Theoretical analysis on the penetration of power line harmonic radiation into the ionosphere

Transcription

1 RADIO SCIENCE, VOL. 37, NO. 6, 1093, doi: /2001rs002486, 2002 Theoretical analysis on the penetration of power line harmonic radiation into the ionosphere Yoshiai Ando and Masashi Hayaawa Department of Electronic Engineering, The University of Electro-Communications, Toyo, Japan Oleg A. Molchanov Institute of Physics of the Earth, Russian Academy of Sciences, Moscow, Russia Received 1 May 2001; revised 25 June 2002; accepted 2 July 2002; published 13 November [1] Radiation at the fundamental and higher harmonic frequencies from power lines affects the environment of Earth s ionosphere and magnetosphere, because such radiation may lead to the enhanced electron precipitation into the ionosphere due to wave-particle interactions in the magnetosphere. In this paper we analyze theoretically the penetration of power line radiation, taing into account the presence of the anisotropic, homogeneous ionosphere. The electromagnetic field can be formulated by using the Fourier transform and field distribution obtained by proper deformation of the integral path with consideration of singularities. We calculate the field distribution as a function of altitude and distance from the source. From the numerical results, we can find several new findings: (1) the effect of the radiation from power line is limited in a relatively narrow region below the cutoff frequency of the guided mode in the Earthionosphere waveguide, while the region affected by the power line radiation becomes much wider if the guided modes exist, (2) the radiated fields are directed almost horizontally, and (3) the altitude distribution of electromagnetic fields changes at the ion gyrofrequency. Comparison with the observed data shows a good agreement with respect to the region affected by power line radiation. INDEX TERMS: 0619 Electromagnetics: Electromagnetic theory; 6934 Radio Science: Ionospheric propagation (2487); 6914 Radio Science: Electromagnetic noise and interference; 6984 Radio Science: Waves in plasma; KEYWORDS: PLHR, ionosphere Citation: Ando, Y., M. Hayaawa, and O. A. Molchanov, Theoretical analysis on the penetration of power line harmonic radiation into the ionosphere, Radio Sci., 37(6), 1093, doi: /2001rs002486, Introduction Copyright 2002 by the American Geophysical Union /02/2001RS [2] Based on the steady accumulation of ground- and satellite-based measurements, radiation from power lines at higher harmonics (power line harmonic radiation; PLHR) of the fundamental has been recently recognized as contamination of Earth s magnetosphere (high energy electrons) by human activity (see the review by Bullough [1995, and references therein]). This PLHR is nown to be occurring mainly on field lines grounded on major industrial areas in the world. The PLHR penetrates through the lower ionosphere and is guided along a field-aligned duct to the magnetospheric equatorial region where the wave-particle interaction taes place, giving rise to wave amplification and stimulation of new emissions [e.g., Bullough, 1983, 1995; Helliwell et al., 1975; Par and Helliwell, 1978; Parrot, 1994; Molchanov et al., 1991; Kiuchi, 1983]. Hence, in order to estimate quantitatively the magnetospheric effect (the wave-particle interaction), we have to estimate the transmission characteristics of PLHR in the ionosphere. But there have been so far proposed very few simple theoretical computations (as presented by [Bullough, 1995]) without including the presence of the ionosphere. [3] In this paper we will theoretically analyze the penetration characteristics of power line harmonic radiations into the anisotropic, homogeneous ionosphere. In reality, the density of charged particles in the ionosphere has complicated profiles along altitude. Here we

