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[1] A modematching method is applied to broadsidecoupled striplines in a shield to obtain a dispersion relation. Higherorder mode analyses are performed, and their cutoff frequencies are evaluated. Good agreement with existing solutions is demonstrated. Higherorder field characteristics are illustrated. The modematching model provides an efficient and stable means of analyzing broadsidecoupled striplines in a shield.
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[2] Rectangular coaxial lines and their derivatives are basic transmission line structures used in transverse electromagnetic (TEM) cells, gigahertz transverse electromagnetic (GTEM) cells, and coupled transmission line (CTL) cells. Various numerical and analytical studies of rectangular coaxial lines have been performed and their propagation characteristics are relatively well known. In particular, CTL cells, first introduced by Yun et al. [1998, 2002, 2003] using two tapered septa, provide relatively uniform field distributions, which are desirable characteristics for EMC testing and probe calibration. Field distributions within CTL cells were numerically investigated due to the structural complexity associated with two septa, which are broadsidecoupled striplines. We note that an understanding of higherorder TE (transverse electric to wave propagation) and TM (transverse magnetic to wave propagation) mode behavior within CTL cells is essential for assessing CTL cell performance in terms of bandwidth and field uniformity. In order to accurately analyze CTL cells, we need a robust analytic scheme, which allows us to efficiently calculate higherorder mode scattering from rectangular septa. The motivation of the present work is to develop such analytic expressions for higherorder modes. We note that various analytic expressions were developed and successfully applied to study wave propagation characteristics of other waveguides and transmission lines [Hyde et al., 2009; Li et al., 2004; Tornero and Melcón, 2004]. A modematching method using the image method and residue calculus was developed to analyze various rectangularshaped transmission lines [Cho and Eom, 2001]. Although the modematching method provides an efficient solution, the use of the image method and residue calculus inevitably entails a lengthy analytical evaluation for the dispersion relation. The purpose of this paper is to develop a modematching formulation for the higherorder TE and TM modes for broadsidecoupled striplines in a shield without recourse to such a complicated analysis. Therefore, the modematching model in this paper should be much easier to apply than the model of Cho and Eom [2001]. In what follows, TE and TM mode analyses are presented and some representative computation results are illustrated.
2. TE Mode Analysis
[3] Consider TE mode (E_{z} = 0) propagation in a CTL cell composed of two identical broadsidecoupled striplines (PEC septa) in a shield, as shown in Figure 1. Assume that a CTL cell has a uniform cross section on the xy plane. In regions I, II, III, IV, and V, the electric vector potentials with an exp (−iωt) time convention are:
where a_{m} = , ζ_{m} = , b_{p} = , η_{p} = , h_{s} = , ψ_{s} = , and m = h = s = 0, 1, 2,…. The boundary conditions at x = −L are given by
The boundary conditions at x = L are given by
We simplify the above boundary conditions using the modematching method. For example, applying (·) cos [a_{n}(y + b)]dy to (6), we obtain
where δ_{mn} is the Kronecker delta, α_{0} = 2, and α_{1} = α_{2} = 1. Expressions I_{1}^{np}, I_{2}^{ns}, I_{3}^{np} are summarized in Appendix A. Similarly from (7)–(13), we obtain simultaneous equations for the unknown modal coefficients A_{m}, B_{p},. The final expressions are in the form of
where their elements are summarized in Appendix A. It should be noted that the diagonal elements (Ψ_{11}^{mn}, Ψ_{22}^{pq},⋯, Ψ_{45}^{sr}, Ψ_{67}^{pq}) are all square matrices. The symbols (, , ⋯) denote column vectors of (A_{m}, B_{p},⋯), respectively. Equations (15) and (16) amount to odd and even mode excitations in the x direction, respectively. By setting the determinants of Φ_{1} and Φ_{2} to zero, we obtain the dispersion relations. The dispersion relation turns out to be simple for numerical evaluation. A rootsearching algorithm must be used to solve the dispersion relation for β. In this paper a bisection method [Press et al., 1992] is used.
3. TM Mode Analysis
[4] A TM mode (H_{z} = 0) analysis is similar to the previous TE mode analysis. The magnetic vector potentials are
where a_{m}, ζ_{m}, b_{p}, η_{p}, h_{s}, ψ_{s} are the same as those in the TE case (m = h = s = 1, 2,⋯). The boundary conditions at x = −L and x = L are given by (6)–(13) with the replacements (E_{y} → E_{z} and H_{z} → H_{y}). The final equations are given by (15) and (16), where their elements are summarized in Appendix B.
4. Numerical Results
[5] Computations are performed to check the rate of convergence of our series solution. We consider a CTL cell discussed by Yun et al. [2003]. Table 1 illustrates the behavior of cutoff frequencies for the TE and TM modes, using two different mode numbers, a and b. Our computational results of mode numbers a and b both agree reasonably with those of Yun et al. [2003]. The difference in the TM_{11} mode between ours and Yun et al. [2003] is about 3%, which may not be critical due to its high cutoff frequency (1.9 GHz). Table 2 shows the convergence rate of ∣A_{m}∣ as m increases. It is seen that the series converges very fast to zero and the model is computationally robust. Next we present explicit field characteristics in Figures 2–5. We show normalized ∣H_{z}∣ distributions of the TE_{01}, TE_{10}, TE_{02} modes in Figure 2, and normalized ∣E_{z}∣ distributions of the TM_{11}, TM_{21} in Figure 3, respectively. It is seen that the first and second indices represent the number of halfwavelength variations in the x and y directions, respectively. Figure 4 illustrates the crosssectional views of electric fields on the xy plane for the TE modes. The crosssectional views of the magnetic fields on the xy plane for the TM modes are plotted in Figure 5. The arrows designate the sizes and directions of the fields. Since the septa have a thickness d, it is also of interest to see the effects of d on cutoff frequencies. Table 3 shows cutoff frequencies versus the different parameters d for five higherorder modes. The cutoff frequencies for the TE modes change by 3% as d varies from 0.2 cm to 1.0 cm, whereas the cutoff frequencies for the TM modes remain almost unchanged.
Table 1. Cutoff Frequencies for TE and TM Modes and Number of Modes Used in Computation^{a}
Here a = 7.5 cm, b = 7.5 cm, L = 4.5 cm, d = 0.0 cm, and h = 4.5 cm. Mode number a: m = 6, p = 3, and s = 6 for TE mode and m = 12, p = 3, and s = 6 for TM mode. Mode number b: m = 16, p = 6, and s = 10 for TE mode and m = 22, p = 5, and s = 7 for TM mode.
TE mode
TE_{01}
799.8
788.6
779.3
TE_{10}
1000.0
1000.1
1001.3
TE_{02}
1188.4
1159.4

