# Point mirroring method (Electrical Machine) Part 2

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### Magnetic field

The magnetic field problem is considered to be linear. Hence, the superposition of partial fields, calculated with the Biot-Savart law, result in the overall three-dimensional field distribution below the line.

In this case each segment of the infinitesimally-thin filament (Fig. 5.14) carries a current The generated flux density of this part of the conductor is The point where the flux density has to be calculated has to be transformed into the co-ordinate system After integrating eq. (5.19) the flux density is calculated with If n is the number of current carrying conductors, superposition of the individual flux densities results in the overall flux density: ### Finite element model

The second numerical field computation method able to compute solutions in this problem class is the finite element method. Special boundary conditions applied to the field problem result in an effective use of this method. The application of open boundary conditions gives the opportunity to discretise the field problem in regions of interest only. This results in lower computational costs. With respect to the computational efforts, only two-dimensional computations are performed with this numerical method. Fig, 5,16. Open boundary model to compute the electric field of a 150 kV AC single system transmission line, (the triangulation of the domain is invisible)

A cross-section of the transmission line is made at the place where the wires are nearest to ground level (Fig. 5.16). Here, the highest field values are expected. A two-dimensional finite element model perpendicular to the line is built. The region of the cross-section is subdivided in triangular finite elements (Fig. 5.17). The potential distribution over each element is approximated by a polynomial. Instead of solving the field equations directly, the principle of minimum potential energy is used to obtain the potential distribution over the whole model. The ratio of the largest size of a finite element to the smallest size in the model of a transmission line is about 10,000. The circular boundary of the model has a radius of about 100m, while the radius of the conductors is a few centimetres. Therefore, special attention must be paid to obtain a regular mesh with well-shaped elements, ensuring an accurate solution of the field problem. Thus, a high degree of discretisation resulting in a large system of equations must be applied. Fig. 5.17 shows a part of the finite element model around one of the phase conductors. The change in the size of the elements in the direction away from a conductor can be noticed (Fig. 5.17).

When computing the electric field strength, it is assumed that the ground plane below the transmission line is an equipotential surface.

Therefore it is not necessary to discretise the ground as indicated in Fig. 5.16. In contrast, in the magnetic field model the ground has the same magnetic properties as the surrounding air and has to be discretised as well. This results in an increased number of finite elements and thus in higher computational costs.

In contrast to the semi-numerical method, where the phase conductors are modelled taking only a few seconds to compute the field quantities, the calculation time of one transmission line on a PC-486 platform using the finite element method is about 30 minutes. Fig. 5.17. Part of the finite element model around a single phase conductor.

### Measurements

The measurement of the electric field strength excited by the transmission line are based on the induced current of the charge oscillations between two halves of an isolated conductive body.

The measurements of the magnetic field strength are based on the electromotive force induced in a coil. Therefore, the probe of the field meter, Holaday Industries model HI-3604, consists both of two circular isolated parallel plates and of a circular coil. To avoid perturbations of the electric field, a fibre optic receiver and a non-conductive tripod to support the field meter are used. Only the rms value of the space component perpendicular to the plane of the probe is measured. The field quantities below the overhead transmission lines are measured at a height of lm above ground level.

### Numerical and experimental results

All computations and measurements on a Belgian 150 kV AC single three-phase system transmission line are performed at the place of the maximum slag.

Because the transmission line is situated in a flat area in Belgium, the ground level is assumed to be even.

To obtain the local field values with reasonable accuracy, a third order finite element solution is necessary. The use of shape functions of third order explains the long computation time. Fig. 5.18 shows the x-and z-component of the rms value of the magnetic field. Good agreement between the measurements and computed data can be stated. The two-dimensional approach overestimates both x- and z-component of the magnetic field. The reason for this lies in the type of approximation of the geometry of the transmission line. In the two-dimensional model a phase conductor of infinite length with constant height above the ground is considered. Therefore, the two-dimensional approach represents the worst case, i.e. the highest values of field strength. Fig. 5.18. Comparison of computed magnetic field distribution and measured data lm above ground a) x-component and b) z-component.

Fig. 5.19 shows the effective value of the z-component of the electric field. The calculations and the measurements show good agreement. Fig. 5.19. Comparison of computed electric field distribution with measured data lm above ground level.

In Fig. 5.20 the three-dimensional electric field distribution below the 150 kV ac transmission line obtained by the PMM method is plotted. The geometrical model of the single phase conductors consists of seven polygon elements per half of the overall span field. The high-voltage pole is located at the global co-ordinates x=0m and y=220m.

As expected, the maximum field values are found in the middle of the field span at the co-ordinates y= 0m. Here, the values of the electric field strength are in the range of 4 kV/m and thus well below the maximum allowed exposure values for the general public, given by the standards in Table 5.2. Fig. 5.20. Electric field distribution below the 150 kV AC high-voltage line computed by the point mirroring method.