2 5-2 ANDO ET AL.: PENETRATION OF POWER LINE HARMONIC RADIATION Figure 1. Configuration for analysis. tae the mean value of them as the zero-th order approximation. We introduce a layer approximation for the simplicity of analysis, so that we can formulate our problem by using the Fourier transform to obtain the electromagnetic fields in the wave number domain. In general, it is difficult to obtain analytically the fields in the space domain by the inverse transform because there are singularities in the wave number plane. This difficulty can be overcome by the recent rapid progress of CPU by computing the fields just only by taing into account the singularities and by proper deformation of the integral path. The details will be shown in section 2. [4] We will present, in section 3, the computational results on the field strength in the ionosphere as a function of altitude and distance from the source at the several harmonics frequencies to examine the PLHR effect on the Earth s environment, and in order to verify the validity of the present method we compare the numerical results with the observed data. Finally, our conclusion is summarized in section Theory of PLHR Penetration Into the Ionosphere 2.1. Formulation of the Problem [5] In this study, we consider the fields in a relatively small region around the source. We then assume that the ground surface of the Earth and the boundary between the air and ionosphere regions are just planes. It was reported that the most possible, significant source of PLHR was zero-phase (unbalanced) harmonic currents in the power line with ground return [Tatnall et al., 1983]. Based on it, we can model the single electric current as the source. Thus, we consider the geometrical configuration of the present problem as shown in Figure 1. [6] There is a semi-infinite perfect conductor below z = 0, and the plasma with the permittivity tensor e r is assumed to extend from z = h + z 0 to infinity. The region with 0 z < h + z 0 is air; that is, the material constants are e 0 and m 0, which are the permittivity and the permeability of free space, respectively. An infinitesimally thin wire in which the current I flows is located at a height of z = z 0, and therefore the current density of this source is represented as J = d(x)d(z z 0 )Iŷ = J y ŷ, where d(x) stands for the Dirac s delta function and ŷ is the unit vector of +y-direction. The Earth s magnetic field(h 0 ) is directed along +zdirection. Note that even if it is the opposite direction, resulting fields have same amplitude, but some components have opposite phase. We adopt e jwt as time factor with the angular frequency w, so the permittivity tensor of plasma is given by [Stix, 1962] where j 2 0 e r ¼ e 0 4 j ; ð1þ w 2 pj 1 ¼ 1 X w j 2 w 2 ; cj 2 ¼ X w cj w 2 pj ; j w w 2 w 2 cj 3 ¼ 1 X j w 2 pj w 2 ; and w pj and w cj are the plasma frequency and gyrofrequency of the particle species j, respectively.

3 ANDO ET AL.: PENETRATION OF POWER LINE HARMONIC RADIATION 5-3 [7] The layer-configuration goes infinitely along y- and x-directions, and assuming the quasi-static current, we may let the fields be independent of y; ¼ 0. Thus, the present problem is reduced to a 2-dimensional one. Moreover, a pair of the Fourier transform can be defined as follows: Z 1 fðþ¼ x ~f ðþ ¼ 1 2p 1 Z 1 ~ fðþe jx d; 1 fðþe x jx dx: ð2þ [8] Fourier-transformed Maxwell s equations in the plasma region are obtained in terms of electric and magnetic fields, ~E and ~H, respectively, as ~E ¼ jwm 0 ~ H x ; j ~E z ¼ jwm 0 ~H y ; ð3bþ j ~E y ¼ jwm 0 ~H z ; ~H ¼ jwe 0 1 ~E x j 2 ~E y ; ~H j ~H z ¼ jwe 0 j 2 ~E x þ 1 ~E y ; ð3eþ j ~H y ¼ jwe 0 3 ~E z ; z j 1 ~E x j 2 ~E y þ ¼ 0; ð3gþ j ~H x ~H ¼ 0: ð3hþ Arranging the above equations, the x and y-components can be expressed in terms of ~E z and ~H z : ~E x ¼ j z 2 wm 0 ~H z þ ; ~E y ¼ wm 0 ~H x ¼ ~H ; ~H y ¼ we 0 3 ~E z : ~H z ; Also, we obtain the differential equations of ~E z and ~H z 2 ~E 2 þ ~ ~H z E z ¼ wm 0 3 ; ð4aþ where 0 2 = w2 e 0 m 0 and r ¼ By letting H z, E z / e jgz, (4b) and (4a) are rewritten in a matrix form: " g r 2 jgwe # jgwm g H ~ z 3 2 ¼ 0: ~E z ð5þ Equation (5) has a nontrivial solution only if the determinant is zero, so g is explicitly given by g 2 ¼ þ 1 2 þ 3 p ffiffiffiffiffiffiffiffiffi w ðþ ð6þ w ð Þ ¼ þ r : ð7þ 3 The second term of the right-hand side of (6) is a 2- valued function with respect to so that g is a 4-valued function of and we have 4 Riemann surfaces. Let the solutions be ±g 1,±g 2, where we define 0 arg[g 1 ]<p and 0 arg[g 2 ]<p, i.e. they