First-Order Symmetric Modes for a Slightly Lossy Coaxial Transmission Line

A complete set of solutions for Maxwell’s equations to first order in the normalized surface impedance z , of the coaxial conductors is found. The resulting characteristic admittance and distributed line parameters are calculated; the distributed line resistance is significantly different from other calculations found in the literature. I . INTRODUCTION E s r l T E the extensive use of coaxial transmission D lines, a complete set of first-order field equations between the conductors for a slightly lossy line does not exist. This is especially curious since there has been considerable effort expended in creating standard lines for research and metrology alike. None of the field expressions found in the literature from Russell in 1909 to Stratton in 1941 to Gardiol in 1987 [1]-[3] displays a set of fields that will satisfy Maxwell’s equations to first order in the skin depth or surface impedance. The purpose of this article is to fill this void. The need for a first-order set of fields with which to calculate the complex admittance of a lossy coaxial open circuit [4] prompted the present study. The calculation required both the principal mode (which is TM in a lossy line and TEM in a lossless line) and the axially symmetric TM waveguide mode fields, and while one might not expect to find the lossy first-order waveguide fields, it came as a surprise that a complete set of the principal mode fields was also not available. The derivation of the fields to be outlined here assumes a vacuum dielectric and an infinitely thick outer conductor. It starts in Section I1 from Stratton’s work [2] with a derivation of the determinantal equation for finding the eigenvalues for both the principal and waveguide modes in a lossy line. The first-order determinantal equation (24) is found at the end of that section, preceded by an equation, (23), for calculating the proportionality constant for the fields intermediate between the center and outer conductor. Section 111 for the infinite conductivity case presents nothing new but is included for completeness and to set the stage for the finite conductivity case in Section IV. The equations for the lossy waveguide modes in this latter section are new, and the principal mode fields include a term missing from the expressions that are found elsewhere in the literature. Equations (45) and (47) are especially noteworthy since they include a new re, term that perturbs the transverse fields. Both the principal and waveguide mode fields in that section satisfy Maxwell’s equations to first order upon substitution. The coaxial line impedances and admittances are calculated from the principal transverse fields, and will differ slightly from those found elsewhere in the literature because of the perturbed principal mode fields. These calculations are carried out in Section V, where the distributed line resistance is significantly different from previous estimates. Results and conclusions are found in the last section. 11. THE DETERMINANTAL EQUATION Fig. 1 shows a cross section of a concentric, infinitely long coaxial transmission line whose outer conductor extends to infinity in the plane of the figure. The inner radius of the outer conductor is b and the radius of the inner conductor is a , and both conductors are constructed from the same material. The direction of propagation is the + z direction, perpendicular to the plane of the cross section. The fields in the three regions have wavenumbers k , , k , , and k , , respectively, and, following Stratton [2], can be written in the form (using the convention while omitting e’”‘) U l ( h , r ) , r < a BHi2’( h , r ) , b < r A Z , ( h , r ) , a < r < b (1)


I. INTRODUCTION
E s r l T E the extensive use of coaxial transmission D lines, a complete set of first-order field equations between the conductors for a slightly lossy line does not exist.This is especially curious since there has been considerable effort expended in creating standard lines for research and metrology alike.None of the field expressions found in the literature from Russell in 1909 to Stratton in 1941 to Gardiol in 1987 [1]- [3] displays a set of fields that will satisfy Maxwell's equations to first order in the skin depth or surface impedance.The purpose of this article is to fill this void.
The need for a first-order set of fields with which to calculate the complex admittance of a lossy coaxial open circuit [4] prompted the present study.The calculation required both the principal mode (which is TM in a lossy line and TEM in a lossless line) and the axially symmetric TM waveguide mode fields, and while one might not expect to find the lossy first-order waveguide fields, it came as a surprise that a complete set of the principal mode fields was also not available.
The derivation of the fields to be outlined here assumes a vacuum dielectric and an infinitely thick outer conductor.It starts in Section I1 from Stratton's work [2] with a derivation of the determinantal equation for finding the eigenvalues for both the principal and waveguide modes in a lossy line.The first-order determinantal equation ( 24) is found at the end of that section, preceded by an equation, (23), for calculating the proportionality constant for the fields intermediate between the center and outer conductor.
Section 111 for the infinite conductivity case presents nothing new but is included for completeness and to set the stage for the finite conductivity case in Section IV.The equations for the lossy waveguide modes in this latter section are new, and the principal mode fields include a term missing from the expressions that are found elsewhere in the literature.Equations ( 45) and (47) are especially noteworthy since they include a new re, term that perturbs the transverse fields.Both the principal and waveguide mode fields in that section satisfy Maxwell's equations to first order upon substitution.
The coaxial line impedances and admittances are calculated from the principal transverse fields, and will differ slightly from those found elsewhere in the literature because of the perturbed principal mode fields.These calculations are carried out in Section V, where the distributed line resistance is significantly different from previous estimates.
Results and conclusions are found in the last section.

