Summary 2. The electromechanical energy conversion

Comparing this expression with the energy balance of the circuit, we find that the total electric
power inlet (on the left of the equation) is divided into three contributions:
• power dissipated in the resistance
2
p p
= Ri
• power related to the magnetic field
• mechanical power
p
µ
di 1 2
= Li + i
dt 2
1 2 dL
p m
= i
2 dt
If we consider that the inductance in the reference frame varies with periodic sinusoidal pattern,
you can highlight an expression for the anisotropy torque:
1 2 dL 1 2 dL dθ
1 2 dL
p m
= i = i = i Ω
2 dt 2 dθ
dt 2 dθ
pm
1 2 dL
Tm
= = i
Ω 2 dθ
Now, suppose you change the old structure so as to also include a winding on the rotating part as
in Figure 2-3.
dL
dt
θ
i 2
v 2
v 1
i 1
Figure 2-3: reluctance and excitation torque in a primitive machine
The equations describing this structure are that of a mutual inductor with variable parameters.
You can then write:
d d
v1
= R1i1
+ ( L1i1
) + ( Lmi2
)
dt dt
d d
v2
= R2i2
+ ( L2i2
) + ( Lmi1
)
dt dt
and developing the derivatives:
di1
dL1
di2
dLm
v1
= R1i1
+ L1
+ i1
+ Lm
+ i2
dt dt dt dt
di2
dL2
di1
dLm
v2
= R2i2
+ L2
+ i2
+ Lm
+ i1
dt dt dt dt
4