have a positive imaginary part. Then with considering the Sommerfeld s radiation condition, ~E z and ~H z are given by where ~E z ¼ Ae jg 1z þ jzbe jg 2z ; ð8aþ ~H z ¼ jy Ae jg 1z þ Be jg 2z ; ð8bþ Y ¼ g wm 0 g 2 ; ð9aþ 1 3 g 2 wm 2 0 Z ¼ 3 g : ð9bþ 3 2 Note that the solutions e jg1z and e jg2z are improper because no reflected wave exists in this model. Therefore, the branch cuts run along the lines =(g 1 ) = 0 and =(g 2 ) = 0 in the complex -plane (=(z) designates the imaginary part pffiffiffiffiffiffiffiffiffi of z). It is possible to ignore the branch cuts due to w ðþbecause g 1 and g 2 are commutative in the equations of the electromagnetic fields which will be derived later. [9] In the air source-free region, the equations of fields are given by letting 1 =1, 2 =0, 3 = 1, then the ~E z and ~H z are uncoupled. Consequently, in z 0 < z < z ~H 2 þ 2 0 r 2 H ~ 2 z z ¼ we 0 ; ð4bþ ~E z ¼ Ce jg 0z þ De jg 0z ; ð10aþ ~H z ¼ Ee jg 0z þ Fe jg 0z ; ð10bþ

4 5-4 ANDO ET AL.: PENETRATION OF POWER LINE HARMONIC RADIATION Figure 2a. Complex -plane and the singularities. where g 0 2 = In0<z < z 0, with taing the boundary conditions at z = 0 into account: ~E z ¼ G cos g 0 z; ~H z ¼ H sin g 0 z: ð11þ The other components can be given by ~E x ¼ j ~H x ¼ ; ~E y ¼ wm 0 ~H z ~H ; ~H y ¼ we 0 E ~ z : And the source current is transformed as follows: ~J y ¼ 1 Z 1 JðyÞe jx dx ¼ I 2p 1 2p d ð z z 0Þ: ð12þ Using usual boundary conditions that the tangential components of fields are continuous except ~H x at z = z 0, where ~H x j z¼z0 þ ~H x j z¼z0 ¼ R z 0 þ ~ z 0 J y dz ¼ I 2p, we get the unnown coefficients A through H, which are given in Appendix A. Taing the inverse Fourier transform of the field components in the wave number domain yields the ones in the space domain Deformation of Integral Path [10] We can obtain the field components in the space domain by the numerical Fourier transform, but the calculation is not straightforward because there are singularities in the complex -plane, and some of integrands do not decay more rapidly than 1 ; that is, the integrals do not converge by direct calculation. Here consider the inverse Fourier transform of (8a): E z ¼ Z 1 1 Ae jg1z þ jzbe jg 2z e jx d; ð13þ where A and ZB tend to be proportional to 1 as! 1. At the frequency lower than the ion gyrofrequency, if increases along the real axis, then either one of g 1 or g 2 is real and another is imaginary. Therefore, ~E z decay as rapidly as 1 ; in other words, the integral of (13) does not converge. In the similar manner it is found that as!1~e x, ~E y, ~H x, ~H y, and ~H z behave lie 1, 1, 0, 2, and 0, respectively. Thus, we cannot directly calculate the inverse Fourier transform of them except ~H y. [11] In order to overcome these difficulties, we need to deform the integral path to converge the inverse Fourier transform [Chew, 1995]. The integral path must be deformed as the integral converges and as it does not cross the branch cuts which occur in connection with g 1 and g 2. We have chosen the proper branch as =(g 1 ) and =(g 2 ) are positive, so that the branch cuts are the lines where =(g 1 ) and =(g 2 ) are equal to zero. Figure 2a shows the schematic of the complex -plane at the frequency p lower than the ion gyrofrequency, where p 1 ¼ ffiffiffiffi pffiffiffiffiffiffiffiffi r0 ; and p 2 ¼ j 3 0, which are the branch points, and the dashed lines signify the branch

5 ANDO ET AL.: PENETRATION OF POWER LINE HARMONIC RADIATION 5-5 Figure 2b. -plane. Deformation of the integral path in complex cuts. Note that in (13) there is no branch for g 0 because it is an even function with respect to g 0. [12] Considering those singularities, we deform the integral path as shown in Figure 2b. In the lower half plane, the factor e jx with x > 0 diverges exponentially. In order to avoid the divergence of the integrand, the slope of the integral path in the lower half plane is limited to q < tan 1 = g 1;2ðz z 0 hþþg 0 h : = ðxþ [13] At the frequency higher than the ion gyrofrequency, if increases along the real axis both g 1 and g 2 are imaginary since the sign of r is negative, so that the integral of (13) necessarily converges on the proper branch because the integrand