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The electrical stability of the electric field antennas in the plasmasphere
Journal of Atmospheric and Solar-Terrestrial Physics 63 (2001) 1217–1224
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The electrical stability of the electric (eld antennas in the plasmasphere E. Kolesnikovaa; ∗ , C. B/eghinb , R. Grardc , C.P. Escoubetc a Radio
and Space Plasma Physics Group, Department of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH, UK b Laboratoire de Physique et de Chimie de l’Environement, CNRS, Orl' eans, France c Space Science Department of ESA, ESTEC, Noordwijk, Netherlands Received 3 December 1999; accepted 18 September 2000
Abstract The long electric dipole antennas mounted on the POLAR satellite are unstable in the plamasphere, at L-shell values of about 2–3, where the Debye length is shorter than 30 cm; the frequency of the instability lies in the vicinity of the ambient electron plasma frequency. Each antenna consists of two spherical sensors, 8 cm in diameter, which are installed at the ends of wire-booms deployed in the spin plane and separated by distances longer than 100 m. Each sphere is electrically decoupled from the wire-boom by a double-stub, a symmetrical arrangement of two cables of 3 m each, whose ac potential is controlled by the output of a preampli(er with unit gain. A passive antenna is stable under all circumstances and the oscillation is necessarily induced by the active elements of the system. The instability is triggered when the Debye length is signi(cantly shorter than the stubs; this phenomenon is due to a capacitive coupling between the sensor and the stubs, when the self-impedance of the latter electrode is inductive. This problem is analysed using a numerical approach based on the surface charge distribution technique c 2001 Elsevier Science Ltd. All rights reserved. developed for an isotropic thermal plasma in the quasi-static approximation. Keywords: Electric (eld antenna; Electrical instability; Plasmasphere
1. Introduction The POLAR satellite was launched on 24 February ◦ 1996 on a orbit inclined at 90 , with apogee and perigee over the northern and southern polar caps, respectively, at geocentric distances of 1.8 and 9 Re. Two of the three electric antennas of the plasma wave instrument oscillate whenever the spacecraft crosses the plasmasphere at L ¡ 3, where the electron density exceeds a few 100 cm−3 . In addition, some rare cases of antenna oscillations were also observed in the boundary layer of the cusp region at a distance of 7–9 Earth radii. The instability gives rise to: (1) a narrow line signal at a few 100 kHz, in the vicinity of the plasma frequency, (2) several harmonics of this line and (3) a broad band noise in the lower-frequency range ∗
Corresponding author. E-mail address: [email protected] (E. Kolesnikova).
(J. Pickett, private communication). This instability is thought to induce a nonlinear response of the preampli(ers and a recti(cation process which also modi(es the Aoating potential of the antenna elements and distorts heavily the quasi-static electric (eld measurements. Similar signatures are observed on the signals collected with the two antennas which lie in the spin plane, at right angles to each other, and have overall lengths of 106 and 136 m. On the other hand, the third antenna parallel to the spin axis which is signi(cantly shorter, ∼15 m tip-to-tip, is always stable. Each of the monopoles which form the antenna array is an active system made of a spherical sensor with built-in preampli(er which drives the ac potential of adjacent stubs. We think that this con(guration, which was meant to improve the quality of the dc measurements, can be unstable due to the coupling between the probe and the stubs in certain plasma conditions.
c 2001 Elsevier Science Ltd. All rights reserved. 1364-6826/01/$ - see front matter PII: S 1 3 6 4 - 6 8 2 6 ( 0 0 ) 0 0 2 2 4 - 8
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E. Kolesnikova et al. / Journal of Atmospheric and Solar-Terrestrial Physics 63 (2001) 1217–1224
We propose here a theoretical explanation of the oscillations, which is valid in a cold (∼1 eV) isotropic or weakly magnetised environment, such as exists in the plasmasphere, where the ratio of the electron gyrofrequency over the plasma frequency does not exceed, say, 0.3. In another paper (Kolesnikova and B/eghin, 2000), we demonstrated that in the cusp region the antenna instability is triggered by the dense suprathermal electron cloud of the solar wind origin. The conditions of instability in the streaming plasma diFer from those in the immobile plasma, but the instability mechanism is basically the same. Finally, it is shown that the new electric antenna system which will Ay on the forthcoming CLUSTER 2 mission should be immune from any instability.
2. Plasma environment
Fig. 1. Empirical relation derived from space observations giving the electron density versus the potential diFerence between a spacecraft and a reference electrode, for given bias currents and an electron mean kinetic energy of 0:5 eV.
2.1. The diagnostic technique No POLAR instrument is dedicated to the analysis of the thermal plasma. We shall therefore apply the method used by Escoubet et al. (1997) on ISEE-1 to derive the electron density from the spacecraft potential. This technique is particularly sensitive in relatively rare(ed plasmas where a sunlit body develops a positive Aoating potential which reAects the balance between the Aows of outgoing photoelectrons and incoming ambient plasma particles. The spacecraft potential is measured with respect to one of the spherical sensors which form the electric antenna array. The positive potential of this electrode is reduced by means of a current source and a signi(cant fraction of the photoelectrons emitted from the surface is injected into the medium. The intensity of this current is less than the saturation level, but much larger, in magnitude, than the net ambient plasma current. As a result, the sensor is maintained at a small positive potential close to that of space and is independent in the (rst approximation, of the plasma environment characteristics. This probe can therefore serve as a reference for monitoring the Auctuations of the spacecraft potential. The plasma current and the spacecraft potential are very sensitive to the electron density but are practically not inAuenced by the temperature, at least in the range of parameters under consideration. This is due to the fact that, for a given density, any increase of the electron Aux due to a variation of the temperature drives the vehicle less positive, but simultaneously reduces the ion sheath size and consequently the spacecraft equivalent cross-sectional area. An empirical relation giving the electron density, Ne , as a function of the potential diFerence between the satellite and the reference probe, Vsp = Vs − Vp , is shown in Fig. 1, for diFerent values of the bias current and a plasma mean kinetic energy of 0.5 eV.
Fig. 2. Variation of the potential of the reference probe with respect to that of the spacecraft, Vs − Vp , around periapsis on 2 July 1996, between 06 and 11 UT.
