3.1 Mathematical Models of Induction Machines
3.1.3.2 Number of Degrees of Freedom in an Induction Motor
The question of the number of the degrees of freedom (2.33) is encountered in systems with lumped parameters whose motion (dynamics) is described by a sys- tem of ordinary differential equations. For the case of an induction machine (Fig.3.4) this means one degree of freedom of the mechanical motion sm = 1, for variable θr denoting the angle of rotor position and the adequate number of the de- grees of freedom se for electric circuits formed by the phase windings. For the case when both the stator and rotor have three phases and the windings are independ- ent, in accordance with the illustration in Fig. 3.4a, the number of electric degrees of freedom is sel = 6. The assumption that electric circuits take the form of phase windings with electric charges Qi as state variables does not exclude the applica- bility of a field model for the calculation of magnetic fluxes ψi linked with the par- ticular windings of the motor. This possibility results from the decoupling of the magnetic and electric fields in the machine and the consideration of electric currents ii=Qi in the machine as sources of magnetic vector potential (2.180), (functions that are responsible for field generation).
a) b)
Fig. 3.4 Diagram with cross-section of induction motor: a) 3-phase stator and rotor wind- ings b) rotor squirrel-cage windings
In this case we have to do with field-circuit models [48], in which the model with lumped parameters describing the dynamics of an electromechanical system (in this study the induction motor) is accompanied by an interactively produced model of the electromagnetic field in which the present flux linkages ψi are
defined. Hence, the model of an induction motor whose diagram is presented in Fig 3.4a has
=7 +
=sm se
s (3.1) degrees of freedom. In this place one can start to think about the state encountered in the windings of a squirrel cage motor (Fig. 3.4b), which does not contain a standard three-phase winding, but has a cage with m = Nr number of bars. The squirrel cage winding responds to the MMFs produced by stator winding current.
The induced EMFs in squirrel cage windings display the same symmetry proper- ties on condition that the squirrel cage of the rotor is symmetric in the range of an- gular span corresponding to a single pole of the stator winding or its total multiple.
Hence, the resulting number of degrees of freedom ssq for a symmetrical squirrel cage winding [101] is expressed by the quotient
u v s
u p m
sq =
2 = (3.2) where: m - the number of bars in the symmetrical cage of a rotor
u,v - relative prime integers
The number of the degrees of freedom of the electric circuits of a rotor’s squirrel- cage winding ssq = u corresponds to the smallest natural number of the bars in a cage contained in a span of a single pole of the stator’s winding or its multiple.
This is done under the silent assumption that the stator’s windings are symmetri- cal. If the symmetry is not actually the case, the maximum number of the degrees of freedom of a cage is equal to
+1
=m
ssq (3.3) which corresponds to the number of independent electric circuits (meshes), in ac- cordance with (2.195), in the cage of a rotor (Fig 3.4b).
For the motor in Fig. 3.4b, we have p = 2, Nr = m = 22, hence the quotient:
v u p
m = = =
2 11 4 22
2 and, as a result, the number of electric degrees of freedom for a squirrel cage winding amounts to ssq = u = 11. This means that in this case the two pole pitches of the stator contain 11 complete slot scales or slot pitches of the rotor, after which the situation recurs. The large number of the de- grees of freedom of the cage makes it possible to account in the mathematical model for the parasitic phenomena [80], for example parasitic synchronic torques.
However, if we disregard deformations of the magnetic field in the air gap and as- sume that it is a plane-parallel and monoharmonic one with the single and basic harmonic equal to ρ = p, then in order to describe such a field we either need only two coordinates or two substitutive phase windings, in most simple cases orthogo- nal ones. For such an assumption of monoharmonic field the number of the degrees of freedom decreases to ssq = 2 regardless of the number of bars in the ro- tor’s cage. In the studies of induction motor drives and its control the principle is to assume the planar and monoharmonic field in the air gap. Nevertheless, at the
stage when we are starting to develop the mathematical models of induction ma- chines, it is assumed for the slip ring and squirrel cage machines that the rotor’s winding is three-phased (as in Fig. 3.4a) for the purposes of preserving a uniform course of reasoning. Hence, as indicated earlier, under the assumptions of a planar and monoharmonic field in the air gap, slip-ring and squirrel-cage motors are equivalent and can be described with a single mathematical model with the only difference that the winding of a squirrel-cage motor is not accessible from outside, in other words, the voltages supplying the phases of the rotor are always equal to zero. According to (2.189) and (2.210), Lagrange’s function for a motor with three phase windings in the stator and rotor can take the form:
∑ ∫
=
+
= 6
1 0 1 2
2
1 ~
) , 0 , 0
~ , , (
k Q
k r k
k r
k
Q d Q
Q J
L θ ψ … … θ (3.4)
and the virtual work (2.198) expressing the exchange of energy is equal to:
∑
=
− +
−
−
= 6
1
) (
) (
k
k k k k r
r
l D u R Q Q
T
A θ δθ δ
δ (3.5)
where:
q = (Q1, Q2,…,Q6, θr) - vector of generalized coordinates J - moment of inertia related to the motor’s shaft, Tl - load torque on the motor’s shaft,
D - coefficient of viscous damping of the revolute motion, Rk - resistance of k-th phase winding,
k k
k i u
Q = , - electric current and supply voltage of k-th phase winding, ψk - magnetic flux linked with k-th winding.
