Rigorous Analytical Expressions for Electromagnetic Parameters of Transmission Lines: Coupled Sliced Coaxial Cable

Technical Feature

Rigorous Analytical Expressions for Electromagnetic Parameters of Transmission Lines: Coupled Sliced Coaxial Cable

This article is a continuation of a previous article that appeared in Microwave Journal and is the first part in the development of accurate closed-form formulas for the primary parameters (inductance [L] and capacitance [C] matrices) and impedances (Z_oe , Z_oo ) of the even- and odd-modes for several configurations of transmission lines (sliced coaxial cables, symmetrical band lines and split ring lines). The analytical expressions, deduced from rigorous analysis by the finite element method (FEM)^1,2 , method of moment (MoM)³ and curve-fitting techniques, can be easily implemented in CAD simulation tools to design components for wireless communication. This study presents accurate and suitable general expressions for all coupled sliced coaxial cables with a wide range of cut depths and an outer to inner conductor radius ratio between 1.4 and 15. The results of the design of an RF coupler using coupled sliced coaxial cables are presented.

N. Benahmed and M. Feham
University of Tlemcen
Tlemcen, Algeria

A directional coupler, using a coupled line composed of two shaped cut coaxial cables, has been analyzed and tested by Djordjevic, et al.³ This type of coupler shows excellent performance in terms of high directivity, very low SWR, good isolation, excellent electromagnetic interference (EMI) shielding, high power handling capability, extremely low cost due to the use of commercial semi-rigid coaxial cables and elimination of a mechanical housing.^4,5

The electrical properties of a lossless coupler using a TEM-mode coupled line can be described in terms of even- (Z_oe ) and odd- (Z_oo ) mode impedances, and its primary parameter matrices [L] and [C],²

where

L_o = proper inductance of the isolated line
C_o = capacitance of the isolated line
M = mutual inductance of the coupled line
g = coupling capacitance of the coupled line

The cross-section of a coupled line with sliced coaxial cables is shown in Figure 1 .

The cable is assumed to be lossless with an inner conductor of radius R_c and an outer conductor of radius R_b .

A material with dielectric constant e_r fills the inside of the cable. A portion of each cable is cut out and two of these cut cables are used to form the coupled line. The cut depth is represented by h on the cross-section and

H. An, et al. presented formulas only for even- and odd-mode characteristic impedances of the coupled line with a sliced coaxial cable.⁶ Their expressions, deduced from a FEM analysis, are valid in the ranges

0 ≤ u ≤ 0.99

and

In this article, formulas are offered for the even- and odd-mode characteristic impedances and also the [L] and [C] matrices of the coupled line with sliced coaxial cables having an outer to inner conductor radius ratio

and a cut depth

0 ≤ u ≤ 0.99

These formulas are deduced from an analysis using two coherent numerical methods: FEM and MoM.

Numerical Results

In order to validate the numerical results, a structure with an outer conductor radius R_b = 1.49 mm, an inner conductor radius R_c = 0.255 mm and a permittivity e_r = 1 is studied. Figures 2 and 3 show a comparison between the FEM and MoM results. Through this comparison, it appears that a good agreement is obtained between the two numerical methods.

The FEM results of the even- and odd-mode characteristic impedances and elements of the [L] and [C] matrices, for different values of conductor radius, are shown in Figures 4 to 7 . Odd-mode characteristic impedance and mutual inductance as a function of cut depth is shown in Figures 8 and 9 .

Derivation of Analytical Expressions

Characteristic Impedances
By curve-fitting to the FEM results, it is found that the even-mode characteristic impedance Z_oe of the coupled line can be expressed by

Z_oe = Z_o + b₁ u + b₂ u² + b₃ u³ (1)

where

Z_o = -31.626 + 45.864r - 5.623r² + 0.354r³ - 0.0085 r⁴
b₁ = 0.862 - 2.982r + 1.841r² - 0.148r³ + 0.0039r⁴
b₂ = -6.862 + 24.858r - 8.728r² + 0.690r³ - 0.018r⁴
b₃ = -20.311 + 4.254r + 4.289r² - 0.404r³ + 0.011r⁴

and the odd-mode characteristic impedance (Z_oo ) is given by

where

a₁ = -37.78 + 50.089r - 6.398r² + 0.417r³ - 0.01r⁴
a₂ = 310.831 - 366.622r + 47.908r² - 3.263r³ + 0.083r⁴
u_o = 1.407 - 0.017r + 0.0024r² - 1.507 10^-4 r³ + 3.554 10^-6 r⁴
du = 0.153 + 0.0045r - 11 10^-4 r² + 8.483 10^-5 r³ -2.21 10^-6 r⁴

Appendix A shows a comparison between the analytical results and those of the literature⁶ for different values of the cut depth.

Through this comparison, it appears that a good correlation is obtained between these analytical results and those already published.

