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The simplest magnetic circuit

The simplest magnetic circuit
The two necessary conditions for the motor to work are current and magnetic field. In other words electric circuit and magnetic circuit are necessary in motor structure. In figure 2, when we remove the coil after magnetizing, the remaining ring magnet forms the simplest magnetic circuit (figure 5).
Because of the remanence, magnetic flux flows out of n-pole through air gap to s-pole and back to n-pole through the magnet, forming a closed loop. If we don’t consider the magnetic flux leakage, and deem the cross section area of air gap being the same as the magnet, we will have the following:
BFe＝Bδ ， μFe* HFe＝μ0*Hδ
Further more, according to Ampère's circuital law we have∮H*dL＝F＝I*W＝0 (There is no current in the ring magnet. I=0)
∮H*dL ＝HFe*LFe+ Hδ*Lδ＝0        (1)
HFe→ Magnetic field density inside magnet         HFe•LFe→ Magnetic potential inside magnet
Hδ→ Magnetic field density of the air gap           Hδ•Lδ→ Magnetic potential of the air gap
From formula (1) we know that the direction of HFe and Hδ are opposite. So we convert Formula (1) as following:
HFe＝－Hδ•Lδ/LFe＝－(Bδ0)*Lδ/LFe ＝ －[Lδ/(μ0*LFe)]*BFe  After conversion, we get:BFe＝－[μ0•LFe/Lδ]*HFe    (2)
Formula (2) is the relation that comes from the magnetic circuit trend. But HFe and BFe need to satisfy the magnetization characteristics of the ring magnet material. Acting as the magnetic source, magnet potential inside the permanent magnet materials is negative. So their magnetization curves locate in the beta quadrant (figure 6). Take ferrite magnet for example, we can get the BFeN and HFeN.at working point and further more, get the value of BδN and H δN.
Line (2)
BFe＝－[μ0*LFe/Lδ]*HFe
Magnetization curve
BFe＝(Br/Hc)*HFe+Br
For better understanding, we introduce magnetic resistance Rm in analogous to electric circuit:
Rm＝L/(μ•S)
Where, L is length of magnetic circuit, S is cross-sectional area of magnetic circuit, μ is p of magnetic medium.
Air gap magnetic potential in figure 5:
Fδ＝HδLδ＝(Bδδ)*Lδ＝[Bδ S/(μδ•S)]*Lδ＝Φδ*[Lδ/(μδ*S)]
Fδ＝Φδ•Rmδ   (3)
Where, Φδ is called air gap magnetic flux, R＝Lδ/(μδ•S）is air gap magnetic resistance.
Above formula is called Hopkinson's law and is analogous to Ohm's Law with resistance replaced by reluctance, voltage by MMF and current by magnetic flux. In electric machinery we always hope to improve air gap magnetic flux to improve the output per unit volume. Formula (3) can be converted to Φδ＝Fδ/R, From this formula we can see that, to achieve this goal we have the following measures.
●Improve Fδ→ Adopt high performance magnet
●Decrease magnetic resistance of magnetic circuit:
a. Shorten length of magnetic circuit
b. Increase cross-sectional area of magnetic circuit (such as add lamination sheets).
c. Increase permeability of magnetic field (use silicon steel with better magnetic performance)
The magnetic circuit in figure 5 can be shown in analogous to electric circuit as figure 7.

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