In this paper, based on the mode analysis of four-conductor lines, the extended mixed-mode chain-parameters and S-parameters of fourconductor lines are estimated using current division factors. The extended mixed-mode chain-parameters of cascaded four-conductor lines are then obtained with mode conversion. And, the extended mixed-mode S -parameters of cascaded four-conductor lines can be predicted from the transformation of the extended chain-parameters. Compared to the extended mixed-mode S-parameters of fourconductor lines, the cross-mode S-parameters are induced in the extended mixed-mode S-parameters of cascaded four-conductor lines, due to the imbalanced current division factors of cascaded two sections. The generated cross-mode S-parameters make the equivalent different- and common-mode conductors not independent from each other again. In addition, a new mode conversion, which applies the imbalanced current division factors, between the extended mixed-mode S-parameters and standard S-parameters is also proposed in this paper. Finally, the validity of the proposed extended mixed-mode S-parameters and mode conversion is confirmed by a comparison of the simulated and estimated results of shielded cable.
Four-conductor lines are widely used in power electronics, such as three-phase power supply circuits. The signals on the multi-conductor lines are conventionally divided into differrential-and common-mode signals, which are used to measure the signal transmission performance of the established circuit. Conventional mixed-mode
The four-conductor lines model used in this paper has three modes: differential-mode-1 (DM1), differential-mode-2 (DM2) and common-mode (CM) [2]. In [2], extended mixed-mode
In this paper, extended mixed-mode
II. MODE ANALYSIS OF FOUR-CONDCUTOR LINES
Shown in Fig. 1, a general structure of four-conductor 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 (
For the lossless lines, the current division factors can be decided by the line capacitances [2].
where,
The line voltage and current of the four-conductor lines, as is well known, satisfy the equations from the per-unit-length 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 two-conductor lines with the equivalent mode capacitance, inductance and characteristic impedance. They are obtained as:
where,
With the above mode transmission line parameters, the terminal voltage and current of each mode can be related through the mode chain-parameters shown in the following.
where, the general expression of chain-parameters for two-conductor lines is shown as:
Extended mixed-mode chain-parameters of four-conductor lines can be decided by collecting each mode chain-parameters. The collection is shown as follows.
As shown in the equation, the extended mixed-mode chain-parameters do not contain the cross-mode chain-parameters, due to the separate mode transmission line parameters. Therefore, each mode chain-parameters are independent from the others, and can be converted to the corresponding mode
The extended mixed-mode
where,
[Fig. 3.] The extended mixed-mode S-parameters for the separate mode conductors described in Fig. 2.
Due to the current direction, the definitions of
In the above equations,
and the conversion from the mode chain-parameters to the mode
where,
As shown in Eq. (6), the new extended mixed-mode
III. CASCADED FOUR-CONDCUTOR LINES
1. The Extended Mixed-Mode S-Parameters
General cascaded four-conductor 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 four-conductor 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 mixed-mode chain-parameters 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
where, Δ
Combining the extended mixed-mode chain-parameters in Eq. (10), Eq. (11) and the mode conversion equations in Eq. (12), the extended mixed-mode chain-parameters 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 cross-mode chain-parameters are induced due to the imbalanced current division factors of the cascaded two four-conductor lines. The extended mixed-mode
Similarity,
The extended mixed-mode
2. The New Mode Conversion between the Extended Mixed-Mode S-Parameters and Standard S-Parameters
The six-port standard
In the equation,
In order to obtain the mode conversion between the extended mixed-mode
where,
Finally, given Eq.(14), Eq.(16), and Eq.(18), the conversion between the extended mixed-mode
Moreover, if both of the cascaded two four-conductor lines are symmetrical (i.e.
Then, the mode conversion in Eq.(19) simplifies to:
With the above mode conversion, the extended mixed-mode
IV. VERICATION OF THE EXTENDED MIXED-MODE
A 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:
By using the simulated inductances and capacitances, the extended mixed-mode
Based on the mode analysis of four-conductor lines, the simulation schematics of each mode
Fig. 10 shows a good consistency between the estimated and simulated results, which verified the validity of the proposed extended mixed-mode
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:
This paper proposed the novel extended mixed-mode