SIMILARITY CRITERIA OF EHF PHYSICAL MODEL OF LONG DISTANCE TRANSMISSION LINE WITH PLATE PHASES

Carson’s equations for computation of frequency dependent parameters of overhead transmission line are modified. The earth is considered to be homogenous. The optimal required number of terms of Carson’s series for applications on computer is determined. Accurate mathematical expressions for resistance and inductance of overhead transmission lines are obtained. Different arrangements of conductors for double circuit transmission lines are studied. The mutual inductance and potential coefficient (mutual capacitance) of such arrangements are computed. Double voltage class type double circuit transmission lines are suggested. The plate type double circuit lines are proposed for power transmission over long distances. This paper presents a mathematical analysis in order to determine the coefficients of similarity criteria of EHF physical model for the given long distance transmission line. The integral analogue method and PI-theorem are used. The independent parameters of EHF physical model are formulated. The scales of four independent parameters of the designed EHF physical model are obtained.


INTRODUCTION
Recently, large blocks of powers are needed to be transmitted over long distances. Hence, the long distance transmission line should be applied. The power transmission may be realised using either controlled or coaxial lines 1'2 as well as the problem of power transmissions over long distances as discussed previously3. It is known that many difficulties will occur if long UHV transmission lines are introduced. The study of electrical performance of such type of lines becomes an interesting subject which may be acheived by physical The earth return effect on long distance lines must be included in the mathematical analysis of transmission line parameters. The effect of a homogenous earth on overhead wires has been studied before5, when the line parameters such as resistance R and inductance L become frequency dependent. The resistance and inductance of a conductor ( Fig. 1)  Results prove that the maximum permissible number of series terms is five. The optimal number of terms must be determined. For this purpose, two, three, four and five terms are considered. The calculations, repeated for each case, show that only two terms of the series is an optimal number for computation of line parameters in order to reduce the computational time. Therefore, the optimal expressions for resistance and inductance may be simplified in the final form:  The results are compared with that of equations (1) and prove the accuracy of the deduced expressions. The aim of the present research is the utilization of this mutual effect so that different arrangements of conductors must be studied (see Fig. 2). The conductor heights and spacing between phases may be varied. The results of calculations are drawn in Fig. 3. From this figure it is shown that, mutual inductance between conductors 1 and 3 ( Fig. 2a) does not depend upon conductor spacing. This value is also weakly depends upon conductor height; for example, at H 8 m, the mutual inductance is 9 mH/km while at H 11 m it becomes 28 mH/km. The mutual inductance between conductors 1 and 6 weakly changes with conductor spacinge as the distance between the two circuits is varied from 1 rn to 4 m. The mutual inductance between conductors 1 and 3 at 8 m height increases from 13 to 18 mH/km and from 36 to 44 mH/km at 11 rn height. Calculations prove that the mutual inductance between conductors 1 and 2 is equal to that between 1 and 5.
The effect of distance between edge conductors for each circuit shows that the rate of rise of the mutual inductance increases as the distance between the two circuits is increased. This will be more efficient if the distance between edge conductors of the upper circuit is decreased as shown in Fig. 3. Therefore, the distance between edge conductors of the upper circuit must be decreased relative to lower one so that the study of potential coefficients, i.e. capacitances of such transmission lines becomes necessary. The results of the computations of mutual potential coefficients for double circuit transmission lines (Fig. 2a) are shown in Fig. 4.
Calculations prove that the mutual capacitance between the two circuits will be increased as the distance between them is decreased. The mutual capacitance between conductors 1 and 6 is reduced due to electric field between conductors 2 and 5. Also, the distance between edge conductors of the upper circuit must be less than that of the lower circuit. This will increase the mutual effect between the two circuits. It can be deduced from Fig. 3 and Fig. 4 that the use of a new earth return path for increasing the mutual effect is more efficient. This means that such a path increases the electric field and thus the mutual capacitance between phases. Therefore, the mutual effect may be utilized for power transmission from one circuit to anothera. The conductor arrangement of Fig. 2 a should be changed to that of Fig. 2b and the optimal distance between the two circuits must be realized. This may be done by decreasing the electric field between conductors 2 and 5. Conductor 2 must be placed above the level of conductors i and 3 (Fig. 2c). This also decreases the distance between edges of the upper circuit. From this analysis, a new arrangement of conductors will be suggested (see Fig. 2d). This increases the mutual capacitance between the two circuits and may be defined as platetype. This type of transmission lines is selected for the next investigations. For this purpose the physical model of such a line must be erected.
The similarity criteria of a physical model for long transmission lines must be deduced. The analysis should be based on the mathematical explaination of actual physical processes or on dimensions of the fundamental parameters. There are two methods for the determination of similarity criteria: the first method is defined as the integral analog while the second is the PI-theorem4. For the integral analog method, all equations of phenomena should be considered, although for the PI-theorem only the fundamental parameters of the system and simulated process must be obtained. Using the integral analog method, steady state characteristic of a 500/750 kV, 1000 km transmission line with plate phases as shown in Fig. 2d should be investigated (see Fig. 5). Voltage V and current I along a section of length dx may be formulated as -dVl/dX (R1 + tOLl)11 + toMI2 -dI1/dx (G1 + joC1) V1 + (g + joK)(V1 V2) -dV2/dx (R2 + 09L2) 12 + joM Ia -dI2/dx (G2 + joC2) V2 + (g + joK)(V2-V1) (7) where 1 and 2 denote the first and second circuits, respectively. The self and mutual capacitances C and K as well as self and mutual inductances L and M and self and mutual conductances G and g can be calculated from the geometrical arrangements of phases. The current can be assumed as a constant along the length dx. The corona power loss should be neglected. The parameters of both circuits will be assumed the same as R1 R2 R, L1 L2 L, C1 C2 C and G1 G2 G. The Using the PI-theorem, these parameters of similarity criteria should be checked. In this case the physical process of steady-state operation may be characterized through all parameters of transmission line. This can be expressed mathematically as follows: f(R, L, C, G, g, K, M, w, x, 11, 12, V1, V2, t) 0 (11) The general number of fundamental parameters is m 14. The matrix of dimensions [D] can be formulated ash: [D] One of the fourth order determinants of matrix (12) will not disappear and contains the dimensions of fundamental parameters. This proves that of the number of independent parameters as well as the number of main units of measurements is equal to 4 4 The parameters of similarity criteria may be calculated by the difference between the quantities of the general number of fundamental parameters rn and the numbers of main units q. It will be equal to 10 as it was derived also by the integral analog method. Therefore, the main parameters are voltage, resistance, capacitance and angular frequency. The parameters of similarity criteria may be formulated as'.
Pi o)M/R, P V2/Vl, P-} K/C, Pio ot P coL/R, P G/coC, P Izx/R/gl/6oC, P x x/coRC, P5 g/coC, The parameters P4 and P7 are dimensionless quantities. Hence, the scales of voltage of first and second circuits mvl and mv2 will be related to scales of self and mutual capacitances mc and mk as follows; mv2/mvl mk/mc 1 Equations (13) of the PI-theorem must be compared with equations (10) for the integral analogue method. The relation between both results can be simplified in the form: P1 1/PP, P2 P, P3 PP/P, P4 P/P, P5 P, P6 P, P7 P, P8 P P, P9 P'4 P, Plo Po 17 (14) This check means that both methods give the same parameters of similarity criteria. Therefore, the integral analogue method is recommeiaded for such simulation.