exponentially decays. In this case, we can calculate the inverse Fourier transform by simply integrating it along the real axis. 3. Numerical Results and Discussion [14] In this section we give the numerical results based on the theory just given. At first, we show an example of numerical results with putting the important point on the behavior of the field distribution around the ion gyrofrequency. Second, we compare the numerical results with the observed data to verify the validity of the present method An Example of Numerical Results [15] Here we calculate the field distribution around the ion gyrofrequency, and observe the behavior of them below and above the frequency. The used parameters are z 0 =10m,h =110 5 m(100 m), m 0 H 0 = T, and I = 1 A. We assumed that the plasma consists of positive ions of one ind and electrons with density m 3, and that mass of the ion (hydrogen) is g. In this case, the ion gyrofrequency, w ci, is rad/sec (762 Hz). These parameters are reasonable because in night the hydrogen ion is dominant among ions which form the ionosphere, at the altitude higher than about 600 m [Bilitza et al., 1993] with the nearly constant density (about m 3 ). [16] Figures 3a 3f show the amplitude of the field components as a function of x and z at the angular frequency w = 2p 100 rad/sec( f = 100Hz) which corresponds to the second harmonics frequency of the power system in the eastern half of Japan. In each figure we plot them over 1000 m with respect to x and z. This is because the layer approximation we adopt may be invalid out of this region because 1000 m on the Earth s surface corresponds to about rad and at x = 1000m the ground surface would pass through about z = 78.5m. All components of the fields are, as you can see from the equations, even or odd functions with respect to x, so that all figures show the symmetry with respect to x =0. [17] From these results, we can observe some peas and valleys along the z-direction in all components, and the pea points of x- and y-components are alternate in position along z-axis. The amplitude scarcely decay with increase of z. On the other hand, the dependence along x-direction is simple; that is, when one moves horizontally away from the source, the amplitudes of x- and y-components decrease simply. The effective region of PLHR in a horizontal plane extend slightly as we increase the observing altitude, but the extension is not so significant in the space shown here. We cannot say that the effect continues up to higher altitude without dissipation though the decay is very small in the present calculation, because we now consider the homogeneous ionosphere and the reflected wave does not exist while in the real ionosphere the plasma parameters depend on altitude and the reflection exists over all region. [18] It is also found that horizontal components of the fields are all strong, while the vertical component is much weaer than them and negligible, that is, the electromagnetic fields are directed horizontally. [19] Figures 4a 4f show the amplitude of the field components for the case of w =2p 200 rad/sec ( f = 200Hz), or the 4-th harmonics. All the other parameters

6 5-6 ANDO ET AL.: PENETRATION OF POWER LINE HARMONIC RADIATION Figure 3. (a) E x at 100Hz in the ionosphere region. (b) E y at 100Hz in the ionosphere region. (c) E z at 100Hz in the ionosphere region. (d) H x at 100Hz in the ionosphere region. (e) H y at 100Hz in the ionosphere region. (f) H z at 100Hz in the ionosphere region. are same as before. We can see that based on the comparison between Figures 3a 3f and 4a 4f the number of peas increases with frequency, and the amplitude of some field components vary, for example, E z becomes larger as the frequency is higher. Other features are the same as the ones with w =2p 100 rad/sec. [20] The previous results are the cases for which the frequencies are lower than the ion gyrofrequency, w ci. Next, we will show the results at the frequency higher than the ion gyrofrequency. Figures 5a 5f represent the numerical results with 2p 800 rad/sec ( f = 800Hz), which corresponds to the 16-th harmonics, and the first harmonics higher than w ci. In contrast to the previous results, Figures 5a 5f do not show any peas and valleys along z-direction, and we can observe a major pea of all components along x-direction. In this case also, the amplitude of fields scarcely decay with an increase in altitude.