Referring to Fig. 5.18, the magnetic flux density of the x- and z-component, generated by the current carrying conductors, are in the range of 0.4-0.6 jiT. According to Table 5.2 those values of the magnetic flux density are far below the allowed limits as well.

Due to the linearity of the problem formulation, calculations of power lines with different types of AC-high-voItage poles carrying multiple three-phase voltage and current systems can be performed. In this case the field components generated by the single systems have to be superposed according to the relative phase angle between the systems and the considered instant of time. Fig. 5.21 shows the results, computed by the semi-numerical technique, of a high-voltage transmission line consisting of six three-phase systems with different voltage level (2×3 80 kV, 2×220 kV, 2×110 kV). For the magnetic flux density it is assumed that each system carries a current of 1000 A. The system with the largest transmission voltage is put at the top of the pole, while the system with the lowest voltage is located below it. Fig. 5.21. a) High-voltage pole construction carrying 6 systems (2×380 kV, 2×220 kV, 2×110 kV) and b) the resulting electric field distribution. (All the three-phase current systems carry a rms. current of 1000 A; the pole is located at the co-ordinate x=0m, y=160m.)

### Comparison of PMM and FEM

Two efficient methods of computing the electric and magnetic fields below AC-high-voltage lines are demonstrated by an example; a Belgian 150 kV AC three-phase single system transmission line. Both methods, PMM and FEM, are compared with respect to accuracy and the required computational effort. To verify the results of the field simulations, measurements of a power line have been carried out, giving good agreement between computed and measured data.

In the PMM model, infinitesimally thin segmented filaments of constant charge or current are approximating the slag of the transmission line to solve the electrostatic and magnetic fields. With reasonable accuracy a three dimensional field distribution can be computed. Relatively low computation times are necessary to compute the three-dimensional field distribution below the power line with the PMM.

Using a standard PC-486/66, the calculation time lies in the range of seconds.

With the finite element method, the distribution of both electric and magnetic field quantities, is computed as well. With respect to the high computational costs when compared to the PMM, a two-dimensional approach in the middle of the span field is chosen. Due to the necessary high discretisation of the problem, the computational costs are in the range of thirty minutes using a PC-486/66. Good agreement between measured data on the Belgian 150 kV line and calculated field distributions by both methods can be stated.

The main problem employing the FEM is the small diameter of the conductors above the large flat conducting ground plane. The difference between the dimensions of a conductor and the field domain of interest is huge, the ratio lying in the range of some 10s. This causes the generation of a large amount of finite elements and thus an enormous computation time, even for the two-dimensional problem, when compared to the efforts necessary for the PMM. A three-dimensional FEM model is difficult to build, due to this huge difference in geometrical dimensions.

This example demonstrates that for problem types, such as the high voltage line, the FEM is not very well suited. In this case, the approach using the PMM is the better choice with respect to computational time, problem dimension (2D/3D) and discretisation problems.

### Effects of fields with low frequency

To evaluate the influence of the transmission line, it is not sufficient to calculate the coupling impedances or capacitances of the line. It is necessary to analyse the actually generated fields in the neighbourhood of the transmission line during the planning phase and to check if given standards for maximum field values are violated.

The interaction of electromagnetic fields with living organisms can be separated into two mechanisms, thermal and non-thermal interactions. Thermal interactions mean the mechanism of the absorption of electromagnetic energy resulting in an increasing temperature. Nonthermal are these interactions where the absorbed energy is not large enough to cause a significant temperature rise. In fields at low frequency, the body does not absorb or negligibly absorbs the wave energy. This implies that biological effects caused by electric or magnetic fields of low frequency fields are non-thermal. Observed non-thermal effects on human beings can be the stimulation of nerves, upright standing skin hair, visual disturbances … , The possible results of these effects may depend on the field characteristics which vary in intensity and frequency.

To judge the mentioned effects, existing technical standards supply the quantities of the electric, magnetic and electromagnetic fields as a function of the frequency. This is important, as the interaction of electromagnetic fields and matter strongly depends on the frequency of the considered field.

Nowadays an increasing sensitivity to ecological problems can be stated. An injurious influence to the health of human beings caused by the direct effect of low frequency electromagnetic fields (50/60 Hz) is scientifically not proven yet. For about twenty-five years research efforts to find a correlation mechanism between the field quantities and their effects on human beings have been going on, without significant success. In this situation, the electric and magnetic field quantities of high-voltage lines have to be examined in order to avoid EMC problems with the environment close to the power transmission line while planning high voltage lines.

A number of standards such as those in preparation by the European Committee for Electrotechnical Standardisation (CENELEC) are based on the known effects for short exposure times. Long term effects are not considered. However, to consider possible as yet undiscovered effects the values for technical fields are reduced by a factor. In Table 5.2 the maximum exposure values for the electric and magnetic field are summarised. The values are distinguished according to the general public and professional workers.

Table 5.2. Maximum exposure values for the electric and magnetic field (ICNIRP).

 exposure  professionals: 8 hours 10 0,5 general public: 5 0,1