Recent Developments of Electrical Drives - Part 45 ppsx

E= −dmThe value of the voltage of the machine operated as generator under load conditions can be calculated by relation 3: V= E − RI − jLσ ωI 3 where V is the voltage on stator windings

III-3.4 ADVANCED MATERIALS FOR HIGH SPEED MOTOR DRIVES

G Kalokiris, A Kladas and J Tegopoulos

Faculty of Electrical and Computer Engineering, National Technical University of Athens,

9, Iroon Polytechneiou Street, 15780 Athens, Greece

tegopoul@power.ece.ntua.g

Abstract The paper presents electrical machine design considerations introduced by exploiting new

magnetic material characteristics The materials considered are amorphous alloy ribbons as well as neodymium alloy permanent magnets involving very low eddy current losses Such advance materials enable electric machine operation at higher frequencies compared with the standard iron laminations used in the traditional magnetic circuit construction and provide better efficiently

Introduction

The impact of innovative materials on the electrical machine design is very important The advantages involved in machine efficiency and performance are important as mentioned in [1,2] These materials enable electric machine operation at high frequencies when supplied

by inverters, compared to the standard iron laminations used in the traditional magnetic circuit construction The features and performance characteristics are analyzed by using field calculations and tested by measurements In this paper, the study of asynchronous and permanent magnet machines based on such materials is undertaken Low losses and high volumic power associated with high speed and converter machine operation are the main advantages of such applications [3–5]

Design procedure

The proposed machine design procedure involves two steps In a first step standard design methodology is used for preliminary design In a second step the method of finite elements

is implemented to calculate the machine efficiency and performance Finally, a prototype is constructed in order to validate and compare the simulated machine characteristics to the corresponding experimental results [6,7]

The method of finite elements, is based on a discretization of the solution domain into small regions In magnetostatic problems the unknown quantity is usually the magnetic vector potential A, and is approximated by means of polynomial shape functions In two-dimensional cases triangular elements can easily be adapted to complex configurations and first order elements exhibit advantages in iron saturation representation [8,9] The size of elements must be small enough to provide sufficient accuracy In this way the differential

S Wiak, M Dems, K Kom˛eza (eds.), Recent Developments of Electrical Drives, 443–450.

2006 Springer.

E= −dm

The value of the voltage of the machine operated as generator under load conditions can

be calculated by relation (3):

V= E − RI − jLσ ωI (3) where V is the voltage on stator windings in V, E is the electromotive force at no load in V,

R is the stator resistance in, L σis the stator leakage inductance in H,ω is the rotor angular

velocity in rad/s, and I is the stator current in A Then the magnetic flux and electromotive forces can be derived by using equations (1) and (2) Furthermore, the total resistance of stator winding can be calculated by the following relations:

R= ρl

whereρ is the electric resistivity of copper, l is the winding total length, and s is the conductor cross-section The winding length l can be estimated from relation (5):

l= 2 × (lax+ lp)× Nw× P (5) where laxis the machine’s axial length in meters, lpis the polar pitch in meters, and Nwis the total number of series connected turns

Results and discussion

The case of a permanent magnet machine has been considered The machine designed has been checked through a 2.5 kW prototype, which has been connected to an appropriate power electronics converter The air-gap width has been chosen 1 mm while a multipole

“peripheral” machine structure has been adopted The geometry of the permanent magnet machine is shown in Fig 1 giving also the mesh employed for the two-dimensional finite element program of the machine involving, approximately 2,100 nodes 4,000 triangular elements

Figure 1 Employed triangular mesh of the one pole part of the permanent magnet machine

constructed

In a first step the no load operating conditions have been examined The corresponding simulated voltage waveform is shown in Fig 2 while the measured one is given in Fig 3, respectively In these figures a good agreement between the simulated and measured results can be observed The simulation results concerning full load voltage of synchronous gen-erator are presented in Fig 4 Fig 5 gives the measured results under the same operating conditions A good agreement can be observed in these figures between the simulated and measured results also in the case of full load

Moreover, measurements were realized for an asynchronous motor, which was supplied

by an inverter with variable frequency The motor is a three phase, four-pole, machine

Figure 2 No load voltage waveform of permanent magnet machine (simulation).

