Design, building and experimental results of a facility to test hysteresis rod parameters
Preprint, Inst. Appl. Math., the Russian Academy of Science

( . )

M.L.Battagliere, F.Grazziani, N.V.Kupriyanova, M.Y.Ovchinnikov, V.I.Penkov
( .., ., .., .., ..)

Russian Academy of Science, Keldysh Institute of Applied Mathematics
. ..

Moscow, 2007

The work was supported by the Russian Foundation for Basic Research (Grants 06-01-00389, 07-01-92001), the Program of the Leading Scientific Schools Support (Grant NSh-2448.2006.1) and Italian Space Agency


A passive magnetic attitude control system is simple, easy to realize, low cost and does not require software development and energy consumption. Due to these features the passive magnetic attitude control system is one of the most common attitude control system implemented on board of small satellites. This type of attitude control system is based on a permanent magnet which provide a restoring torque to align the satellite with the Earth magnetic field direction and an energy dissipation system which consists on a set of magnetic hysteresis rods. Predicting of performances in orbit and the obtainable accuracy with this type of system is not so easy. The main problem is represented by necessity of an accurate modelling of the hysteresis rod magnetization and by the evaluation of the magnetic parameters of rods. The purpose of this work is to design, develop and verify a facility to test rods hysteresis in the laboratory.

, , . . , , , , . . , .


Passive Magnetic Attitude Control System (PMACS) is one of the most common attitude control system implemented on board of small satellites launched in orbit to perform low cost or high risk scientific experiments [1,2]. The aim of a nanosatellite use is to test in orbit new technology and to check ideas under a limited budget and short time for a development [3, 4, 5, 6]. Students of the School of Aerospace Engineering of University La Sapienza of Rome and Moscow Institute of Physics and Technology participate in development of PMACS [7, 8].

Choice of attitude control system is based on mission requirements, mission profile, requirements of the scientific instrumentation and requirements of satellite angular motion. In many cases, in actual fact the microsatellite or nanosatellite does not require high precision of orientation or complicated manoeuvres during the flight. Additionally, it can be preferable to avoid movable parts and to limit consume of energy and resources of the satellite and at long last another important requirement can be that the system is as low cost as possible. PMACS satisfies all these requirements [1, 9]. Usually it consists of a strong permanent magnet and energy dissipation system based on magnetic hysteresis rods. Axis of the permanent magnet is aligned with the oriented axis of the satellite in order to provide a restoring torque which aligns the satellite with Earth magnetic field. During motion of the satellite along the orbit the magnetic field direction changes, consequently the satellite oscillates around the direction of the magnetic field. To reduce amplitude of the oscillations a set of hysteresis rods in the equatorial plane of the permanent magnet is used as an energy dissipating system [8, 10]. This system provides also an initial dissipation to reduce oscillations due to ejection from a launcher [8]. The rods are made from magnetically soft material and their parameters depend strongly on manufacturing process and heat treatment; so it is not so easy to predict theoretically their behaviour. Rods efficiency depends on their volume, material used, technique of heat treatment, elongation (i.e. ratio of the length of rod to its diameter) and also on their arrangement on board of the satellite. In fact, it is well known that hysteresis rods influence mutually each other (demagnetization effect) [11] and that the best arrangement of hysteresis rods in the satellite body is to place these rods far from permanent magnet as much as possible; but these requirements are not satisfied when size of satellite is too small as in the case of a nanosatellite. In other hand, for the previous ten years a great increase of nanosatellite production tool place, mostly at universities and with Cubesat program coming after 2000 [12].