7 ANDO ET AL.: PENETRATION OF POWER LINE HARMONIC RADIATION 5-7 Figure 4. (a) E x at 200Hz in the ionosphere region. (b) E y at 200Hz in the ionosphere region. (c) E z at 200Hz in the ionosphere region. (d) H z at 200Hz in the ionosphere region. (e) H y at 200Hz in the ionosphere region. (f ) H z at 200Hz in the ionosphere region. [21] We examined the results with more several harmonics above w ci. The results showed us that the tendency of the field distribution were similar to the results with w =2p 800 rad/sec, and the strength only varied. [22] Additionally speaing, the time to calculate the points data of the above figures is about 2- minutes by using PC with 1.7 GHz CPU. The Fourier integrals calculated here converge rapidly by choosing the exponentially decaying path, and we did not encounter any instability in computing. [23] Based on the numerical results presented before, we can summerize the penetration characteristics of PLHR in the following. [24] First, the horizontal region where the electromagnetic field is strong is limited to, for the present

8 5-8 ANDO ET AL.: PENETRATION OF POWER LINE HARMONIC RADIATION Figure 5. (a) E x at 800Hz in the ionosphere region. (b) E y at 800Hz in the ionosphere region. (c) E z at 800Hz in the ionosphere region. (d) H x at 800Hz in the ionosphere region. (e) H y at 800Hz in the ionosphere region. (f ) H z at 800Hz in the ionosphere region. parameter, ±200 m from the source, and does not extend with higher altitude. This distance corresponds to nearly twice the lower boundary of the ionosphere. [25] Second, the fields have nearly only horizontal component at these frequencies, and the vertical component of the electric field is negligible. [26] Third, the field distribution depends on frequency, in particular, the ion gyrofrequency plays a significant role in the distribution. In the context of polarization at x =0, where the wave vector is along z-direction, theelectric field is elliptically polarized, and above the ion gyrofrequency, apart from the phase, the major axis is fixed at a direction (because of the presence of only one mode of propagation(whistler mode)). But below the ion gyrofrequency the direction of the major axis changes depending on altitude, and the change becomes more rapid as the frequency is

9 ANDO ET AL.: PENETRATION OF POWER LINE HARMONIC RADIATION 5-9 Figure 6. Max B -f characteristics. Figure 7b. 3.4Hz. Distribution of B at frequencies 1.9 higher. This is due to the wave interference of the two possible modes of propagation Comparison With Observed Data [27] In order to verify the validity of our calculation, we compare the numerical results with the observed data. Here, we adopt the following values as parameters: (1) The height of the lower boundary of the ionosphere is 80 m. (2) The height of the power line is 10 m. This value is a mean height of a typical high voltage (HV) line (<70V) [Tatnall et al., 1983]. (3) The electron and ion density is m 3 [Bilitza et al., 1993]. (4) The mass of ion is g. We assume that the oxygen ion, O +, is dominant in the observation points [Bilitza et al., 1993; Richmond, 1995]. (5) The Earth s magnetic field is T. We may simulate the situation of the diurnal ionosphere by setting the parameters as above. In day, the oxygen ion is dominant up to the altitude 1000 m and the mean value of the electron and the oxygen ion and the value at the altitude 600 m all are approximately m 3 [Bilitza et al., 1993]. [28] First, we calculate the dependence of the maximum of the magnetic flux density, Max B, on frequency at the fixed altitude 600 m, which corresponds to the low altitude of the orbits of the observation satellite, Ariel 4. Figure 6 shows the numerical result of Max B -f characteristics with the source current 1A, where 1g designates Tesla. And also we show the distribution of B at some frequencies with altitude 600 m as a function of x in Figures 7a 7c. [29] It is found that there are some peas at about 1.9 Hz and the multiples. We can see from Figures 7a 7c that the shape of field distribution changes greatly across Figure 7a. 2.0Hz. Distribution of B at frequencies 1.6 Figure 7c. 3.8Hz. Distribution of B at frequencies 3.4