Figure 4 Full load voltage waveform of permanent magnet machine (simulation).

Time (sec)

200 150 100 50 0 50 –100 –150 –200

0.00E+00 2.50E-00 5.00E-00 7.50E-00 1.00E-02 1.25E-02

Figure 5 Full load voltage waveform of permanent magnet machine (measurement).

a

b

Figure 6 Simulated field distribution in the machine under low load conditions (a) Fundamental

supply frequency of 300 Hz, (b) switching frequency of 10 kHz

supplied at a frequency of 400 Hz, at a voltage of 208 V while the nominal, speed is 10.800 rpm The motor was tested under no load and low load operating conditions, for various frequencies

Fig 6(a) shows the field distribution in the machine supplied at fundamental frequency

of 300 Hz, while Fig 6(b) gives the field distribution at the switching frequency of 10 kHz Fig 7 presents the respective measured phase voltage and current time variations

Fig 8 shows the field distribution in the machine supplied at fundamental frequency

of 100 Hz, while Fig 9 presents the respective measured phase voltage and current time variations at the switching frequency of 10 kHz

Table 1 presents the measured and simulation results under no load conditions with a switching frequency of 1 kHz Table 2 presents the same results under low load conditions Table 3 presents the results related to a switching frequency of 10 kHz

The simulated torque Tsis calculated by the relation:

Ts= Ft· rg (6) where Ftis the total circumferential tangential force in N and rgis the middle air-gap radius

in m The Maxwell’s stress tensor is calculated by relation (7):

Ft= 1

μ0



C

BnBtdl L0 (7)

b

Figure 7 Measured supply quantities in the machine for supply frequency of 300 Hz, under low load

conditions (a) Phase voltage time variation, (b) phase current time variation

Figure 8 Simulated field distribution in the machine, fundamental supply frequency of 100 Hz under

low load conditions

a

b

Figure 9 Measured supply quantities in the machine for supply frequency of 100 Hz under low load

conditions (a) Phase voltage time variation, (b) phase current time variation

Table 1 Measured and simulation results under no load conditions and fs= 1 kHz

f 1 fundamental (Hz) I 1 measured (A) V 1 measured (V) T measured (Nm) V 1 simulated (V) T simulated (Nm)

Table 2 Measured and simulation results under low load conditions and fs= 1 kHz

f 1 fundamental (Hz) I 1 measured (A) V 1 measured (V) T measured (Nm) V 1 simulated (V) T simulated (Nm)

Table 3 Measured and simulation results under low load conditions and fs= 10 kHz

f 1 fundamental (Hz) I 1 measured (A) V 1 measured (V) T measured (Nm) V 1 simulated (V) T simulated (Nm)

[1] M.R Dubois, H Polinder, J.A Ferreira, Contribution of permanent-magnet volume elements

to no-load voltage in machines, IEEE Trans Magn., Vol 39, No 3, pp 1784–1792, 2003 [2] T Higuchi, J Oyama, E Yamada, E Chiricozzi, F Parasiliti, M Villani, Optimization procedure

of surface permanent magnet synchronous motors, IEEE Trans Magn., Vol 33, No 2, pp 1943–

1946, 1997

[3] A Toba, T Lipo, Generic torque maximizing design methodology of surface permanent magnet Vernier machine, IEEE Trans Ind Appl., Vol 36, No 6, pp 1539–1546, 2000

[4] G Tsekouras, S Kiartzis, A Kladas, J Tegopoulos, Neural network approach compared to sensitivity analysis based on finite element technique for optimization of permanent magnet generators, IEEE Trans Magn., Vol 37, No 5/1, pp 3618–3621, 2001