Ten years ago about 14 university spacecrafts had been launched [13]. Nowadays about 62 university-built spacecrafts have already been launched (and, at least, other 25 are scheduled to be launched in 2006/2007) by about 50 university teams in various countries of the world [14]. This trend in the space research and university teaching activity, as well as the necessity to develop an accurate modelling of the hysteresis rods magnetization to predict performance in orbit of PMACS suggested idea to develop a simple facility to perform tests in laboratory of the hysteresis rods[1] [15]. Main goal of these experimental activity carried out at the Keldysh Institute of Applied Mathematics of Russian Academy of Science (KIAM RAS) is to analyze the behaviour of hysteresis rods of two different kinds in presence of external magnetic field along the rod and to compare these results with the results obtained using the model already developed at KIAM RAS and incorporated to the developing of the PMACS of TNS-0 nanosatellite [15]. In this issue the design, realization and test of the facility is described and preliminary results of the tests carried out on the hysteresis rods are shown.



The facility to test hysteresis rods magnetization is based on a simple circuit. There is a transformer which provides the power supply. The transformer is connected with a variable resistor in parallel with a solenoid. The transformer has been modified to provide a variable voltage of about () in input of frequency. Variable resistor provides a maximum resistance of about and it is utilized in the scheme to change the value of the current which flows in the circuit. In this way it is possible to test behaviour of hysteresis rods at changing of magnetic field applied. After that, there is a solenoid realized with a copper wire rolled up at plastic cylindrical body with a diameter of . Resistor and main coil have been arranged on a wood base. Main parameters of the facility are resumed in the Table 1 and shown in Fig.1. Facility under realization and final arrangement are visible in Fig.2. In the Table 1 the resistance of the coil-1 has been calculated on the basis of well known formulas [16] and measured with a tester


. (2.1)


All symbols in this expression have already been explained in the list of symbols at the Appendix 3. To size coil-1 the following formulas [16] have been used


, (2.2)


where , the Earth magnetic field intensity and , the length of coil-1 are known; in this way there are two independent parameters, and , and the current can been modified using the variable resistor.


Fig.1. Main parameters of coil


Table 1. Main parameters of coil-1 and of the facility to test hysteresis rods magnetization



Resistor (Rp)

Measured Resistance of coil-1

Resistivity of wire

Section of wire


Length of wire for coil-1

inside of the solenoid


inside of the solenoid



Inductance of coil-1


Fig.2. Facility during the realization (left) and final assembling (right)

Number of loops for coil-1 has been chosen considering also that solenoid had to cover whole length of hysteresis rod to evaluate rod parameters changing position of the second coil. As a reference we considered for values variable from and ( and ) on the Earth surface; value of Earth magnetic field in LEO (Low Earth Orbit), for example at of height, is over the equator and over the poles. Maximum value of magnetic field inside of the solenoid () is approximately . It corresponds to . In this case hysteresis rod is saturated. We can estimate the intensity of saturation for the available rods for testing is approximately . Using variable resistor the value of current which flows in the solenoid and then the value of inside of coil-1 are changed.


Fig.3. An example of hysteresis loop for a ferromagnetic material (left). In the diagram the values of saturation and coercitive fields for hysteresis rod-1 available for test in laboratory are shown. The trend of magnetic permeability of a ferromagnetic material when B changes is presented (right)


A typical hysteresis loop which shows the relation between magnetizing field and induction for a ferromagnetic material is available (Fig.3). Main notations are: is a saturation induction, is a residual induction, is a coercitive force. Shape of this cycle depends on an amplitude of , material, its remagnetization history. Area inside of the cycle is proportional to dissipated energy during a cycle [16]. In the right the trend of magnetic permeability depending of is shown. The permeability begins by an initial value which corresponds at inclination of curve of first magnetization in the origin and grows until maximum value. After that it decreases and inclines to the value of an initial permeability [16]. After such a brief summary of main parameters of the ferromagnetic material and non-linearity of the magnetic permeability an idea of this experimental activity is discussed.

