ELEKTRYKA 1 Zeszyt 3 (15) Rok LVI Krzysztof BUDNIK, Wojciech MACHCZYŃSKI Instytut Elektrotechniki i Elektroniki Przemysłowej, Politechnika Poznańska MAGNETIC FIELD OF UNDERGROUND CABLES Summary. The paper presents a theoretical study of the calculation of magnetic fields in vicinity of underground electric power lines. Exact and simplified methods of the determination of the magnetic flux density are presented. For a straight underground conductor the exact method bases on the Fourier transform technique, whereas the simplified method is based on the Biot-Savart law. Keywords: magnetic field, underground cables, vector potential, magnetic flux density, Fourier transform, Biot-Savart law POLE MAGNETYCZNE KABLI PODZIEMNYCH Streszczenie. W pracy przedstawiono teoretyczne rozwaŝania, dotyczące obliczania pola magnetycznego w pobliŝu podziemnych kabli elektroenergetycznych. Zaprezentowano metodę dokładną, opartą na rozwiązaniu analitycznym uzyskanym z wykorzystaniem transformaty Fouriera, oraz metodę uproszczoną bazującą na prawie Biota-Savarta. Słowa kluczowe: pole magnetyczne, kable podziemne, potencjał wektorowy, indukcja magnetyczna, transformata Fouriera, prawo Biota-Savarta 1. INTRODUCTION Transmission of the electric power is accompanied with generation of low frequency electromagnetic fields. Nowadays of special concern is the possibility of detrimental environmental effects arising from the electrical and magnetic fields formed adjacent to the transmission lines. These fields may affect both operation of near electric and electronic devices and appliances and also various living organisms. The paper presents a theoretical study of the calculation of magnetic fields in vicinity of underground electric power lines. Exact and simplified methods of the determination of the magnetic field are presented. For a straight underground conductor the exact method bases on the Fourier transform technique, whereas the simplified method is based on the Biot-Savart law.
8 K. Budnik, W. Machczyński. MAGNETIC FIELD CALCULATIONS.1. Exact method (Fourier transform technique) Exact magnetic field computation techniques assume that the current carrying power line conductors are straight horizontal wires of infinite length [1-6]. Consider an infinitely long current carrying conductor placed at depth d under the earth surface, Fig. 1. The current I flows in direction of the x-axis and varies with the time as exp(jωt) where ω is the radian frequency. The x, y plane is considered to be the earth surface. It is assumed that the earth is an isotropic, homogeneous medium of finite conductivity γ. The magnetic permeability of the soil and of the air is µ. The displacement currents in both regions: the air and the earth are neglected. z µ=µ γ 1 = ε 1 =ε d y x I underground conductor µ=µ γ =const. ε =const. Fig. 1. An infinitely long current carrying conductor under the earth surface Rys. 1. Nieskończenie długi przewód podziemny z prądem The vector potential of the electromagnetic field has the x-component only denoted A x (y, which satisfies the following equations: where: the Laplace equation in the air: the Helmholtz equation in the earth: A A Ax + x = z (1) y z A x y + δ Dirac delta function, x z k A k = jωµ γ, j = 1 x = Iδ ( y) δ ( z d) z µ ()
Magnetic field underground 9 The vector potential A in the air can be obtained if the Fourier transform is used and the boundary conditions in the system considered expressing the continuity of the normal component of the magnetic flux density and the tangential components of the magnetic as well as electric intensities are taken into account: Ax 1) = Ax ) (3) 1 Ax1( y,) 1 Ax ) = (4) µ z µ z Hence the x-component of the vector potential in the air can be written in the form: A x1 d µ I e = π u + cos( uy) du z u u + k + k uz It should be noted that the formula (5) is the same that obtained by Sunde []. The magnetic flux density B r can be obtained from the equation: In the case considered: r B = rot A r B = 1y Ax z x, z y 1 A ( y where 1 x, 1 z are the unit vectors in the direction x and z respectively. become: From eqns (7) and (5) two components of the magnetic flux density in the air (z ) B y B d µ I ue = π u + Inserting to equations (8) and (9): u + k cos( uy) du z z d µ I ue = π u + u + k uz sin( uy) du z u u + k + k uz we have for the y - component of the magnetic flux density (5) (6) (7) (8) (9) u = k n (1) B I k k d n + j k nz µ ne y ) π = cos( k ny dn n + n + j (11)