[5] D.C Aliprantis, S.A Papathanassiou, M.P Papadopoulos, A.G Kladas, “Modeling and Control

of a Variable-Speed Wind Turbine Equipped with Permanent Magnet Synchronous Generator”, International Conference on Electrical Machines, Helsinki, Finland, 2000, pp 558–562 [6] N.A Demerdash, J.F Bangura, A.A Arkadan, A time-stepping coupled finite element-state space model for induction motor drives, IEEE Trans Energy Convers., Vol 14, No 4,

pp 1465–1477, 1999

[7] T.M Jahns, Motion control with permanent magnet AC machines, IEEE Proc., Vol 82, No 8,

pp 1241–1252, 1994

[8] C Marchand, Z Ren, A Razek, “Torque Optimization of a Buried Permanent Magnet Syn-chronous Machine by Geometric Modification using FEM”, Proc EMF’94, Leuven, Belgium,

1994, pp 53–56

[9] H.C Lovatt, P.A Watterson, Energy stored in permanent magnets, IEEE Trans Magn., Vol 35,

No 1, pp 505–507, 1999

III-3.5 IMPROVED MODELING OF THREE-PHASE TRANSFORMER

ANALYSIS BASED ON NONLINEAR B-H

CURVE AND TAKING INTO ACCOUNT

ZERO-SEQUENCE FLUX

B Kawkabani and J.-J Simond

Laboratory for Electrical Machines, Swiss Federal Institute of Technology,

EPFL-STI-ISE-LME, ELG Ecublens, CH-1015 Lausanne, Switzerland

basile.kawkabani@epfl.ch

Abstract The present paper deals with a new approach for the study of the steady-state and transient

behavior of three-phase transformers This approach based on magnetic equivalent circuit-diagrams,

takes into account the nonlinear B-H curve as well as zero-sequence flux The nonlinear B-H curve is

represented by a Fourier series, based on a set of measurement data For the numerical simulations, two methods have been developed, by considering the total magnetic flux respectively the currents as state variables Numerical results compared with test results and with FEM computations confirm the validity of the proposed approach

Introduction

Traditionally in most of power system studies, the modeling of a three-phase transformer is

reduced to its short-circuit impedance The B-H curve introduced in some improved models

and based on a set of measurement data, is approximated generally by several straight-line

segments connecting the points of measurements But apparently, such B-H curve obtained is

not smooth at the joints of the segments, and the slopes of the straight lines, representing the permeability, are discontinuous at these joints Moreover, in the set of differential equations considering the currents as state variables, one needs the expressions of the derivatives of the inductances vs the currents, which is impossible by using the above mentioned procedure

For that reason in the present study, the nonlinear B-H curve (or U-I curve) is represented

by a Fourier series technique [1], based on the set of measurement data An analytical

ex-pression of a smooth B-H curve connecting the discrete measurement points can be defined.

By using a magnetic equivalent circuit-diagram representing the three-phase transformer, all the self and mutual inductances can be expressed analytically in function of the magnetic reluctances of the cores These inductances (and their derivatives) can be determined pre-cisely using the predetermined series Fourier representation, and adapted at each integration step in the numerical simulations

S Wiak, M Dems, K Kom˛eza (eds.), Recent Developments of Electrical Drives, 451–460.

2006 Springer.

Fourier series representation of B-H curve (U = f (I ) curve)

A set of N + 1 discrete measurement data U n and I n or B n -H nof a three-phase transformer

(n = 0, 1, 2, 3, , N) is given For the sake of making use of the Fourier series, a mirror image of this set of data is made about the U or B axis (Fig 1).

One has:

f (H ) = a0+∞

k=1

a k · cos(ξ k · H) (1) with:

a0= 1

Hmax

·

N



n=1



B n · (H n − H n−1)−1

2 · α n · (H n − H n−1)2



(2)

and

a k= 2

Hmax

·

N



n=1

⎡

⎢

⎢

⎢

⎢

α n

ξ k

· sin (ξ k · H n−1)· (H n − H n−1)+ 1

ξ2 · α n · (cos (ξ k · H n)− cos (ξ k · H n−1))+

B n

ξ k

· (sin (ξ k · H n)− sin (ξ k · H n−1))

⎤

⎥

⎥

⎥

⎥

(3)

ξ k = k · π

Hmax

for the k th term (5)

α n = B n − B n−1