10 5-10 ANDO ET AL.: PENETRATION OF POWER LINE HARMONIC RADIATION Figure 8. B distribution in wider region at 3.8 Hz. those frequencies. In Figure 7a the fields distribute only around x = 0 below 1.8Hz, while the fields spread to a wider region above 1.9Hz. This indicates that there exists a guided wave above this frequency, so that the fields distribute along x-direction with little damping. Actually, the integrands of the spectral integrals in section 2, e.g. (13), have the poles corresponding to the guided wave modes. Therefore, the peas in Figure 6 correspond to the lower cutoff frequencies of guided waves of dominant and higher modes propagating between the ionosphere and the Earth s surface [Hayaawa et al., 1994]. Similarly, there appears another guided mode above 3.8Hz as shown in Figures 6 and 7c. [30] We examined the dependence of these pea frequencies on the ionospheric and geometrical parameters numerically. It is found that only the height of the boundary between the ionosphere and the air region, or the width of the waveguide influences the pea frequencies. [31] In order to estimate how wide a region the PLHR affects, we calculated the field distribution in a wider range up to x = 10 Mm at 3.8 Hz, which is shown in Figure 8. As you can see, the region where the field is effective spreads up to about x =1 1.5Mm when the guided mode exists, though the fields distribute within about 200 m without guided modes. [32] Here we tae the observation data by Bullough [1995] to compare with the above numerical results. In his paper, the data observed by the satellite, Ariel 4 around Winnipeg, Canada, are shown in his Figure That figure shows the percentage occurrence of 3.2Hz signal (with 1.0Hz passband). Prominent are the data in the north hemisphere in (northern) summer and the percentage occurrence of the signal intensities is 70 and 80dB above g 2 /Hz at about 330 ± 30 in invariant coordinates at the observed altitude, 600 m. This angle corresponds to 2.9Mm along the azimuthal direction on the Earth s surface. From the numerical results we have estimated the region illuminated by the PLHR is within Mm, or 2 3 Mm in both sides of the source. The power line in Winnipeg runs 500 m to northeast direction pffiffiffi so that the affected region becomes about 2 times along azimuthal direction. With taing into account the above, our estimation with respect to the region affected by PLHR is very reasonable. [33] In Figure of Bullough [1995], the signal intensities which seem to be caused by radiation from the Winnipeg power line is about db above g 2 / Hz as pea, and db as minimum. In order to compare those data, we convert Figure 6 into the data in terms of the field intensity as shown in Figure 9. From this figure, we can find that the field intensities due to power line is about 20dB on the average, and about 35dB for the largest case. It is noted that this result is for the case that the current is 1A uniformly all over frequencies, so does not simulate real situation. It is very difficult to estimate the harmonic current flowing into the power line as a function of frequency, and, to our nowledge, such research has not been reported. Referring to Tatnall et al. [1983], we adopt A as the current, or adding db to the field intensities, in the frequency band 2.7 to 3.7 Hz in the present case. Moreover, the field intensities depend on the height of the source very much. We can estimate it approximately as follows: the height of the source, z 0, is much smaller than h, and over the region where the contribution of the integration in (13) is dominant, g 0 z 0 1 and g 0 h 1, so that expanding sin g 0 z 0 around zero and cos g 0 (z 0 + h) and tan g 0 (z 0 + h) around g 0 h in (A1) and (A2), we see that A and B are proportional to z 0. Here we set z 0 = 10m, but several times higher lines are possible, for which we may add 14dB (50m). However, our calculation results do not agree with the Figure 9. The magnetic field intensity.

