General information on the electric machines

The goal of this page is to explain and show how an electric machine functions and produces a couple.

Static circuit

That is to say a magnetic circuit surrounded by a winding comprising NR whorls supplied with a tension

u \,

. One notes

\ varphi \,

flow by whorl and

\ Phi = NR \ varphi \,

the total flow embraced by the reel.

One can make the electric diagram are equivalent according to with a resistance R which symbolizes the losses in the cables. and a fem $e= {D \ Phi \ over dt} \,$ to see Lenz's law.

thus one can write.

$u=Ri+ {D \ Phi \ over dt} \,$

By multiplying this equation by $idt \,$ one obtains:

$u.i.dt=R.i^2.dt+N.i.d \ varphi \,$

Energy balance

Thus one feeds a magnetic circuit with a tension U, the circuit consumes a We power, One obtains heat W_th (the cables heat) and the remainder is magnetic energy. thus $dW_e=dW_ {HT} +dW_ {m} \,$

Repronons the formula higher $u.i.dt=R.i^2.dt+N.i.d \ varphi \,$ One can identify $dW_e=u.i.dt \,$ the consumption and $dW_ {HT} = R.i^2.d T \,$ the thermal losses.

By identification one from of deduced that $dW_m=Nid \ varphi \,$ . Thus:

$W_m = \ int {Nest \ varphi} \,$

If it is considered that the circuit is indeformable then $dS=0 \,$ with $S \,$ = surface delimited by the cicuit.

$\ varphi = B.S \ Rightarrow D \ varphi = S.dB+dS.B \ Rightarrow dW_m=N.i.S.dB \,$

$Ni = \ int {H.dl} =Hl$

thus one from of deduced $dW_m=H.l.S.dB=H.dB.V \,$ with $V= l.S = \,$ Volume

thus $W_m= \ int {H.dB.V}$

Linear case: It is considered that the material not-is saturated.

thus $\ Phi =Li \,$ and $B= \ driven. H \,$

$W_m= \ frac {1} {2}. \ Phi .i \,$ if $\ Phi=L.i \,$ then $Wm= \ frac {1} {2}. L.i^2$

$\ frac {W_m} {V} = \ frac {1} {2}. B.H= \ frac {1} {2}. \ driven. H^2= \ frac {B^2} {2 \ driven}$

one poses $W_m+W'_m= \ Phi .i=N. \ varphi .i \,$ with:

$W_m \,$ = magnetic energy
$W'_m \,$ = Co-energy

in the linear case = $W_m=W'_m= \ Phi .i /2 \,$

Deformable or dynamic circuit

As the circuit is moving one has mechanical energy in addition to thermal energy and Magnétique energy.

Thus: $dW_e=dW_ {HT} +dW_ {meca} +dW_m \,$ , with:

$dW_e=u.i.dt= (Ri+N \ frac {D \ varphi} {dt}) .i.dt=R.i^2.dt+N.i.d \ varphi \,$
$dWth= R.i^2.dt \,$
$dW_ {meca} = F.dx \,$ (linear displacement) or $dW_ {meca} = C.d \ theta \,$ (rotation)

Moreover one neglects the losses iron and frictions.

thus one obtains:

R.i.dt+N.i.d \ varphi=Fdx+dW_m \,

F= (- \ frac {dW_m} {dt}) \ varphi=cste \,

like $W_n+W'_m = \ varphi. N.i \,$

Elementary machines

Particular case

Stator smoothes Smooth Rotor

Stator smoothes Projecting Rotor

Projecting stator smooth Rotor

Projecting stator Projecting Rotor

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