2.2. Measurements The Auctuation of the spacecraft potential around periapsis is illustrated in Fig. 2 for the orbit represented in Fig. 3. POLAR enters and leaves the plasmasphere at 06:37 and 08:30 UT on the down leg and at 09:20 and 10:31 UT on the up leg of the orbit; crosses the polar cap from 08:30 to 9:20 UT. The drastic variations of JV = −Vsp observed at those times are associated with the large plasma density gradient which characterises the plasmapause, auroral cavity and the polar-cap density depletions. It can be veri(ed from the wave spectrum and quasi-static electric (eld data that the spacecraft relative potential variations observed between 09:32 and 10:00 (hatched area) are not due to a local rarefaction of the plasma but to a recti(cation of the ac signal induced by the antenna instability. The dashed line
E. Kolesnikova et al. / Journal of Atmospheric and Solar-Terrestrial Physics 63 (2001) 1217–1224
Fig. 3. Projection of the POLAR orbit on 2 July 1996 in the YZ-plane of the geocentric solar magnetospheric coordinate system, superimposed upon a few (eld lines of the Tsyganenko 96 model.
is evidence of the fact that the average spacecraft potential Vsp is smaller — and the electron density larger — during the second crossing of the plasmasphere, when the instability is observed. The spacecraft potential measurement is not meaningful when the antenna is unstable but the level, which immediately precedes the onset or follows the end of the oscillation yields the density threshold above which this phenomenon takes place. The bias current was 173.8 nA on 2 July 1996 and it is seen with reference to Figs. 1 and 2 that an electron mean kinetic energy of 0.5 eV yields an electron density of 370 cm−3 . A statistical study performed over 411 plasmaspheric crossings reveals that instabilities are observed during 64% of these crossings; the average threshold electron density is 264 cm−3 , corresponding to a Debye length of 38.3 cm, if we assume a mean kinetic energy of 0.5 eV.
3. Antenna and amplier 3.1. Con
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Fig. 4. Con(guration of a monopole with interconnection and equivalent circuits. Table 1 Dimensions of the monopoles (m) Element
Spin plane Length
Stub Guard Boom
3 0.1 60:7=45:7
Spin axis Diameter 10−3
2× 2 × 10−3 2 × 10−3
Length
Diameter
0.646 8:9 × 10−2 5.2
6 × 10−3 1:7 × 10−2 2 × 10−2
1.8 m, the overall separations between the centres of the spherical sensors are estimated to be about 100 and 130 m in the spin plane and 13.8 m along the spin axis (Harvey et al., 1995). The boom and cable braids are electrically connected to the spacecraft structure; the dc potentials of the monopole elements (sensor, stubs, guards and screens) are oFset in various ways, in order to control the Aow and exchange of photoelectrons emitted from sunlit surfaces, reduce the signal at spin frequency induced by solar irradiation and improve the quality of quasi-static electric (eld measurements. 3.2. Equivalent circuit The ac signal collected by each sphere is fed to a built-in preampli(er with a gain |A| close to unity and a phase shift ’ negligible at low frequencies (Fig. 5). The adjacent stubs are connected to the preampli(er output through a high-pass (lter and play the role of electric guards, which reduces the coupling between the sensor and the boom and improves the quality of ac (eld measurements in a vacuum environment (Fig. 4b).
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M Fig. 5. Transfer function of the preampli(er (L. Ahlen, private communication).
As the dc potentials of the sensors, stubs and guards play no role in the ac amplifying chain which is only considered here, we will ignore the actual bias dc current in the equivalent circuit given in Fig. 4c; Is and Ig are the ac currents emitted from the sphere and the stubs; Vs and V0 are the signals at the input and output of the ampli(er; Vg is the feedback voltage applied to the stubs; Vb , the voltage of the boom which is electrically connected to the spacecraft structure, serves as reference; Ci = 2 pF is the equivalent input capacitance of the electronic circuit; Cc =2 pF is the stray coupling capacitance between the input and output of the preampli(er; Cg =220 nF and Rg =100 kN form a high-pass (lter through which the output signal is applied to the stubs. The components of the equivalent circuit represented in Fig. 4c are all located within the monopole (sensor and stubs); the coupling eFect which take place in the surrounding plasma, between the individual elements of the monopole and between these elements and the boom (structure included), are not indicated in this diagram.
Fig. 6. Open-loop con(gurations of the amplifying circuit.
4. Stability of an active monopole
where ! is the working angular frequency and
4.1. Nyquist criterion
V0 = |A|exp(i’) Vi
The two monopoles of one single antenna are considered as separate entities, since each one is (tted with its own ampli(er and will, in practice, oscillate at frequencies which are not rigorously identical. We study the stability using the Nyquist criterion, which states that a system is not stable if the locus of the open-loop gain in the complex plane circles the point of abscissa +1 when the working frequency varies from zero to in(nity. The circuit is opened at the front-end of the electronic circuit, as shown in Fig. 6a. No current Aows at this location, due to the large input impedance of the preampli(er, and the load connected to the sensor is not modi(ed when the loop is so opened. The transfer function of the loop is Vs V 0 Vs f(!) = = ; (1) Vi Vi V 0
(2)
is the gain of the ampli(er (Fig. 5). If the ampli(er is ideal (|A| = 1; ’ = 0), the transfer function reduces to f(!) =
Vs ; V0
(3)
this ratio is de(ned by the passive elements of the equivalent circuit and the network which represents the external coupling between the sphere (S), the stubs (G) and the boom and structure (B), through the ionised environment. The contribution of the ambient plasma to the feedback process is singled out in Fig. 6b. With an ideal preampli(er (A = 1), no stray capacitances (Ci = Cc = 0) and a direct link between output and stubs (Cs = ∞), we can write V0 = Vg and f(!) =
Vs : Vg
(4)
E. Kolesnikova et al. / Journal of Atmospheric and Solar-Terrestrial Physics 63 (2001) 1217–1224
Fig. 7. Real part of the normalised potential pro(le around a point charge (after Kolesnikova and B/eghin, 2000).
