Motor/generator

[0002] This invention relates to the structure of a motor/generator.

[0003] Tokkai Hei 8-340663 published by the Japanese Patent Office in 1996 discloses a motor/generator comprising two rotors and one stator coaxially arranged in three layers wherein one of the rotors is driven as a motor and the other is driven as a generator.

[0004] In this motor/generator, wo sets of coils are installed in the stator for providing a magnetic field for respective rotors, and two sets of inverters, i.e., current controllers, are provided for controlling respective sets of coils.

[0005] However, in the case of this motor/generator, since two sets of coils and inverters are required, current losses such as copper losses or switching losses are large.

[0006] It is therefore an object of this invention to reduce current losses of a motor/generator having the construction described in the prior art.

[0007] In order to achieve the above object, this invention provides a motor/generator comprising a first rotor comprising plural magnetic poles and supported free to rotate, a second rotor comprising plural magnetic poles and supported free to rotate coaxially with the first rotor, a stator fixed co-axially with the first rotor, and a coil unit comprising plural coils disposed at equal angular intervals on the stator, wherein the coil unit forms plural rotating magnetic fields of equal number to the number of magnetic poles of the first rotor according to a first alternating current and forms plural rotating magnetic fields of equal number to the number of magnetic poles of the second rotor according to a second alternating current.

[0008] The motor/generator further comprises an electrical circuit for supplying a composite electrical current comprising the first alternating current and the second alternating current to the coil unit, and a circuit for compensating a torque fluctuation produced in either of the two rotors due to the non -uniformity of the magnetic field accompanying the relative rotation of the first rotor and second rotor.

[0009] The details as well as other features and advantages of this invention are set forth in the remainder of the specification and are shown in the accompanying drawings.

[0010] FIG. 1 is a schematic cross-sectional view of a motor/generator according to this invention.

[0011] FIG. 2 is similar to FIG. 1, but showing a second embodiment of this invention.

[0012] FIG. 3 is a schematic diagram of a control circuit according to this invention.

[0013] FIG. 4 is a schematic cross-sectional view of an inverter according to this invention.

[0014] FIG. 5 is a schematic cross-sectional view of a motor/generator used as a model for theoretically analyzing the principle of this invention.

[0015] FIGS. 6A-6C are diagrams showing a variation of the flux density in the motor/generator of FIG. 8.

[0016] FIG. 7 is a schematic cross-sectional view of a motor/generator having a magnetic pole number ratio of 2:1 wherein a stator has coils on its inner and outer circumference, conceived for the purpose of describing composite current flow.

[0017] FIGS. 8A-8D are diagrams showing a change of magnetic flux density in the motor/generator of FIG. 7.

[0018] FIG. 9 is similar to FIG. 7, but showing a case where outer and inner coils are integrated.

[0019] FIGS. 10A and 10B are diagrams showing a distribution of alternating current to drive the motor/generator of FIG. 9.

[0020] FIG. 11 is similar to FIG. 7, but showing a motor/generator with a magnetic pole number ratio of 3:1.

[0021] FIGS. 12A-12D are diagrams showing a change of magnetic flux density in the motor/generator of FIG. 11.

[0022] FIGS. 13A-13C are diagrams describing magnetic force interference in the motor/generator of FIG. 11.

[0023] FIG. 14 is similar to FIG. 11, but showing a case where outer and inner coils are integrated.

[0024] FIGS. 15A and 15B are diagrams showing a distribution of alternating current driving the motor/generator of FIG. 14.

[0025] FIG. 16 is a schematic cross-sectional view of a motor/generator according to a third embodiment of this invention.

[0026] FIG. 17 is similar to FIG. 16, but showing a fourth embodiment of this invention.

[0027] FIG. 18 is similar to FIG. 16, but showing a fifth embodiment of this invention.

[0028] Referring to FIG. 1 of the drawings, a motor/generator has a three-layer construction comprising a cylindrical stator 2 and rotors 3, 4 arranged with a predetermined clearance outside and inside the stator 2.

[0029] The stator 2, outer rotor 3 and inner rotor 4 are arranged coaxially. The stator 2 is fixed to a casing 21 of a motor/generator 1 as shown in FIG. 3. The outer rotor 3 is fixed to a frame 5, and the frame 5 rotates relative to the casing 21 via an axis 18. The inner rotor 4 rotates relative to the casing 21 via an axis 19.

[0030] The inner rotor 4 comprises permanent magnets having N poles and S poles respectively arranged on each of two semicircles, as shown in FIG. 1. The outer rotor 3 comprises permanent magnets having a pole number three times that of the rotor 4 with three S poles and N poles alternately arranged at 60 degree intervals.

[0031] When the magnetic pole number ratio of the outer rotor 3 and inner rotor 4 (referred to hereafter simply as magnetic pole number ratio) is 3:1 the rotating magnetic fields relative to the inner rotor 4 has no effect on the rotational torque of the outer rotor 3, as described later.

[0032] The rotating magnetic fields relative to the outer rotor 3 does have an effect on the rotation torque of the inner rotor 4, and causes a torque fluctuation thereof. However, as will be clear from the following theoretical analysis, this torque fluctuation may be expressed as a function of the phase difference (ω₁-ω₂) of the outer rotor 3 and inner rotor 4, and by adding an amplitude modulation to the alternating current supplied to the coils 6 of the stator 2, the torque fluctuation may be eliminated beforehand and the inner rotor 4 can be rotated under a fixed rotation torque. The stator 2 comprises three of the coils 6 per magnetic pole of the outer rotor 3, i.e., a total of eighteen coils. These coils 6 are arranged at equal intervals on the same circumference.

[0033] For convenience, the numbers #1-#9 are appended to these eighteen coils, as shown in FIG. 1.

[0034] To distinguish from part numbers, the symbol # will be appended before a number when referring to a coil.

[0035] The coils 6 are wound around six cores 7. The cores 7 are arranged in a circumferential direction across gaps 9. To reduce the total number of cores 7, the cores are split into three by slits 8 and three coils 6 are wound on one of the cores 7.

[0036] The following currents I₁-I₁₈are passed into these eighteen coils.

[0037] I₁=Ia+Id

[0038] I₂=Ic

[0039] I₃=Ib

[0040] I₄=Ia+If

[0041] I₅=Ic

[0042] I₆=Ib

[0043] I₇=Ia+Ie

[0044] I₈=Ic

[0045] I₁₀=)

[0046] I₁₁=I_c(=I)

[0047] I₁₂=Ib(=I)

[0048] I₁₃=(=)

[0049] I₁₄=Ic(=I)

[0050] I₁₅=Ib(=I₆)

[0051] I₁₆=(=)

[0052] I₁₇=Ic(=I)

[0053] I₉=Ib

[0054] I₁₈=Ib(=I)

[0055] Herein, current numbers with underlining show currents in the reverse direction.