THE DETERMINANTAL EQUATION
Fig. 1 shows a cross section of a concentric, infinitely long coaxial transmission line whose outer conductor extends to infinity in the plane of the figure.The inner radius of the outer conductor is b and the radius of the inner conductor is a , and both conductors are constructed from the same material.The direction of propagation is the + z direction, perpendicular to the plane of the cross section.The fields in the three regions have wavenumbers k , , k , , and k,, respectively, and, following Stratton [2], can be written in the form (using the convention while omitting e'"')  in (1).The < signs in ( 1 ) are a manifestation of the fact that the radial electric fields begin and end on the surface charge densities generated at the conductor-dielectric interfaces, while the G signs in (2) and (3) express the need for continuity in the tangential electric and magnetic fields at these same interfaces.J, and H,@' ( i = 0,l) are Bessel functions of the first and third kinds (Hankel functions), and where N, is a Bessel function of the second kind, or Neumann function, G is a constant, and x represents any of the arguments in ( 1 ) or (2).For each of the three regions ( i = 1,2,3), where p and E , are the magnetic permeability and electric permittivities.The permeability p is assumed to be identical for the three nonmagnetic regions.The conductivity a, is the same in regions 1 and 3 and vanishes for region 2. Equation (7) with u2 = 0 gives the wavenumber in the vacuum region 2: and where A is the free-space wavelength.
approximated by (IzJ -=z 1 ) The wavenumbers in regions 1 and 3 ( i = 1,3) can be and Here qn is the free-space impedance m, and is the skin depth of the conductors.The + sign in (12) corresponds to the + sign in e + J w t .
Using the fact that y = jk in ( 5 ) and the first term of (10) leads to the approximation for h , and h , ( i = !, 3 ) : Then, by using z , = 0 .0 0 0 0 2 3 6 ~e J " / 4 ( 1 4 ) it is straightforward to show that where p is the resistivity in pfl.cm, and f is the frequency in GHz.Equation (15) shows that the terms within the parentheses in (10) and (13) can be dropped without significant error, leading to the useful series of approximations which are accurate well into the visible part of the frequency spectrum.

Equations for Determining G and h( = h , )
Boundary conditions on the E and H fields require that their tangential components be continuous at both r = a and r = b.Equating the E components at r = a and the H components at r = a, dividing the two results, and using (61, (8), and (16) leads to (with h = h , ) Performing a similar operation on the components at and where respectively.The second terms in both ( 19) and (20) were discarded in arriving at (21) and ( 22) since they will lead rrh 2G

F , ( r , h ) = --Z , ( h r )
Equations ( 5) and ( 8) have been used to arrive at the last two expressions in (29).It is straightforward to show from (30) and (31) that F,(r,O)= l / r and that F,,(r,O)= 0, yielding the usual TEM solution for the principal mode.

-J , ( W N " ( h b ) -Jl(hb)N"(ha)]
(24 where only first-order terms in z , were retained.Solving the determinantal equation (24) for the eigenvalues h will lead to the principal TM (TEM for infinite conductivity) mode and the axially symmetric TM waveguide modes.
The principal wave will be TM for the lossy line since the second expression in (2) is nonzero in this case.Solutions of the determinantal equation will be considered next, the infinite conductivity case being presented first. (34)

B. Wmeguide Modes
To get the first-order waveguide modes, h,, from (38) and (39) is inserted into (23) for h to get G; , (see (32) for G,), which, in turn, is inserted into (1143).The result is reduced to first oider in z , (second order in h,,) and leads to E,,, = A , , [ Z l ( k , , r ) + A Z l ( r n , r ) ] e -r ~~~z (50) H,, = A m y , Z , ( k , , r ) 1 + -+ A Z , ( rn, r  Comparison with Stratton's results shows that he obtains only the last term of (571, indicating that his first-order equations are slightly in error.This discrepancy will cause errors in the distributed line parameters [4], to be discussed next.