The model in this form already contains two simplifications, i.e. it disregards iron losses associated with magnetization of the core and changes of the windings’
resistances following a change in their temperature.
From the above the equations of motion for electric variables follow in the form:
= − =1,…,6
∂
− ∂
⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂ u R Q k
Q L Q
L dt d
k k k k k
(3.6)
with the capacitors missing from the system =0
∂
∂ Qk
L .
Whereas, according to (3.4) and using the designation for currents im=Qm, we obtain
l r l
l k
l i
l k
r k
k k
i d i
i i i
i Q i
L
l ~
) , 0 , , 0
~, , , , (
) , 0 , , 0 , , , (
1 1 6
1 0 1
θ ψ
θ ψ
…
…
…
…
− +
=∑ ∫∂∂
+
+
∂ =
∂
(3.7)
If k-th winding were the final one, as for k = n = 6, then k(1, ,n, r)
k
i Q i
L =ψ … θ
∂
∂ (3.8)
The resulting equation takes the form
l r l
l k
l i
l k
r k
k k r n k
i d i
i i i
i Q i
i L i
l ~
) , 0 , , 0
~, , , , (
) , 0 , , 0 , , , ( )
, , , (
1 1 6
1 0
1 1
θ ψ
θ ψ
θ ψ
…
…
…
…
…
− +
∑ ∫= ∂∂
+
+
∂ =
= ∂
(3.9)
From the comparison of (3.8) –(3.9) it results that for the simplicity of notation we should treat the equation in (3.7), which is currently considered, as the final one.
In this case the equation for the circuits of an induction motor takes the form
( k i in r ) uk Rkik
dt
d ψ (1,…, ,θ ) = − (3.10) After the differentiation of the left-hand side we obtain
6 , , 1
6
1
= …
−
∂ = +∂
∂
∑∂
=
k
i R dt u
di
i k k k
m
r r
k m m
k θ
θ ψ ψ
(3.11)
The left-hand side expressions (3.11) denote electromotive forces induced in k-th phase winding as a result of the variations in time of flux linkage. The first terms are derived from the variations of the currents and are called electromotive forces of transformation, while the final term is related to the angular speed of the rotor
r r =Ω
θ and is called the electromotive force of rotation. The equation for the torque expressed with variable θr takes the following form
r l r r
D L T
L dt
d θ
θ ∂θ =− −
− ∂
⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂
and, consequently,
( ) k l r k
i
r k
k k r
r i i i di T D
dt J
d θ θ kψ … … θ =− − θ
∂
− ∂ ∑∫
= − ~
) , 0 , , 0
~, , , (
6
1 0
1
1 ,
Which, in case of constant moment of inertia J, can be denoted alternatively as
r l e
r T T D
Jθ = − − θ (3.12) where:
k k
i
r k
k k r
e i i i di
T
k ~
) , 0 , , 0
~, , , (
6
1 0
1
∑∫ 1
= ∂ −
= θ∂ ψ … … θ (3.13)
is the electromagnetic torque produced by the induction motor. In spite of the fact that the resulting equations of motion are stated for a system with lumped parame- ters and in this case for 7 variables corresponding to 7 degrees of freedom of the motor, they can find a very broad application. This results from the general form of the flux linkage associated with the particular phase windings ψk = (i1,…,in, θr).
It could be gained by various methods accounting for the saturation and various engineering parameters of the magnetic circuit. For a squirrel-cage motor, for the case if one needed to account for the existing parasitic torques, it would be neces- sary to abandon the starting assumption of the monoharmonic image of the field in the air gap and, hence increase, the number of equations for the phases of the rotor from 3 to ssq, as it results from (3.2).