Coupled-line analysis using MPCM

The idea presented here is that a coupled-line structure can be formed, assuming two concentric rectangles γ and β, if the inner rectangle (β-circuit) is extruded longitudinally from either side of the outer one (γ-circuit), resulting into two patches namely α, as seen in Fig. 12(b). Therefore, by adjusting port localization, and thereby modifying the PCM, coupled lines can be created and analyzed using desegmentation of two rectangular patches γ and β, as depicted in Fig. 1(a); where P_a represents input and output ports on α. Similarly, Q and R denote common (red) ports on the interface between α and β, and P_b (green) ports in the middle of β, respectively. The resulting coupled line is designed on Rogers RO4003 substrate with ɛ_r = 3.55 and thickness h = 1.6 mm, and analyzed using the MPCM. The results are plotted against those obtained from the commercially available full-wave software, HFSS, as illustrated in Fig. 1(b). As seen, a very good agreement is evident between the results, for different spacing values (S ₁ and S ₂) (see Fig. 1(b)); hence validating the MPCM technique to analyze coupled-line structures. Referring to Fig. 1(b), note that the emphasis here has not been on whether or not a good coupled line has been designed. In other words, a coupled line with not-so-good S-parameters response (particularly S ₁₁) has been designed intentionally, so a meaningful comparison can be made, and that a better contrast (higher resolution) can be made between the two responses.

Fig. 1.

(a) Illustration of coupled lines made by desegmentation of two planar rectangular patches, for input (IP) and output (OP) ports; W (line width of rectangular patch) = 1.5 mm, L (line length of rectangular patch) = 48 mm; and (b) S-parameters plot of the coupled-line structure obtained by MPCM and HFSS for two different spacing values (S ₁ = 0.2 mm and S ₂ = 0.7 mm) between adjacent planar rectangular patches.

Multi-section coupled-line structure can be designed by connecting coupled-line pairs using the segmentation process (see Fig. 2). This requires calculating the impedance matrix for every coupled line separately using desegmentation relation (A6), and then linking up rectangles using segmentation relation (A4). Evidently, design of multi-section coupled-lines in commercially-available software (be it ADS, HFSS, CST, etc.) is an iterative process and involves complex, processor-intensive and time-consuming optimizations. In this paper, a new approach is presented, which is manifested in the form of an open-source tool, that significantly reduces such complexity. As shall be explained in the following section, the proposed tool is based on simultaneously utilizing the MPCM and impedance calculations of single coupled-line section. Relying on simple, fast, and accurate lumped-element equivalent circuit analysis, equivalent circuit lumped element values are calculated and optimum associated coupled-line geometric parameters are generated, producing the final design layout very quickly.

Fig. 2.

(a) Equivalent bi-section planar circuit model (with N ports), and (b) layout of the cascaded multi-conductor coupled-line structure.

Extracting equivalent circuit values using TLM

Figure 3(a) depicts the structure of a simple two-conductor coupled-line with line width, W, spacing between two lines, S, length of coupled lines, l. Equivalent circuit model of this coupled-line for an infinitesimal line section Δx is shown in Fig. 3(b); where L, C, R and G are per-unit-length inductance, capacitance, resistance and conductances, respectively. Also, L_m and C_m represent mutual coupling inductance and capacitance, respectively.

Fig. 3.

(a) Layout and (b) equivalent circuit model of a two-conductor coupled-line.

Referring to Fig. 3(b), the idea is that using the relations obtained from MPCM (i.e. (A5) and (A6)) and those obtained from TLM, a set of simultaneous equations can be obtained based on Z parameters, from which lumped-element parameters L, C, L_m, and C_m can be calculated as shall be seen.

