Radio Science

Higher-order modes for broadside-coupled striplines in a shield

Authors


Abstract

[1] A mode-matching method is applied to broadside-coupled striplines in a shield to obtain a dispersion relation. Higher-order mode analyses are performed, and their cutoff frequencies are evaluated. Good agreement with existing solutions is demonstrated. Higher-order field characteristics are illustrated. The mode-matching model provides an efficient and stable means of analyzing broadside-coupled striplines in a shield.

1. Introduction

[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 broadside-coupled striplines. We note that an understanding of higher-order 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 higher-order mode scattering from rectangular septa. The motivation of the present work is to develop such analytic expressions for higher-order 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 mode-matching method using the image method and residue calculus was developed to analyze various rectangular-shaped transmission lines [Cho and Eom, 2001]. Although the mode-matching 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 mode-matching formulation for the higher-order TE and TM modes for broadside-coupled striplines in a shield without recourse to such a complicated analysis. Therefore, the mode-matching 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 (Ez = 0) propagation in a CTL cell composed of two identical broadside-coupled striplines (PEC septa) in a shield, as shown in Figure 1. Assume that a CTL cell has a uniform cross section on the x-y plane. In regions I, II, III, IV, and V, the electric vector potentials with an exp (−iωt) time convention are:

equation image
equation image
equation image
equation image
equation image

where am = equation image, ζm = equation image, bp = equation image, ηp = equation image, hs = equation image, ψs = equation image, and m = h = s = 0, 1, 2,…. The boundary conditions at x = −L are given by

equation image
equation image
equation image
equation image

The boundary conditions at x = L are given by

equation image
equation image
equation image
equation image

We simplify the above boundary conditions using the mode-matching method. For example, applying equation image (·) cos [an(y + b)]dy to (6), we obtain

equation image

where δmn is the Kronecker delta, α0 = 2, and α1 = α2equation image = 1. Expressions I1np, I2ns, I3np are summarized in Appendix A. Similarly from (7)(13), we obtain simultaneous equations for the unknown modal coefficients Am, Bp,equation image. The final expressions are in the form of

equation image
equation image

where their elements are summarized in Appendix A. It should be noted that the diagonal elements (Ψ11mn, Ψ22pq,⋯, Ψ45sr, Ψ67pq) are all square matrices. The symbols (equation image, equation image, ⋯) denote column vectors of (Am, Bp,⋯), respectively. Equations (15) and (16) amount to odd and even mode excitations in the x direction, respectively. By setting the determinants of Φ1 and Φ2 to zero, we obtain the dispersion relations. The dispersion relation turns out to be simple for numerical evaluation. A root-searching algorithm must be used to solve the dispersion relation for β. In this paper a bisection method [Press et al., 1992] is used.

Figure 1.

Cross section of broadside-coupled striplines in a shield: region I, −a < x < −L, −b < y < b; region II, −L < x < L, h + d < y < b; region III, −L < x < L, −h < y < h; region IV, −L < x < L, −b < y < −hd; region V, L < x < a, −b < y < b.

3. TM Mode Analysis

[4] A TM mode (Hz = 0) analysis is similar to the previous TE mode analysis. The magnetic vector potentials are

equation image
equation image
equation image
equation image
equation image

where am, ζm, bp, ηp, hs, ψs 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 (EyEz and HzHy). 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 TM11 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 ∣Am∣ 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 ∣Hz∣ distributions of the TE01, TE10, TE02 modes in Figure 2, and normalized ∣Ez∣ distributions of the TM11, TM21 in Figure 3, respectively. It is seen that the first and second indices represent the number of half-wavelength variations in the x and y directions, respectively. Figure 4 illustrates the cross-sectional views of electric fields on the x-y plane for the TE modes. The cross-sectional views of the magnetic fields on the x-y 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 higher-order 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.

Figure 2.

Normalized ∣Hz∣ distributions versus x (m) and y (m) at the respective cutoff frequencies. (a) TE01, (b) TE10, and (c) TE02. The parameters used are the same as those in Table 1.

Figure 3.

Normalized ∣Ez∣ distributions versus x (m) and y (m) at the respective cutoff frequencies. (a) TM11 and (b) TM21. The parameters used are the same as those in Table 1.

Figure 4.

Electric fields on the x-y plane for TE modes at the respective cutoff frequencies. (a) TE01, (b) TE10, and (c) TE02. The parameters used are the same as those in Table 1.

Figure 5.

Magnetic fields on the x-y plane for TM modes at the respective cutoff frequencies. (a) TM11 and (b) TM21. The parameters used are the same as those in Table 1.

Table 1. Cutoff Frequencies for TE and TM Modes and Number of Modes Used in Computationa
ModesCutoff Frequencies (MHz)
Our Theory 
Mode Number aMode Number bYun et al. [2003]
  • 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   
   TE01799.8788.6779.3
   TE101000.01000.11001.3
   TE021188.41159.4-
   TE111386.31384.11383.1
   TE211887.11870.3-
   TE202000.12000.1-
TM mode   
   TM111927.71928.71980.2
   TM212561.72563.7-
   TM313371.13372.9-
   TM123428.53431.8-
Table 2. Convergence Behavior of Modal Coefficient Am for TE01 Modea
mAm
  • a

    Here a = 7.5 cm, b = 7.5 cm, L = 4.5 cm, d = 0.3 cm, and h = 4.5 cm.

10.47668
30.029151
56.9127 × 10−4
75.9672 × 10 −4
91.1068 × 10−4
118.2871 × 10−6
136.7749 × 10−6
156.839 × 10−7
Table 3. Cutoff Frequencies Versus Parameter d for Five Modesa
d (cm)Cutoff Frequencies (MHz)
TE01TE10TE02TM11TM21
  • a

    Other parameters used in computation are the same as those for Table 1.

0.2788.8991.81139.31929.62566.3
0.4790.7983.11123.91930.12567.8
0.6794.7973.91113.31930.42568.5
0.8803.2964.11110.61930.52568.8
1.0814.3953.61112.21930.52570.0

5. Conclusion

[6] A mode-matching model for the higher-order TE and TM modes for broadside-coupled 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 mode-matching approach can be further extended to the analysis of three-dimensional CTL cells with tapering structures. While this paper dealt with broadside-coupled transmission lines, a similar mode-matching method can be used to analyze more complicated structures including various edge-coupled striplines.

Appendix A: Integrals I1np, 2ns, 3np, and Matrix Elements of TE Mode

[7] 

equation image

where un = sin [an(b + h)] and vn = sin [an(bh)].

equation image

Appendix B: Integrals I1np, I2ns, 3np, and Matrix Elements of TM Mode

[8] 

equation image
equation image

Acknowledgments

[9] This work was supported by the IT Research and Development program of MKE/IITA (2008-F-014-01, Study on Electromagnetic Compatibility for Protecting Electromagnetic Environment in Ubiquitous Society).

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