Drive system efficiency - Propulsion Systems for Hybrid Vehicles - page 398

8.1.1 Core losses
Iron losses in soft ferromagnetic materials are classically separated into hysteresis
and eddy current components, P h and P e [2]. It can be said that hysteresis loss is
caused by localized irreversible changes as magnetizing intensity is cycled within
the confines of the material's saturation level of induction. When there are no
minor hysteresis loops to contend with, the classical Steinmetz equation for core
loss applies:

P core ¼ P h þ P e
P core ¼ k h M core fB a

ð 8 : 1 Þ

2

þ k e M core ð fB Þ

where the coefficients k h and k e are in W/kg and core mass is in kg. The hysteresis
component exponent on induction is in the range 1.6

2.2. Some authors also
add an additional term to (8.1) to account for high frequency harmonic losses to
account for inverter drive switching frequency components in M/G voltage. Others
prefer to condense the entire expression given by (8.1) into a single term with a
coefficient easily extracted from manufacturer data sheets on the particular sheet
steel used. Equation (8.2) illustrates this format:

< a <

a b

B
B s

w
w 0

P core ¼ k core M core

ð 8 : 2 Þ

a ¼ 1 : 9
b ¼ 1 : 6

where k core is the data sheet value for core loss at the normalized induction, B 0 , and
excitation frequency, w 0 . For a typical good quality steel used in hybrid M/G, such
as Tempel M19, the value of the loss coefficient, k core = 1.5 10 5 W/kg at B 0 =1T,
and w 0 =2 p (400).
In hybrid propulsion systems, the switching converter does introduce voltage
(and current) harmonics that contribute additional core losses through the excitation
of minor hysteresis loops in the magnetic steels employed in the designs. This effect
can be accounted for if the component loss coefficients given in (8.1) are made
functions of the induction. This augmented coefficient, k e K ( B ), has the property of
adding an additional excitation loss component to the total core loss expression:

P h ¼ k h M core fB a K ð B Þ

B X

n

c h

K ð B Þ¼ 1 þ d B ð 8 : 3 Þ

i

P h ¼ P hc þ P h exc

where d B is the incremental induction change due to an excursion about a minor
hysteresis loop. Figure 8.2 illustrates the major and minor hysteresis loops used in
the context of this discussion.