[0056] In the equations for these composite currents, I₁₀-I₁₈are the reverse of 1₁-I₉l. Hence, in a combination where the magnetic pole number ratio is 3:1, the phase is reversed over half the circumference, and a nine-phase alternating current (half of eighteen-phase) may be used.

[0057] This is why the coil numbers #1-#9 are used. For example, alternating currents of the same phase but opposite direction are passed through each of the two coils #1. Similar processing is performed for the other coils #2-#9.

[0058] Next, the above current settings will be described in detail referring to FIG. 11.

[0059] For the purpose of comparison with FIG. 1, this diagram shows a case where specific coils d, f, e are provided on the inner circumference of the stator 2 stator generating rotating magnetic fields relative to the inner rotor 4, and specific coils a, c, b are provided on the outer circumference of the stator 2 for generating rotating magnetic fields relative to the outer rotor 3.

[0060] To combine these two groups of coils into the one group shown in FIG. 1, the current passed into the coil d is taken up by the coil a which is in the vicinity of the coil d, the current passed into the coil f is taken up by the coil a which is in the vicinity of the coil f, and the current passed into the coil e is taken up by the coil a which is in the vicinity of the coil e. Further the current passed through the coil d is taken up by the coil a which is in the vicinity of the coil d, the current passed into the coil f is taken up by the coil a which is in the vicinity of the coil f, and the current passed into the coil e is taken up by the coil a which is in the vicinity of the coil e.

[0061] The equations for the composite currents I₁-I₁₈mentioned above are mathematical expressions of this concept. However, the current may be set by other methods as described hereafter.

[0062] When this current setting is applied, two rotating magnetic fields can be formed simultaneously, i.e., rotating magnetic fields relative to the inner rotor 4 and rotating magnetic fields relative to the outer rotor 3, despite the use of one series of coils. The above current setting of Id, If, Ie is performed in synchronism with the rotation of the inner rotor 4 and the current setting of Ia, Id, Ib is performed in synchronism with the rotation of the outer rotor 3. A phase advance or delay is set relative to the direction of the torque, but this is the same as the case of an ordinary synchronous motor.

[0063] Next, referring to FIG. 3, the case will be described of a controller supplying the composite currents I₁-I₁₈mentioned above to the stator 2.

[0064] The controller comprises an inverter 12 to convert a direct current of a battery power supply 11 into an alternating current.

[0065] Since the total number of coils is eighteen, an inverter having a phase number equal to the total number of coils is required, i.e., an inverter which produces an eighteen-phase alternating current, but as the eighteen-phase current reverses every 180 degrees, an inverter which generates nine-phase alternating current may be used. Consequently, a smaller number of transistors and diodes is required to construct the inverter 12.

[0066] The inverter 12 corresponds to a common three-phase bridge type inverter converted to a nine-phase inverter as shown in FIG. 4, and it comprises eighteen transistors Tr1-Tr18 and the same number of diodes.

[0067] An ON/OFF signal supplied to each gate of the inverter 12, i.e., base of the transistors, is a pulse width modulation (PWM) signal.

[0068] Rotation angle sensors 14 and 13 which detect the phases of the rotors 3 and 4 are fitted. The signals from these sensors 13, 14 are input into a control circuit 15. A PWM signal is generated in the control circuit 15 based on positive or negative torque command values input to the control circuit 15 for specifying the desired rotational torque to be exerted on the outer rotor 3 and inner rotor 4.

[0069] In this way, in this motor/generator 1, two rotors 3 and 4 and one stator 2 are arranged coaxially in three layers, common coils 6 are provided in the stator 2, and a composite current is passed in the coils 6 so as to generate rotating magnetic fields inside and outside which are equal in number to the number of magnetic poles of the rotors 3 and 4. Therefore, when one of the rotors 3 and 4 is driven as a motor and the other is driven as a generator, a current differential between the motor drive power and the power generated by the generator may be passed into the coils 6, and thereby largely reduce current losses compared to the case when coils are provided separately for the rotors 3 and 4.

[0070] Further, as the two rotors 3 and 4 can be controlled by a single inverter 12, the cost of the inverter can be reduced, and as the power switching transistor capacitance of the inverter is reduced, switching efficiency improves.

[0071] On the other hand, regarding the constant torque fluctuation produced in the rotation of the inner rotor 4 due to the non-uniformity of the magnetic field accompanying the relative rotation of the two rotors 3 and 4, this torque fluctuation may be eliminated beforehand by adding an amplitude modulation to the component of the composite current which generates rotating magnetic fields relative to the outer magnets, and torque fluctuation is thereby suppressed.

[0072] Next, a second embodiment of the invention will be described referring to FIG. 2.

[0073] In the aforesaid first embodiment, there were six of the cores 7, but in this embodiment, all the coils 6 are wound on an identical core 7. However, slits 22 and 23 of predetermined width are provided at a fixed interval in the core 7 to largely increase the magnetic resistance so as to avoid interference between the magnetic fluxes generated by the coils 6. With this construction, the number of steps for manufacturing the motor/generator is less than in the aforesaid first embodiment.

[0074] In the first and second embodiments, the magnetic pole number ratio is 3:1, but according to theoretical analysis, except in the case where the magnetic pole number ratio is 2:1, one of the rotors 3, 4 or the rotating magnetic fields which drive that rotor causes a fluctuation in the rotation of the other rotor. This torque fluctuation is prevented by adding an amplitude modulation to the alternating current. This processing is also based on theoretical analysis.

[0075] Next, the theoretical analysis of the driving forces acting on the rotors will be performed with respect to the magnetic pole number ratios.

[0076] (1) N(2p—2p) type

[0077] First, describing the notation N(2p—2p), 2p on the left represents the number of magnetic poles of the permanent magnets of the outer rotor 3, and 2p on the right represents the number of magnetic poles of the permanent magnets of the inner rotor 4. Accordingly, the magnetic pole number ratio N(2p—2p) means a motor /generator wherein the magnetic polymer ratio of the outer rotor 3 and inner rotor 4 is 1:1.

[0078] N is a positive integer. If N is 1, the magnetic pole number of both the outer rotor 3 and inner rotor 4 is two, and if N is 2, the magnetic pole number of both the outer rotor 3 and inner rotor 4 is four. FIG. 5 shows the case of a motor/generator wherein N is 1.

