2.2 Loss evaluation
2.2.2 Iron losses
Iron losses are probably the most complex loss component of all. The foundations of the loss evaluation in an iron core were laid originally by C.P Steinmetz in On the law of Hysteresis in 1892. According to (Steinmetz, 1984), iron losses can be evaluated by an analytical equation
^{)= Ÿ eH .¡+ Ÿ)¢eH ,
2.20
where
- κh is the hysteresis coefficient, - κe is the eddy current coefficient, - f is the affecting frequency and
- eH is the peak value of flux density in the material.
The equation describes how much energy is lost per cycle in the material. The power loss can be obtained by multiplying the Eq. 2.20 by frequency. The equation implies that the losses are proportional to the area inside the hysteresis loop. The hysteresis loop defines the maximum flux density limits for given field strengths ending to a certain value in both of the odd quadrants, which are the saturation limits for the magnetic flux density. Although this equation is quite a robust approach, it gives a generalized idea of the phenomenon in the material. According to the Steinmetz equation, the hysteresis losses are proportional to a material-specific hysteresis constant, and the applied flux density to a certain power, which varies between 1.6 and 2 in the literature.
The latter part presents the eddy current losses in relation to a certain eddy current factor and to the frequency of the phenomenon. Many other attempts to determine the magnitude of core losses have been presented ever since. A modernized version of iron loss hand calculation relies on the material data provided by electric sheet manufacturers. In (Vogt, 1983), the equation for iron losses is given in the form
•‹$ )= {)Ÿw£w .¤ ¥H.¤¦ ,
2.21
where
- υu1.5 is the specific total loss of the material at 1.5 T and the frequency under observation [W/kg],
- mFe is the core mass and
- κu is the geometry-related coefficient given in Table 2.7.
The loss formulation Ÿw£w .¤can be replaced by a more extensive formula, which describes the harmonic hysteresis losses and the eddy current losses separately
Ÿw w .¤= ,¤Ÿ §¢/ 1892. According to (Steinmetz, 1984), iron losses can be evaluated by an analytical equation
^{) = Ÿ eH .¡+ Ÿ)¢eH ,
2.20
where
- κh is the hysteresis coefficient, - κe is the eddy current coefficient, - f is the affecting frequency and
- eH is the peak value of flux density in the material.
The equation describes how much energy is lost per cycle in the material. The power loss can be obtained by multiplying the Eq. 2.20 by frequency. The equation implies that the losses are proportional to the area inside the hysteresis loop. The hysteresis loop defines the maximum flux density limits for given field strengths ending to a certain value in both of the odd quadrants, which are the saturation limits for the magnetic flux density. Although this equation is quite a robust approach, it gives a generalized idea of the phenomenon in the material. According to the Steinmetz equation, the hysteresis losses are proportional to a material-specific hysteresis constant, and the applied flux density to a certain power, which varies between 1.6 and 2 in the literature.
The latter part presents the eddy current losses in relation to a certain eddy current factor and to the frequency of the phenomenon. Many other attempts to determine the magnitude of core losses have been presented ever since. A modernized version of iron loss hand calculation relies on the material data provided by electric sheet manufacturers. In (Vogt, 1983), the equation for iron losses is given in the form
•‹$ )= {)Ÿw£w .¤ ¥H.¤¦ ,
2.21
where
- υu1.5 is the specific total loss of the material at 1.5 T and the frequency under observation [W/kg],
- mFe is the core mass and
- κu is the geometry-related coefficient given in Table 2.7.
The loss formulation Ÿw£w .¤ can be replaced by a more extensive formula, which describes the harmonic hysteresis losses and the eddy current losses separately
Ÿw w .¤= ,¤Ÿ §¢/
50¨ + ) ,¤Ÿ)§¢/
50¨
2.22
where
- κh is the hysteresiscoefficient, - κe is the eddy currentcoefficient,
- vh1,5 is the hysteresis core loss component [W/kg] and - ve1,5 is the eddy current core loss component [W/kg].
Table. 2.7. Geometry-related core loss coefficients for Eqs. 2.21 and 2.22 (Vogt, 1983).
