Voltage transformation matrix

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. 

However, the current transformation matrix B is not equal to the voltage transformation matrix A Transposing the first equation of Equation 1.144 gives... 

This shows that the current transformation matrix can be found from the voltage transformation matrix. In general, D is assumed to be an identity matrix. Under this assumption ... 

In the case of a completely transposed three-phase line, any of the transformation matrices explained in Section 1.4.4.1 can be used. The current transformation matrix is the same as the voltage transformation matrix. Let us apply the traveling-wave transformation ... 

Figure 1.31 shows the frequency and time dependence of the voltage transformation matrix of an untransposed vertical single-circuit line. It is clear from the figure that the frequency dependence is greater than the time dependence of the transformation matrix. The maximum deviation from the average value is about 10% for the time dependence and about 30% for the frequency dependence. Also, the frequency and time dependencies are much greater in the vertical line case than in the horizontal line case. 

The voltage transformation matrix A for a symmetrical two-conductor system is given by... 

Assume that the sample cable in Section 3.2.4 is buried as a single-phase cable. Find the impedance and admittance matrices for the single-phase example cable using EMTP. Use the Bergeron model and calculate the impedance and admittance matrices at 1 kHz. From the impedance and admittance matrices found in (1), find the phase constants for the earth-return mode and the coaxial mode using the voltage transformation matrix... 

This same explanation can be given by applying the transformation matrices in Equations 1.176, Equations 1.177 and 1.178 in the case of a completely transposed line. When a line is untransposed, the transformation matrices are no longer useful — except Equation 1.179, which can be used as an approximation of a transformation matrix of an untransposed horizontal line. In the case of an untransposed line, the transformation matrix is frequency-dependent as explained in Section 1.5.1 thus, the modal voltage and current distributions vary as the frequency changes. Also, the current distribution differs from the voltage distribution. 

Transform these modal voltages into actual phasor voltages by using the transformation matrix [A] ... 

Modal Distribution of Multiphase Voltages and Currents 1.4.4.1 Transformation Matrix... 

Matrix Particle growth Stability and crack formation Dissolution of y-LiA102 in the electrolyte Phase transformation from y to a variety Changes in the microstructure Increase in the ionic resistivity Decrease in the cell voltage Decrease in the cell life... 

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