[0079] (1—1) Basic formulae

[0080] In FIG. 5, if the permanent magnets of the outer rotor 3 (hereafter abbreviated as outer magnets) m₁and the permanent magnets of the inner rotor 4 (hereafter abbreviated as inner magnets) m₂are replaced by equivalent coils, the magnetic flux densities B₁, B₂of the permanent magnets may be represented by the following equations (1) and (2).

[0081] B₁=Bm₁·sin (ω₁·t−θ)=μ·Im₁·sin (ω₁·t−θ)   (1)

[0082] B₂=Bm₂·sin (ω₂·t+α−θ)=μ·Im₂·sin (ω₁·t+α−θ)   (2)

[0083] where,

[0084] Bm₁, Bm₂=magnetic flux amplitudes,

[0085] μ=magnetic permeability,

[0086] Im₁=equivalent direct current of outer magnets,

[0087] Im₂=equivalent direct current of inner magnets,

[0088] ω₁−rotational angular velocity of outer magnets,

[0089] ω₂=rotational angular velocity of inner magnets,

[0090] α=phase difference of outer and inner magnets (when t=0), and

[0091] t=elapsed time from time when phase of outer magnets and stator coil coincided.

[0092] If the current passed through the stator is a three-phase alternating current, the magnetic flux density Bc due to the stator coil is expressed by the following equation (3). $\begin{array}{cc}\begin{array}{c}\mathrm{Bc}=\mu ·n·\\ \left\{\mathrm{Ica}\left(t\right)·\sin \theta +\mathrm{Icb}\left(t\right)·\sin \left(\theta -\frac{2\pi }{3}\right)+\mathrm{Icc}\left(t\right)·\sin \left(\theta -\frac{4\pi }{3}\right)\right\}\end{array}& \left(3\right)\end{array}$

[0093] where, n=coil constant.

[0094] In equation (3), Ica(t), Icb(t), Icc(t) are currents which are different in phase by 120 degrees.

[0095] The variation of the aforesaid magnetic flux densities B₁, B₂and Bc is shown in FIGS. 6A-6C. The magnetic flux density changes as a sine-wave, and a total magnetic flux density B at an angle θ is expressed by the following equation (4). $\begin{array}{cc}\begin{array}{c}B=B_1+B_2+\mathrm{Bc}\\ =\mu ·{\mathrm{Im}}_1·\sin \left({\omega }_1·t-\theta \right)+\mu ·{\mathrm{Im}}_2·\sin \left({\omega }_2·t+\alpha -\theta \right)+\mu ·n·\\ \left\{\mathrm{Ica}\left(t\right)·\sin \theta +\mathrm{Icb}\left(t\right)·\sin \left(\theta -\frac{2\pi }{3}\right)+\mathrm{Icc}\left(t\right)·\sin \left(\theta -\frac{4\pi }{3}\right)\right\}\end{array}& \left(4\right)\end{array}$

[0096] Here, let the torque acting on the outer rotor 3 be τ₁. If the force which acts on a semicircle of the outer rotor 3 is f₁, the force which then acts the other semicircle is also f₁. Accordingly, the force acting on the whole circumference is 2f₁, and the torques, may be expressed by the following equation.

[0097] τ₁=2f₁·r₁

[0098] where, r₁=distance to outer magnets from center shaft of outer rotor.

[0099] Here, the force f₁is a drive force which occurs when a direct current Im₁is generated in a magnetic field of magnetic flux density B. From the above equation, it is seen that there is a directly proportional relation between the torque τ₁and the drive force f₁. As an equivalent direct current is formed for each semicircle, f₁is given by the following equation.

[0100] f₁=Im₁·B

[0101] where, θ=ω₁·t_o.

[0102] From this equation and equation (4), f₁may be expressed by the following equation (5). $\begin{array}{cc}\begin{array}{c}{f}_1={\mathrm{Im}}_1·[[\mu ·{\mathrm{Im}}_2·\sin \left({\omega }_2·t+\alpha -{\omega }_1·t\right)+\mu ·n·\\ \{\mathrm{Ica}\left(t\right)·\sin \left({\omega }_1·t\right)+\mathrm{Icb}\left(t\right)·\sin \left({\omega }_1·t-\frac{2\pi }{3}\right)+\\ \mathrm{Icc}\left(t\right)·\sin \left({\omega }_1·t-\frac{4\pi }{3}\right)\}]\end{array}& \left(5\right)\end{array}$

[0103] Similarly, if the force acting on the semicircle of the inner rotor 4 is f₂, the force acting on the whole rotor is 2f₂, so the torque τ₂acting on the inner magnets m₂may be expressed by the following equation.

[0104] τ₂=2f₂·r₂

[0105] where, r₂=distance from center axis of inner rotor 4 to the inner magnets m₂.,

[0106] Here, the force f₂is the drive force due to an equivalent direct current Im₂in a magnetic field of magnetic flux density B. As an equivalent direct current is formed for each semicircle, f₂is given by the following equation.

[0107] f₂=Im₂·B

[0108] where, θ=ω₂·t+α.

[0109] From this equation and equation (4), f₂may be expressed by the following equation (6) $\begin{array}{cc}\begin{array}{c}{f}_2={\mathrm{Im}}_2·[\mu ·{\mathrm{Im}}_1·\sin \left({\omega }_1·t-{\omega }_2·t-\alpha \right)+\mu ·n·\\ \{\mathrm{Ica}\left(t\right)·\sin \left({\omega }_2·t+\alpha \right)+\mathrm{Icb}\left(t\right)·\sin \left({\omega }_2·t+\alpha -\frac{2\pi }{3}\right)+\\ \mathrm{Icc}\left(t\right)·\sin \left({\omega }_2·t+\alpha -\frac{4\pi }{3}\right)\}]\end{array}& \left(6\right)\end{array}$

[0110] (1-2) When external rotating magnetic fields are applied In order to pass currents in the coils a, b, c each of which has a phase difference of β with respect to the rotating outer magnets m₁, alternating currents Ica(t), Icb(t), Icc(t) in equation (3) are set by the following equations (7A)-(7C). $\begin{array}{cc}\mathrm{IIca}\left(t\right)=\mathrm{Ic}·\cos \left({\omega }_1·t-\beta \right)& \text{(7A)}\\ \mathrm{Icb}\left(t\right)=\mathrm{Ic}·\cos \left({\omega }_1·t-\beta -\frac{2\pi }{3}\right)& \text{(7B)}\\ \mathrm{Icc}\left(t\right)=\mathrm{Ic}·\cos \left({\omega }_1·t-\beta -\frac{4\pi }{3}\right)& \text{(7C)}\end{array}$

[0111] where,

[0112] Ic=amplitude of alternating currents, and

[0113] β=phase difference.