Assuming a lossless line, impedance (Z) and admittance (Y) matrices of the coupled line of Fig. 3(a) can be given as [Reference Schutt-Aine and Mittra19]:

(1)$$Z = \left[{\matrix{ {\,jL\omega } & {\,jL_m\omega } \cr {\,jL_m\omega } & {\,jL\omega } \cr } } \right], \;$$

(2)$$Y = \left[{\matrix{ {\,j( C + C_m) \omega } & {-jC_m\omega } \cr {-jC_m\omega } & {\,j( C + C_m) \omega } \cr } } \right], \;$$

where ω is the angular frequency.

Applying Kirchhoff's voltage and current laws (KVL and KCL) to the equivalent circuit of Fig. 3(b), one hasdV/dx = −ZI anddI/dx = −YV, which (after differentiation of the latter and substitution into the former) gives d ²V/dx ² = ZYV(x denotes the infinitesimal TL section along the direction of propagation). Substituting V in d ²V/dx ² = ZYVby its exponential form V ∝ e ^γx gives γ ²UV = ZYV, where γ is the propagation constant and U is the unit matrix.

Substituting Z and Y from (1) and (2) into γ ²UV = ZYVand solving it gives:

(3)$$\left\vert {\left[{\matrix{ {\gamma^2} & 0 \cr 0 & {\gamma^2} \cr } } \right] + \omega^2\left[{\matrix{ {L( C + C_m) -L_mC_m} & {L_m( C + C_m) -LC{}_m} \cr {L_m( C + C_m) -LC{}_m} & {L( C + C_m) -L_mC_m} \cr } } \right]} \right\vert = 0. $$

Given γ = jβ, phase constant β can be obtained from (3) as:

(4a)$${\beta _1} = \omega \sqrt {C(L + {L_m})} $$

(4b)$$\beta _2 = \omega \sqrt {( C + 2C_m) ( L-L_m) } . $$

For the coupled line, the Telegrapher's Equations can result in input and output voltages and currents (V ₁, I ₁ and V ₂, I ₂) of [Reference Schutt-Aine and Mittra19]:

(5)$$\left[{\matrix{ {V_1} \cr {V_2} \cr } } \right] = a_1^ + ( e^{{-}j\beta _1x} + \Gamma _1e^{\,j\beta _1x}) \left[{\matrix{ 1 \cr 1 \cr } } \right] + a_2^ + ( e^{{-}j\beta _2x} + \Gamma _2e^{\,j\beta _2x}) \left[{\matrix{ 1 \cr {-1} \cr } } \right]. $$

Given dV/dx = −ZI and substituting Z by (1), the current matrix can be obtained as:

(6)$$\left[{\matrix{ {I_1} \cr {I_2} \cr } } \right] = \displaystyle{{a_1^ + \beta _1} \over {( L + L_m) \omega }}( e^{{-}j\beta _1x}-\Gamma _1e^{\,j\beta _1x}) \left[{\matrix{ 1 \cr 1 \cr } } \right] + \displaystyle{{a_2^ + \beta _2} \over {( L-L_m) \omega }}( e^{{-}j\beta _2x}-\Gamma _2e^{\,j\beta _2x}) \left[{\matrix{ 1 \cr {-1} \cr } } \right]. $$

Now that V and I matrices are found (from (5) and (6)), considering the boundary condition and symmetry of impedance matrix by substituting β in voltage and current relations, four main elements of impedance matrices Z of a lossless coupled line can be found. As such one can obtain Z ₁₁ matrix at point x = −l by dividing (5) by (6):

