) belong to the symmetric TM waveguide modes [l], [5], which are of no interest in the present work. A simple and rapidly convergent iteration scheme for finding the root h of (28) can be obtained with the following seven steps: make an initial estimate for h using (32) in (35) ,(h,a) and R,(h,b) from the results of step 2 and (29); calculate the right and left sides of the second equality in (28) from the results in steps 1, 2, and 3; calculate the difference SG between the calculations in step 4 and the correction to h using (12); I n -+ - add the correction Sh to the h estimated in step 1; finally, iterate steps 2 through 6 until the desired accuracy in h is achieved.
Three iterations are sufficient to produce an h accurate to approximately 14 significant figures for the numerical example in Section V. The formula in (30) was obtained by differentiating the difference between the two sides of the last equation in (31) with respect to h (note that the logarithm of h cancels when taking the difference).
Once h is found, h , and h, can be calculated from (12) after the propagation constant y is calculated from (15).

IV. APPROXIMATIONS
The exact expressions found in the previous two sections are inconvenient to use in many practical applications because of the effort and time required for their calculation, so accurate approximations are often necessary. Some of the more important approximations are presented in this section.
The following two approximations (6) for G can be obtained by dropping the k i terms in the denominators of (28): where if k , = k,, then h , is proportional to the reciprocal of ki or the normalized surface impedance of the conductors (see (A5)). This approximation gives good accuracy only at the upper, microwave end of the line's frequency range. This is the solution that has survived since 1941 and that is in common usage today [l]. The positive square roots in (35) and (37) are taken when h and h , are required.
Equation (31) is a "small h" approximation to G whereas the corresponding microwave expression is a first-order approximation in the metallic surface impedance of the conductors, approximating R , and R ,     for the calculations are u2 = ' 7 11.1 = 11.2 = 11.3 = for Other line sizes are bemeen the conductors, a feature that is evident in the approximate equation (40) also. The local minima, however, are seen to approach 0 at r = 1.79 (a value easily similar to the ones shown below. derived from (40)) as the frequency is increased. Fig. 4 is a graph of the phase angle of E, between the The magnetic field described by (19), (22), and (25) is conductors. The phase remains relatively constant with plotted in Fig. 2 for various frequencies as a function of frequency until a transition region around 1.79 is reached, the radius r . The magnitude is normalized by its maxi-at which point it abruptly changes by approximately 180". mum value at r = a and the radius by the inner conductor This phase reversal is sufficient to ensure that there is a radius a. The fields between r = a and r = b at the positive average Poynting flux into the conductors at both various frequencies differ a small amount because of r = a and r = b, accounting for the inner and outer conconductor loss, but the vertical scale of the graph is too ductor losses.    Table I illustrates the rapidity with which the iteration scheme for finding the root h of the determinantal equation discussed in Section I11 converges. The real and imaginary parts of h shown in the first row are the equation (35) estimates. The convergence is seen to be 15 significant figures in three iterations.

A. Fields, h, Skin Depths
The skin depth is defined as that distance into the conductors at which the magnitudes of H4 or E, fall to l / e (see the dashed levels in Fig. 2) of their respective values at the surface of the conductors. It is generally different for the two conductors, but this difference is not significant in the microwave frequency region of the line. The outer conductor skin depth is generally less than the planar approximation because the fields are spreading out as r increases so it requires a shorter distance for the l / e falloff to occur. An opposite effect holds for the inner conductor until a crossover at approximately 7 kHz. The crossover and the fact that both inner and outer conductor skin depths become constant as the frequency decreases can be seen by examining Fig. 2, an exercise left to the reader.

