Pseudo-Differential Models for Propagation and Dissipation

Pseudo-Differential Models for Propagation and Dissipation Phenomena in Electrical Machine Windings P. Bidan, C. Neacsu, T. Lebey Laboratoire de G´enie Electrique - Universit´e Paul Sabatier 118 route de Narbonne, 31068 Toulouse cedex 4 (France) [email protected]

Keywords: Electrical machine modelling, transmission line, pseudo-differential operators, diffusive representation.

influence the voltage shapes.

Abstract To accurately describe the voltage distribution among the coils of the stator of an electrical machine during transient phases, a convenient pseudo-differential operator is introduced in a transmission-line model. This enables to take into account energy losses due to skin effect. By use of the diffusive representation, a finite-dimension state-space realization is built and efficient time-domain simulations are then performed.

1 Introduction Variable speed drives using AC motors are commonly realized by switching inverter power supplies (fig.1). Modern pulse-width modulation (PWM) inverters generate voltage impulses with a short rise time on the inputs of the winding phases of the motor. This short rise time causes travelling wave phenomena which have much more important effects than with low frequency AC supplies. Due to wave propagation and reflections, the voltage distribution among the winding phases is highly nonlinear and voltage peaks higher than D.C. bus appear. The major consequence is the increase of voltage stresses between wires. This can lead to a premature ageing and to the failure of insulation. Many papers discuss the impact of PWM waveforms on the motor windings and various models are proposed to predict the voltage distribution among the coils of the stator [3]-[8], [10]. The difficulty of modelling is due to the complexity of the stator windings, of the iron core geometry and of various electromagnetic phenomena. Fundamental description requires an analysis by way of partial differential equations. At last, the simulation by means of finite element implementation is delicate and complex. A more simple way assumes that a stator winding may be described by a particular transmission line [10], [5], [6], but one of the main difficulty is to accurately describe the losses during propagation, which significantly

Figure 1: Variable speed drive of an AC motor. A classical transmission line model with constant parameters, may be used from an infinitesimal point of view (figure 2). Several resistors are usually introduced to take into account the different losses: resistive losses in wire (a part of r), dielectric losses in insulation (Ri ) and losses in iron frame (a part of r for the hysteresis losses and R for the eddy currents). Indeed, due to the large frequency range excitation generated by voltage steps, transient skin effects appear in wire and particularly in the iron core [10], [4]. It is well known that skin effects are governed by a diffusion equation with specific boundary conditions. In our case, due to the frequency dependence, constant resistors may not be used to represent the corresponding losses with a sufficient accuracy.

Figure 2: Transmission line. The aim of this paper is to show how this losses may be

taken into account using a convenient pseudo-differential operator. In order to build numerical simulations, this infinitedimension pseudo-differential model is approximated using the so-called diffusive representations introduced by G. Montseny, J. Audounet and D. Matignon [1], [9]. In section 2, the approach is first illustrated on the simple case of a coil with a flaky iron core with circulating eddy currents. The admittance of this coil is of pseudo-differential type, with diffusive representation. Then in section 3, a pseudo-differential operator is introduced in the transmission line model of the winding, to represent the distributed serial admittance of the line, leading to a 1-D partial pseudodifferential equation (PPDE). The frequency response of this admittance is evaluated through input impedance measurements, and a finite-dimension diffusive realization of it is then identified by Hilbertian techniques. Finally in section 4, time-domain simulations of the proposed model are compared to those obtained through measurements.

the general solution of which is given by: B(z, s) = α(s)e

√ µσs

√

+ β(s)e−

µσs

.

(3)

Assuming that the induction is quasi-uniform on the outside surface of the core, i.e. B( 2b , s) = B(− 2b , s) = B0 (s), we B0 (s) obtain α = β = 2 cosh( and (3) becomes: b√ µσs) 2

B(z, s) = B0 (s)

√ cosh(z µσs) . √ cosh( 2b µσs)

(4)

Figure 4 shows the frequency response of B(z,s) B0 (s) , describing the distribution of induction along the cross section of the core. It is well known that skin effect appearing on induction is due to the presence of eddy currents in the core (see fig. 3).

2 Pseudo-differential model of a coil with iron losses: an ideal case 2.1

Modelling

We consider a coil of n turns with an iron core strip shaped of toroidal form (fig. 3).

