ФИЗИКА МЕТАЛЛОВ И МЕТАЛЛОВЕДЕНИЕ, 2015, том 116, № 8, с. 818-824
^ ЭЛЕКТРИЧЕСКИЕ ^^^^^^^^^^^^
И МАГНИТНЫЕ СВОЙСТВА
DETERMINATION OF THE MAGNETIC LOSSES IN LAMINATED CORES UNDER PULSE WIDTH MODULATION VOLTAGE SUPPLY
© 2015 г. N. Vidal, K. Gandarias, G. Almandoz, and J. Poza
Faculty of Engineering, University of Mondragon, Arrasate-Mondragon, Gipuzkoa, 20500 Spain
e-mail: firstname.lastname@example.org Поступила в редакцию 21.02.2014 В окончательном варианте — 01.04.2014
In the laminated ferromagnetic cores employed in transformers and electrical machines energy losses occur resulting in a warming effect and efficiency decreases. Normally, manufacturers only provide iron losses data when a sinusoidal voltage supply is applied, but the actual operating characteristics of electrical machines include non-sinusoidal supplies, in particular pulse width modulation (PWM). This information can be experimentally obtained, but only measuring systems that have function generators with arbitrarily programmable waveforms allow measurements in presence of higher harmonics. Therefore, having an analytical tool to obtain the most accurate estimation of the magnetic losses is of great interest in addressing the design of electric machines. This paper validates an analytical expression based procedure, which delivers results with acceptable accuracy under all operating conditions for the estimation of losses in laminated cores. In addition, it investigates the influence of the modulation amplitude and the switching frequency of the PWM signals in the magnetic losses of soft magnetic materials. For this purpose, non-oriented fully processed electrical steel strips have been measured in a commercial AC permeameter using a single strip tester.
The majority of AC motors are fed through non sinusoidal supply, in particular using pulse width modulation (PWM) inverters, but the magnetic material characteristics usually reported in material data sheets only include information about the energetic behavior of magnetic materials supplied with sinusoidal voltage, typically at 50 or 60 Hz. A PWM supply leads to an increase of the specific losses in the laminated steel, so motor designers usually have to use their own experience or Finite Element analysis [1, 2] to estimate the increase of the iron losses due to PWM supply. This information can be experimentally obtained, but only apparatus with function generators with arbitrarily programmable waveforms, allow measurements in presence of higher harmonics. Such an item of equipment is usually not available for motor designers. Therefore, having an analytical tool to obtain the most accurate estimation of the magnetic losses is of great interest in addressing the design of electric machines. Appropriate modeling is widely used for prediction of the magnetic properties of soft magnetic materials. It is an important tool widely employed in sensor or electrical machine design [3, 4].
Several authors have presented different engineering approaches to solve this problem [5—9]. For this work, the computational model presented in [10, 11] has been chosen due to its simplicity and minimal experimental data requirements.
In this paper, the analytical procedure chosen is described and several tests are performed. Comparative
analysis of the analytically predicted and the experimental results confirm the validity of the employed method. In addition, the influence of the modulation amplitude and the switching frequency of the PWM signals in the magnetic losses of soft magnetic materials are studied.
The experimental results reported here have been obtained on non-oriented fully processed electrical steel strips usually employed in electrical machine cores. For this purpose, a commercial AC permeameter with a feedback yoke has been used. In this paper, experimental data were obtained using a single strip tester while Boglietti et al. measured magnetic losses using an Epstein frame  and a toroidal core [6, 13].
II. ANALYTICAL PROCEDURE
The analytical method employed in this work for predicting iron losses in soft magnetic materials with both sinusoidal and PWM voltage supply was presented in  and is based on the Boglietti model .
A. Analytical Procedure with Sinusoidal Supply
The classical losses Ph and anomalous losses Pec are grouped into one single term. In this way the losses are given by
P = Ph + Pec Ph = k„ (f, B )/B2 and (1)
PeC = keC(f, B)(/B)2.
To simplify the calculations, the frequency is taken like a parameter and the variation of kh and kec is considered only as function of the induction B for different frequency intervals. Third-degree polynomials are employed:
(B) = £ khnB, kec(B) = £ kec„Bn. (2)
j = 0
If it is assumed that at low frequencies the losses are essentially only of hysteresis type, the coefficient kh can be estimated by
kh = P /fBp,
where f is a low frequency (between 5 and 15 Hz) and BP is the induction's maximum value of the hysteresis loop.