In fact, the role of the stray capacitances is not predominant and that of the connecting (lter is negligible, because the cut-oF frequency, (2Rs Cs )−1 = 7 Hz, lies well below the range of interest. 4.2. Mechanism of the instability We shall (rst describe in simple terms some aspects of the interaction between an electrode carrying an alternating charge and a warm and isotropic plasma, in the quasi-static approximation. We shall then provide a qualitative explanation of the mechanism responsible for the instability of an active monopole, based upon considerations about the impedance of a body and the potential distribution in its environment. It will be concluded that an active system develops an instability whenever three basic conditions are ful(lled: (1) the self-impedance of the stubs is inductive, (2) the Debye length is much smaller than the distance between the stubs and the spacecraft structure, (3) the Debye length is
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at the same time much larger than both the size of the sensor (the radius of the sphere) and the separation between the sensor and the stubs (Fig. 4). In other words, the instability is observed at frequencies below the electron plasma frequency, within a range of Debye lengths which depends on the con(guration of the antenna. The complex ac potential around an electrode is de(ned by the alternating charges distributed over its surface or induced in its environment. Fig. 7, for example, represents the real part of the potential around a sphere much smaller that D (a point charge), as a function of the normalised radial distance, = r=D , when the working frequency, f, is smaller than the plasma frequency, fp , by 3% (N = 0:97); the imaginary part of the potential is small and is ignored in the (rst approximation. The potential gradient at short distances ( ¡ 0:5) is controlled by the charge carried by the body as in a vacuum, whereas at larger distances the eFect of the charge induced in the plasma must be taken into account. The ratio of the real part of the potential to the charge, V=q, is positive, i.e. the impedance with respect to in(nity of a body with dimensions much smaller than D remains a capacitance even when N ¡ 1. The variation of the potential along a (eld line which intersects the surface of a body may therefore take one of the forms qualitatively represented in Fig. 8: (1) when D is in(nite the medium behaves like a vacuum; (2) when D is (nite, but much larger than the size of the body, the pro(le resembles that illustrated in Fig. 7; (3) when the size of the body is much larger than D , the eFect of the induced charges is preponderant and the surface potential is negative, but the polarity of the electric (eld remains unchanged in the close vicinity of the surface; (4) when D = 0, the potential variation is monotonic as in (1), but the polarity is reversed because the permittivity of a cold plasma is negative when N ¡ 1. The impedance with respect to the in(nity of the body is capacitive in (1) and (2) because the charge and potential are in phase, inductive in (3) and (4) because the charge and potential are out of phase. Let us now focus our attention on curve (3) and let us place a point Aoating electrode (sensor) at a distance of
Fig. 8. Schematic representation of the potential along (eld lines intersecting the surface of an electrode, for various Debye lengths.
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Fig. 9. Simpli(ed open-loop circuit illustrating the plasma coupling between the sensor, the stubs and the spacecraft structure.
the order of or less than D from the surface of the body (stubs); the potential of the sensor, Vs , and that of the stubs, Vg , are such that Vs =Vg ¿ 1. If the sensor and the stubs are, respectively, connected at the input and output of an ideal ampli(er with unit gain and in(nite input impedance (Fig. 6b), then the conditions for instability are ful(lled. The open-loop con(guration of this idealised circuit is shown in Fig. 9; the coupling between the stubs and the sensor is represented by a capacitance 1 de(ned by the stubs and the equipotential surface on which the sensor is located; the global impedance of the stubs is inductive and symbolically represented by the series combination of 1 and a negative capacitance −2 (|2 | ¡ 1 ). The input impedance of the preampli(er is in(nite (Is = 0) and the open-loop gain given by Eq. (4), Vs =Vg = 1 =(1 − 2 ), is necessarily larger than unity. For a more rigorous treatment of this problem, the reader is referred to the work of Kolesnikova and B/eghin (2000).
where vjk is the ac potential induced at the surface element j by an oscillating unit charge located in the surface element k. The coePcient vjk are functions of (1) the working frequency f, (2) the density and temperature (or electron plasma frequency fp and Debye length D ) of the ambient medium and (3) the separation between the grid elements j and k. We shall use the exact values of the potential Vjk which have been formulated by B/eghin (1995) for an isotropic Maxwellian plasma. The set of Eq. (5) is expressed in terms of n charges and n potentials, but the electrical state of the system is further constrained by the relations which reAect the way in which the n surface elements are connected. The sensor, stubs and boom, whose surfaces are approximated by the (nite elements with indices 1 to l; l + 1 to m and m + 1 to n, respectively, are equipotential and we can write Vs (i = 1; : : : ; l); Vi = Vg (i = l + 1; : : : ; m); (6) Vb (i = m + 1; : : : ; n): The boom and spacecraft potential Vb is taken as a reference, and estimating a ratio such as Vs =Vg amounts to evaluating the potential of the sphere Vs for any given value of the stubs potential Vg . The eFective number of unknowns is therefore reduced to n+1. A complementary set of relations, involving N equations with N − 1 additional unknowns and constraining further the charge carried by the electrodes, is clearly required to solve this problem. The derivatives of these charges, i.e. the currents, depend upon the internal connections which link the electrodes and on the con(guration of the built-in electronic circuit.
5. Numerical approach
5.2. Interconnection equations
5.1. The surface charge distribution method
The charges carried by the sphere (S) and the stubs (G) can be expressed as the integrals of the emitted currents Is and Ig de(ned in Fig. 6a, namely
The plasma is again assumed to be homogeneous and isotropic, and the Debye length is much larger than the diameters of the spherical sensor and cylindrical stubs and boom. Electrostatic interactions between the monopole and its environment (ion sheath, wake, photoelectrons) are neglected; it is therefore assumed that the outer surfaces of the electrodes do not interfere with the motion of the plasma particles and can be approximated by transparent and conductive meshes. The coupling eFect induced by the plasma is computed in the quasi-static approximation with the surface charge distribution method of B/eghin and Kolesnikova (1998). The surface of the whole system is divided into n elements; each grid element carries a discrete charge qj and develops a potential Vj . The potentials and charges distributed over this network are related by n equations of the form Vj =
n k=1
qk vjk ;
(5)
l
qk = −i
k=1 m k=l+1
Is ; !
(7)
Ig !