Figure 3: Coil with iron core strip-shaped. From Maxwell equations, assuming that the iron permeability (µ) and conductivity (σ) of iron are constant and neglecting propagation phenomena in all directions (no dis− → placement currents), the induction B verifies the diffusion − → → − ∂ equation ∆ B − µσ ∂t B = 0 inside the tore. Due to the − → symmetry of the system, the induction B is oriented and uniform according to x direction. Assuming that the depth (b) of strip is small in comparison with its width (a) and neglecting the induction leakage outside the core, the induction is quasiuniform according to y direction. Then the induction inside the core verifies the one-dimension diffusion equation: ∂B(z, t) ∂ 2 B(z, t) − µσ = 0. ∂z 2 ∂t ¿From Laplace transform (s = equal to zero, (1) becomes

∂ ∂t )

(1)

with initial conditions

∂ 2 B(z, s) − µσsB(z, s) = 0, ∂z 2

(2)

Figure 4: Induction distribution through the cross section of the core. The induction flux through a cross section of the core is then Z 2b ZZ → − → − B(z, s) dz (5) Φ(s) = B · dS = a −b

’2 “ 2a b√ =√ B0 (s) tanh µσs . µσs 2

(6)

Using the Maxwell-Ampere equation in integral form for the H− → P − → → − magnetic field H (i.e. H · d l = I), we can explain the induction B0 (s) considering a closed line (of length l) on the outside surface of the core: H0 (s)l = ni(s)

(7)

where i is the current in the coil. With B0 = µH0 , we obtain: nµi(s) (8) B0 (s) = . l Using (8) in (5), the total flux of induction through the coil is given by: 2aµn2 ΦT (s) = nΦ(s) = √ tanh l µσs = L0

tanh

q

s

q s ω0



’

“ b√ µσs i(s) 2

i(s),

(9) (10)

ω0

ab 4 , and L0 = n2 µ the coil inductance b2 µσ l at low frequencies. Using the Maxwell-Faraday equation in T integral form at terminals of the coil (i.e. v(t) = dΦ dt ), the symbolic expression of the coil voltage is given by: where ω 0 =

v(s) = sΦT (s) = L0 ω 0 tanh

q

s ω0

‘q

s ω0

i(s).

(11)

The impedance of the coil is then for expressed: q ‘ q v(s) s s Z(s) = = L0 ω 0 tanh ω0 ω0 , i(s)

(12)

and the admittance: H(s) =

1 1 q ‘ q . = s s Z(s) L0 ω 0 tanh ω0 ω0

(13)

ω0

corresponds only to an external point of view, because ΦT is not the induction flux simply associated to the current i: ΦT also includes the induction flux generated by eddy currents. Remark 2.2 The same expressions may be obtained for a coil with a core sheet-shaped if l  a  b. In this case, the approximation of the infinite-length coil has to be applied. Considering the input-output transfer: voltage (v) → current (i) of the coil, the admittance then appears as the symbol of a (convolutive) pseudo-differential operator of the form: ” “q •−1 ’q 1 1 ∂ 1 ∂ = tanh . ω 0 ∂t ω 0 ∂t L0 ω 0

Diffusive realization and representation of pseudodifferential operators

The concept of ”diffusive representation”, introduced in [1], [9], is devoted to the transformation of pseudodifferential causal operators into standard memoryless (infinite-dimensional) input-output state representations [7]. We consider here only time-invariant operators. Definition 2.1 [9] The diffusive representation H(ξ) of an ∂ operator of symbol H( ∂t ) is solution, when it exists, of the integral equation: Z +∞ H(ξ) H(jω) = dξ, ω ∈ R, t > 0. (15) jω +ξ 0 In [9], some examples of diffusive representations are given. In the case of the studied coil, operator (14) associated to the admittance (13) has the following diffusive representation: " # +∞ X 1 2 2 δ(ξ − ω 0 n π ) . (16) δ(ξ) + 2 H(ξ) = L0 n=1 Theorem 2.1 [9] The (causal) correspondence g(t) := ∂ H( ∂t )f (t) is obtained by the input(f )-output(g) state-space ”diffusive realization”:  ∂ψ(ξ, t)   = −ξψ(ξ, t) + f (t), ξ > 0, ψ(ξ, 0) = 0 ∂t Z +∞  g(t) :=  H(ξ)ψ(ξ, t) dξ, 0

(17)

where H is the diffusive representation of H.

Remark 2.1 From (9) the symbolic inductance √ of the coil s tanh ω T (s) may be defined as L(s) = Φi(s) = L0 √ s 0 , but it

∂ H( ∂t )

2.2

(14)

Example 2.1 Using (16), realization (17) is a time-domain model of the coil where (see fig. 3) input is the voltage v and output the current i. ∂ Remark 2.3 The ”impulse response” associated to H( ∂t ) is given by [9]: Z +∞ (18) h(t) = e−tξ H(ξ) dξ = (L H)(t). 0

2.3 Finite-order approximation of the diffusive realization From classical quadrature methods, finite-order convergent approximations of (17) are of the form [9]:  ·  ψ (t) = −ξ ψ (t) + f (t), ξ > 0, ψ (0) = 0  k k k k K (19) P  Hk ψ k (t) .  ge(t) := k=1

An experimental study [2] has confirmed the validity of this model. This operator asymtotically behaves like a half i− 12 h ∂ order integrator, L01ω0 ω10 ∂t , for high frequencies. Despite its interest, such a formulation is not helpful for timedomain simulations.