1 K. Budnik, W. Machczyński The key aspect of magnetic flux density evaluation is to calculate the infinite integral in eqn.(11), where the exponent in the exponential function in the integrand is complex and is given by k d n + j. In consequence the real and imaginary parts of the integrand are oscillatory functions, what can lead to numerical problems by evaluation of the integral. Taking into account that n + 1 = j( n n + n + j j ) (1) and denoting: n + j = a + jb (13) where: 4 n + 1 + n a = (14) one becomes 4 n + 1 n b = (15) µ I k By = j π k nz k da ne e k nz k da j ne e [( n a) cos( k db b sin( k db) ] cos( k ny) dn + [( n a) sin( k db + b cos( k db) ] cos( k ny) dn (16) Similarly we obtain the z-component of the magnetic flux density in the form Bz = µ I k j π k nz k da ne e k nz k da j ne e [( n a) cos( k db b sin( k db) ] sin( k ny) dn + [( n a)sin( k db + b cos( k db) ] sin( k ny) dn (17) The integrals in the formulas have to be solved numerically. Figures - 4 show a typical dependency of the real and imaginary integrand parts on the integral variable n.
Magnetic field underground 11 Extensive investigations of the integrand with different geometrical and electrical parameters of the system shown in Fig. 1, have led to determination of the integral upper limit. In practical cases, for f = 5 Hz, commonly depths of the conductors buried in the earth with typical earth conductivity, the upper limit can be not greater then 5. Finally, the magnitude of the magnetic flux density: B ( ) B ) ( y B ( ), = y + z z (18) Fig.. Real and imaginary parts of the integrand J y for different earth conductivity and d=1 m, y=, z=1 m Rys.. Część rzeczywista i urojona funkcji podcałkowej J y w funkcji konduktywności gruntu dla d=1 m, y=, z=1 m
1 K. Budnik, W. Machczyński Fig. 3. Real and imaginary parts of the integrand J y for different coordinate z and d=1 m, y=, γ=1-3 S/m Rys. 3. Część rzeczywista i urojona funkcji podcałkowej J y w funkcji współrzędnej z dla d=1 m, y=, γ=1-3 S/m
Magnetic field underground 13 Fig. 4. Real and imaginary parts of the integrand J y for different coordinate y and d=1 m, z=1 m, γ=1-3 S/m Rys. 4. Część rzeczywista i urojona funkcji podcałkowej J y w funkcji współrzędnej y dla d=1 m, z=1 m, γ=1-3 S/m.. Simplified method (Biot-Savart law) The magnetic field in the observation point P(x,y, produced by a current path c as in Fig. 5 can be computed using the Biot-Savart law [1, ]:
14 K. Budnik, W. Machczyński µ I( dl 1 B( x, y, = 4π r r ) c (19) where I is a phasor current, the vector element dl coincides with the direction of the current I, 1 r is a unit vector in the direction of the vector r, r is the distance between the source point P s (X, Y, Z) and the observation point P(x, y, and µ is the magnetic permeability of the vacuum. I dl 1 r P s (X,Y,Z) r P(x,y, c Fig. 5. Current path generating the magnetic field Rys. 5. Pole magnetyczne przewodu z prądem If the current carrying conductor is straight and infinitely long as shown in Fig. 1, it follows from eqn.(19) that B µ = π I y + z () 3. EXAMPLE Consider a system of three conductors shown in Fig.6. The conductors of the underground power cable (separate cores) are buried in flat configuration at depth d = 1, m. The conductors for the three phases are separated by the distance a =,5 m. We wish now to calculate the magnetic flux density in the air at the frequency f = 5 Hz, at the height z = 1, m above ground level, for currents: I 1 = 1 A, I = 1. e -j 1 A, I 3 = 1. e j 1 A. The calculations have been curried out using the MatLab according to accurate relationships (16) and (17) as well as to simplified relation ().