(8)
qk = −i
which can be rewritten l
qk = V0 Cc − Vs (Cc + Ci );
k=1 m k=l+1
q k = V 0 C g − Vg C g −
i !Rg
(9) :
(10)
A new variable, V0 , has been introduced and the number of independent relations between the elementary charges qk has been increased by 2; the ratio Vs =V0 can therefore be
E. Kolesnikova et al. / Journal of Atmospheric and Solar-Terrestrial Physics 63 (2001) 1217–1224
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Fig. 10. Modulus of the open-loop gain versus normalised frequency and Nyquist diagram with D = 28 cm.
derived from the set of Eqs. (5) and (6) and (9) and (10). The transfer function of the open-loop circuit is obtained by multiplying this ratio by the gain of the ampli(er. If we simplify the equivalent circuit as in Fig. 6b, the sphere (S) is Aoating, does not sustain any current, carries no charge and Eqs. (9) and (10) are replaced by l
qk = 0:
(11)
k=1
6. Numerical results 6.1. Open-loop gain The equivalent circuit of the system is illustrated in Fig. 6a; the boom has the maximum length given in Table 1 and the presence of the spacecraft body is ignored. The boom and stubs are divided into cylindrical elements, 0.1 D in length, and the sphere is approximated by a number of octogons and pentagons with dimensions much less than D . The open-loop gain modulus vs. frequency and the associated Nyquist diagram are plotted in Fig. 10 for D =28 cm and fp = 200 kHz. The diagram circles the point of abscissa +1 and the circuit is therefore unstable at a normalised frequency N ¡ 1. Additional simulation runs have demonstrated the following points: (1) standing charge distributions with pseudo-sinusoidal waveforms and wavelengths of several 10 D develop along the stubs and the boom; (2) a dipole with short stubs is more stable than one with long stubs; an instability takes place when the net charge of the stubs is in anti-phase with its potential; (3) the standing wave has a vanishingly small amplitude at large distances from the sensor; it is concluded that extending the boom length inde(nitely would not improve
Fig. 11. Boundaries between plasma regions where an active antenna is stable (upper left) and unstable (lower right) for eFective lengths of 130 m (full line) or 13.8 m (dashed line) (after Kolesnikova and B/eghin, 2000).
the stability and that the eFect of the spacecraft body is negligible; (4) the supporting boom has a stabilizing aFect, i.e. a dipole with no boom (an academic situation) is prone to oscillate for a larger Debye length; (5) the built-in ampli(er adds a negative phase shift and reduces the open-loop gain; a circuit including an ideal ampli(er with A = 1 is unstable for a shorter Debye length; (6) neglecting the internal stray capacitances (Ci = Cc = 0) and assuming that the time constant (Rs Cs )−1 is zero has no signi(cant eFect on the open-loop gain. The stabilities of the longest and shortest monopoles carried by POLAR, namely those with stubs of 3 and 0.646 m which are, respectively, perpendicular and parallel to the spin axis (Table 1), are compared in Fig. 11. Each curve divides the plane which characterises the ionised environment, temperature versus plasma frequency, into upper and lower regions where the antennas are respectively stable and unstable. The
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Fig. 12. Con(guration of the CLUSTER 2 monopoles.
fact that the long antenna is occasionally unstable, but that the short antenna is always stable indicates that D is sometimes less than about 30 cm but always longer than, say, 8 cm along the orbit of POLAR. 7. Corrective action The experience gained with POLAR has been taken into account in the design of the electric antennas which have been developed for CLUSTER 2. This ESA magnetospheric mission was launched in mid-2000 and consists of four identical spacecraft, each of which carries two electric antennas, 93 m in overall length, mounted in the spin plane at right angle to each other. The con(guration of each monopole is illustrated in Fig. 12. The spherical sensor has a diameter of 8 cm and is connected to the input of the preampli(er by a thin wire, 1.5 m in length and 0.3 mm in diameter. The box which contains the preampli(er has a height of 3 cm and has the same diameter as the sphere. The preampli(er housing and the (rst 1.5 m of cable are connected to the output of the preampli(er through a (lter. The shield of the inner section of the cable-boom is maintained at the potential of the spacecraft structure; the diameter of the cable is 2.7 mm. The equivalent circuit of this monopole is not too diFerent, in principle, from that shown in Fig. 4c. There is, however, a major diFerence in the (lter; a bypass capacitor of about 20 000 pF is connected in parallel with the resistor Rg , which implies that the amplitude of the signal applied to the stubs is practically reduced to zero when the frequency exceeds a few 100 Hz and that no instability can develop at the plasma frequency. 8. Conclusion A quite simple model of the performance of the POLAR antennas in conjunction with the surface charge distribution
method, can explain the mechanism responsible for the instability observed near the plasma frequency, when the spacecraft crosses the plasmasphere. The conditions for instability are ful(lled while the coupling between the stubs and the sensor is capacitive, but the self-impedance of the stubs is inductive. This is achieved for certain values of the Debye length relative to the stubs size and the boom length. The numerical simulation shows that the long monopoles of the POLAR satellite must oscillate when D is less than about 30 cm, a quantity to be compared with an experimental value of 38 cm derived from the spacecraft potential measurements. It is concluded that an active antenna in an ionised environment is potentially unstable, unless the feedback potential applied to the stubs or guards is reduced to zero in the frequency range where an instability can develop. Acknowledgements E.K. was supported during the course of this study by the ESA Grant and PPARC Grants PPA=G=O=1997=00254 and PPA=G=O=1999=00181. We thank O. Storey for his help and support in evaluating this paper. References B/eghin, C., 1995. Series expansion of electrostatic potential radiated by a point of source in isotropic Maxwellian plasma. Radio Science 30, 307–322. B/eghin, C., Kolesnikova, E., 1998. The surface-charge distribution approach for modelization of quasistatic electric antennas in isotropic thermal plasma. Radio Science 33, 503. Escoubet, C.P., Pedersen, A., Schmidt, R., Lindquist, P.A., 1997. Density in the magnetosphere inferred from ISEE-1 spacecraft potential. Journal of Geophysical Research 102, 17,595–17,609. Harvey, P., Mozer, F.S., Pankow, D., Wygant, J., Maynard, N.C., Singer, H., Sullivan, W., Anderson, P.B., PfaF, R., Aggson, T., Pedersen, A., FRalthammar, C.-G., Tanskannen, P., 1995. The electric (eld instrument on the Polar satellite. Space Science Review 71, 583–596. Kolesnikova, E., B/eghin, C., 2000. The instability problem of the electric (eld antennas on the Polar spacecraft. Radio Science, accepted for publication.