Working principle of this facility is based on induction law of Newman-Faraday-Lenz, that is, while a variable current flows in the coil-1 it generates in a secondary coil an induced electromagnetic force. The secondary coil (coil-2) is used as a sensor which moves along the solenoid (coil-1). We see signals from coils on oscilloscope screen. Channel-1 of oscilloscope is connected to the coil-1, channel-2 is connected to the coil-2. During the first step of the work the idea is to observe Lissajouss pictures (Appendix 1) generated by these two signals and to establish the relation between inclination of Lissajouss ellipse which depends on difference of phase of two signals and rod permeability along its length. Nevertheless, the non-linearity of the ferromagnetic material (Fig.3) which composes the rod limits the possibility to see a regular ellipse of Lissajous.

At the end of this section the problem of sizing of secondary coil and choice of its placing with respect to the coil-1 is considered.

Magnetic field inside of coil-1 is calculated using the formulas [16]


where the magnetic permeability of vacuum is equal to . Formulas (2.3) is valid for a solenoid with (length of solenoid is much more than radius of solenoid cross section) and not near extremities of solenoid. Induced flux in the secondary coil is expressed by [16]


, (2.4)

where is the magnetic field induction inside of the coil-1, is a number of loops of coil-2 and is the cross section of coil-2. Induced electromagnetic force in the coil-2 is evaluated on the basis of the following set of formula [16]


. (2.5)

Presence of factor reduces remarkably the amplitude of signal of the coil-2 (≈mV). It means that it is important to evaluate in way in order to get a visible signal on the screen of the oscilloscope considering also the presence of electromagnetic noise due to electrical network at within the laboratory. To watch a signal on the channel-2 we fixed its amplitude , that is, and at least. On the basis of this requirement the number of loops of coil-2 has been fixed at about 1500, so that . Test demonstrated that hysteresis rod amplifies this signal in about 20 times. Main parameters of coil-2 are sketched in the Table 2.

Table 2. Main parameters of coil-2 of the facility for testing of hysteresis rods magnetization

Measured resistance of coil-2,

Resistivity of wire

Cross section


Length of wire for coil-2

Section of wire

Inductance of coil-2


The next step has been to establish the better way to arrange the secondary coil with respect to the first coil to carry out measurements. Signal of the coil-2 depends also on position of coil-2 with respect to the field generated by coil-1. Signal is maximum when plane of the coils is perpendicular to force lines of field generated by coil-1 and minimum while the plan is parallel to the force lines. To maximize the factor of the mutual induction the scheme of arrangement sketched in Fig.4 has been chosen.


Fig.4. Scheme of arrangement of the coil-2 with respect to the coil-1


To move coil-2 along the coil-1 in a very simple way we measure punctual properties of the hysteresis rods placed inside of cylindrical support of coil-1.


A first elementary test has been performed to check functioning of the facility. This test allowed us, also, to confirm variation of the rod parameters along its length. A compass has been placed near the coil-1 without to supply the facility and without a rod inside the core of solenoid. Magnetic needle of compass finds ones bearings in the magnetic north direction (approximately 11.5 degrees far from geographic north direction). In this test we are not interested in the quantitative results but qualitative analysis in only. For this test it has been necessary to put a diode to supply facility with a continue current because we need of a constant field which changes direction of the compass needle. In this way we verify that facility works generating a magnetic field inside of the solenoid. To do this a check alimentation has been switched on. During this experience the value of variable resistor was fixed at which corresponds to . This value was comparable with value of Earth magnetic field which lies on the Earth surface approximately in the range from to . The compass needle did not change in a remarkable way but to verify the correct working of facility it was enough to put one hysteresis rod inside of core of solenoid: suddenly compass needle changed its position moving of about 15 degrees. In the pictures 1 and 3 of Fig.5 one sees this angular motion of the compass needle.


Fig.5. First test of working of the facility


Moving hysteresis rod inside of coil-1 great variations of the angle have been measured. This result confirmed the interest in the goal of this work, that is, to try to explain in which way parameters of the hysteresis rod change along its length and to develop a modelling useful for next applications on board of small satellites.