TE_{11}
1386.3
1384.1
1383.1
TE_{21}
1887.1
1870.3

TE_{20}
2000.1
2000.1

TM mode
TM_{11}
1927.7
1928.7
1980.2
TM_{21}
2561.7
2563.7

TM_{31}
3371.1
3372.9

TM_{12}
3428.5
3431.8

Table 2. Convergence Behavior of Modal Coefficient A_{m} for TE_{01} Mode^{a}
m
∣A_{m}∣
a
Here a = 7.5 cm, b = 7.5 cm, L = 4.5 cm, d = 0.3 cm, and h = 4.5 cm.
1
0.47668
3
0.029151
5
6.9127 × 10^{−4}
7
5.9672 × 10 ^{−4}
9
1.1068 × 10^{−4}
11
8.2871 × 10^{−6}
13
6.7749 × 10^{−6}
15
6.839 × 10^{−7}
Table 3. Cutoff Frequencies Versus Parameter d for Five Modes^{a}
d (cm)
Cutoff Frequencies (MHz)
TE_{01}
TE_{10}
TE_{02}
TM_{11}
TM_{21}
a
Other parameters used in computation are the same as those for Table 1.
0.2
788.8
991.8
1139.3
1929.6
2566.3
0.4
790.7
983.1
1123.9
1930.1
2567.8
0.6
794.7
973.9
1113.3
1930.4
2568.5
0.8
803.2
964.1
1110.6
1930.5
2568.8
1.0
814.3
953.6
1112.2
1930.5
2570.0
5. Conclusion
[6] A modematching model for the higherorder TE and TM modes for broadsidecoupled striplines in a shield was developed. The model yields fast convergent series, and is computationally efficient and robust. The analytic formulation in this paper provides a viable tool for designing optimum CTL cells, which can be used for various EMC testing purposes. A modematching approach can be further extended to the analysis of threedimensional CTL cells with tapering structures. While this paper dealt with broadsidecoupled transmission lines, a similar modematching method can be used to analyze more complicated structures including various edgecoupled striplines.
Appendix A: Integrals I1np, 2ns, 3np, and Matrix Elements of TE Mode
[7]
where u_{n} = sin [a_{n}(b + h)] and v_{n} = sin [a_{n}(b − h)].
Appendix B: Integrals I1np, I2ns, 3np, and Matrix Elements of TM Mode
[8]
Acknowledgments
[9] This work was supported by the IT Research and Development program of MKE/IITA (2008F01401, Study on Electromagnetic Compatibility for Protecting Electromagnetic Environment in Ubiquitous Society).