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The electrical stability of the electric (eld antennas in the plasmasphere E. Kolesnikovaa; ∗ , C. B/eghinb , R. Grardc , C.P. Escoubetc a Radio
and Space Plasma Physics Group, Department of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH, UK b Laboratoire de Physique et de Chimie de l’Environement, CNRS, Orl' eans, France c Space Science Department of ESA, ESTEC, Noordwijk, Netherlands Received 3 December 1999; accepted 18 September 2000
Abstract The long electric dipole antennas mounted on the POLAR satellite are unstable in the plamasphere, at L-shell values of about 2–3, where the Debye length is shorter than 30 cm; the frequency of the instability lies in the vicinity of the ambient electron plasma frequency. Each antenna consists of two spherical sensors, 8 cm in diameter, which are installed at the ends of wire-booms deployed in the spin plane and separated by distances longer than 100 m. Each sphere is electrically decoupled from the wire-boom by a double-stub, a symmetrical arrangement of two cables of 3 m each, whose ac potential is controlled by the output of a preampli(er with unit gain. A passive antenna is stable under all circumstances and the oscillation is necessarily induced by the active elements of the system. The instability is triggered when the Debye length is signi(cantly shorter than the stubs; this phenomenon is due to a capacitive coupling between the sensor and the stubs, when the self-impedance of the latter electrode is inductive. This problem is analysed using a numerical approach based on the surface charge distribution technique c 2001 Elsevier Science Ltd. All rights reserved. developed for an isotropic thermal plasma in the quasi-static approximation. Keywords: Electric (eld antenna; Electrical instability; Plasmasphere
1. Introduction The POLAR satellite was launched on 24 February ◦ 1996 on a orbit inclined at 90 , with apogee and perigee over the northern and southern polar caps, respectively, at geocentric distances of 1.8 and 9 Re. Two of the three electric antennas of the plasma wave instrument oscillate whenever the spacecraft crosses the plasmasphere at L ¡ 3, where the electron density exceeds a few 100 cm−3 . In addition, some rare cases of antenna oscillations were also observed in the boundary layer of the cusp region at a distance of 7–9 Earth radii. The instability gives rise to: (1) a narrow line signal at a few 100 kHz, in the vicinity of the plasma frequency, (2) several harmonics of this line and (3) a broad band noise in the lower-frequency range ∗
Corresponding author. E-mail address: [email protected] (E. Kolesnikova).
(J. Pickett, private communication). This instability is thought to induce a nonlinear response of the preampli(ers and a recti(cation process which also modi(es the Aoating potential of the antenna elements and distorts heavily the quasi-static electric (eld measurements. Similar signatures are observed on the signals collected with the two antennas which lie in the spin plane, at right angles to each other, and have overall lengths of 106 and 136 m. On the other hand, the third antenna parallel to the spin axis which is signi(cantly shorter, ∼15 m tip-to-tip, is always stable. Each of the monopoles which form the antenna array is an active system made of a spherical sensor with built-in preampli(er which drives the ac potential of adjacent stubs. We think that this con(guration, which was meant to improve the quality of the dc measurements, can be unstable due to the coupling between the probe and the stubs in certain plasma conditions.
c 2001 Elsevier Science Ltd. All rights reserved. 1364-6826/01/$ - see front matter PII: S 1 3 6 4 - 6 8 2 6 ( 0 0 ) 0 0 2 2 4 - 8
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E. Kolesnikova et al. / Journal of Atmospheric and Solar-Terrestrial Physics 63 (2001) 1217–1224
We propose here a theoretical explanation of the oscillations, which is valid in a cold (∼1 eV) isotropic or weakly magnetised environment, such as exists in the plasmasphere, where the ratio of the electron gyrofrequency over the plasma frequency does not exceed, say, 0.3. In another paper (Kolesnikova and B/eghin, 2000), we demonstrated that in the cusp region the antenna instability is triggered by the dense suprathermal electron cloud of the solar wind origin. The conditions of instability in the streaming plasma diFer from those in the immobile plasma, but the instability mechanism is basically the same. Finally, it is shown that the new electric antenna system which will Ay on the forthcoming CLUSTER 2 mission should be immune from any instability.
2. Plasma environment
Fig. 1. Empirical relation derived from space observations giving the electron density versus the potential diFerence between a spacecraft and a reference electrode, for given bias currents and an electron mean kinetic energy of 0:5 eV.
2.1. The diagnostic technique No POLAR instrument is dedicated to the analysis of the thermal plasma. We shall therefore apply the method used by Escoubet et al. (1997) on ISEE-1 to derive the electron density from the spacecraft potential. This technique is particularly sensitive in relatively rare(ed plasmas where a sunlit body develops a positive Aoating potential which reAects the balance between the Aows of outgoing photoelectrons and incoming ambient plasma particles. The spacecraft potential is measured with respect to one of the spherical sensors which form the electric antenna array. The positive potential of this electrode is reduced by means of a current source and a signi(cant fraction of the photoelectrons emitted from the surface is injected into the medium. The intensity of this current is less than the saturation level, but much larger, in magnitude, than the net ambient plasma current. As a result, the sensor is maintained at a small positive potential close to that of space and is independent in the (rst approximation, of the plasma environment characteristics. This probe can therefore serve as a reference for monitoring the Auctuations of the spacecraft potential. The plasma current and the spacecraft potential are very sensitive to the electron density but are practically not inAuenced by the temperature, at least in the range of parameters under consideration. This is due to the fact that, for a given density, any increase of the electron Aux due to a variation of the temperature drives the vehicle less positive, but simultaneously reduces the ion sheath size and consequently the spacecraft equivalent cross-sectional area. An empirical relation giving the electron density, Ne , as a function of the potential diFerence between the satellite and the reference probe, Vsp = Vs − Vp , is shown in Fig. 1, for diFerent values of the bias current and a plasma mean kinetic energy of 0.5 eV.
Fig. 2. Variation of the potential of the reference probe with respect to that of the spacecraft, Vs − Vp , around periapsis on 2 July 1996, between 06 and 11 UT.