B. Approximations
The microwave and full range approximations differ by the R , Aj and R , = j approximations used in deriving the former. Figs. 6 and 7 are graphs of the magnitudes and angles of R , and R , as a function of frequency. They show that the magnitudes begin to diverge from their microwave approximations at about 1 MHz and the angles begin to diverge somewhere between 1 and 10 MHz. Thus, for a 7 mm line the microwave approximations start to fail as the frequency drops below about 1 MHz. Using  The dotted y curves in Fig. 8 also show how the real and imaginary parts of the propagation constant vary in the limits of high and low frequencies. The curves are linear in these limits, indicating a power law dependence with frequency for both real and imaginary components. In the high-frequency limit the phase angle varies linearly with frequency while the loss component varies as the square root of the frequency. In the low-frequency limit both components vary as the square root of the frequency. These results can be compared with [7, fig. 5.191 and explain how both a and p approach zero in that figure.
The error in the full range approximations of the fields in region 2 is greatest at the highest usable line frequency. Table I1 shows a comparison between the exact and approximate field values for a radius of 1 . 5~ and a frequency of 18 GHz. (The magnitudes have arbitrary units and the phases are in units of degrees, while the magnitude error is a relative error and the phase error has the 1319 units of degrees.) The errors decrease rapidly as the frequency decreases.

C. Conductor Currents
It proves interesting to calculate the total conductor currents since we are now in possession of the exact fields. The first equation in (2)     currents are not in general equal; this is true even when the conductors are made of identical materials since the magnitude of the ratio in (53) is still not unity because of conductor losses. In the case of vanishingly small losses ( p = 0 or (T =CO>, however, both sets of equations, (52) and (531, result in Z3 = -I,. Fig. 9 illustrates how the magnitude of the field ratio in (53) diverges from unity. This variation is due exclusively to conductor loss since the conductors are assumed to be constructed of the same material for the calculation. The magnitude of the ratio in (53) corresponds to the circled points at the right in the figure for the various frequencies indicated. Examination of these points reveals that the ratio is equal to 1 to within 12 parts per million (ppm) at 18 GHz, 5 ppm at 10 GHz, and 0.5 ppm at 2 GHz, showing that the current ratio rapidly approaches unity as the frequency decreases. Table 111 also shows this effect.
where (15) has been used to obtain the second expression. Fig. 10 is a graph of the magnitude and phase of this impedance. It is interesting to note that the magnitude is not equal to the free-space impedance (377 Kl) at all frequences, decreasing from a value of approximately 20000 R at 1 Hz to the free-space value between 1 kHz and 1 MHz. Equally interesting is the phase of Z,. The magnitude continues to increase as f -l / ' as the frequency decreases while the phase levels out at a negative 45". The increase in magnitude is due to the fact that the magnitude of the radial electric field remains constant with frequency while the magnetic field magnitude falls off as f '1,.
The variation with frequency seen in Fig. 10 vanishes as it should when the line loss disappears, the remaining constant value being the free-space impedance with zero phase. Being able to calculate the exact y provides the first real check on the accuracy of the microwave and full range approximations to the propagation constant. The f curves in fig. 8 show that the full range approximation is highly accurate over the entire usable frequency range of the line while the microwave approximation suffers a large relative error at frequencies below about 1 MHz. It should be noted, however, that the y o loss and angle error curves show absolute errors less than 2X1OP6 dB/cm and l o p 6 deg/cm respectively at these lower frequencies, values that may be sufficiently small for most practical purposes even though the relative errors are large.
Approximate expressions for the fields between the conductors were presented that satisfy Maxwell's equations to first order in h2. The errors in these expressions are shown in Table 11.
It is often assymed that the principal mode wave impedance for the fields between the conductors is equal to the free-space value m. Fig. 10 shows that this assumption is significantly in error below approximately 1 MHz when the line is lossy. are sufficiently accurate to be used without concern, and make calculations like (37) easier because the right side of (Al) does not depend on the solution h of the determinantal equation (28). The corresponding error from replacing h, with k , will be discussed first and then the surface impedance.
From (121, and calculations show (see Fig. 12) that the second factor on the right is equal to 1 to better than six parts in 10'. Thus, for practical purposes, where m is the magnitude ad=. The function on the right side can be accurately approximated by polynomial fitting if necessary, making its calculation quick and easy.