[0114] The drive force f₁, f₂is calculated by substituting equations (7A)-(7C) in equations (5)-(6). $\begin{array}{c}{f}_1={\mathrm{Im}}_1·\left\{\mu ·{\mathrm{Im}}_2·\sin \left({\omega }_2·t+\alpha -{\omega }_1·t\right)\right\}+\mu ·n·\mathrm{Ic}·\\ \{\cos \left({\omega }_1·t-\beta \right)·\sin \left({\omega }_1·t\right)+\\ \cos \left({\omega }_1·t-\beta -\frac{2\pi }{3}\right)·\sin \left({\omega }_1·t-\frac{2\pi }{3}\right)+\\ \cos \left({\omega }_1·t-\beta -\frac{4\pi }{3}\right)·\sin \left({\omega }_1·t-\frac{4\pi }{3}\right)\}\end{array}$

[0115] Here, the above equation may be rewritten using the formula

[0116] cos(a+b)={fraction (1/2+L )}·{sin (2a+b)−sin (b)}.

[0117] $\begin{array}{cc}\begin{array}{c}{f}_1={\mathrm{Im}}_1·[\mu ·{\mathrm{Im}}_2·\sin \left({\omega }_2·t+\alpha -{\omega }_1·t\right)+\mu ·n·\mathrm{Ic}·\\ [\frac{1}{2}·\left\{\sin \left(2{\omega }_1·t-\beta \right)+\sin \beta \right\}+\\ \frac{1}{2}·\left[\sin \left\{2\left({\omega }_1·t-\frac{2\pi }{3}\right)-\beta \right\}+\sin \beta \right]+\\ \frac{1}{2}·\left[\sin \left\{2\left({\omega }_1·t-\frac{4\pi }{3}\right)-\beta \right\}+\sin \beta \right]]]\\ ={\mathrm{Im}}_1·[\mu ·{\mathrm{Im}}_2·\sin \left({\omega }_2·t+\alpha -{\omega }_1·t\right)+\frac{1}{2}·\mu ·n·\mathrm{Ic}·\\ [3\sin \beta +\sin \left\{2\left({\omega }_1·t-\frac{2\pi }{3}\right)-\beta \right\}+\\ \sin \left\{2\left({\omega }_1·t-\frac{4\pi }{3}\right)-\beta \right\}]]\\ ={\mathrm{Im}}_1·[\mu ·{\mathrm{Im}}_2·\sin \left({\omega }_2·t+\alpha -{\omega }_1·t\right)+\frac{1}{2}·\mu ·n·\mathrm{Ic}·\\ \left\{3\sin \beta +\sin \left(2{\omega }_1·t-\frac{4\pi }{3}-\beta \right)+\sin \left(2{\omega }_1·t-\frac{8\pi }{3}-\beta \right)\right\}]\\ ={\mathrm{Im}}_1·[\mu ·{\mathrm{Im}}_2·\sin \left({\omega }_2·t+\alpha -{\omega }_1·t\right)+\frac{1}{2}·\mu ·n·\mathrm{Ic}·\\ \left\{3\sin \beta +\sin \left(2{\omega }_1·t-\beta -\frac{2\pi }{3}\right)+\sin \left(2{\omega }_1·t-\beta -\frac{4\pi }{3}\right)\right\}]\\ =-{\mathrm{Im}}_1·\left[\mu ·{\mathrm{Im}}_2·\sin \left\{\left({\omega }_2-{\omega }_1\right)·t-\alpha \right\}-\frac{3}{2}·\mu ·n·\mathrm{Ic}·\sin \beta \right]\end{array}& \left(8\right)\end{array}$

[0118] Equation (8) has a form wherein the first term which is a torque fluctuation amount due to the effect of the magnetic field of the inner agnets is added to the second term which is a constant torque.

[0119] Also, f₂may be rewritten by the following equation. $\begin{array}{c}{f}_2={\mathrm{Im}}_2·B\\ ={\mathrm{Im}}_2·[\mu ·{\mathrm{Im}}_1·\sin \left({\omega }_1·t-{\omega }_2·t-\alpha \right)+\mu ·n·\mathrm{Ic}·\\ \{\cos \left({\omega }_1·t-\beta \right)·\sin \left({\omega }_2·t+\alpha \right)+\\ \cos \left({\omega }_1·t-\frac{2\pi }{3}-\beta \right)·\sin \left({\omega }_2·t-\frac{2\pi }{3}+\alpha \right)+\\ \cos \left({\omega }_1·t-\frac{4\pi }{3}-\beta \right)·\sin \left({\omega }_2·t-\frac{4\pi }{3}+\alpha \right)\}]\end{array}$

[0120] Here, the above equation may be rewritten using the formula

[0121] cos(a)·sin(b)=½·{sin(a+b)−sin(a−b)}.