Inductance Matrix Per Unit Length

The proper and mutual inductances of the coupled line are given by Equations 3 and 4, respectively.

for 1.4 ≤ r ≤ 6
L_o1 = 436.432 - 366.538e^{-(r-1.4)/3.1212}
A = 27.802 - 23.536e^{-(r-1.4)/2.5891}
v = 0.8115 - 0.12441e^{-(r-2)/2.4837}
u_o = 0.7396 - 0.140e^{-(r-1.4)/1.9622}
for 6 < r ≤ 15
L_o1 = 585.577 - 437.491e^{-(r-1.4)/6.3446}
A = 31.106 - 21.724e^{-(r-2)/3.5100}
v = 0.84299 - 0.153e^{-(r-2)/2.7368}
u^o = 0.75584 - 0.11663e^{-(r-2)/2.7368}

M_o1 = 6.41196 - 9.790r + 2.229r² - 0.2574r³ + 0.01429r⁴ - 3.04337 10^-4 r⁵
A = 41.40657 + 28.596r - 1.88142r² + 0.05831r³
v = 0.35977 + 0.12777e^{-(r-1.4)/2.0479} + 0.12901e^{-(r-1.4)/13.71907}
for 1.4 ≤ r ≤ 9
u_o = 1.05019 + 0.32048e^{-(r-1.4)/0.748}
for 9 < r ≤ 15
u_o = 0.52158 + 0.12912r - 0.01035r² + 2.81944 10^-4 r³

Capacitance Matrix Per Unit Length
Equations 5 and 6 give the proper and coupling capacitance of the coupled line with sliced coaxial cables, respectively.

A = 20.49514 + 97.89073e^{-(r-1.4)/0.4082} + 44.34461e^{-(r-1.4)/3.25406}
B₁ = -14.38252 + 8.21613r - 2.11056r² + 0.26037r³ - 0.01521r⁴ + 3.38153 10^-4 r⁵
B₂ = 8.41414 + 3.64185r - 1.00611r² + 0.08854r³ - 0.00249r⁴ - 9.01856 10^-6 r⁵
B₃ = -84.99654 + 11.33169r - 0.24693r² - 0.06647r³ + 0.00411r⁴ - 3.29262 10^-5 r⁵
B₄ = -14.88927 + 61.27891e^{-(r-0.99433)/19.259} + 67.37928e^{-(r-0.99443)/1.32918}

for 1.4 ≤ r ≤ 7
g₁ = -18.10709 + 18.82952r - 8.41344r² + 1.85398r³ - 0.19828r⁴ + 0.0082r⁵
u_o = 1.40387 - 0.36507r + 0.17369r² - 0.04274r³ + 0.00525r⁴ - 2.54706 10^-4 r⁵
v = 1.01161 - 0.75568r + 0.37576r² - 0.09433r³ + 0.01168r⁴ - 5.67111 10^-4 r⁵
A = 386.55698 - 445.76406r + 223.82846r² - 56.34847r³ + 6.99263r⁴ - 0.34033r⁵
for 7 < r ≤ 15
g₁ = -0.47174 - 0.47528e^{-(r-7)/4.4014}
u_o = 4.00282 - 1.24415r + 0.21668r² - 0.01879r³ + 8.08443 10^-4 r⁴ - 1.37917 10^-5 r⁵
A = 7.12991 + 16.23608e^{-(r-7)/9.6403} + 6.11658e^{-(r-7)/1.78277}

Appendix B shows a comparison between the analytical and numerical results for R_c = 0.255 mm, R_b = 0.51 mm, u = 0.7 and e_r = 1.

The above expressions are valid for airlines. For dielectric filled coaxial lines with dielectric constant of e_r , the expressions are

C'_o = C_o e_r

and

L = L_o

The relative errors between numerical and analytical results are less than 2 peR_c ent over a wide range, indicating good accuracy of the closed-form expressions for the coupled line with sliced coaxial cables.

Directional Coupler Design

Figure 10 shows the structure of a directional coupler using the coupled line with cut coaxial cables. The parameters of this coupler, operating in the 300 MHz to 3 GHz frequency range, using the coupled line with sliced coaxial cable, are

Z_oe = 48.35 W
Z_oo = 36.03 W

The features of the coupled line obtained from the proposed formulas are

• Outer conductor radius R_b = 0.51 mm
• Inner conductor radius R_c = 0.255 mm
• Dielectric constant e_r = 1
• Conductivity s = 5.65 10⁷ (Wm)^-1
• Coupler length 1 = 40 mm

The simulated response of the designed coupler is shown in Figure 11 .

In the frequency range 0.6 to 3 GHz, the coupling is less than 22 dB and the minimum directivity is 17 dB, which permit the designed coupler to have good coupling and isolation over a large frequency range.

Conclusion

This article presents a set of accurate closed-form formulas for the primary parameter ([L], [C]) matrices and the impedances (Z_oe , Z_oo ) of the even- and odd-modes for the coupled sliced coaxial cables. These expressions deduced from the FEM and MoM are valid in a wide range of outer to inner conductor radius ratios.

As an application of this study, a directional coupler, operating over a broad frequency range using the coaxial coupled line, has been successfully designed.

References
1. N. Benahmed, M. Feham and M. Kameche, "Finite Element Analysis of Planar Couplers," Applied Microwave & Wireless , Vol. 12, No. 10, October 2000.
2. N. Benahmed and M. Feham, "Finite Element Analysis of RF Couplers with Sliced Coaxial Cable," Microwave Journal , Vol. 2, No. 2, November 2000, pp. 106-120.
3. A.R. Djordjevic, D. Darc, M.C. Goran and T.K. Sarkan, Circuit Analysis Models for Multiconductors Transmission Lines , Artech House Inc., Norwood, MA, 1997.
4. H. An, O. Monti, R.G. Bossio and K. Wu, "A Novel Type of Low Cost High Performance Coaxial Cables Coupler," 25th European Microwave Conference , 1995.
5. H. An, R.G. Bossio and K. Wu, "Ultra-wide Band Directional Couplers with Coaxial Cable," Canadian Conf. Electr. and Comp. Eng ., 1995.
6. H. An, T. Wang, R.G. Bossio and K. Wu, "Accurate Closed-form Expression for Characteristic Impedance of Coupled-line with Sliced Coaxial Cable," IEE Proceedings , 1995.