A second test has been performed to check the instrumentation available in the laboratory with aim to avoid uncertainties in the analysis of the results of tests on the hysteresis rods during the next experimental activity. The test has been carried out using a RLC circuit (Fig.6) with the main goal to check Lissajouss pictures visible on the screen of the oscilloscope depending of different kind of signals in input. The part RC of circuit has been sized (Appendix 2) in way to have two signals with the same amplitude to obtain a Lissajouss circle. Three resistances has been soldered in series in order to obtain a total resistance of about . Capacitor has been chosen to obtain a signal with the same amplitude () of the resistance (). Its capacity is equal to . As inductance has been chosen during the tests it has been possible to establish that the coil-1 behaved as a resistance because its inductance is very small. In fact a calculation of this inductance demonstrated that its value is approximately equal to , whereas its measured resistance is . In any case this fact did not restrict results of the test because the idea was to use signals with different phases to confirm theoretically expected Lissajouss pictures and it was done with a RC circuit.


Fig.6. Circuit RLC used to test operative way X-Y in the oscilloscope


Results obtained by connecting channel-1 of the oscilloscope with resistance and channel-2 with capacitor are shown in Fig.7.                   Amplitude of signal is about  the same  ( and capacitive reactance ) and they have a difference of phases of 90.

Voltages and measured with oscilloscope correspond at theoretical values. Corresponding Lissajouss picture is an ellipse very close to a circle with small disturbances.


Fig.7. Results for RC circuit: signals with respect to line (left) and corresponding Lissajouss picture (right)


The same test has been carried out by connecting channel-1 and channel-2 of the oscilloscope with the same resistance. In this case Lissajouss picture expected is a line with an inclination of 45. Results are visible in Fig.8.


Fig.8. Channel-1 and channel-2 connected at the same signal in input


After, channel-1 has been connected with capacitor and channel-2 with coil-1. Theoretically we exepect a difference of phases and Lissajouss picture is a line with an inclination of 135 but here this coil behaves as a resistance because . Results obtained with measurements confirm it (Fig.9).


Fig.9. Channel-1 connected at capacitor and channel-2 connected at coil-1


In Fig.8 we see two signals with where the second signal has an amplitude quite less then one of the first signal (and ). Correspondent Lissajouss picture is an ellipse without inclination with respect to axis X and Y. At the end, a test has been carried out with a small inductance () and a resistance (). Result confirmed a difference of phase of 90. These simple tests allowed us to confirm a correct working of the oscilloscope (after a compensation of the probes) and will be useful in the analysis of signal related with hysteresis rods.



It is well known [16] that in the extremities of a solenoid when a current flows there is an edge effect which halves the value of internal field. Neglecting values of field in the extremities of the solenoid (), measurements have been carried out from of the length of the solenoid to to verify the uniformity of inducting field in the central part of the coil-1. Voltage applied at the solenoid is corresponding to a current and to a field inside of the coil-1. Results available in Fig.10 show that there is not a very uniform behaviour of internal field along the coil-1. Probably these results depend on non-uniform distribution of coils during manufacturing process both coil-1 and coil-2. In any case these results have to be taken into account in the analysis of the measurements related to the hysteresis rods.


Fig.10. Voltage values measured with coil-2 along the solenoid without rod inside



Tests have been performed on two different hysteresis rods. The first rod named rod-1 has a length of and a section of with a rectangular shape. This type of rod has been utilized on board of MUNIN nanosatellite [10] of Institute of Space Physics of Kiruna (Sweden) launched on 21th of November, 2000 from Vandenberg Air Force Base located on the Central Coast of California with the Delta 7000 Launch Vehicle. Rod-1 has been manufactured with molybdenum permalloy of the 79NM specification; its composition includes 79% of Ni, 4% of Mo and 17% of Fe [10]. Main parameters of the rod are available in Table 3.