[0122] $\begin{array}{cc}\begin{array}{c}{f}_2={\mathrm{Im}}_2·[\mu ·{\mathrm{Im}}_1·\sin \left({\omega }_1·t-{\omega }_2·t-\alpha \right)+\mu ·n·\mathrm{Ic}·\frac{1}{2}·\\ \{\sin \left({\omega }_1·t-\beta +{\omega }_2·t+\alpha \right)-\sin \left({\omega }_1·t-\beta -{\omega }_2·t-\alpha \right)+\\ \sin \left({\omega }_1·t-\frac{2\pi }{3}-\beta +{\omega }_2·t-\frac{2\pi }{3}+\alpha \right)-\\ \sin \left({\omega }_1·t-\frac{2\pi }{3}-\beta -{\omega }_2·t+\frac{2\pi }{3}-\alpha \right)+\\ \sin \left({\omega }_1·t-\frac{4\pi }{3}-\beta +{\omega }_2·t-\frac{4\pi }{3}+\alpha \right)-\\ \sin \left({\omega }_1·t-\frac{4\pi }{3}-\beta -{\omega }_2·t+\frac{4\pi }{3}-\alpha \right)\}]\\ ={\mathrm{Im}}_2·[\mu ·{\mathrm{Im}}_1·\sin \left({\omega }_1·t-{\omega }_2·t-\alpha \right)+\mu ·n·\mathrm{Ic}·\frac{1}{2}·\\ [\sin \left\{\left({\omega }_1+{\omega }_2\right)·t+\alpha -\beta \right\}-\sin \left\{\left({\omega }_1-{\omega }_2\right)·t-\alpha -\beta \right\}+\\ \sin \left\{\left({\omega }_1-{\omega }_2\right)·t-\frac{4\pi }{3}+\alpha -\beta \right\}-\\ \sin \left\{\left({\omega }_1-{\omega }_2\right)·t-\alpha +\beta \right\}+\\ \sin \left\{\left({\omega }_1+{\omega }_2\right)·t-\frac{8\pi }{3}+\alpha -\beta \right\}-\\ \sin \left\{\left({\omega }_1-{\omega }_2\right)·t-\alpha -\beta \right\}]]\\ ={\mathrm{Im}}_2·[\mu ·{\mathrm{Im}}_1·\sin \left({\omega }_1·t-{\omega }_2·t-\alpha \right)-\frac{3}{2}·\mu ·n·\mathrm{Ic}·\\ \sin \left\{\left({\omega }_1-{\omega }_2\right)·t-\alpha -\beta \right\}+\mu ·n·\mathrm{Ic}·\frac{1}{2}·\\ [\sin \left\{\left({\omega }_1+{\omega }_2\right)·t+\alpha -\beta \right\}+\\ \sin \left\{\left({\omega }_1-{\omega }_2\right)·t+\alpha -\beta -\frac{2\pi }{3}\right\}+\\ \sin \left\{\left({\omega }_1+{\omega }_2\right)·t+\alpha -\beta -\frac{4\pi }{3}\right\}]]\\ =\mu ·{\mathrm{Im}}_2·[{\mathrm{Im}}_1·\sin \left\{\left({\omega }_1-{\omega }_2\right)·t-\alpha \right\}-\frac{3}{2}·n·\mathrm{Ic}·\\ \sin \left\{\left({\omega }_1-{\omega }_2\right)·t-\alpha -\beta \right\}\end{array}& \left(9\right)\end{array}$

[0123] (1-3) When inner rotating magnetic fields are applied

[0124] In order to pass currents in the coils a, b, c each of which has a phase difference of γ with respect to the rotating inner magnets m₂, alternating currents Ica(t), Icb(t), Icc(t) in the above equation (3) are set by the following equations (10a)-(10C). $\begin{array}{cc}\mathrm{IIca}\left(t\right)=\mathrm{Ic}·\cos \left({\omega }_2·t-\gamma \right)& \text{(10A)}\\ \mathrm{Icb}\left(t\right)=\mathrm{Ic}·\cos \left({\omega }_2·t-\gamma -\frac{2\pi }{3}\right)& \text{(10B)}\\ \mathrm{Icc}\left(t\right)=\mathrm{Ic}·\cos \left({\omega }_2·t-\gamma -\frac{4\pi }{3}\right)& \text{(10C)}\end{array}$

[0125] where,

[0126] Ic=amplitude of alternating currents, and

[0127] γ=phase difference.

[0128] The drive force f₁, f₂is calculated by substituting equations (10A)-(10C) in equations (5)-(6). $\begin{array}{c}{f}_1={\mathrm{Im}}_1·[\mu ·{\mathrm{Im}}_2·\sin [{\omega }_2·t+\alpha -{\omega }_1·t+\mu ·n·\mathrm{Ic}·\\ \{\cos \left({\omega }_2·t-\gamma \right)·\sin \left({\omega }_1·t\right)+\\ \cos \left({\omega }_2·t-\gamma -\frac{2\pi }{3}\right)·\sin \left({\omega }_1·t-\frac{2\pi }{3}\right)+\\ \cos \left({\omega }_2·t-\gamma -\frac{4\pi }{3}\right)·\sin \left({\omega }_1·t-\frac{4\pi }{3}\right)\}]]\end{array}$

[0129] Here, the above equation may be rewritten using the formula

[0130] cos(a)·sin(b)=½·{sin(a+b)−sin(a−b)}

[0131] $\begin{array}{cc}\begin{array}{c}{f}_1={\mathrm{Im}}_1·[\mu ·{\mathrm{Im}}_2·\sin \left({\omega }_2·t+\alpha -{\omega }_1·t\right)+\frac{1}{2}·\mu ·n·\mathrm{Ic}·\\ \{\sin \left({\omega }_2·t-\gamma +{\omega }_1·t\right)-\sin \left({\omega }_2·t-\gamma -{\omega }_1·t\right)+\\ \sin \left({\omega }_2·t-\gamma -\frac{2\pi }{3}+{\omega }_1·t-\frac{2\pi }{3}\right)-\\ \sin \left({\omega }_2·t-\gamma -\frac{2\pi }{3}-{\omega }_1·t+\frac{2\pi }{3}\right)+\\ \sin \left({\omega }_2·t-\gamma -\frac{4\pi }{3}+{\omega }_1·t-\frac{4\pi }{3}\right)-\\ \sin \left({\omega }_2·t-\gamma -\frac{4\pi }{3}+{\omega }_1·t+\frac{4\pi }{3}\right)\}]+\\ ={\mathrm{Im}}_1·[\mu ·{\mathrm{Im}}_2·\sin \left\{\left({\omega }_2-{\omega }_1\right)·t+\alpha \right\}+\frac{1}{2}·\mu ·n·\mathrm{Ic}·\\ [\sin \left\{\left({\omega }_2+{\omega }_1\right)·t-\gamma \right\}-\sin \{\left({\omega }_2-{\omega }_1\right)·t-\gamma )+\\ \sin \left\{\left({\omega }_2+{\omega }_1\right)·t-\gamma -\frac{4\pi }{3}\right\}-\sin \left\{\left({\omega }_2-{\omega }_1\right)·t-\gamma \right\}+\\ \sin \left\{\left({\omega }_2+{\omega }_1\right)·t-\gamma -\frac{8\pi }{3}\right\}-\sin \left\{\left({\omega }_2-{\omega }_1\right)·t-\gamma \right\}]]\\ ={\mathrm{Im}}_1·[\mu ·{\mathrm{Im}}_2·\sin \left\{\left({\omega }_2-{\omega }_1\right)·t+\alpha \right\}-\frac{3}{2}·\mu ·n·\mathrm{Ic}·\\ \sin \left\{\left({\omega }_2-{\omega }_1\right)·t-\gamma \right\}+\frac{1}{2}·\mu ·n·\mathrm{Ic}·\\ [\sin \left\{\left({\omega }_2+{\omega }_1\right)·t-\gamma \right\}+\sin \left\{\left({\omega }_2+{\omega }_1\right)·t-\gamma -\frac{2\pi }{3}\right\}+\\ \sin \left\{\left({\omega }_2+{\omega }_1\right)·t-\gamma -\frac{4\pi }{3}\right\}]]\\ =-\mu ·{\mathrm{Im}}_1·[{\mathrm{Im}}_2·\sin \left\{\left({\omega }_1-{\omega }_2\right)·t-\alpha \right\}-\frac{3}{2}·n·\mathrm{Ic}·\\ \sin \left\{\left({\omega }_1-{\omega }_2\right)·t+\gamma \right\}]\end{array}& \left(11\right)\end{array}$

[0132] Equation (11) shows that a torque fluctuation occurs only in the outer magnets.

