Mode Analysis of Cascaded FourConductor Lines Using Extended MixedMode S Parameters
 Author: Zhang Nan, Nah Wansoo
 Publish: Journal of electromagnetic engineering and science Volume 16, Issue1, p57~65, 31 Jan 2016

ABSTRACT
In this paper, based on the mode analysis of fourconductor lines, the extended mixedmode chainparameters and
S parameters of fourconductor lines are estimated using current division factors. The extended mixedmode chainparameters of cascaded fourconductor lines are then obtained with mode conversion. And, the extended mixedmode S parameters of cascaded fourconductor lines can be predicted from the transformation of the extended chainparameters. Compared to the extended mixedmodeS parameters of fourconductor lines, the crossmodeS parameters are induced in the extended mixedmodeS parameters of cascaded fourconductor lines, due to the imbalanced current division factors of cascaded two sections. The generated crossmodeS parameters make the equivalent different and commonmode conductors not independent from each other again. In addition, a new mode conversion, which applies the imbalanced current division factors, between the extended mixedmodeS parameters and standardS parameters is also proposed in this paper. Finally, the validity of the proposed extended mixedmodeS parameters and mode conversion is confirmed by a comparison of the simulated and estimated results of shielded cable.

KEYWORD
Cascaded , Current Division Factors , Extended MixedMode SParameters , FourConductor Lines , Mode Analysis

