Self Inductance of a Solenoid with a Permanent-Magnet Core

Transcription

1 1 Probem Sef Inductance of a Soenoid with a Permanent-Magnet Core Kirk T. McDonad Joseph Henry Laboratories, Princeton University, Princeton, NJ (March 3, 2013; updated October 19, 2018) Deduce the sef inductance L of a soenoida coi of N turns of radius r and ength r when its core is a cyinder with permanent-magnetization density M parae to the soenoid axis. Consider an osciatory current I(t) in the coi, and show that the EMF across the coi, due to the permanent magnet, has the character of a capacitance rather than an inductance. 2 Soution In the imit of arge /r the magnetic fied due to the permanent magnetization density M = M ẑ is (in SI units) 1 μ 0 M = μ 0 M ẑ (inside), B M (1) 0 (outside). In the quasistatic approximation, where radiation is negected, it seems reasonabe to suppose that the magnetic fied due to the current I(t) is 2 μ 0 NI ẑ/ (inside), B I (2) 0 (outside). The energy stored in the magnetic fied B I, which is significant ony inside the voume πr 2 of the cyinder, is given by B 2 U I = I dvo 1 μ 0 πn 2 r 2 I 2. (3) 2μ 0 2 Stored energy of the form (3) hods whenever the magnetic fied is due to eectrica currents that have started up from zero. This exampe incudes permanent magnetism, which can be thought of as due to permanent supercurrents that do not change as the current I in the rest of the circuit varies. We reca that the force on a sma magnetic dipoe m in an externa magnetic fied B ext is 3 F = (m B ext) = U m, where U m = m B ext. (4) 1 See, for exampe, pp of [1]. 2 The direction of the z-axis is chosen such that eq. (2) hods. Then, for negative M, B M is antiparae to B I. 3 See,forexampe,p.87of[2]. 1

2 If the permanent magnet in the present exampe is hed at rest with respect to the surrounding coi (with any fixed reation between the directions of the magnetization M and the fied B I of the coi), no work is done by the sum of the forces (4) on the magnetic dipoes in the magnet as the magnetic fied increases from zero. Athough the quantity U m of eq. (4) changes in this process, this quantity does not correspond to a change in energy of the system. Hence, we can say that to within a constant the tota energy stored in the system (other than in the power source of the eectrica currents) is simpy that given by eq. (3), which can be written in the famiiar form where the sef inductance L 0 of the coi is U I = 1 2 LI 2, (5) L = μ 0πN 2 r 2, (6) which is the same as if the permanent magnet were not present. 2.1 The Permanent Magnet Can Move with Respect to the Coi It coud aso be that the permanent magnet is not fixed in pace with respect to the coi as the atter is energized. We suppose that the permanent magnet is sma enough that it ies within the uniform fied region of the coi, such that the force on the magnetic is negigibe. Then, the center of mass of the magnet does not move with respect to the coi. 4 However, if the tota magnetic moment MV M,whereV M is the voume of the magnet, is not parae to the magnetic fied B I of the coi, the permanent magnet experiences a torque, τ = MV M B I = B MV M μ 0 B I, (7) and wi rotate as a consequence if the motion of the magnet is unconstrained. If the cyindrica magnet were constrained to rotate ony about its symmetry axis, which has a fixed direction (parae to the magnetization M), then the torque (7) has no component aong the axis of rotation and does not infuence the rotation of the magnet; the quantity U M = MV M B I does not have the significance of a stored energy that can affect the sef inductance of the coi. For a more interesting case, suppose instead that the cyindrica magnet is constrained to rotate about an axis perpendicuar to its symmetry axis (which is parae to the magnetization M), with the axis of rotation being fixed perpendicuar to the symmetry axis of the coi, as sketched in the figure on the next page. Let θ be the variabe ange between M and B I and be the moment of inertia of the magnet about the its (fixed) axis of rotation. 5 4 If the center of mass of the magnet did move, but the magnet is sma compared to the coi, the fux of magnetic fied through the coi due to the permanent magnet does not change, and there woud be no effect on the circuit. This is consistent with the concusion that the center of mass of the magnet does not move. 5 Ange θ =90 in the figure. 2