Table 3. Main parameters of the rod

Initial magnetic permeability


Maximum magnetic permeability

Coercitive force

, [A/m]

Residual induction

, [T]

Elongation of rod-1







Second hysteresis rod named rod-2 has a length of a section of with a rectangular shape and elongation of p=65. For the rod main parameters are not available but parameters of the material are available in [10, 17]. Information about elongation allows us to do some considerations, that is, the parameter p for rod-2 is very far with respect to the usual optimal values of elongation which lie in the range 200-300 [10]. In our work for reasons of briefness and to give a complete view of the results obtained with experimental tests of the rod-1 it will be refereed only diagrams and results for this rod. Results of rod-2 confirm the same behaviour of the rod-1. The shape of the signal is similar but much more irregularities appear and values of amplitude are greater. Tests have been performed on the hysteresis rod-1 changing the value of the current inside the coil-1. A summary of some values is available in Table 4 where is the voltage applied at coil-1, is a current which flows through the solenoid, and are, respectively, the magnetic field intensity and magnetic induction flux inside of the coil-1. In the first case () rod is not in saturation. In the second one the rod approaches the saturation. In the third case rod is in saturation. In the last case we evaluated behaviour of the rod in a limit condition when ( is saturation field intensity).


Table 4. Main values of different configurations for test on the hysteresis rod-1


















For voltage values , and it has been verified that signal shape generated by coil-2 in the oscilloscope without rod inside of the solenoid follows cosine law as it is evaluated in the relation (2.5) assuming . The shape of signal related at the rod-1 when and is sketched in Fig.11. Preliminary results of measuring showed that the magnetic field is maximum in the centre of the rods and minimum at the extremity and it should be noted that there is no important changing in the shape (only in the amplitude) of the signal when the position of the coil-2 varies along the rod-1. Tests have been repeated with the facility arranged in a metallic box to verify if noise occurs in these measurements but results obtained are the same.

To confirm this result about magnetization of the rod tests have been repeated with a different configuration (Fig.11). The goal was to check either the field in the extremities of rod is in different times less with respect to the value in the centre of rod due to the characteristics of magnetization along the rod or this is an effect of the field in extremities of the solenoid and an effect related to results showed in Fig.9.

Fig.11. Results of test on the hysteresis rod-1: amplitude of signal is minimum at the extremities and maximum in the centre of rod


In this configuration the coil-2 has been placed in different positions along the solenoid and the rod has been inserted in way to do measurements at its extremity. Scheme of this test with coil-2 arranged in the middle of solenoid (at ) is shown in Fig.12. Results demonstrated that this trend can be related to the characteristic of the magnetization of the rod.


Fig.12. Results of test on the hysteresis rod-1: coil-2 has been placed at 10 cm along the solenoid with respect to the left extremity and hysteresis rod has been inserted in the coil-1 in a way to have the left extremity inside of coil-2


Measurements have been completed by changing inclination of coil-1 with respect to Earth magnetic field whose direction has been specified with the compass. Measurements have been obtained for a random inclination (Series-I: random ), in the perpendicular direction of coil-1 with respect to Earth magnetic field direction (Series-II: normal ) and in the same direction of Earth magnetic field (Series-III: parallel ). In Fig.13, the diagram of the results obtained with different set of measurements in laboratory is sketched.

These results show that there is no a visible effect of the relative position with respect to the Earth magnetic field direction. Small differences in the measurements can depend on casual errors due to, for example, small differences in the position of coil-2 along coil-1 or in the readings of oscilloscope. When voltage applied to coil-1 increases the shape of signal changes. This changing is visible in Fig.14. Results for hysteresis rod-1 when and are available in Fig.15 and Fig.16 respectively.

Fig.13. Trend of magnetic field inside of rod-1: X axis represents the length of hysteresis rod, Y axis represents experimentally values obtained during tests in laboratory along the rod-1 for induced voltage in coil-2. Diagram shows 3 set of measurements with different positions of the facility with respect to Earth magnetic field direction


Fig.14. Results of test on the hysteresis rod-1: amplitude of signal is maximum in the centre of rod; in this picture we see signal corresponding at coil-2 placed at about 10.5 cm with respect to the extremity of left of the rod when

Results for confirm the same behaviour of these two previous cases. At this step of work we can not establish if this changing depends by saturation of the rod or not. There are not visible effects of the relative position between rod and Earth magnetic field.