V. CHARACTERISTIC ADMITTANCE AND DISTRIBUTED LINE PARAMETERS
The characteristic admittance and, hence, the distributed line parameters depend upon the transverse fields E, and H6, and since the first-order fields differ from those usually found in the literature [l], [2], the resulting admittance and line parameters will also differ.As the fields found in the previous section are complete firstorder solutions to Maxwell's equations, it is assumed that the expressions derived in this section are the more correct.
The series impedance per unit length 9 and the shunt admittance per unit length 9 of a transmission line are related to the characteristic admittance Yo and the propagation constant r (48) by the equations r = ro9 9 = ror (58) 9 = 9 + j w J 9 = 9 + j w P . ( where 9, 2, 9, and G are the distributed resistance, inductance, conductance, and capacitance of the line respectively. The characteristic admittance is calculated from by using ( 45) and (47) and by reducing the result to first order: for 50 R and 75 R lines respectively.Note that Fo in (63) is not the same as the Fo(r, h ) in ( 31) and (44).
The distributed inductance and capacitance for the lossless line are given by the usual equations: To get the first-order parameters, the first-order propagation constant (48) and characteristic admittance (61) are inserted into (58); the results are reduced to first order and then identified with 9, 2, S, and via (59).The  The first-order line parameters, (65)-(68), differ from those used in practice by the corrections associated with FO.That is, setting F0 = O in these equations reduces them to the usual expressions.Of particular note is the nonvanishing conductance 9 even for the lossless dielectric assumed in the derivation.This is due to a quadrature contribution in the field perturbation term occurring both in the line voltage and in the charge density on the surface of the conductors and is associated with the re, terms in (45) and (47).
The corrections to the usual line parameters are, for the most part, small, the one exception being the distributed resistance.All of the corrections depend upon the relative magnitude of k2a2Fo, which for the 14 mm, 7 mm, and 3.5 mm coaxial lines is plotted in Fig. 2 from 1 to 100 GHz, showing that the quantity remains below approximately 0.1 for all three lines.The quantity dok2aZFo is plotted in Fig. 3 per the square root of the resistivity (reciprocal of the conductivity) in pR.cm.The plots shown give the correction for the real part and the negative imaginary part of the characteristic impedance Yo.For a resistivity of 9 pus1 .cm, the magnitude of the corrections is below 0.003% to 0.006% for all of the lines.The correction for the resistance is shown in Fig. 4. The corrections can get as high as 6% at the upper end of the respective frequency bands.Fig. 5 shows that the relative inductance correction is not significant.It is below 0.003% to 0.006% for all of the lines shown.
The shunt conductance for a line with lossless dielectric vanishes, so the correction in Fig. 6 is plotted in absolute siemens instead of percent.The lossy line is seen to produce a conductance of about 0.0001 S/cm for a resistivity of 1 p a -c r n .
The final plot, Fig. 7, shows the relative correction for the capacitance in percent.All the lines sustain a correction of about 0.01% or less for a 1 p a s c m line resistivity.

VI. SUMMARY AND CONCLUSIONS
Maxwell's equations for a slightly lossy coaxial transmission line have been solved to first order in the normalized surface impedance z , for the principal and symmetric waveguide modes.The correctness of the principal mode equations, (434471, or the waveguide mode equations, (50)-( 52), can be proved by inserting them into  Plot of J in pS/cm as a function of frequency from 1 to 100 GHz.The three curves shown correspond to the dimensions for 14 mm, 7 mm, and 3.5 mm transmission lines.The curves are discontinued at the highest usable frequencies in the various lines.to rederive the characteristic impedance and the dism I

ACKNOWLEDGMENT
The author is grateful to S. Perera for checking the final equations.

