Multiconductor System

In a multiconductor system separated by nonconductive mediums, capacitance (C) is the proportionality constant between the charge (q) on each conductor and the voltage (V) between each conductor. The total equilibrium system charge is zero. Capacitance is dependent on conductor geometry, conductor spatial relationships, and the material properties surrounding the conductors. 

To analyze a transient in a distributed-parameter line, a traveling-wave theory is explained for both single- and multiconductor systems. A method to introduce velocity difference and attenuation in the multiconductor system is described together with field test results. Impedance and admittance formulas of unusual conductors, such as finite-length and vertical conductors, are also explained. 

It should be noted that the impedance and admittance in this equation become a matrix when a conductor system is composed of multiconductors. Remember that a single-phase cable is, in general, a multiconductor system because the cable consists of a core and a metallic sheath or a screen. In an overhead conductor, no conductor internal admittance y exists, except a... 

Pollaczek derived a general formula that can deal with earth-return impedances of overhead conductors, underground cables, and multiconductor systems composed of overhead and underground conductors in the following form [7,13] ... 

Equations 1.40 through 1.42 hold true for the multiconductor system shown in Figure 1.17, provided that the coefficients Z, Y, R, L, G, and C are now matrices and the variables V and I are vectors of the order n in an n-conductor system. 

Let us define matrix P as a product of series impedance matrix Z and shunt admittance matrix 7 for a multiconductor system ... 

As discussed in Section 1.4.1, analysis of a multiconductor system requires a number of computations of functions. The application of eigenvalue theory makes it easy to calculate matrix functions. This is a major advantage of eigenvalue theory. 

This equation shows that each mode is independent of the other modes therefore, a multiconductor system can be treated as a single-conductor system in a modal domain. The solutions in a modal domain can be found by n operations, whereas solving Equation 1.138 in an actual domain requires time complexity of o(n ) since the coefficient matrix is an n x n matrix. Matrix A is called the voltage transformation matrix as it transforms the voltage in a modal domain to that in an actual domain. 

The unknown coefficients Vf and in the general solution expressed as Equation 1.110 are determined from boundary conditions. There are many approaches to obtain voltage and current solutions in a multiconductor system. The most well-known method is the four-terminal parameter (F-parameter) method of two-port circuit theory. The impedance parameter (Z-parameter) and the admittance parameter (Y-parameter) methods are also well known. It should be noted that the F-parameter method is not suitable for application in high-frequency regions, while the Z- and F-parameter methods are not suitable in low-frequency regions because of the nature of h5q)erbolic functions. 

Vj and Vj. are the voltage vectors at the sending and receiving ends in a multiconductor system... 

The coefficients F -F in a multiconductor system are obtained in the same manner as in Equation 1.87, considering a matrix form from Equations 1.110 and 1.111 ... 

Note that F and F4 are not the same in a multiconductor system, but they are the same in the case of a single-conductor system. 

In a multiconductor system, the transformation matrix A is also frequency dependent. Frequency dependence is significant in the cases of an untransposed vertical overhead line and an underground cable. In the former, more than 50% difference is observed between Afj (i, /th element of matrix A) at 50 Hz and 1 MHz. In an untransposed horizontal overhead line, the frequency dependence is less noticeable. 

For a multiconductor system, this equation is applied to each modal wave. 

Let us consider the vertical multiconductor system illustrated in Figure 1.61. In the same manner as the finite-length horizontal conductor, the following impedance formula is obtained ... 

Ametani, A. and A. Ishihara. 1993. Investigation of impedance and line parameters of a finite-length multiconductor system. Trans. lEE Jpn. B-113(8) 905-913. 

Ozawa, J. et al. 1985. Lighming surge analysis in a multiconductor system for substation insulation design. IEEE Trans. Power Appl. Syst. 104 2244. 

In a multiconductor system, the transformation matrix A is also frequency dependent. The frequency dependence is significant in the case of an... 

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