Fig.15. Results for rod-1 for . Diagram shows three sets of measurements with different positions of the facility with respect to Earth magnetic field direction


Fig.16. Results for rod-1 for . Diagram shows three sets of measurements with different positions of the facility with respect to Earth magnetic field direction


A number of mathematical models are available to describe the hysteresis loop of soft magnetic material. We consider a simple approximating relation to simulate magnetization curve of hysteresis [10]:



where . Here sign + is used for the ascending branch (right side of the loop) and sign is used for the descending portion (left side) of the loop, , is the coercitive force, is the residual induction of the rod and is the saturation flux density of permeable rod. Considering this approximation on the basis of Newman-Faraday-Lenz law the following relation for induced signal in coil-2 is obtained




where and are the numbers of loop of coil-1 and coil-2 respectively, is the length of solenoid, is the section of coil-2, () is the pulsation and is the amplitude of current which flows through the solenoid. Graphic obtained by relation (7.2) using MATLAB tools is available in Fig.17. In this relation all material parameters are known except which is variable in the range for 4-79NM permalloy [17]. Usually it can be estimated considering for this material (it means that ) or evaluated from hysteresis loop (). Graphical representation shows that the signal shape obtained using model (7.1) gives a good approximation for the signal shape visible in the oscilloscope (while the road is in saturation) but the voltage values are very different. This result depends on parameters , and because we used material parameters in this simulation (road parameters can be very different).

Fig.17. Mathematical results of signal in the oscilloscope obtained considering

a simple approximating model (7.1) for hysteresis


If we do not put rod inside of the solenoid a mathematical relation which describes signal induced in coil 2 is more simple and it follows a cosine law


. (7.3)

On the basis of this analysis we can conclude that the typical shape of signal obtained for both rods depends on hysteresis properties. To confirm these results it is necessary to repeat the tests on the others roads and to compare results.

Changing value of current in the software used to simulate this signal a signal shape changes simulating the signal shape in the oscilloscope. To try to obtain a better analytical representation of the signal obtained during the tests in laboratory when voltage and we changed empirically the function (7.2). Using two parameters ( and ) a correct approximation for the shape of this signal has been obtained. For amplitude it is necessary to multiply by a reduction factor because we used, in this preliminary simulation, material parameters. Results of this first empirical approximation while are available in Fig.18. They can be compared with picture shown in Fig.11.


Fig.18. Analytical representation of measured signal obtained using a simple mathematical model for hysteresis loop of rod-1, and introducing factors and .


While current in the software is varied the shape of this function changes as the signal in the oscilloscope. Results for and are visible in Fig.19.


Fig.19. Analytical representation of measured signal obtained considering simple mathematical model for hysteresis loop of rod-1, , for (left) and (right)




Due to certain limitations we do not have the possibility to expose all experimental results obtained in the framework of this activity. In this first part we described all phases of sizing and development of a facility to carry out tests in laboratory on hysteresis rods. We described and showed tests carried out to verify correct working of this facility and after that we showed test results of the hysteresis rods. We verified experimentally that the magnetizing field in the rod is maximum in the centre of the rod length and minimum in the extremities. Changing the value of current inside of inducing coil (coil-1) we verified hysteresis rod behaviour in different configurations: normal working, approaching to the saturation and saturation. Effect of the relative position between hysteresis rod and Earth magnetic field direction has been also evaluated in all working conditions of the facility. In the last we determinate an analytical representation of signal obtained using coil-2 when hysteresis rod is arranged inside of the solenoid. This relation has been evaluated on the basis of theoretical considerations for two simple models of the hysteresis but it seems to approximate real signal shape only for values of current which correspond to of the rod. To obtain simulation of signal in normal working conditions of the rod and approaching to the saturation we modified empirically previous relations. Comparison between experimental results and pictures showed a good approximation but a more accurate analysis to interpret behaviour of signal related to rod when voltage applied to solenoid changes are requested. Tests to evaluate effects of a permanent magnet on the hysteresis rods have been also carried out. The next step of this work will be to interpret all experimental results and to try to develop a modelling for the magnetic permeability of the rod depending by its length.