I. INTRODUCTION
Fourconductor lines are widely used in power electronics, such as threephase power supply circuits. The signals on the multiconductor lines are conventionally divided into differrentialand commonmode signals, which are used to measure the signal transmission performance of the established circuit. Conventional mixedmode
S parameters are commonly used to interpret the characteristics of the existing mode signals [1]. The conventional mixedmodeS parameters include the differentialmodeS parameters, commonmodeS parameters and differrentialcommon crossmodeS parameters. The differentialcommon crossmodeS parameters are induced due to the asymmetry of the signal lines.The fourconductor lines model used in this paper has three modes: differentialmode1 (DM1), differentialmode2 (DM2) and commonmode (CM) [2]. In [2], extended mixedmode
S parameters are proposed to characterize the three mode signals of fourconductor lines. The extended mixedmodeS parameters are estimated from the equivalent independent mode transmission line parameters, which are obtained based on the mode analysis with current division factors [2]. Compared with the conventional mixedmodeS parameters, there are no crossmodeS parameters existing in the proposed extended mixedmodeS parameters, regardless of whether the four conductor lines are symmetrical or asymmetrical.In this paper, extended mixedmode
S parameters are further applied to a cascaded fourconductor lines model. In general, the two parts of the cascaded fourconductor lines have different current division factors. In other words, the cascaded fourconductor lines are imbalanced. However, these different current division factors will induce crossmodeS parameters in the extended mixedmodeS parameters, which are different from the extended mixedmodeS parameters of fourconductor lines. We note that, unlike the conventional mixedmodeS parameters in which the crossmodeS parameters occur due to the asymmetry of the conductor lines, the crossmodeS parameters of the cascaded fourconductor lines occur due to the different current division factors of the cascaded two parts. In the following sections, this paper will introduce extended mixed modeS parameters of the cascaded fourconductor lines based on mode analysis in detail.II. MODE ANALYSIS OF FOURCONDCUTOR LINES
Shown in Fig. 1, a general structure of fourconductor lines is composed of three signal conductors and one ground conductor. The currents on the signal conductors can be decomposed into three mode currents: DM1 (
d 1), DM2 (d 2) and CM (c ) [2]. As shown in Fig. 1, a current division factorh _{1} is used to divide the same direction currents of DM1. And the other two current division factorsh _{2} andh _{3} are used for the currents of CM. With the current division factors, the mode transformations of the voltage and current on the fourconductor lines are given by the following collections:For the lossless lines, the current division factors can be decided by the line capacitances [2].
where,
C_{ii} andC_{ij} are the self and mutualcapacitances of the fourconductor lines. For the symmetrical fourconductor lines, the above current division factors become:h _{1} = 1⁄2, andh _{2} =h _{3} = 1⁄3.The line voltage and current of the fourconductor lines, as is well known, satisfy the equations from the perunitlength line inductances and capacitances [3]. We take the equations of mode transformation in Eq. (1) and the expression of current division factors in Eq. (2) into the line voltage and current equations, each mode can be considered as an independent twoconductor lines with the equivalent mode capacitance, inductance and characteristic impedance. They are obtained as:
where,
μ andε are the effective permeability and permittivity of the fourconductor lines, respectively. They can be estimated through the inductances and capacitances of the fourconductor lines, i.e. =LC με .I With the above mode transmission line parameters, the terminal voltage and current of each mode can be related through the mode chainparameters shown in the following.
where, the general expression of chainparameters for twoconductor lines is shown as:
Extended mixedmode chainparameters of fourconductor lines can be decided by collecting each mode chainparameters. The collection is shown as follows.
As shown in the equation, the extended mixedmode chainparameters do not contain the crossmode chainparameters, due to the separate mode transmission line parameters. Therefore, each mode chainparameters are independent from the others, and can be converted to the corresponding mode
S parameters.The extended mixedmode
S parameters of the fourconductor lines are generated by combining each modeS parameters, are given in Eq. (6) below [2].where,
a_{mi} andb_{mi} , as shown in Fig. 3, are the normalized port waves of them (m =d 1,d 2,c ) mode equivalent circuits. They are normalized by the port terminal voltage, current, and port reference impedance. For port1 of them mode, the normalizations are:[Fig. 3.] The extended mixedmode Sparameters for the separate mode conductors described in Fig. 2.
Due to the current direction, the definitions of
m mode at port2 are:In the above equations,
Z _{0}_{m} is the port reference impedance. Based on the mode analysis in Fig. 1, the port reference impedances are equal toand the conversion from the mode chainparameters to the mode
S parameters can be derived by plugging Eq. (7) into Eq. (5):where,
Z _{0} is the port reference impedance, commonlyZ _{0} =50 Ω.As shown in Eq. (6), the new extended mixedmode
S parameters do not contain the crossmodeS parameters, the same with the extended mixedmode chainparameters. The new extended mixedmodeS parameters are completely different from the conventional mixedmodeS parameters in [1]. In the conventional mixedmodeS parameters, the crossmode parameters will be induced when the conductors are asymmetrical. In the following section, the extended mixedmodeS parameters for the cascaded fourconductor lines are further described in detail.III. CASCADED FOURCONDCUTOR LINES
1. The Extended MixedMode SParameters
General cascaded fourconductor lines, in which the cascaded two parts have different current division factors, are shown in Fig. 4. The line voltage and current are continuous on the cascaded fourconductor lines. The mode voltage and current, however, are discontinuous due to the different current division factors. As shown in Fig. 5, the equivalent mode conductors of the cascaded two parts can be obtained by processes similar to those described in Section II. Then, the extended mixedmode chainparameters of the connected equivalent mode conductors are given as:
In addition, note that the mode conversion occurs at the connection of the two equivalent mode circuits. The voltage or current sources from the mode conversion are induced at the connection points [4]. The mode conversions at
z_{a} =l_{a} (z_{b} = 0) are expressed as:where, Δ
h is the difference of the current division factors, i.e., Δh_{i} =h_{bi} h_{ai} , (i = 1,2,3). For the balanced cascaded fourconductor lines, (i.e., Δh_{i} = 0), the above voltage and current conversion matrices become the unit matrix.Combining the extended mixedmode chainparameters in Eq. (10), Eq. (11) and the mode conversion equations in Eq. (12), the extended mixedmode chainparameters of the equivalentmode circuits in Fig. 5, which are the multiplication of each connected part, are denoted as in Eq. (13). It is worth noting that the crossmode chainparameters are induced due to the imbalanced current division factors of the cascaded two fourconductor lines. The extended mixedmode