[0133] Also, f₂may be rewritten by the following equation. $\begin{array}{c}{f}_2={\mathrm{Im}}_2·[\mu ·{\mathrm{Im}}_1·\sin \left({\omega }_2·t-{\omega }_1·t-\alpha \right)+\mu ·n·\mathrm{Ic}·\\ \{\cos \left({\omega }_2·t-\gamma \right)·\sin \left({\omega }_2·t+\alpha \right)+\\ \cos \left({\omega }_2·t-\gamma -\frac{2\pi }{3}\right)·\sin \left({\omega }_2·t+\alpha -\frac{2\pi }{3}\right)+\\ \cos \left({\omega }_2·t-\gamma -\frac{4\pi }{3}\right)·\sin \left({\omega }_2·t+\alpha -\frac{4\pi }{3}\right)\}]\end{array}$

[0134] Here, the above equation may be rewritten using the formula

[0135] cos(a)·sin(b)=½·{sin(a+b)−sin(a−b)}.

[0136] $\begin{array}{cc}\begin{array}{c}{f}_2={\mathrm{Im}}_2·[\mu ·{\mathrm{Im}}_1·\sin \left({\omega }_1·t-{\omega }_2·t-\alpha \right)-\frac{3}{2}·\mu ·n·\mathrm{Ic}·\\ \sin \left(-\alpha -\gamma \right)+\frac{1}{2}·\mu ·n·\mathrm{Ic}·\{\sin (2·{\omega }_2·t+\alpha -\gamma \}+\\ \sin \left(2·{\omega }_2·t+\alpha -\gamma -\frac{2\pi }{3}\right)+\sin \left(2·{\omega }_2·t+\alpha -\gamma -\frac{4\pi }{3}\right)\}]\\ =\mu ·{\mathrm{Im}}_2·\left[{\mathrm{Im}}_1·\sin \left\{\left({\omega }_1-{\omega }_2\right)·t-\alpha \right\}+\frac{3}{2}·n·\mathrm{Ic}·\sin \left(\alpha +\gamma \right)\right]\end{array}& \left(12\right)\end{array}$

[0137] Equation (12) has a form wherein the first term which is a torque fluctuation amount due to the effect of the magnetic field of the inner magnets is added to the second term which is a constant torque.

[0138] (1-4) When the outer rotating magnetic fields and inner rotating magnetic fields are applied together

[0139] The above Ica(t), Icb(t), Icc(t) are set to pass a current through the coils 6 in synchronism with the outer magnets and inner magnets. $\begin{array}{cc}\mathrm{Ica}\left(t\right)=\mathrm{Ic}·\cos \left({\omega }_1·t-\beta \right)+{\mathrm{Ic}}_2·\cos \left({\omega }_2·t-\gamma \right)& \text{(13A)}\\ \mathrm{Icb}\left(t\right)=\mathrm{Ic}·\cos \left({\omega }_1·t-\beta -\frac{2\pi }{3}\right)+{\mathrm{Ic}}_2·\cos \left({\omega }_2·t-\gamma -\frac{2\pi }{3}\right)& \text{(13B)}\\ \mathrm{Icc}\left(t\right)=\mathrm{Ic}·\cos \left({\omega }_1·t-\beta -\frac{4\pi }{3}\right)+{\mathrm{Ic}}_2·\cos \left({\omega }_2·t-\gamma -\frac{4\pi }{3}\right)& \text{(13C)}\end{array}$

[0140] The drive forces f₁, f₂are calculated by the following equations (14), (15). $\begin{array}{cc}\begin{array}{c}{f}_1={\mathrm{Im}}_1·[\mu ·{\mathrm{Im}}_2·\sin \left({\omega }_2·t+\alpha -{\omega }_1·t\right)+\mu ·n·\\ [\left\{\mathrm{Ic}·\cos \left({\omega }_1·t-\beta \right)+{\mathrm{Ic}}_2·\cos \left({\omega }_2·t-\gamma \right)\right\}·\sin \left({\omega }_1·t\right)+\\ \left\{\mathrm{Ic}·\cos \left({\omega }_1·t-\beta -\frac{2\pi }{3}\right)+{\mathrm{Ic}}_2·\cos \left({\omega }_2·t-\gamma -\frac{2\pi }{3}\right)\right\}·\\ \sin \left({\omega }_1·t-\frac{2\pi }{3}\right)+\{\mathrm{Ic}·\cos \left({\omega }_1·t-\beta -\frac{4\pi }{3}\right)+\\ {\mathrm{Ic}}_2·\cos \left({\omega }_2·t-\gamma -\frac{4\pi }{3}\right)\}·\sin \left({\omega }_1·t-\frac{4\pi }{3}\right)]]\\ ={\mathrm{Im}}_1·[\mu ·{\mathrm{Im}}_2·\sin \left({\omega }_2·t+\alpha -{\omega }_1·t\right)+\mu ·n·\\ \{\mathrm{Ic}·\cos \left({\omega }_1·t-\beta \right)·\sin \left({\omega }_1·t\right)+{\mathrm{Ic}}_2·\cos \left({\omega }_2·t-\gamma \right)·\\ \sin \left({\omega }_1·t\right)+\mathrm{Ic}·\cos \left({\omega }_1·t-\beta -\frac{2\pi }{3}\right)·\sin \left({\omega }_1·t-\frac{2\pi }{3}\right)+\\ {\mathrm{Ic}}_2·\cos \left({\omega }_2·t-\gamma -\frac{2\pi }{3}\right)·\sin \left({\omega }_1·t-\frac{2\pi }{3}\right)+\\ \mathrm{Ic}·\cos \left({\omega }_1·t-\beta -\frac{4\pi }{3}\right)·\sin \left({\omega }_1·t-\frac{4\pi }{3}\right)+\\ {\mathrm{Ic}}_2·\cos \left({\omega }_2·t-\gamma -\frac{4\pi }{3}\right)·\sin \left({\omega }_1·t-\frac{4\pi }{3}\right)\}]\\ ={\mathrm{Im}}_1·[\mu ·{\mathrm{Im}}_2·\sin \left({\omega }_2·t+\alpha -{\omega }_1·t\right)+\mu ·n·\\ [\mathrm{Ic}·\{\cos \left({\omega }_1·t-\beta \right)·\sin \left({\omega }_1·t\right)+\cos \left({\omega }_1·t-\beta -\frac{2\pi }{3}\right)·\\ \sin \left({\omega }_1·t-\frac{2\pi }{3}\right)+\cos \left({\omega }_1·t-\beta -\frac{4\pi }{3}\right)·\\ \sin \left({\omega }_1·t-\frac{4\pi }{3}\right)\}+{\mathrm{Ic}}_2·\{\cos \left({\omega }_2·t-\gamma \right)·\sin \left({\omega }_1·t\right)+\\ \cos \left({\omega }_2·t-\gamma -\frac{2\pi }{3}\right)·\sin \left({\omega }_1·t-\frac{2\pi }{3}\right)+\\ \cos \left({\omega }_2·t-\gamma -\frac{4\pi }{3}\right)·\sin \left({\omega }_1·t-\frac{4\pi }{3}\right)\}]]\\ ={\mathrm{Im}}_1·[\mu ·{\mathrm{Im}}_2·\sin \left({\omega }_2·t+\alpha -{\omega }_1·t\right)+\mu ·n·\\ \left[\mathrm{Ic}·\left(\frac{3}{2}·\sin \beta \right)+{\mathrm{Ic}}_2·\left[\frac{3}{2}·\sin \left\{\left({\omega }_1-{\omega }_2\right)t+\gamma \right\}\right]\right]]\end{array}& \left(14\right)\end{array}$