The work was supported by the Russian Foundation for Basic Research (Grants N 06-01-00389 and N 07-01-92001), the Program of the Leading Scientific Schools Support (Grant N NSh-2448.2006.1) and Italian Space Agency.



[1] V.A.Sarychev and M.Yu. Ovchinnikov, Magnetic Attitude Control Systems for Artificial Earth Satellites. Survey of Science and Technology. Series: Exploration of Space, Moscow, 1985, VINITI, v.23, 104p.

[2] V.V.Beletskiy and A.A.Khentov, Rotational Motion of a Magnetized Satellite. Moscow: Nauka. Main Office of Physics and Mathematics Literature, 1985.

[3] M.Ovchinnikov, V.Pen'kov, O.Norberg and S.Barabash, Attitude Control System for the First Swedish Nanosatellite MUNIN, Acta Astronautica, 2000, V.46, Issue 2-6, pp.319-326.

[4] M.Yu.Ovchinnikov, V.I.Penkov, A.A.Ilyin and A.S.Selivanov, Magnetic Attitude Control Systems for the Nanosatellite Series, Extended Abstracts for 5th International Symposium of IAA "Small Satellites for Earth Observation", 4-8 April, 2005, Berlin, Germany, Paper IAA-B5-1201,pp.429-432

[5] F.Graziani, F.Santoni, F.Piergentili, F.Bulgarelli, M.Sgubini, M.Ronzitti, M.L.Battagliere, UNISAT microsatellites: an affordable way to test in orbit innovative technologies and ideas, 23rd AIAA International Communications Satellite Systems Conference, (ICSSC-2005), 25-28 Settembre, Roma

[6] M.Yu.Ovchinnikov, A.A.Ilyin, N.V.Kupriyanova, V.I.Penkov, Attitude dynamics of the first Russian nanosatellite TNS-0, Acta Astronauctica, 2007, Vol.61, Issue 1-6, 277-285

[7] F.Graziani, M.Ferrante, G.B.Palmerini, F.Santoni, P.Tortora, UNISAT Program, a University Tool for Space Education, paper IAF-00-P.2.07, 51st International Astronautical Congress, 2-6 Oct 2000, Rio de Janeiro, Brasil

[8] F.Santoni, M.Zelli, Passive Magnetic Attitute Stabilization of the Unisat-4 microsatellite, 57th International Astronautical Congress, Valencia, Spain, 2-6th October 2006

[9] Ovchinnikov M.Yu., Methods to Obtain the Principal Parameters of Simple ACS for Small Satellites, Acta Astronautica, March 1998, V.43, Issue 11-12, pp.597-605

[10] M.Yu.Ovchinnikov and V.I.Penkov, Passive Magnetic Attitude Control System for the Munin Nanosatellite, Cosmic Research, Vol.40, No. 2, 2002, pp 142-156

[11] A.P.Kovalenko, Magnetic Attitude Control Systems of Space Vehicles, M., Mechanical Engineering, 1975, 248 p


[13] M.Swartwout, University-Class Satellites: from marginal utility to Disruptive Research Platforms, SSC04-II-5, 18th Annual AIAA/USU, Conference on Small Satellites, Logan, UT, 9-12 August 2004

[14] M.Swartwout, Twenty (plus) year of University-Class Spacecraft: a review of what was, an understanding of what is and look at what should be next, 20th Annual AIAA/USU, August 2006