2.2. Measurements The Auctuation of the spacecraft potential around periapsis is illustrated in Fig. 2 for the orbit represented in Fig. 3. POLAR enters and leaves the plasmasphere at 06:37 and 08:30 UT on the down leg and at 09:20 and 10:31 UT on the up leg of the orbit; crosses the polar cap from 08:30 to 9:20 UT. The drastic variations of JV = −Vsp observed at those times are associated with the large plasma density gradient which characterises the plasmapause, auroral cavity and the polar-cap density depletions. It can be veri(ed from the wave spectrum and quasi-static electric (eld data that the spacecraft relative potential variations observed between 09:32 and 10:00 (hatched area) are not due to a local rarefaction of the plasma but to a recti(cation of the ac signal induced by the antenna instability. The dashed line
E. Kolesnikova et al. / Journal of Atmospheric and Solar-Terrestrial Physics 63 (2001) 1217–1224
Fig. 3. Projection of the POLAR orbit on 2 July 1996 in the YZ-plane of the geocentric solar magnetospheric coordinate system, superimposed upon a few (eld lines of the Tsyganenko 96 model.
is evidence of the fact that the average spacecraft potential Vsp is smaller — and the electron density larger — during the second crossing of the plasmasphere, when the instability is observed. The spacecraft potential measurement is not meaningful when the antenna is unstable but the level, which immediately precedes the onset or follows the end of the oscillation yields the density threshold above which this phenomenon takes place. The bias current was 173.8 nA on 2 July 1996 and it is seen with reference to Figs. 1 and 2 that an electron mean kinetic energy of 0.5 eV yields an electron density of 370 cm−3 . A statistical study performed over 411 plasmaspheric crossings reveals that instabilities are observed during 64% of these crossings; the average threshold electron density is 264 cm−3 , corresponding to a Debye length of 38.3 cm, if we assume a mean kinetic energy of 0.5 eV.
3. Antenna and amplier 3.1. Con
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Fig. 4. Con(guration of a monopole with interconnection and equivalent circuits. Table 1 Dimensions of the monopoles (m) Element
Spin plane Length
Stub Guard Boom
3 0.1 60:7=45:7
Spin axis Diameter 10−3
2× 2 × 10−3 2 × 10−3
Length
Diameter
0.646 8:9 × 10−2 5.2
6 × 10−3 1:7 × 10−2 2 × 10−2
1.8 m, the overall separations between the centres of the spherical sensors are estimated to be about 100 and 130 m in the spin plane and 13.8 m along the spin axis (Harvey et al., 1995). The boom and cable braids are electrically connected to the spacecraft structure; the dc potentials of the monopole elements (sensor, stubs, guards and screens) are oFset in various ways, in order to control the Aow and exchange of photoelectrons emitted from sunlit surfaces, reduce the signal at spin frequency induced by solar irradiation and improve the quality of quasi-static electric (eld measurements. 3.2. Equivalent circuit The ac signal collected by each sphere is fed to a built-in preampli(er with a gain |A| close to unity and a phase shift ’ negligible at low frequencies (Fig. 5). The adjacent stubs are connected to the preampli(er output through a high-pass (lter and play the role of electric guards, which reduces the coupling between the sensor and the boom and improves the quality of ac (eld measurements in a vacuum environment (Fig. 4b).
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M Fig. 5. Transfer function of the preampli(er (L. Ahlen, private communication).
As the dc potentials of the sensors, stubs and guards play no role in the ac amplifying chain which is only considered here, we will ignore the actual bias dc current in the equivalent circuit given in Fig. 4c; Is and Ig are the ac currents emitted from the sphere and the stubs; Vs and V0 are the signals at the input and output of the ampli(er; Vg is the feedback voltage applied to the stubs; Vb , the voltage of the boom which is electrically connected to the spacecraft structure, serves as reference; Ci = 2 pF is the equivalent input capacitance of the electronic circuit; Cc =2 pF is the stray coupling capacitance between the input and output of the preampli(er; Cg =220 nF and Rg =100 kN form a high-pass (lter through which the output signal is applied to the stubs. The components of the equivalent circuit represented in Fig. 4c are all located within the monopole (sensor and stubs); the coupling eFect which take place in the surrounding plasma, between the individual elements of the monopole and between these elements and the boom (structure included), are not indicated in this diagram.
Fig. 6. Open-loop con(gurations of the amplifying circuit.
4. Stability of an active monopole
where ! is the working angular frequency and
4.1. Nyquist criterion
V0 = |A|exp(i’) Vi
The two monopoles of one single antenna are considered as separate entities, since each one is (tted with its own ampli(er and will, in practice, oscillate at frequencies which are not rigorously identical. We study the stability using the Nyquist criterion, which states that a system is not stable if the locus of the open-loop gain in the complex plane circles the point of abscissa +1 when the working frequency varies from zero to in(nity. The circuit is opened at the front-end of the electronic circuit, as shown in Fig. 6a. No current Aows at this location, due to the large input impedance of the preampli(er, and the load connected to the sensor is not modi(ed when the loop is so opened. The transfer function of the loop is Vs V 0 Vs f(!) = = ; (1) Vi Vi V 0
(2)
is the gain of the ampli(er (Fig. 5). If the ampli(er is ideal (|A| = 1; ’ = 0), the transfer function reduces to f(!) =
Vs ; V0
(3)
this ratio is de(ned by the passive elements of the equivalent circuit and the network which represents the external coupling between the sphere (S), the stubs (G) and the boom and structure (B), through the ionised environment. The contribution of the ambient plasma to the feedback process is singled out in Fig. 6b. With an ideal preampli(er (A = 1), no stray capacitances (Ci = Cc = 0) and a direct link between output and stubs (Cs = ∞), we can write V0 = Vg and f(!) =
Vs : Vg
(4)
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Fig. 7. Real part of the normalised potential pro(le around a point charge (after Kolesnikova and B/eghin, 2000).