Magnetic field underground 15 z I 1 I I 3 d y a a Fig. 6. Underground current carrying conductors of the three phase power cable Rys. 6. Podziemny trójfazowy kabel elektroenergetyczny The results of calculations are shown in Fig. 7 and Fig.8, respectively. It should be noted, that the results of calculations are practically identical. It should be however pointed out, that the simplified method based on the Biot-Savart law is adequate for homogeneous soils having a relative permeability equal to unity and for low frequency values [8]. On the contrary, the accurate method can be used in more complex cases. Fig. 7. Profile of the module of the magnetic flux density obtained using the accurate formulas Rys. 7. Rozkład modułu wektora indukcji magnetycznej wyznaczony z zaleŝności dokładnych
16 K. Budnik, W. Machczyński Fig. 8. Profile of the module of the magnetic flux density obtained using the simplified formula Rys. 8. Rozkład modułu wektora indukcji magnetycznej wyznaczony z zaleŝności przybliŝonych 4. FINAL REMARKS From the equations presented the quasi-stationary magnetic field produced by currents flowing in conductors of an underground power cable can be calculated. The accurate formulas are derived by a general solution of the electromagnetic field equations, using a methodology based on the vector potential approach. The simplified formula follows from the Biot-Savart law and gives results with considerably accuracy, comparing with the exact method. The results derived may be applied in practical EMI problems related to earth return circuits laid along power cable in the same right of way. BIBLIOGRAPHY 1. Carson J.R.: Wave propagation in overhead wires with ground return. Bell System Technical Journal, 196, No. 5, s. 539-554.. Sunde E.D.: Earth conduction effects in transmission system. New York, Dover 1968. 3. Krakowski M.: Obwody ziemnopowrotne. WNT, Warszawa 1979. 4. Machczyński W.: Oddziaływania elektromagnetyczne na obwody ziemnopowrotne rurociągi podziemne. Wydawnictwo Politechniki Poznańskiej, Poznań 1998. 5 Guide on the influence of high voltage AC power systems on metallic pipelines. CIGRE, 1995.
Magnetic field underground 17 6. ITU-T, Directives concerning the protection of telecommunication lines against harmful effects from electric power and electrified railways lines; Vol. II. Calculating induced voltages and currents in practical cases, Vol. III. Capacitive, inductive and conductive coupling: Physical theory and calculation methods: International Telecommunication Union, Geneva 1989. 7. Adhikari S., Holbert K.E., Karady G.G., Dyer M.L.: Modelling Magnetic Fields Generated by Single-Phase Distribution Cables. 39 th North American Power Symposium, NAPS 7, p. 85 91. 8. Malo Machado V., Almeida M.E., Guerreiro das Neves M.: Accurate magnetic field evaluation due to underground power cables, Euro. Trans. Electr. Power (8). Published online in Wiley InterScience, (www.interscience.wiley.com) DOI: 1.1/etep.96. Wpłynęło do Redakcji dnia 1 września 1 r. Recenzent: Prof. dr hab. inŝ. Zygmunt Piątek Omówienie Przesył energii elektrycznej jest związany z generowaniem pola elektromagnetycznego niskiej częstotliwości. Obecnie szczególne zainteresowanie budzi moŝliwość niekorzystnego oddziaływania pola na organizmy Ŝywe oraz infrastrukturę techniczną znajdujące się w pobliŝu linii elektroenergetycznych wysokiego napięcia. W pracy przedstawiono teoretyczne rozwaŝania, dotyczące wyznaczenia rozkładu pola magnetycznego w pobliŝu podziemnych kabli elektroenergetycznych. Zaprezentowano metodę dokładną, opartą na rozwiązaniu analitycznym uzyskanym z wykorzystaniem transformaty Fouriera, oraz metodę uproszczoną bazującą na prawie Biota-Savarta. Uzyskane wyniki mogą mieć praktyczne znaczenie w analizie oddziaływań elektromagnetycznych podziemnych kabli elektroenergetycznych na pobliskie obwody ziemnopowrotne.