S parametersof the cascaded fourconductor lines can also be transformed from the extended mixedmode chainparameters. The transformation between the extended mixedmode chainparameters andS parameters, however, cannot be done as in Eq. (9), where that the extended chainparameters were separated into three independent twoport mode chainparameters. That is because each mode parameters in Eq. (13) are associated with each other by the crossmode parameters. Because of the space limitation, the transformation between them is not shown in here. It can be performed by emulating the method of twoport transformation in Eq. (9). The ports of each mode are denoted in Fig. 6, and the definition of the extended mixedmodeS parameters is shown as follows.Similarity,
a_{mi} andb_{mi} are the mode normalized port waves. Their normalizations are:The extended mixedmode
S parameters shown in Eq. (14) are composed of the selfmodeS parameters and the crossmodeS parameters. The selfmode S parameters describe the mode current flowing on the cascaded fourconductor lines, while the crossmodeS parameters are used to describe the mode conversions occurring at the connection of the cascaded structures. Besides, the cascaded fourconductor lines shown in Fig. 4 can also be labelled with the normalized standard port waves, shown in Fig. 7. Then, the sixport standardS parameters are induced to present the signal transmission on the cascaded fourconductor lines. The mode conversion between the standardS parameters and the extended mixedmodeS parameters is derived in the following section.2. The New Mode Conversion between the Extended MixedMode SParameters and Standard SParameters
The sixport standard
S parameters of the cascaded fourconductor lines are defined as:In the equation,
a_{i} andb_{i} are the normalized standard waves at porti (i = 1, 2, …, 6). Their definitions are:In order to obtain the mode conversion between the extended mixedmode
S parameters and the standardS parameters, we substitute the voltages and current mode transformation in Eq. (1) into the definition equations of port waves in Eq. (15) and Eq. (17). Then, a conversion between the normalized standard port waves (a_{i} ,b_{i} ) and the mode port waves (a_{mi} ,b_{mi} ) is obtained as:where,
_{em,1} andM _{em,2} are the new mode conversion matrices shown below. Compared to the mode conversion matrix in [5], the new conversion matrices proposed in Eq.(18) contain the different current division factors, which consider the divided mode current on the asymmetrical signal lines.M Finally, given Eq.(14), Eq.(16), and Eq.(18), the conversion between the extended mixedmode
S parameters and the standardS parameters is:Moreover, if both of the cascaded two fourconductor lines are symmetrical (i.e.
h _{a1} =h _{b1} = 1/2 andh _{a2} =h _{a3} =h _{b2} =h _{b3} = 1/3), the mode conversion matrices listed above become:Then, the mode conversion in Eq.(19) simplifies to:
With the above mode conversion, the extended mixedmode
S parameters can also be directly converted from standardS parametersusing the current division factors. To confirm the validity of the new proposal in this paper, the estimated standardS parameters and extended mixedmodeS parameters of the shielded cable, used as an example of cascaded fourconductor lines, were compared with the simulated results.IV. VERICATION OF THE EXTENDED MIXEDMODE
S PARAMETERS FOR THE CASCADED FOURCONDCUTOR LINESA shielded cable is shown in Fig. 8, in which three cascaded inner conductors are used as the signal lines and the outside shield is the ground. In this model, the inner conductors and the outside shield are perfect conductors. A dielectric material with a permittivity of 2 is fills in the shield. The cascaded two parts are simulated by Ansoft Q3D to separately estimate the inductances and capacitances. As shown in Fig. 8, the radius of the cascaded signal lines, though of different sizes, are all symmetrical with the outside shield. The current division factors of the two cascade two parts are balanced. They are:
h _{a1} =h _{b1} = 1/2 andh _{a2} =h _{a3} =h _{b2} =h _{b3} = 1/3. Therefore, there are no crossmodeS parameters in the extended mixedmodeS parameters of the shielded cable.By using the simulated inductances and capacitances, the extended mixedmode
S parameters of the shielded cable were first estimated by following the above processes. The extended mixedmodeS parameters can also be simulated by 3D EM software (CST Microwave Studio) in the frequency range of 1 GHz to 10 GHz.Based on the mode analysis of fourconductor lines, the simulation schematics of each mode
S parameters, composing the extended mixedmodeS parameters, are shown in Fig. 9. Note that because there are no crossmodeS parameters, each modeS parameters are independent and can be simulated separately. The simulated and estimated extended mixedmodeS parameters are compared in Fig. 10.Fig. 10 shows a good consistency between the estimated and simulated results, which verified the validity of the proposed extended mixedmode
S parameters. Moreover, as shown in Fig. 10, theS parameters of DM1 are the same as theS parameters of DM2. That is due to the symmetrical signal lines of the shielded cable. For the symmetrical fourconductor lines, the characteristic impedances of DM1 and DM2 in Eq. (3) have a relationship that:Z _{Cd2} = (4/3)Z _{Cd1}. And the mode port reference impedances in Eq.(8) have the same relationship, i.e.,Z _{0d2} = (4/3)Z _{0d1}. Besides, the standardS parameters of the shielded cable were also simulated by the Microwave Studio and were also estimated by the simplified conversion equation in Eq. (21). The good match shown in Fig. 11 confirmed the validity of the mode conversions between the extended mixedmodeS parameters and the standardS parameters.Furthermore, a shielded cable with imbalanced current division factors was also analyzed. As shown in Fig. 12, the signal lines in the second part of the shielded cable are still symmetrical, but the signal lines in the first part change to asymmetrical. With the simulated inductances and capacitances, the current division factors of the first part become:
h _{a1} = 0.2838,h _{a2} = 0.182 andh _{a3} = 0.3156. Therefore, the crossmodeS parameters were induced due to the imbalanced current division factors. It is worth noting that the selfmodeS parameters of the imbalanced shielded cable cannot be simulated by the schematics in Fig. 9. That is because the selfmodeS parameters are not independent of each other, due to the existence of the crossmodeS parameters. Therefore, only the estimated extended mixedmodeS parameters of the shielded cable are just shown in Fig. 13. Unlike the selfmodeS parameters of the balanced shielded cable in Fig. 10, the selfmodeS parametersshown in Fig. 12(a) are completely different due to the asymmetrical parts in the shielded cable. And, as shown in Fig. 12(b), the mode conversion occurring between DM1 and CM is the lowest. Finally, the standardS parameters of the imbalanced shielded cable were estimated and compared with the simulated results in Fig. 14. The completely identical results, once again, confirmed the effectiveness of the proposed extended mixedmodeS parameters and the mode conversion for the cascaded fourconductor lines.V. CONCLUSION
This paper proposed the novel extended mixedmode
S parameters of cascaded fourconductor lines, based on the independent mode analysis of the general fourconductor lines with current division factors. Compared to the extended mixedmodeS parameters of fourconductor lines, the crossmodeS parameters are generated by the different current division factors of the cascaded two fourconductor lines. In addition, two new conversion matrices, containing the different current division factors, were also proposed in this paper. The validity of the extended mixedmodeS parameters and the conversion matriceswas verified by a comparison of the estimated and simulated results of a balanced shielded cable and an imbalanced shielded cable.