[0141] Equation (14) has a form wherein a torque fluctuation is added to a constant torque according to a rotation phase difference β relative to the outer magnets m₁. $\begin{array}{c}{f}_2={\mathrm{Im}}_2·[\mu ·{\mathrm{Im}}_1·\sin \left({\omega }_1·t-{\omega }_2·t-\alpha \right)+\mu ·n·\\ [\left\{\mathrm{Ic}·\cos \left({\omega }_1·t-\beta \right)+{\mathrm{Ic}}_2·\cos \left({\omega }_2·t-\gamma \right)\right\}·\sin \left({\omega }_2·t+\alpha \right)+\\ \left\{\mathrm{Ic}·\cos \left({\omega }_1·t-\beta -\frac{2\pi }{3}\right)+{\mathrm{Ic}}_2·\cos \left({\omega }_2·t-\gamma -\frac{2\pi }{3}\right)\right\}·\\ \sin \left({\omega }_2·t+\alpha -\frac{2\pi }{3}\right)+\{\mathrm{Ic}·\cos \left({\omega }_1·t-\beta -\frac{4\pi }{3}\right)+\\ {\mathrm{Ic}}_2·\cos \left({\omega }_2·t-\gamma -\frac{4\pi }{3}\right)\}·\sin \left({\omega }_2·t+\alpha -\frac{4\pi }{3}\right)]]\\ ={\mathrm{Im}}_2·[\mu ·{\mathrm{Im}}_1·\sin \left({\omega }_1·t-{\omega }_2·t-\alpha \right)+\mu ·n·\\ \{\mathrm{Ic}·\cos \left({\omega }_1·t-\beta \right)·\sin \left({\omega }_2·t+\alpha \right)+{\mathrm{Ic}}_2·\cos \left({\omega }_2·t-\gamma \right)·\\ \sin \left({\omega }_2·t+\alpha \right)+\mathrm{Ic}·\cos \left({\omega }_1·t-\beta -\frac{2\pi }{3}\right)·\sin \left({\omega }_2·t+\alpha -\frac{2\pi }{3}\right)+\\ {\mathrm{Ic}}_2·\cos \left({\omega }_2·t-\gamma -\frac{2\pi }{3}\right)·\sin \left({\omega }_2·t+\alpha -\frac{2\pi }{3}\right)+\\ \mathrm{Ic}·\cos \left({\omega }_1·t-\beta -\frac{4\pi }{3}\right)·\sin \left({\omega }_2·t+\alpha -\frac{4\pi }{3}\right)+\\ {\mathrm{Ic}}_2·\cos \left({\omega }_2·t-\gamma -\frac{4\pi }{3}\right)·\sin \left({\omega }_2·t+\alpha -\frac{4\pi }{3}\right)\}]\\ ={\mathrm{Im}}_2·[\mu ·{\mathrm{Im}}_1·\sin \left({\omega }_1·t-{\omega }_2·t-\alpha \right)+\mu ·n·\\ [\mathrm{Ic}·\{\cos \left({\omega }_1·t-\beta \right)·\sin \left({\omega }_2·t+\alpha \right)+\cos \left({\omega }_1·t-\beta -\frac{2\pi }{3}\right)·\\ \sin \left({\omega }_2·t+\alpha -\frac{2\pi }{3}\right)+\cos \left({\omega }_1·t-\beta -\frac{4\pi }{3}\right)·\\ \sin \left({\omega }_2·t+\alpha -\frac{4\pi }{3}\right)\}+{\mathrm{Ic}}_2·\{\cos \left({\omega }_2·t-\gamma \right)·\sin \left({\omega }_2·t+\alpha \right)+\\ \cos \left({\omega }_2·t-\gamma -\frac{2\pi }{3}\right)·\sin \left({\omega }_2·t+\alpha -\frac{2\pi }{3}\right)+\\ \cos \left({\omega }_2·t-\gamma -\frac{4\pi }{3}\right)·\sin \left({\omega }_2·t+\alpha -\frac{4\pi }{3}\right)\}]]\end{array}$