In fact, the role of the stray capacitances is not predominant and that of the connecting (lter is negligible, because the cut-oF frequency, (2Rs Cs )−1 = 7 Hz, lies well below the range of interest. 4.2. Mechanism of the instability We shall (rst describe in simple terms some aspects of the interaction between an electrode carrying an alternating charge and a warm and isotropic plasma, in the quasi-static approximation. We shall then provide a qualitative explanation of the mechanism responsible for the instability of an active monopole, based upon considerations about the impedance of a body and the potential distribution in its environment. It will be concluded that an active system develops an instability whenever three basic conditions are ful(lled: (1) the self-impedance of the stubs is inductive, (2) the Debye length is much smaller than the distance between the stubs and the spacecraft structure, (3) the Debye length is
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at the same time much larger than both the size of the sensor (the radius of the sphere) and the separation between the sensor and the stubs (Fig. 4). In other words, the instability is observed at frequencies below the electron plasma frequency, within a range of Debye lengths which depends on the con(guration of the antenna. The complex ac potential around an electrode is de(ned by the alternating charges distributed over its surface or induced in its environment. Fig. 7, for example, represents the real part of the potential around a sphere much smaller that D (a point charge), as a function of the normalised radial distance, = r=D , when the working frequency, f, is smaller than the plasma frequency, fp , by 3% (N = 0:97); the imaginary part of the potential is small and is ignored in the (rst approximation. The potential gradient at short distances ( ¡ 0:5) is controlled by the charge carried by the body as in a vacuum, whereas at larger distances the eFect of the charge induced in the plasma must be taken into account. The ratio of the real part of the potential to the charge, V=q, is positive, i.e. the impedance with respect to in(nity of a body with dimensions much smaller than D remains a capacitance even when N ¡ 1. The variation of the potential along a (eld line which intersects the surface of a body may therefore take one of the forms qualitatively represented in Fig. 8: (1) when D is in(nite the medium behaves like a vacuum; (2) when D is (nite, but much larger than the size of the body, the pro(le resembles that illustrated in Fig. 7; (3) when the size of the body is much larger than D , the eFect of the induced charges is preponderant and the surface potential is negative, but the polarity of the electric (eld remains unchanged in the close vicinity of the surface; (4) when D = 0, the potential variation is monotonic as in (1), but the polarity is reversed because the permittivity of a cold plasma is negative when N ¡ 1. The impedance with respect to the in(nity of the body is capacitive in (1) and (2) because the charge and potential are in phase, inductive in (3) and (4) because the charge and potential are out of phase. Let us now focus our attention on curve (3) and let us place a point Aoating electrode (sensor) at a distance of
Fig. 8. Schematic representation of the potential along (eld lines intersecting the surface of an electrode, for various Debye lengths.
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Fig. 9. Simpli(ed open-loop circuit illustrating the plasma coupling between the sensor, the stubs and the spacecraft structure.
the order of or less than D from the surface of the body (stubs); the potential of the sensor, Vs , and that of the stubs, Vg , are such that Vs =Vg ¿ 1. If the sensor and the stubs are, respectively, connected at the input and output of an ideal ampli(er with unit gain and in(nite input impedance (Fig. 6b), then the conditions for instability are ful(lled. The open-loop con(guration of this idealised circuit is shown in Fig. 9; the coupling between the stubs and the sensor is represented by a capacitance 1 de(ned by the stubs and the equipotential surface on which the sensor is located; the global impedance of the stubs is inductive and symbolically represented by the series combination of 1 and a negative capacitance −2 (|2 | ¡ 1 ). The input impedance of the preampli(er is in(nite (Is = 0) and the open-loop gain given by Eq. (4), Vs =Vg = 1 =(1 − 2 ), is necessarily larger than unity. For a more rigorous treatment of this problem, the reader is referred to the work of Kolesnikova and B/eghin (2000).
where vjk is the ac potential induced at the surface element j by an oscillating unit charge located in the surface element k. The coePcient vjk are functions of (1) the working frequency f, (2) the density and temperature (or electron plasma frequency fp and Debye length D ) of the ambient medium and (3) the separation between the grid elements j and k. We shall use the exact values of the potential Vjk which have been formulated by B/eghin (1995) for an isotropic Maxwellian plasma. The set of Eq. (5) is expressed in terms of n charges and n potentials, but the electrical state of the system is further constrained by the relations which reAect the way in which the n surface elements are connected. The sensor, stubs and boom, whose surfaces are approximated by the (nite elements with indices 1 to l; l + 1 to m and m + 1 to n, respectively, are equipotential and we can write Vs (i = 1; : : : ; l); Vi = Vg (i = l + 1; : : : ; m); (6) Vb (i = m + 1; : : : ; n): The boom and spacecraft potential Vb is taken as a reference, and estimating a ratio such as Vs =Vg amounts to evaluating the potential of the sphere Vs for any given value of the stubs potential Vg . The eFective number of unknowns is therefore reduced to n+1. A complementary set of relations, involving N equations with N − 1 additional unknowns and constraining further the charge carried by the electrodes, is clearly required to solve this problem. The derivatives of these charges, i.e. the currents, depend upon the internal connections which link the electrodes and on the con(guration of the built-in electronic circuit.
5. Numerical approach
5.2. Interconnection equations
5.1. The surface charge distribution method
The charges carried by the sphere (S) and the stubs (G) can be expressed as the integrals of the emitted currents Is and Ig de(ned in Fig. 6a, namely
The plasma is again assumed to be homogeneous and isotropic, and the Debye length is much larger than the diameters of the spherical sensor and cylindrical stubs and boom. Electrostatic interactions between the monopole and its environment (ion sheath, wake, photoelectrons) are neglected; it is therefore assumed that the outer surfaces of the electrodes do not interfere with the motion of the plasma particles and can be approximated by transparent and conductive meshes. The coupling eFect induced by the plasma is computed in the quasi-static approximation with the surface charge distribution method of B/eghin and Kolesnikova (1998). The surface of the whole system is divided into n elements; each grid element carries a discrete charge qj and develops a potential Vj . The potentials and charges distributed over this network are related by n equations of the form Vj =
n k=1
qk vjk ;
(5)
l
qk = −i
k=1 m k=l+1
Is ; !
(7)
Ig !