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[Fig. 1.] Three modes definition of the fourconductor lines: M1 (d1), DM2 (d2), and CM (c).

[Fig. 2.] Equivalent circuits of the three separate modes: (a) DM1, (b) DM2, and CM (c).

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[Fig. 3.] The extended mixedmode Sparameters for the separate mode conductors described in Fig. 2.

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[Fig. 4.] The structure of the cascaded fourconductor lines.

[Fig. 5.] The equivalent mode circuits of the cascaded fourconductor lines in Fig. 4.

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[Fig. 6.] Definition of the port variables for the extended mixedmode Sparameters of equivalent mode circuits in Fig. 5.

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[Fig. 7.] Definition of sixport standard Sparameters for Fig. 3.

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[Fig. 8.] Shielded cable with balanced current division factors.

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[Fig. 9.] Simulation schematics of selfmode Sparameters. (a) DM1, (b) DM2, and (c) CM.

[Fig. 10.] The comparison of simulated and estimated extended mixedmode Sparameters for the shielded cable in Fig. 8.

[Fig. 11.] The comparison of standard Sparameters for the shielded cable in Fig. 8.

[Fig. 12.] Shielded cable with imbalanced current division factors.

[Fig. 13.] The estimated extended mixedmode Sparameters for the imbalanced shielded cable in Fig. 12. (a) The selfmode Sparameters and (b) the crossmode Sparameters.

[Fig. 14.] The estimated and simulated standard Sparameters of shielded cables in Fig. 12.