[0142] Here, the above equation may be rewritten using the formula

[0143] cos(a)·sin(b)=½·{sin(a+b)−sin(a−b)}

[0144] $\begin{array}{cc}\begin{array}{c}{f}_2={\mathrm{Im}}_2·[\mu ·{\mathrm{Im}}_1·\sin \left({\omega }_1·t-{\omega }_2·t-\alpha \right)+\mu ·n·\\ [\mathrm{Ic}·[\frac{1}{2}·\{\sin \left({\omega }_1·t-\beta +{\omega }_2·t+\alpha \right)-\\ \sin \left({\omega }_1·t-\beta -{\omega }_2·t-\alpha \right)\}+\frac{1}{2}·\\ \{\sin \left({\omega }_1·t-\beta -\frac{2\pi }{3}+{\omega }_2·t+\alpha -\frac{2\pi }{3}\right)-\\ \sin \left({\omega }_1·t-\beta -\frac{2\pi }{3}-{\omega }_2·t-\alpha +\frac{2\pi }{3}\right)\}+\frac{1}{2}·\\ \{\sin \left({\omega }_1·t-\beta -\frac{4\pi }{3}+{\omega }_2·t+\alpha -\frac{4\pi }{3}\right)-\\ \sin \left({\omega }_1·t-\beta -\frac{4\pi }{3}-{\omega }_2·t-\alpha +\frac{4\pi }{3}\right)\}]+{\mathrm{Ic}}_2·[\frac{1}{2}·\\ \left\{\sin \left({\omega }_2·t-\gamma +{\omega }_2·t+\alpha \right)-\sin \left({\omega }_2·t-\gamma -{\omega }_2·t-\alpha \right)\right\}+\\ \frac{1}{2}·\{\sin \left({\omega }_2·t-\gamma -\frac{2\pi }{3}+{\omega }_2·t+\alpha -\frac{2\pi }{3}\right)-\\ \sin \left({\omega }_2·t-\gamma -\frac{2\pi }{3}-{\omega }_2·t-\alpha +\frac{2\pi }{3}\right)\}+\frac{1}{2}·\\ \{\sin \left({\omega }_2·t-\gamma -\frac{4\pi }{3}+{\omega }_2·t+\alpha -\frac{4\pi }{3}\right)-\\ \sin \left({\omega }_2·t-\gamma -\frac{4\pi }{3}-{\omega }_2·t-\alpha +\frac{4\pi }{3}\right)\}]]]\\ ={\mathrm{Im}}_2·[\mu ·{\mathrm{Im}}_1·\sin \left({\omega }_1·t-{\omega }_2·t-\alpha \right)+\mu ·n·\\ [\mathrm{Ic}·\{\frac{1}{2}·\{\sin \left({\omega }_1·t-\beta +{\omega }_2·t+\alpha \right)-\\ \sin \left({\omega }_1·t-\beta -{\omega }_2·t-\alpha \right)\}+\frac{1}{2}·\\ \{\sin \left({\omega }_1·t-\beta +{\omega }_2·t+\alpha -\frac{4\pi }{3}\right)-\\ \sin \left({\omega }_1·t-\beta -{\omega }_2·t-\alpha \right)\}+\frac{1}{2}·\\ \{\sin \left({\omega }_1·t-\beta +{\omega }_2·t+\alpha -\frac{8\pi }{3}\right)-\\ \sin \left({\omega }_1·t-\beta -{\omega }_2·t-\alpha \right)\}+{\mathrm{Ic}}_2·[\frac{1}{2}·\\ \left\{\sin \left({\omega }_2·t-\gamma +{\omega }_2·t+\alpha \right)-\sin \left({\omega }_2·t-\gamma -{\omega }_2·t-\alpha \right)\right\}+\\ \frac{1}{2}·\{\sin \left({\omega }_2·t-\gamma +{\omega }_2·t+\alpha -\frac{4\pi }{3}\right)-\\ \sin \left({\omega }_2·t-\gamma -{\omega }_2·t-\alpha \right)\}+\frac{1}{2}·\\ \{\sin \left({\omega }_2·t-\gamma +{\omega }_2·t+\alpha -\frac{8\pi }{3}\right)-\\ \sin \left({\omega }_2·t-\gamma -{\omega }_2·t-\alpha \right)\}]]]\\ ={\mathrm{Im}}_2·[\mu ·{\mathrm{Im}}_1·\sin \left({\omega }_1·t-{\omega }_2·t-\alpha \right)+\frac{1}{2}·\mu ·n·\mathrm{Ic}·\\ \{\sin \left({\omega }_1·t-\beta +{\omega }_2·t+\alpha \right)-\sin \left({\omega }_1·t-\beta -{\omega }_2·t-\alpha \right)+\\ \sin \left({\omega }_1·t-\beta +{\omega }_2·t+\alpha -\frac{4\pi }{3}\right)-\\ \sin \left({\omega }_1·t-\beta +{\omega }_2·t-\alpha \right)+\\ \sin \left({\omega }_1·t-\beta +{\omega }_2·t+\alpha -\frac{8\pi }{3}\right)-\\ \sin \left({\omega }_1·t-\beta -{\omega }_2·t-\alpha \right)\}+\frac{1}{2}·\mu ·n·{\mathrm{Ic}}_2·\\ \{\sin \left({\omega }_2·t-\gamma +{\omega }_2·t+\alpha \right)-\sin \left({\omega }_2·t-\gamma +{\omega }_2·t-\alpha \right)+\\ \sin \left({\omega }_2·t-\gamma +{\omega }_2·t+\alpha -\frac{4\pi }{3}\right)-\\ \sin \left({\omega }_2·t-\gamma -{\omega }_2·t-\alpha \right)+\\ \sin \left({\omega }_2·t-\gamma +{\omega }_2·t+\alpha -\frac{8\pi }{3}\right)-\\ \sin \left({\omega }_2·t-\gamma -{\omega }_2·t-\alpha \right)\}]\\ ={\mathrm{Im}}_2·[\mu ·{\mathrm{Im}}_1·\sin \left({\omega }_1·t-{\omega }_2·t-\alpha \right)+\frac{1}{2}·\mu ·n·\mathrm{Ic}·\\ [-3·\sin \left\{\left({\omega }_2-{\omega }_1\right)t-\alpha -\beta \right\}+\frac{1}{2}·\mu ·n·{\mathrm{Ic}}_2·\\ \left\{-3·\sin \left(-\alpha -\gamma \right)\right\}]]\\ ={\mathrm{Im}}_2·[\mu ·{\mathrm{Im}}_1·\sin \left({\omega }_1·t-{\omega }_2·t-\alpha \right)-\frac{3}{2}·\mu ·n·\mathrm{Ic}·\\ \sin \left\{\left({\omega }_2-{\omega }_1\right)·t-\alpha -\beta \right\}+\frac{3}{2}·\mu ·n·{\mathrm{Ic}}_2·3·\sin \left(\alpha +\gamma \right)]\end{array}& \left(15\right)\end{array}$