The procedure followed for the estimation of losses under sinusoidal apply consists of the following steps:
1. Measure the iron losses corresponding to at least four induction values (i.e. 0.1, 0.5, 1 and 1.5 T) for a low frequency (i.e. 10 Hz).
2. Determine experimental kh with (3) for all four points.
3. Compute the terms khn from (2).
4. Measure the iron losses corresponding to four induction values, for the mean frequency of the range of interest (i.e. 200 Hz for the range 0—400 Hz).
5. Determine experimental kec for all four points with the expression
keC = ( P - khfBp ) /fBp,
P n Phsin + X Pecsin,
where Ph sin is the hysteresis losses with sinusoidal supply related to the fundamental component; Pecsin is the eddy-current losses with sinusoidal supply related to the fundamental component; <V> is the voltage mean rectified value; Vef is the voltage rms value; < Vfund> is the mean rectified value of the fundamental voltage; Vef fund is the rms value of the fundamental voltage; and Ph sin and Pecsin are calculated by (1) and (2) for the fundamental component:
P = P
-'sin ± '
Ph sin = kh(f und, Bfund )ffund Bfund ,
Pecsin = kec(ffund, Bfund)(fundBfund) , 3
kh(Bfund) = £ khnBfund,
n = 0 3
c(Bfund) = £ kecnBfund.
i = 0
where fr is the mean frequency (in this case, 200 Hz).
6. Compute the terms kecn from (2).
7. The loss coefficients are estimated with (2) for any other induction level.
8. The total iron losses P under sinusoidal supply are computed with (1) for any level of induction and frequency.
Therefore, this analytical method involves some measurements. For the experimental determination of parameters of (3) and (4) the hysteresisgraph described in section III has been employed.
B. Analytical Procedure with PWM Supply
The analytical method employed neglects the effect of the minor loops and allows the estimation of the iron losses as a function of the voltage supply provided and its fundamental value. This method is based on the equations:
The different coefficients are obtained following the steps 1—8 of procedure described previously for the sinusoidal supply.
III. MAGNETIC LOSSES MEASUREMENTS
A. AC Hysteresis and Loss Measurement
The AC permeameter employed here is Rema-comp C—200 from Magnet-Physik. For carrying out a measurement, it is necessary to supply the sample with magnetizing (primary) windings and secondary (flux sensing) windings. These windings can either directly wound on the sample, typical for ring shaped samples, or a coil system can be used. In this particular case, a fixture that allows measurements of strips has been used. This fixture is composed of a laminated core, a primary winding with 100 turns to measure the field generation and a secondary with 200 turns to measure the induction. The fixture employed is laminated to avoid eddy currents and its permeability is high compared to the permeability of the specimen. The block diagram in Fig. 1 shows a single sheet specimen and the measuring system. The control program adjusts a function generator which excites an external power amplifier to reach the selected level of magnetic field strength or flux density. The measuring frequency can be selected. The piece of equipment has an oscillo-graphical recording method and works as follows. The magnetic field strength H is determined directly from the voltage drop UH = RI across a measuring resistor R with low inductance. I is the magnetizing current supplied by the power amplifier. This voltage and the voltage induced in the secondary windings are simultaneously sampled by two fast analog to digital converters
VIDAL h gp.
Single sheet tester
Preamplifier Analog/digital converter
POWER AMPLIFIER FREQUENCY GENERATOR
Fig. 1. Block diagram of the measuring system.
(ADC). The secondary voltage U = —d^/dt is then numerically integrated to give the magnetic flux density B.
The procedure followed by the system to record the losses in function of the induction P(B) in the specimen is as follows. First the maximum amplitude to achieve a selected Bmax is established and then the specimen is demagnetized. 50 hysteresis loops are measured, starting with 1/50 of the amplitude that was obtained for excitation to Bmax. The losses P are calculated. The respective peak values, either of B, and the accompanying losses are stored [14, 15].
B. The Waveform of the Function Generator
The waveform of the function generator which is used to control the power amplifier can be selected by the operator. Our function generator allowed for user defined waveforms, so, apart from the standard predefined sinusoidal waveforms, PWM signals can be used. When creating and storing the signals, some requirements must be followed, i.e., the file must contain 16384 values for one period exactly, one poin
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