∂ with ge(t) ' H( ∂t )f (t). Note that in practice, small dimensions (about 10-30 ξ k ) are most of time sufficient for precise approximations. This point is essential, particularly for long-memory operators, the standard approximations of which are traditionally based on the convolutive expression, which leads to algorithms of prohibitive complexity.

2.4 Optimal identification of diffusive representations ∂ Another way to directly obtain an approximation of H( ∂t ) is to solve a particular optimization problem [9]. Let us con€  sider a column vector H = H(jω m ) m=1:M of measured ∂ frequency response of H( ∂t ). From (19), we can define the “ ’K P Hk column vector HK = , in which the jω m +ξ k

k=1

m=1:M

column vector H := (Hk )k=1:K must be found. The vector HK may be written in condensed form as: HK = GH,

(20)

where G is a M × K matrix defined by : Gm,k =

1 . jω m + ξ k

(21)

The problem of searching the vector H which minimizes the Euclidean distance between the two vectors H and HK :  2 2  min H − HK  = min H − GH

H∈RK

H∈RK

(22)

has a unique solution classically given by: −1

H = [Re (G∗ G)]

€  Re G∗ H ,

(23)

Figure 5: Diffusive identification of coil admittance: (- -) theoretical, (—–) identified.

with G∗ the conjugate transpose of the matrix G. Remark 2.4 The set {ξ k } is chosen according to specific informations on H. Figure 5 exhibits the comparison between the frequency response of the theoretical coil admittance (13) (for L0 = 1H and ω 0 = 1rd/s) and the frequency response of an identified finite-dimension diffusive realization. For the identification procedure, {ω m }m=1:M is a geometric sequence between 10−2 rd/s and 103 rd/s with M = 400; {ξ k }k=1:K−1 is a geometric sequence between 10−3 rd/s and 104 rd/s with K = 25 and ξ K = 0 rd/s. The vector H is then: H = (−2.227 10−3 , 2.413 10−3 , −1.037 10−3 , 3.186 10−4 , −1.362 10−4 , 1.170 10−4 ,

− 1.761 10−4 , 3.663 10−4 , −9.039 10−4 ,

2.462 10−3 , −7.214 10−3 , 2.358 10−2 ,

− 9.399 10−2 , 1.784, 4.839 10−1 , 1.172, 2.580, 1.404, 6.383, 2.109, 4.857, 27.92, 29.05, 19.50, 1.001)T .

Remark 2.5 In [2], the same approach has been successfully used in a simpler case, from measurements.

3 Pseudo-differential modelling of transmission lines for machine winding description 3.1

1-D partial pseudo-differential model of a general transmission line

∂ In the symbolic domain (s = ∂t ), the general equations of a linear transmission line of length L (see fig.2) are:

 ∂v(x, s)   = Z(x, s) i(x, s)   ∂x     ∂i(x, s) = Y (x, s) v(x, s) ∂x

(24)

with v(0, s) = E(s) , v(L, s) = ZL (s) i(L, s), and ZL the impedance at the end of the line. From definition of the symbol, (24) may be rewritten in the time-domain following the specific notation of pseudo-differential operators:    Z(x, ∂ ) i(x, t) = ∂v(x, t)  ∂t  ∂x   ∂i(x, t)  ∂  Y (x, ∂t , ) v(x, t) = ∂x

∂ with v(0, t) = E(t), v(L, t) = ZL ( ∂t )i(L, t).

(25)

3.2 Diffusive realizations of the pseudo-differential operators of the transmission line The aim of this section is to build a ”time-local realization” of model (25) for the input-output system represented on figure 6, by use of diffusive representations. Assuming that the 1 1 symbols H(x, s) = Z(x,s) , W (x, s) = Y (x,s) and ZL (s) admit diffusive representations H, W and ZL respectively, system (25) admits the following overall realization:   ∂ψ(ξ, x, t) = −ξψ(ξ, x, t) + ∂v(x, t)    ∂t ∂x        ∂ϕ(ξ, x, t) ∂i(x, t)  = −ξϕ(ξ, x, t) + (26) ∂t Z +∞ ∂x     i(x, t) := H(x, ξ)ψ(ξ, x, t) dξ    0 Z  +∞     v(x, t) := W(x, ξ)ϕ(ξ, x, t) dξ, 0

t > 0, x ∈]0, L[, with boundary conditions:

  v(0, t) = E(t),    ∂χ(ξ, t) = −ξχ(ξ, t) + i(L, t) ∂t Z +∞     v(L, t) := ZL (ξ)χ(ξ, t) dξ,