Machine type Teeth Yoke manufacturer. However, the specific total loss can be fitted to the existing manufacturer data. For example, M270-35A by Surahammars Bruks has specific losses according to Table 2.8.
Table. 2.8. M270-35A specific loss data according to the Surahammars data sheet.
- vh1,5 is the hysteresis core loss component [W/kg] and - ve1,5 is the eddy current core loss component [W/kg].
Table. 2.7. Geometry-related core loss coefficients for Eqs. 2.21 and 2.22 (Vogt, 1983).
Machine type Teeth Yoke manufacturer. However, the specific total loss can be fitted to the existing manufacturer data. For example, M270-35A by Surahammars Bruks has specific losses according to Table 2.8.
Table. 2.8. M270-35A specific loss data according to the Surahammars data sheet.
Equation 2.29 can be fitted to the existing data in Table 2.8 by selecting the hysteresis and eddy-current components vh1.5 and ve1.5 as 1.50 and 0.97 W/kg, respectively. The additional eddy current coefficient needs a value of its own for each frequency. The best fitting is achieved with the coefficient values 1.0, 0.7, 0.6, and 0.5 for the frequencies 50, 100, 200 and 400 Hz, respectively.
The fitting results are presented in Fig. 2.5.
Fig. 2.5. Specific losses of M270-50A according to Table 2.8 data fitted with the core loss coefficients of Table 2.7.
In the case of known harmonic content in the flux density, the material-specific constants κe can be calculated more accurately by the equation
Ÿ)=∑ teHª &
eH
2.23
Equation 2.23 can be used together with the finite-element-based design programs. A similar approach has also been presented in (Pyrhönen et al., 2008) for the total core losses by equation
•{)= κ«,;eH {)
2.24
where κh,e is the material-specific total loss per kg, including hysteresis and eddy currents, and mFe is the mass of the iron circuit. The equation can be used to determine losses in each individual part
Equation 2.29 can be fitted to the existing data in Table 2.8 by selecting the hysteresis and eddy-current components vh1.5 and ve1.5 as 1.50 and 0.97 W/kg, respectively. The additional eddy current coefficient needs a value of its own for each frequency. The best fitting is achieved with the coefficient values 1.0, 0.7, 0.6, and 0.5 for the frequencies 50, 100, 200 and 400 Hz, respectively.
The fitting results are presented in Fig. 2.5.
Fig. 2.5. Specific losses of M270-50A according to Table 2.8 data fitted with the core loss coefficients of Table 2.7.
In the case of known harmonic content in the flux density, the material-specific constants κe can be calculated more accurately by the equation
Ÿ)=∑ teHª &
eH
2.23
Equation 2.23 can be used together with the finite-element-based design programs. A similar approach has also been presented in (Pyrhönen et al., 2008) for the total core losses by equation
•{)= κ«,;eH {)
2.24
where κh,e is the material-specific total loss per kg, including hysteresis and eddy currents, and mFe is the mass of the iron circuit. The equation can be used to determine losses in each individual part
of the machine, but the problem is that the value κh,e determined by the material manufacturer is valid only for sinusoidally varying flux densities. This problem is usually solved by using empirically determined loss coefficients presented in Table 2.9.
Table. 2.9. Core loss coefficient κh,e according to the machine type for Eq. 2.31.
Machine type Yoke Teeth
Synchronous machine 1.7–2.5 1.5–1.8
Asynchronous machine 1.7–2.5 1.5–1.8
Approaches to analytically solve eddy current and hysteresis losses are numerous. Common to all these is that the equations are more or less based on the classical definitions for eddy current and hysteresis losses presented above. The problem is that with the modern frequency-converter-fed systems, the loss evaluation and material properties are not accurate for classical equations.
Experimental studies have shown a significant increase in core losses with the PWM excitation (Tutkun, 2002), (Boglietti et al., 1991), (Boglietti et al., 2003) . The actual power dissipation in frequency converter drives is considerably higher than for the pure sinusoidal waveform case owing to the harmonic waves in the supply current waveform. The current causes the same distortion to the flux produced in the coils, and thus, increases the amount of induced iron losses.
To minimize the PWM effect, the modulation index should be close to unity and the switching frequency above 5 kHz (Boglietti et al., 1993), (Boglietti et al., 1995).