Fractional order modeling of rotor skin effect in induction machines

Abstract

This paper presents a fractional Park model which is intended to characterize skin effect of squirrel cage induction machines. Usual modeling of skin effect is based on a ladder network; it is demonstrated that a fractional impedance is an alternative to this ladder network. The fractional Park’s model has been validated by output error identification. A methodology has been proposed to select the more appropriate fractional model of the rotor, respecting the diffusive nature of skin effect.

Appendices

Appendix 1

Consider a fractional order system such as:

$$ Y(s) = H_{n}(s) E(s) $$

(24)

where

$$ H_{n}(s) = \frac{b_{0}}{a_{0} + s^{n}} = L\bigl\{ h_{n}(t) \bigr\} $$

(25)

This system can be represented by a frequency distributed model (refer to [23, 31]):

$$ \left\{\begin{array}{@{}l} \frac{\partial x(w,t)}{\partial t}=-w x(w,t)+e(t)\\[2mm] y(t)=\int^{\infty}_{0}\mu(w)x(w,t)\,dw \end{array}\right. $$

(26)

and

$$ h_n(t)=\int^{\infty}_{0}\mu(w)e^{-wt}\,dw $$

(27)

with

$$ \mu(w) = \frac{b_{0}\frac{\sin(n\pi)}{\pi} w^{n}}{w^{2n} + 2a_{0}\cos (n\pi)w^{n} + a_{0}^{2}} $$

(28)

where 0<n<1, μ(w) is called frequency weighting function.

This continuous frequency weighted model is not directly usable. A practical model is obtained by frequency discretization of μ(w), where the function μ(w) is replaced by a multiple step function with K steps (see Fig. 24).

Fig. 24

Frequency discretization of μ(w)

For an elementary step, the height is μ(w _k ), and the width is Δw _k . Let c _k be the weight of the kth element.

After a frequency discretization of μ(w) we obtain

$$ h_{n}(t) \cong\sum_{k = 1}^{K} \mu(w_{k})e^{ - w_{k}t} \Delta w_{k} $$

(29)

Let us define

$$ \mu(w_{k})\Delta w_{k} = c_{k} $$

(30)

thus

$$ h_{n}(t) \cong\sum_{k = 1}^{K} c_{k}e^{ - w_{k}t} $$

(31)

where $c_{k}e^{ - w_{k}t}$ is the impulse response of a first order system $\frac{c_{k}}{s+w_{k}}$.

This means that h _n (t) is an infinite (K≫1) sum of impulse responses and

$$ H_{n}(s) = L\bigl\{ h_{n}(t) \bigr\} \cong\sum _{k = 1}^{K} \frac{c_{k}}{s + w_{k}} $$

(32)

We are interested by the electrical case where

$$ y(t) = i(t)\qquad e(t) = u(t) $$

(33)

Let

$$ U(s)=Z_n(s)I(s) $$

(34)

or

$$ I(s)=\frac{1}{Z_n(s)}U(s)=\frac{b_0}{a_0+s^n}U(s) $$

(35)

then

$$ Z_n(s)=\frac{a_0}{b_0}+\frac{1}{b_0}s^n $$

(36)

we can write that i(t) is the sum of K elementary currents i _k (t):

$$ i(t)=\sum^K_{l=1}i_k(t) $$

(37)

where

$$ I_{k}(s) = \frac{1}{Z_{k}(s)}U(s) $$

(38)

with for each elementary impedance:

$$ Z_{k}(s) = R_{k} + l_{k}s $$

(39)

then

$$ I_{k}(s) = \frac{1}{R_{k} + l_{k}s}U(s) = \frac{\frac{1}{l_{k}}}{s + \frac{R_{k}}{l_{k}}}U(s) $$

(40)

If we define

$$ \left\{\begin{array}{@{}l} c_k=\frac{1}{l_k}\\[2mm] w_k=\frac{R_k}{l_k} \end{array}\right. $$

(41)

then

$$ I_{k}(s) = \frac{c_{k}}{s + w_{k}}U(s) $$

(42)

Finally

(43)

which shows that

$$ \frac{1}{Z_{n}(s)} \cong\sum_{k = 1}^{K} \frac{1}{R_{k} + l_{k}s} $$

(44)

Appendix 2

Consider heat transfer in a plane wall as represented on Fig. 25 (refer to Benchellal [8]).

Fig. 25

Heat transfer in a plane wall

ϕ(x,t) (heat flux) and T(x,t) (temperature) verify the heat Partial Differential Equation:

$$ \frac{\partial T(x,t)}{\partial t} = \alpha\frac{\partial ^{2}T(x,t)}{\partial x^{2}} $$

(45)

$$ \phi(x,t) = - \lambda\frac{\partial T(x,t)}{\partial x}\quad\mbox{with}\ \alpha= \lambda/\rho c $$

(46)

The boundary conditions are ϕ(0,t) and T(L,t)=cte, ϕ(0,t) is considered as the input of the system, T(0,t) is the output.

Using Laplace transform, we get

$$ \frac{T(0,s)}{\phi(0,s)} = H(s) = \frac{1}{\lambda S\sqrt{\frac{s}{\alpha}}} \frac{1 - e^{ - 2L\sqrt{\frac{s}{\alpha }} }}{1 + e^{ - 2L\sqrt{\frac{s}{\alpha }} }} $$

(47)

where S: surface of faces A and B.

If s→∞ or equivalently t→0, then

$$ \mathop{H(s)}\limits_{s \to \infty } \to\frac{\sqrt{\alpha}}{\lambda Ss^{0.5}} $$

(48)

which is the characteristic feature of a diffusive phenomenon.

Practically, H(s) can be approximated by $H_{n}(s) = \frac{b_{0}}{a_{0} + s^{n}}$ (n free) at low and medium frequencies or by $H_{n_{1},n_{2}}(s) = \frac{b_{0} + b_{1}s^{n_{1}}}{a_{0} + a_{1}s^{n} + s^{n_{1} + n_{2}}}$ (n ₁ free, n ₂=0.5) on a wide frequency range (see [8] for more details).