( 3 )Conductor k 3 h 3 whereFig. 1 .
Fig. 1.Cross sectional view of a coaxial line with an infinite outer condi:,:ior Th(.adii of the inner and outer conductors are a and b.The regions shown are: 1) the inner conductor; 2) the airdielectric space be..deen the conductors; and 3) the outer conductor.The quantities h, and k , (i= 1,2,3) for the three regions are defined in the text.
r = b leads to It now remains to solve (17) and (18) simultaneously for the parameter h after they have been somewhat simplified.The arguments k a / z , and k b / z , are large enough to make use of the asymptotic expansions of the right sides 1F'T.T. TRANSACTIONS O N MIC'ROWAVF.TIIEORY ANI) TEC'HNIQUFS, VOL.. 38, NO. 11, NOVEMBER 1990 of (17) and (18).Thus, A .Principal Mode Only the fields between the conductors ( a < r < b ) are of interest, so only the second equations in (1)-(3) need be evaluated.It will prove convenient (but not essential) to replace the constant A in (1) and (2) with -A a h /2G because of a logarithmic divergence in G as h approaches 0. The resulting equations take the form E, = AFl( r , O ) e -y z (27) , , waveguide modes where the eigenvalues k,,, are solutions to Jo(k,,a)No(k,,b) -Jo(k,,b)No(k,,,a) = 0. (26) The value of G, one for each mode, is then obtained from (23) with z , = 0. where IV.FINITE CONDUCTIVITY The determinantal equation (24) can be solved for the case of finite conductivity (z, f 0) by expanding the Bessel functions in a Taylor series about the infinite conductivity solutions.Expanding about the principal mode solution h = 0 and retaining only the first-order zs terms gives (37) jkz,( 1 + b / a ) b In ( b / a ) h2 = The waveguide mode solution h,, corresponding to k , is h, = k , + 6 k , The principal mode fields are obtained from (27) and (28) with F,(r,O) and FO(r,O) replaced by F,(r,h) and Fo(r, h).Evaluation of the constant -.srh /2G in (30) and (31) is accomplished by expanding (23) in powers of h and keeping only the first order terms: (41) where c is the Napierian base (2.718) raised to the power of Euler's constant and is generated in the expansion.The next term to appear on the right side of (41) is third order in h and has been discarded.Using the last expression for -.rrh/2G in (30) and (4) leads to1 + re, Fl( r , h ) = -The constant c dropped out of (30) when the expression was reduced to first order in z , (second order in h).Using (41) in (31) leads to Using (42) and (44) in the field equations leads to A ( 1 + re,) e-rr E, satisfy Maxwell's equations to first order in z , and differ from the usual expressions found in the literature [2] for the transverse fields by the term re,.
( rn, r ) = [ r Z ; ( k , r ) + G:,N,( k , r ) ] 6 k , (53) rZ(,( k , r ) + G:,NO( k , r)+ --7 Z , ( k , r ) 6 k , (54) L: 3 ' 1The functions Z , ( k , r ) and Z / ( k , r ) ( i = 0,l) are calculated from the formula (4) with the first-order G, in place of G, and Z / ( x ) = d Z , ( x ) / d u .The quantity 6k,, is defined in (391, and SY, / Y, is obtained from (36).The correctness of these equations can also be verified by insertion into Maxwell's equations and reducing the results to first order in z,.The only other treatment of the slightly lossy coaxial line similar to that presented here of which the author is aware is that byStratton  in 1941 [2, pp.545-5541, where he treats the E, and H, principal mode fields.It is possible to rewrite (42) and (43) in the form h 2 r 2 h2r2 r jkr'z, r F , ( r , h ) -l = ---In- a ) l n ( b / a ) -' [ b ] 21n ( b / a ) b / a + l 2 a Y is the wave admittance (29) and Po is the characteristic admittance for the lossless line.F,, equals 0.35 and 1.26 of k2a2F,, as a function of frequency f r o m 1 t o 1 0 0 GHz.The t h r e e curves shown c o r r e s p o n d t o the d i m e n s i o n s f o r 14 mm, 7 mm.and 3.5 mm transmission lines.

Fig. 3 .
Fig. 3. Plot of dl,k2a2Fl, per 6 in percent as a function of frequency from 1 to 100 GHz.The thee curves shown correspond to the dimensions for 14 mm, 7 mm, and 3.5 mm transmission lines.

Frequency (GHz)Fig. 5 .
Fig.5.Plot of -A l/l' per 6 in percent as a function of frequency from 1 to 100 GHz.The three curves shown correspond to the dimensions for 14 mm, 7 mm, and 3.5 mm transmission lines.The curves are discontinued at the highest usable frequencies in the various lines.

Fig. 7 .
Fig. 7. Plot of A 6 / 8 per 6 in percent as a function of frequency from 1 to 100 GHz.The three curves shown correspond to the dimensions for 14 mm, 7 mm, and 3.5 mm transmission lines.

r ) , b < r A Z , ( h , r ) , a < r < b
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