3 The rotationa equation of motion of the magnet is d 2 θ dt = τ = B MV M μ 0 NI(t) sin θ = NB MV M I(t) sin θ (8) 2 μ 0 If the initia ange is θ 0 and we write θ = θ 0 + ϑ, then the equation of motion (8) can be written as d 2 ϑ dt = NB MV M I(t) (sin θ 2 0 cos ϑ +cosθ 0 sin ϑ). (9) In an AC circuit where the coi is in series with a resistor R and the current is I(t) =I 0 cos ωt, the equation of motion of the permanent magnet becomes d 2 ϑ dt = NB MV M I 0 cos ωt(sin θ 2 0 cos ϑ +cosθ 0 sin ϑ). (10) This has the osciatory soution, NB M V M I 0 cos θ 0 =0, sin θ 0 = ±1, ϑ = ϑ 0 cos ωt, ϑ 0 =sinθ 0, (11) ω 2 for sma ϑ 0. The votage source V (t) does mechanica work on the rotating magnet at rate P mech = d dt ϕ 2 = τ ϕ NB ( ) MV M I 0 cos ωt NB M V M I 0 sin θ 0 ω sin θ 0 cos ωt ω 2 = N 2 BM 2 V M 2 I ω cos ωt sin ωt, (12) so the tota (instantaneous) power provided by the source is, recaing eq. (5), 6 P = VI = I 2 R + du I dt + P mech = I 2 R + LII + N 2 BM 2 V M 2 I ω cos ωt sin ωt I [ ( mech = I IR + L N ) ] 2 BMV 2 M 2 I. (13) ω We might aso consider the interaction magnetic-fied energy, U int B I B M V m /μ 0, which is roughy equa and opposite to the magnetic-moment energy U M = MV M B I = B I B M V m /μ 0 U int.since these energies argey cance, the anaysis eading to eq. (15) overestimates the effective capacitance. Difficuties in evauating force and energy in systems with permanent magnets are discussed, for exampe, in [3, 4, 5]. 3

4 If we now switch to compex notation, writing V = V 0 e iωt, I = I 0 e iωt with a I 0 compex constant, then I = iωi and the impedance Z of the system is Z = V I = R + iωl + N 2 B 2 M V 2 M iω 2 R + iωl + 1 iωc. (14) We can say that the effect of the osciating permanent magnet on the system is not so much to change the sef inductance (6) of the coi, 7 but to give it an effective capacitance, C = 2 N 2 B 2 M V 2 M. (15) The system behaves ike a series R-L-C circuit, with resonant anguar frequency ω 0 = 1 LC = B M V M μ0 πr 2, (16) at which frequency the magnitude of the current is maxima, with I 0 = V 0 /R. Eectromechanica resonances have been observed in the apparatus described in [6] (private communication, David J. Jefferies). The capacitance induced by a rotating magnetic fied underies the sensor described in [7]. The magnet can aso make fu rotations, driven by the AC power source, in which case the system is a kind of singe-phase motor, as first demonstrated by Bay [8] in References [1] K.T. McDonad, Eectricity and Magnetism, Lecture 8, [2] K.T. McDonad, Eectricity and Magnetism, Lecture 7, [3] H.C. Lovatt and P.A. Watterson, Energy Stored in Permanent Magnets, IEEE Trans. Mag. 35, 505 (1999), [4] P. Campbe, Comments on Energy Stored in Permanent Magnets, IEEE Trans. Mag. 36, 401 (2000), 7 As the permanent magnet rotates, the fux of its magnetic fied through the coi varies, and an EMF is induced in the circuit (as pointed out by Pei-Hsun Jiang). For θ = θ 0 + ϑ with sma ϑ, this fux is proportiona to ϑ, sotheemf is proportiona to ϑ, which is proportiona to I/ω, recaing eqs. (11)-(12). Whie this EMF (reactance) is associated with magnetism, its dependence on current and frequency is that of a capacitive, rather than an inductive, reactance. It is certainy not associated with the sef inductance of the circuit, since the permanent magnet is not part of the nomina eectric circuit. And, since the magnet is permanent, rather than an eectromagnetic, it is not to be associated with a mutua inductance in the usua sense. 8 Variants of fywhees have ong been used as energy storage devices in mechanica systems, and aso in eectromechanica systems such as motor-generator sets, athough such fywhees are generay not magnets. 4

5 [5] S. Sanz et a., Evauation of Magnetic Forces in Permanent Magnets, IEEE Trans. App.. Semi. 20, 846 (2010), [6] M.N. Wybourne et a., Frequency-crossing phonon spectrometer techniques, Rev. Sci. Instr. 50, 1634 (1979), [7] F.-M. Hsu et a., A Nove Magnetic-Induced Capacitive-Sensing Rotation Sensor, IEEE Sensors, 1 (2012), [8] W. Bay, A Mode of producing Arago s Rotation, Proc. Phys. Soc. London 3, 115 (1879), 5