(8)
qk = −i
which can be rewritten l
qk = V0 Cc − Vs (Cc + Ci );
k=1 m k=l+1
q k = V 0 C g − Vg C g −
i !Rg
(9) :
(10)
A new variable, V0 , has been introduced and the number of independent relations between the elementary charges qk has been increased by 2; the ratio Vs =V0 can therefore be
E. Kolesnikova et al. / Journal of Atmospheric and Solar-Terrestrial Physics 63 (2001) 1217–1224
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Fig. 10. Modulus of the open-loop gain versus normalised frequency and Nyquist diagram with D = 28 cm.
derived from the set of Eqs. (5) and (6) and (9) and (10). The transfer function of the open-loop circuit is obtained by multiplying this ratio by the gain of the ampli(er. If we simplify the equivalent circuit as in Fig. 6b, the sphere (S) is Aoating, does not sustain any current, carries no charge and Eqs. (9) and (10) are replaced by l
qk = 0:
(11)
k=1
6. Numerical results 6.1. Open-loop gain The equivalent circuit of the system is illustrated in Fig. 6a; the boom has the maximum length given in Table 1 and the presence of the spacecraft body is ignored. The boom and stubs are divided into cylindrical elements, 0.1 D in length, and the sphere is approximated by a number of octogons and pentagons with dimensions much less than D . The open-loop gain modulus vs. frequency and the associated Nyquist diagram are plotted in Fig. 10 for D =28 cm and fp = 200 kHz. The diagram circles the point of abscissa +1 and the circuit is therefore unstable at a normalised frequency N ¡ 1. Additional simulation runs have demonstrated the following points: (1) standing charge distributions with pseudo-sinusoidal waveforms and wavelengths of several 10 D develop along the stubs and the boom; (2) a dipole with short stubs is more stable than one with long stubs; an instability takes place when the net charge of the stubs is in anti-phase with its potential; (3) the standing wave has a vanishingly small amplitude at large distances from the sensor; it is concluded that extending the boom length inde(nitely would not improve
Fig. 11. Boundaries between plasma regions where an active antenna is stable (upper left) and unstable (lower right) for eFective lengths of 130 m (full line) or 13.8 m (dashed line) (after Kolesnikova and B/eghin, 2000).
the stability and that the eFect of the spacecraft body is negligible; (4) the supporting boom has a stabilizing aFect, i.e. a dipole with no boom (an academic situation) is prone to oscillate for a larger Debye length; (5) the built-in ampli(er adds a negative phase shift and reduces the open-loop gain; a circuit including an ideal ampli(er with A = 1 is unstable for a shorter Debye length; (6) neglecting the internal stray capacitances (Ci = Cc = 0) and assuming that the time constant (Rs Cs )−1 is zero has no signi(cant eFect on the open-loop gain. The stabilities of the longest and shortest monopoles carried by POLAR, namely those with stubs of 3 and 0.646 m which are, respectively, perpendicular and parallel to the spin axis (Table 1), are compared in Fig. 11. Each curve divides the plane which characterises the ionised environment, temperature versus plasma frequency, into upper and lower regions where the antennas are respectively stable and unstable. The
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Fig. 12. Con(guration of the CLUSTER 2 monopoles.
fact that the long antenna is occasionally unstable, but that the short antenna is always stable indicates that D is sometimes less than about 30 cm but always longer than, say, 8 cm along the orbit of POLAR. 7. Corrective action The experience gained with POLAR has been taken into account in the design of the electric antennas which have been developed for CLUSTER 2. This ESA magnetospheric mission was launched in mid-2000 and consists of four identical spacecraft, each of which carries two electric antennas, 93 m in overall length, mounted in the spin plane at right angle to each other. The con(guration of each monopole is illustrated in Fig. 12. The spherical sensor has a diameter of 8 cm and is connected to the input of the preampli(er by a thin wire, 1.5 m in length and 0.3 mm in diameter. The box which contains the preampli(er has a height of 3 cm and has the same diameter as the sphere. The preampli(er housing and the (rst 1.5 m of cable are connected to the output of the preampli(er through a (lter. The shield of the inner section of the cable-boom is maintained at the potential of the spacecraft structure; the diameter of the cable is 2.7 mm. The equivalent circuit of this monopole is not too diFerent, in principle, from that shown in Fig. 4c. There is, however, a major diFerence in the (lter; a bypass capacitor of about 20 000 pF is connected in parallel with the resistor Rg , which implies that the amplitude of the signal applied to the stubs is practically reduced to zero when the frequency exceeds a few 100 Hz and that no instability can develop at the plasma frequency. 8. Conclusion A quite simple model of the performance of the POLAR antennas in conjunction with the surface charge distribution
method, can explain the mechanism responsible for the instability observed near the plasma frequency, when the spacecraft crosses the plasmasphere. The conditions for instability are ful(lled while the coupling between the stubs and the sensor is capacitive, but the self-impedance of the stubs is inductive. This is achieved for certain values of the Debye length relative to the stubs size and the boom length. The numerical simulation shows that the long monopoles of the POLAR satellite must oscillate when D is less than about 30 cm, a quantity to be compared with an experimental value of 38 cm derived from the spacecraft potential measurements. It is concluded that an active antenna in an ionised environment is potentially unstable, unless the feedback potential applied to the stubs or guards is reduced to zero in the frequency range where an instability can develop. Acknowledgements E.K. was supported during the course of this study by the ESA Grant and PPARC Grants PPA=G=O=1997=00254 and PPA=G=O=1999=00181. We thank O. Storey for his help and support in evaluating this paper. References B/eghin, C., 1995. Series expansion of electrostatic potential radiated by a point of source in isotropic Maxwellian plasma. Radio Science 30, 307–322. B/eghin, C., Kolesnikova, E., 1998. The surface-charge distribution approach for modelization of quasistatic electric antennas in isotropic thermal plasma. Radio Science 33, 503. Escoubet, C.P., Pedersen, A., Schmidt, R., Lindquist, P.A., 1997. Density in the magnetosphere inferred from ISEE-1 spacecraft potential. Journal of Geophysical Research 102, 17,595–17,609. Harvey, P., Mozer, F.S., Pankow, D., Wygant, J., Maynard, N.C., Singer, H., Sullivan, W., Anderson, P.B., PfaF, R., Aggson, T., Pedersen, A., FRalthammar, C.-G., Tanskannen, P., 1995. The electric (eld instrument on the Polar satellite. Space Science Review 71, 583–596. Kolesnikova, E., B/eghin, C., 2000. The instability problem of the electric (eld antennas on the Polar spacecraft. Radio Science, accepted for publication.