THE INDEPENDENT PARAMETERS
The coefficients of similarity are equal to 4, and all parameters of the physical model can be expressed through these four coefficients. They should be defined as independent parameters. The scales of these four independent parameters will be selected for the simulation of the proposed long distance transmission with plate phases. The selection should be based on the practicability of the measured values in laboratory. Firstly, the scale of frequency me will be assumed.
The operating frequency is suggested as 5 MHz so that the scale of frequency becomes 10 -5 and the length of EHF physical model may be deduced. The length scale ml should be related to scale of angular frequency mo. In this case, the frequency criteria (ot idem) will be achieved. The scale of length may be formulated as a function of scales of resistance and inductance or capacitance (mr and m or m) as ml 1//mr mc mo 1/m,o /mL mc The EHF model will have a length of 10 m while the scale of wave velocity m" can be expressed by m" 1//mL mc It is known that the propagation velocity on overhead type of lines is very closed to light velocity7. The scale of length may be related to the scale of time mt as ma mt. The other three scales must be selected according to the practicability application of EHF physical model.

CONCLUSIONS
The deduced expressions for frequency dependent parameters of transmission lines reduce the computational time with the same accuracy. The mutual effect between conductors and between circuits of double circuit lines can be utilized. The proposed new earth path increases the electric field between phases. The distance between edge conductors of upper circuit of transmission line should be shorter than that of lower one.
The obtained arrangement of conductors (plate phases) gives the maximum transmitted power for long transmissions. The determined parameters of similarity criteria for plate lines can be used successfully. This line can be simulated by EHF physical models.
For modeling of long distance lines, the integral analog method is recommended. The suggested EHF model is practicable for steady state investigation.