(27)

0

under consideration. This gives:  ∂ψ(ξ, x, t) ∂v(x, t)   = −ξψ(ξ, x, t) +   ∂t ∂x     Z +∞ ∂ψ(ξ, x, t) λ ∂v(x, t) = H(ξ) dξ − v(x, t)   ∂t ∂x β  0  Z +∞     i(x, t) := H(ξ)ψ(ξ, x, t) dξ, 0

(30) t > 0, x ∈]0, L[, with boundary and initial conditions (27), (28). 3.3.2 Identification of the diffusive realization of admittance The frequency response of the input impedance (Zin ) of a stator-winding phase (between input terminal and iron frame) has been measured (see figure 7) on an equipped induction machine. Two cases have been considered whether the middle point is connected (called Zinc and corresponding to ZL = 0 in (24)) or not connected (called Zinnc and corresponding to ZL = +∞ in (24)) to the iron frame. The characteristic impedance is given by the well-known relation: p Zc (jω) = Zinc (jω)Zinnc (jω). (31)

and initial conditions

E(0) = 0, ψ(ξ, x, 0) = 0, ϕ(ξ, x, 0) = 0, χ(ξ, 0) = 0. (28)

Figure 6: Input-output approach. In order to get numerical solutions, this model may be discretized on the x and ξ-variables, following classical schemes. 3.3 Application to a machine winding 3.3.1 Assumptions and basic model As considered by many authors [5], [10], [6], a winding is supposed to be an homogeneous transmission line i.e., from [24], the operator H(x, s) and W (x, s) do not depend on ∂ ) is of diffusive type, of the x. We also assume that H( ∂t form nearby (14). On the other hand W (s) is assumed, for simplicity, to be: W (s) =

1 , β > 0, λ > 0, βs + λ

(29)

which is one of the most simple way to represent the dielectric behavior of insulating material in the frequency range

Figure 7: Input impedance of a winding phase: middle point connected (- -) or not (—) to the frame. (-·-) characteristic impedance. Remaining that, from (24) in an homogeneous case, the characteristic impedance is defined as s Z(jω) 1 v(jω) Zc (jω) = , (32) = =p i(jω) Y (jω) H(jω)Y (jω)

H(jω) may be evaluated from the relation: H(jω) =

1 , Zc2 (jω)Y (jω)

(33)

if Y (jω) is known. With this end in view, Y (jω) is evaluated in the form (29): Y (jω) =

1 = βs + λ, W (jω)

(34)

using the measurements of Zinnc (jω) at low frequencies. The impedance H(jω) evaluated is given in figure 8.

t > 0, x ∈]0, L[. Equation (35) has been discretized on the c c x-variable and implemented into Matlab -Simulink . In an experimental configuration presented in figure 9, figures 10 and 11 show, on top, the comparison between the simulations and the measurements of the voltage v(x, t) in two cases: the first one with middle point not-connected (fig. 10 , Sw open) and the second one with middle point connected (fig. 11 , Sw closed) to the iron frame. The response of an optimized standard model (a transmission line of type of figure 2) is visible for comparison at the bottom. On each part, E is the input voltage (at x = 0) and vi the voltage at point xi (see fig. 9). The excellent accuracy between simulations and measurements highlights the pertinency of the 1-D partial pseudo-differential model (25) and its diffusive version (30), (35). Note the long-memory behaviors, visible in the case of middle point connected to the iron frame, very ill-fitted by the standard model.

Figure 9: Test configuration for step-response comparison.

References

Figure 8: Comparison between the admittance H measured (- -) and identified (—). Finally, we have identified a finite-dimension diffusive realization of H(jω) (see figure 8), following the method presented in part 2.4.

4 Numerical time-simulations versus measurements In order to obtain time-domain simulation, and using a finitedimension diffusive realization of H(jω), the basic model of the winding (30) becomes:  ∂v(x, t) ∂ψ k (x, t)   = −ξ k ψ k (x, t) +    ∂t ∂x       K ∂v(x, t) X ∂ψ k (x, t) λ (35) = − v(x, t) Hk  ∂t ∂x β   k=1   K  X    t) i(x, := Hk ψ k (x, t),   k=1

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Figure 10: Step-response of the winding, middle point notconnected to the iron frame (top: diffusive model; bottom: standard model).

Figure 11: Step-response of the winding, middle point connected to the iron frame (top: diffusive model; bottom: standard model).

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