[15] V.A.Sarychev, V.I.Pen'kov, M.Yu.Ovchinnikov, Mathematical Model of a Hysteresis, Based on Magnet-Mechanical Analogy, J. Matematicheskoe Modelirovanie, 1989, v.1, N 4, pp.122-133

[16] D.Sette, M.Bertolotti, Lezioni di Fisica 2: Elettromagnetismo e ottica, Masson 1998, Milano, printed in Italy

[17] R.M.Bozort, Ferromagnetism, Van Nostrand Comapany, Reinhold, New York 1951


Let us regard two simple harmonic motions and, at first, we consider that they have the same frequency. We can chose origin of coordinate system to have initial phase equal to zero along X axis; equation for coordinate is


. (A.1.1)

Equation for coordinate is


, (A.1.2)

where is the difference of phase between oscillations and , is the frequency of signals and and are the amplitudes; we assume . Trajectory of particle is limited by line of equation and . When two motions are in phase and equation of trajectory is


. (A.1.3)


This equation is represents by PQ in Fig.A.

Fig.A. Composition of simple harmonic motions with the same frequency and perpendicular directions. Trajectory depends by difference of phase of two signals


If then and equation of trajectory is


. (A.1.4)


This is equation is represented by RS in Fig.A. It means that for and we have a linear polarization. If then , trajectory of particle is an ellipse with equation


. (A.1.5)

Particle moves along this ellipse in clockwise direction. To check this characteristic we evaluate velocity of particle in the point A


, (A.1.6)

when (or ) trajectory of particle is again an ellipse with axis parallel to coordinate axes but particle moves along this in anticlockwise direction. It means that for and there is an elliptical polarization in the composition of two simple harmonic motions. If trajectory becomes a circle and we say that there is a circular polarization for the composition of two signals. For a generic value of trajectory is again an ellipse but with axes inclined with respect to the coordinate axes. Changing coordinate system one can demonstrate the relation between inclination of main axes of ellipse with respect to coordinate axes and difference of phase of the two signal is


. (A.1.7)

Some possible trajectories for different (difference of phase) are available in Fig.B. These trajectories are called Lissajouss Figures.


Fig.B. Possible trajectories for different ∆δ between two perpendicular harmonic signals with the same frequency


Trajectory is an enclosed curve while two signals have the same frequencies. If then the trajectory is an open curve and shape of trajectory depends on the ratio and difference of phase . Lissajouss figures for different ratio and ∆δ are sketched in Fig.C.


Fig.C. Trajectories for different between two harmonic perpendicular signals with different ratio




Let us consider an RLC circuit supplied by an alternative voltage . Circuit equation is [16]


. (A.2.1)

This is a linear differential equation similar to the equation of forced oscillations in a mechanical system. Its solution is the sum of a homogenous solution of the associate equation and of a particular integral. Homogenous solution describes transitory. We are interested in the regime solution described by particular integral. Particular solution is a current


, (A.2.2)

where is the difference of phase and . To calculate and it needs to consider the impedance of circuit. Then the value of current is


, (A.2.3)

where is the resistance, is the inductive reactance (, where is the inductance) and is the capacitive reactance (, where is the capacity). To calculate the phase the following relation is used


. (A.2.4)

Voltages for R, L and C are






voltage peak (maximum amplitude)

voltage peak to peak


magnetic dipole intensity

number of loops of coil-1

number of loops of coil-1

cross section of wire

diameter of wire

diameter of cross section of coil-1

diameter of cross section of coil-2

length of coil-1

length of coil-2

length of wire for coil 1

length of wire for coil 2

resistivity of wire

cross section of coil-1

cross section of coil-2

resistance of potentiometer

resistance of coil-1

resistance of coil-2

magnetic field induction

magnetic field intensity

Earth magnetic field intensity

magnetic flux

induced electromotive force

magnetic permeability of vacuum

capacitive reactance

inductive reactance

[1] Idea to develop and build a laboratory facility for hyteresis rod parameters exploration in combination with advanced numerical model was suggested by Dr.Vladimir Penkov.