$$\eqalign{& Z_{11} = \displaystyle{{V_1( x = {-}l) } \over {I_1( x = {-}l) }}\vert {_{condition} } =\cr & \displaystyle{\displaystyle{a_1^ + ( e^{\,j\beta _1l} + \Gamma _1e^{{-}j\beta _1l}) + a_2^ + ( e^{\,j\beta _2l} + \Gamma _2e^{{-}j\beta _2l}) } \over \matrix{{a_1^ + \sqrt {( {C/( {L + L_m} ) } ) } ( e^{\,j\beta _1l}-\Gamma _1e^{{-}j\beta _1l}) +}\cr {a_2^ + \sqrt {( {( {C + 2C_m} ) /( {L-L_m} ) } ) } ( e^{\,j\beta _2l}-\Gamma _2e^{{-}j\beta _2l}) }}} \cr & _{condition}\matrix{ {I_1( x = 0) = 0} \cr {I_2( x = 0) = 0} \cr {I_2( x = {-}l) = 0} \cr } \Rightarrow \Gamma _1 = \Gamma _2 = 1\& \displaystyle{{a_2^ + } \over {a_1^ + }} =\cr & \displaystyle{{\sqrt {( {C/( {L + L_m} ) } ) } {^ \ast} \sin \beta _1l} \over {\sqrt {( {( {C + 2C_m} ) /( {L-L_m} ) } ) } {^\ast} \sin \beta _2l}}} $$

which simplifies to:

(7)$$\eqalign{Z_{11} = \displaystyle{\matrix{{\sqrt {( {( {L + L_m} ) /C} ) } {^ \ast} \cos \beta _1l \, {^ \ast} \, \sin \beta _2l +}\cr \displaystyle{{\sqrt {( {( {L-L_m} ) /( {C + 2 {^ \ast} C_m} ) } ) } {^ \ast} \sin \beta _1 l \,{^ \ast} \, \cos \beta _2l}} \over \displaystyle{2j \, {^ \ast} \, \sin \beta _1l \, {^ \ast} \, \sin \beta _2l}}, \;}}$$

where β ₁ and β ₂ have already been defined in terms of L and C parameters in (4).

Similarly, Z ₁₂ is obtained as:

$$\eqalign{& Z_{12} = \displaystyle{{V_1( x = {-}l) } \over {I_1( x = 0) }}\vert {_{conditions} } =\cr & \displaystyle{{a_1^ + ( e^{\,j\beta _1l} + \Gamma _1e^{{-}j\beta _1l}) + a_2^ + ( e^{\,j\beta _2l} + \Gamma _2e^{{-}j\beta _2l}) } \over {a_1^ + \sqrt {( {C/( {L + L_m} ) } ) } ( 1-\Gamma _1) + a_2^ + \sqrt {( {( {C + 2C_m} ) /( {L-L_m} ) } ) } ( 1-\Gamma _2) }} \cr & _{Condition\matrix{ {} & {} \cr } }\matrix{ {I_2( x = 0) = 0} \cr {I_1( x = {-}l) = 0} \cr {I_2( x = {-}l) = 0} \cr } } $$

which simplifies to:

(8)$$Z_{12} = Z_{21} = \displaystyle{\matrix{{\sqrt {( {( {L + L_m} ) /C} ) } {^ \ast} \cos \beta _1l \, {^ \ast} \sin \beta _2l-}\cr {\sqrt {( {( {L-L_m} ) /( {C + 2 \, {^ \ast} C_m} ) } ) } \, {^ \ast} \sin \beta _1l \,{^ \ast} \cos \beta _2l}} \over {2j \, {^ \ast} \sin \beta _1l \, {^ \ast} \sin \beta _2l}}. $$

And Z ₁₃ is obtained as:

$$\eqalign{& Z_{13} = \displaystyle{{V_1( x = {-}l) } \over {I_2( x = {-}l) }}\vert {_{conditions} } =\cr& \displaystyle{{a_1^ + ( e^{\,j\beta _1l} + \Gamma _1e^{{-}j\beta _1l}) + a_2^ + ( e^{\,j\beta _2l} + \Gamma _2e^{{-}j\beta _2l}) } \over \matrix{{a_1^ + \sqrt {( {C/( {L + L_m} ) } ) } ( e^{\,j\beta _1l}-\Gamma _1e^{{-}j\beta _1l}) -}\cr{a_2^ + \sqrt {( {( {C + 2C_m} ) /( {L-L_m} ) } ) } ( e^{\,j\beta _2l}-\Gamma _2e^{{-}j\beta _2l}) }}} \cr & _{Condition\matrix{ {} & {} \cr } }\matrix{ {I_1( x = 0) = 0} \cr {I_2( x = 0) = 0} \cr {I_1( x = {-}l) = 0} \cr } } $$

which simplifies to:

(9)$$ \! \! Z_{13} = Z_{31} \!\! = \displaystyle{{\sqrt {( {( {L + L_m} ) /C} ) } {^ \ast} \! \sin \beta _2l + \sqrt {( {( {L-L_m} ) /( {C + 2\, {^ \ast} C_m} ) } ) } \,{^ \ast} \! \sin \beta _1l} \over {2j \, {^\ast} \sin \beta _1l \, ^ \ast \sin \beta _2l}}. $$

And finally, Z ₁₄ is obtained as:

$$\eqalign{& Z_{14} = \displaystyle{{V_1( x = {-}l) } \over {I_2( x = 0) }}\vert {_{conditions} } =\cr & \displaystyle{{a_1^ + ( e^{\,j\beta _1l} + \Gamma _1e^{{-}j\beta _1l}) + a_2^ + ( e^{\,j\beta _2l} + \Gamma _2e^{{-}j\beta _2l}) } \over {a_1^ + \sqrt {( {C/( {L + L_m} ) } ) } ( 1-\Gamma _1) -a_2^ + \sqrt {( {( {C + 2C_m} ) /( {L-L_m} ) } ) } ( 1-\Gamma _2) }} \cr & _{Condition\matrix{ {} & {} \cr } }\matrix{ {I_1( x = 0) = 0} \cr {I_1( x = {-}l) = 0} \cr {I_2( x = {-}l) = 0} \cr } } $$

which simplifies to:

(10)$$Z_{14} = Z_{41} = \displaystyle{{\sqrt {( {( {L + L_m} ) /C} ) } {^ \ast} \sin \beta _2l-\sqrt {( {( {L-L_m} ) /( {C + 2 {^ \ast} C_m} ) } ) } {^ \ast} \sin \beta _1l} \over {2j \, {^ \ast} \. \sin \beta _1l \, {^ \ast} \sin \beta _2l}}. $$

Therefore, the Z matrix of the coupled line with four ports under the symmetry, reciprocity, and unitary conditions can be written as:

(11)$$Z = \left[{\matrix{ {Z_{11}} & \cdots & {Z_{14}} \cr \vdots & \ddots & \vdots \cr {Z_{41}} & \cdots & {Z_{44}} \cr } } \right]. $$

The impedance matrix of (11) can be determined if the equivalent circuit parameters L, C, L_m, and C_m are known. However, since these parameters are all unknown for start, one has to use four simultaneous equations of (7)–(10). Interestingly, left-hand-side (LHS) of (7)–(10) correspond to the first element of Z_α given in (A6) obtained from the MPCM analysis for four external ports (p_a) associated to ports 1 to 4 of the coupled line of Fig. 3(a); that is:

$$Z_{\,p_a\gamma }-Z_{\,p_ap_b}[ {Z_{\,p_b\gamma }-Z_{\,p_b\beta }} ] ^{{-}1}Z_{\,p_bp_a} = \left[{\matrix{ {Z_{\,p_{a1}p_{a1}}} & \cdots & {Z_{\,p_{a1}p_{a4}}} \cr \vdots & \ddots & \vdots \cr {Z_{\,p_{a4}p_{a1}}} & \cdots & {Z_{\,p_{a4}p_{a4}}} \cr } } \right]$$

These elements are then substituted in LHS of relations (7)–(10) to solve four simultaneous equations (using the Gradient method) and to extract the four unknowns L, C, L_m, and C_m. In other words, Z ₁₁, Z ₂₁, Z ₃₁, and Z ₄₁ which are obtained from MPCM analysis are substituted in (7)–(10) obtained from equivalent circuit analysis; and solved to give values of L, C, L_m, and C_m. These values are then tabulated in a look-up table and plotted in Fig. 5.