11 ANDO ET AL.: PENETRATION OF POWER LINE HARMONIC RADIATION 5-11 observed one and would be much less even if the source current was estimated too high. The reasons of this disagreement is considered that the observed field is not only due to direct radiation, and it is the sum of the direct radiation and the additional amplification due to waveparticle interaction in the magnetospheric equator [Hayaawa et al., 1977]. On the other hand, if the radiated field tae the largest value estimated above, it becomes about 65dB, which is about 28dB above the Ariel4 receiver noise (37dB above g 2 /Hz). We can conclude that in the ionosphere the radiation from power lines may grow beyond natural and other human noises, and then induce new emissions in the magnetosphere. 4. Conclusion [34] We have analyzed the penetration of power line radiation into the anisotropic ionosphere by using the layer-configuration model. The fields can be formulated by using the Fourier transform and we have calculated them by taing the numerical inverse transform with deformation of the integral path. We have examined the amplitude of field components at some harmonics frequencies. As results, we can predict the effects of PLHR as follows: (1) The field distribution depends on the frequency. The effect of PLHR is predominant in horizontally defined without guided wave modes. If guided waves exist, the region affected expands up to several Mega meters for the case of the present parameters. (2) The fields are directed almost horizontally. (3) The ion gyrofrequency is also a characteristic frequency to change the behavior of the field. (4) The direct radiation from power lines is much greater than other noise. However, the field observed by the satellites consists of not only the direct radiation from the power line, but also other sources, for example, due to wave-particle interaction in the magnetosphere. Appendix A: The Coefficients in the Field Equations [35] The coefficients to be determined in section 2 are given by A ¼ I U 2 sin g 0 z 0 e jg 1ðz 0 þhþ ; ða1þ 2p D 0 cos g 0 ðz 0 þ hþ B ¼ I 2p jz 1 U 1 sin g 0 z 0 e jg 2ðz 0 þhþ D 0 cos g 0 ðz 0 þ hþ ; ða2þ C ¼ I sin g 0 z 0 e jg0 ðz0þhþ 4pg 0 D 0 cos g 0 ðz 0 þ hþ g 0 3 ðs 1 S 2 Þ f1 þ j tan g 0 ðz 0 þ hþg; ða3þ D ¼ I sin g 0 z 0 e jg0 ðz0þhþ 4pg 0 D 0 cos g 0 ðz 0 þ hþ g 0 3 ðs 1 S 2 Þ f1 þ j tan g 0 ðz 0 þ hþg; ða4þ E ¼ ji sin g 0 z 0 e jg0 ðz0þhþ 4pg 0 D 0 cos g 0 ðz 0 þ hþ fð g 0 þ g 1 ÞYU 2 ðg 0 þ g 2 ÞZ 1 U 1 g; ða5þ F ¼ ji sin g 0 z 0 e jg0 ðz0þhþ 4pg 0 D 0 cos g 0 ðz 0 þ hþ fð g 0 g 1 ÞYU 2 where ðg 0 g 2 ÞZ 1 U 1 g; ða6þ G ¼ I g 0 3 ðs 2 S 1 Þtan g 0 z 0 e jg 0h ; 2pg 0 D 0 cos g 0 ðz 0 þ hþ ða7þ References H ¼ ji g 0 ðyu 2 Z 1 U 1 2pg 0 D 0 cos g 0 ðz 0 þ hþ Þe jg 0h D 0 ¼ YV 1 U 2 Z 1 V 2 U 1 U 1 ¼ S 1 þ jg 0 3 tan g 0 ðz 0 þ hþ; U 2 ¼ S 2 þ jg 0 3 tan g 0 ðz 0 þ hþ; V 1 ¼ g 0 jg 1 tan g 0 ðz 0 þ hþ; V 2 ¼ g 0 jg 2 tan g 0 ðz 0 þ hþ; S 1 ¼ 3 g 1 2 wm 1 0 Y ; 1 S 2 ¼ 3 g 2 2 wm Z : ; ða8þ Bilitza, D., K. Rawer, L. Bossy, and T. Gulyaeva, International reference ionosphere Past, present, future, Adv. Space Res., 13(3), 3 23, Bullough, K., Satellite observations of power line harmonic radiation, Space Sci. Rev., 35, , Bullough, K., Power line harmonic radiation: Sources and environmental effects, in Handboo of Atmospheric Electrodynamics, vol. 2, edited by H. Volland, pp , CRC Press, Boca Raton, Fla., Chew,W.C.,Waves and Fields in Inhomogeneous Media, IEEE Press, Piscataway, N. J., Hayaawa, M., K. Bullough, and T. R. Kaiser, Properties of storm-time magnetospheric VLF emissions deduced from the Ariel 3 satellite and ground-based observations, Planet. Space Sci., 25, , Hayaawa, M., K. Ohta, and K. Baba, Wave characteristics of twee atmospherics deduced from the direction finding measurement and theoretical interpretations, J. Geophys. Res., 99, 10,733 10,743, 1994.

12 5-12 ANDO ET AL.: PENETRATION OF POWER LINE HARMONIC RADIATION Helliwell, R. A., J. P. Katsufrais, T. F. Bell, and R. Raghuram, VLF line radiation in the Earth s magnetosphere and its association with power system radiation, J. Geophys. Res., 80, , Kiuchi, H., Overview of power-line radiation and its coupling to the ionosphere and magnetosphere, Space Sci. Rev., 35, 33 41, Molchanov, O. A., M. Parrot, M. M. Mogilevsy, and F. Lefeuvre, A theory of PLHR emissions to explain the weely variation of ELF data observed by a low-latitude satellite, Ann. Geophys., 9, , Par, C. G., and R. A. Helliwell, Magnetospheric effects of power line radiation, Science, 200(4343), 727, Parrot, M., Observations of power line harmonic radiation by the low-altitude Aureol-3 satellite, J. Geophys. Res., 99, , Richmond, A. D., Ionospheric electrodynamics, in Handboo of Atmospheric Electrodynamics, vol. 2, edited by H. Volland, pp , CRC Press, Boca Raton, Fla., Stix, T. H., The Theory of Plasma Waves, John Wiley, New Yor, Tatnall, A. R. L., J. P. Matthews, K. Bullough, and T. R. Kaiser, Power-line harmonic radiation and the electron slot, Space Sci. Rev., 35, , Y. Ando and M. Hayaawa, Department of Electronic Engineering, The University of Electro-Communications, Chofugaoa, Chofu, Toyo, Japan. (ando@whistler. ee.uec.ac.jp) O. A. Molchanov, Institute of Physics of the Earth, Russian Academy of Sciences, Moscow, Russia.