Note that, considering that modal analysis has been used in the parallel-coupled lines, effectively frequency dispersion has been considered in calculations. Moreover, referring to (4), the terms $e^{{-}j\beta _1x}$ and $e^{{-}j\beta _2x}$indicate two different propagation constants in the coupled line, which is indeed a manifestation of frequency dispersion. Moreover, since the above relations are deduced from Maxwell's equations, which itself takes frequency dispersion into account, then the proposed approach also takes account of frequency dispersion. A fact that is approved by almost identical results obtained from both full-wave simulator (HFSS) and the proposed technique, as shall be seen.

To verify the validity of the proposed method, the coupled-line designed using MPCM (seen in Fig. 1(a)) is assumed on Rogers RO4003 substrate (h = 0.25 mm, ɛ_r = 3.55, TanD = 0.0027), with line width W = 1.5, spacing S = 0.8, and length l = 48 (all in mm). Utilizing (A6), and (7)–(10), the lumped element values associated with the specified geometries are determined to be, L = 432.55nH, C = 42.86pF, L_m = 125.37nH, and C_m = 12.15pF. As such, relevant S-parameters are produced by simple Z-to-S conversion. The same geometry is also analyzed in the full-wave simulator ANSYS HFSS in order to compare the results with the presented method.

As depicted in Fig. 4, a very good match is evident between results obtained from HFSS and the proposed method; confirming the validity and accuracy of the proposed technique.

Fig. 4.

S-parameters of the analyzed coupled-line obtained from HFSS, MPCM and proposed tool.

Regardless of the technique used, analysis of multi-section coupled-line structures becomes very complicated, be it using PCM, MPCM or well-known methods such as Moments or FEM; due to the large number of analytical ports involved (in MPCM) (see Fig. 2(a)), meshing, and iteration frequency (in HFSS). However, the beauty of the proposed approach is that it is inherently simple, as the MPCM method is used once only at the beginning of the analysis, and that the rest of analysis relies on solving straight-forward simple TL equations; thus, offering the ability to rapidly generate the optimal layout. Therefore, the presented technique is extended to offer a simplified procedure for the design of multi-section coupled-lines; where the analysis of the overall structure would involve analysis of individual coupled-line sections from (11).

In order to provide a comprehensive dataset for the developed tool, calculations have been carried out for a range of different values of S (assuming fixed TL widths), for which relevant L, C, L_m, and C_m have been extracted from the combination of MPCM and equivalent circuit method (i.e. equations (7)–(10)) and recorded in the form of a look-up table. Thus, for a coupled line with fixed line widths, only the spacing S must be specified, according to which L, C, L_m, and C_m are extracted immediately. Figure 5 is a graphical illustration of electrical versus physical parameters of the coupled-line analyzed earlier in this paper, plotted from the look-up table. Note that limited data have been shown in the figure for illustration purposes only.

Fig. 5.

Plot of change of L, L_m, C, and C_m change of S at 0.6–1.6 GHz for W = 1.5 mm, L = 48 mm on Rogers RO4003 with ɛ_r = 3.55, and h = 0.25 mm.

Also, note how the extracted coupling parameters in Fig. 5 are plotted in terms of line spacing, and that loss is independent of the spacing, and so, excluding loss does not reduce the generality of the problem.

Evidently, such information can be very useful in that for any desired spacing, associated lumped-element values (L, C, L_m, and C_m) can be extracted immediately, and vice versa. The data become particularly useful if more than one pair of coupled-line section (a multi-section coupled-line) is present; since the same procedure can be applied to the other pairs of coupled-line structure too (assuming minimum or no mutual coupling between pairs of coupled-line). These extracted lumped-element values can be substituted in (7)–(10), yielding in Z matrix (relation (11)) for each coupled-line section. The Z matrices shall be converted into ABCD matrices, then multiplied, to result in the overall Z matrix (and eventually the S-parameters) of the multi-section coupled-line.