P ROPERTIES OF ELECTRICAL STEELS

Fig. 1.20 shows the DC B-H curve in the first quadrant for two steels. The lower curve is a typical electrical motor steel having 1.5% Silicon to increase the resistivity to limit eddy-current losses. The saturation flux-density of such steels (i.e. the flux-density at which the incremental permeability becomes equal to µ₀) is typically about 2.1T. The upper curve is for a cobalt- iron alloy with a saturation flux-density of about 2.3T. This material is much more expensive than normal electrical steel, and is only used in special applications such as highly rated aircraft generators, where light weight and high power density are at a premium. The maximum permeability of electrical steels is of the order of 5,000 µ0, and usually occurs between 1 and 1.5 T.

In Fig. 1.20, the total permeability of the electrical steel at 2.0T is about 2.0/3,000 which is approximately 530 µ0.

P ' C_hf B_pkn

%

%

%

%

Fig. 1.21 Typical form of variation of losses in electrical steel, versus frequency and flux-density

Losses. Under AC conditions, a power loss

arises in electrical steel as shown in Fig. 1.21, which indicates increasing loss as the frequency and flux-density increase. The loss is attributed to

(a) hysteresis;

(2) eddy-currents; and (3) “anomalous loss”.

The hysteresis component is associated with the changing magnitude and direction of the magnetization of the domains, while the eddy- current loss is generated by induced currents. Eddy-currents can be inhibited by laminating the steel, so that the eddy-currents become resistance limited and the loss is then inversely proportional to the resistivity. If the eddy- currents are resistance-limited the loss is also proportional to 1/t2, where t is the lamination

thickness. At higher frequencies the resistance limited condition is lost, and the losses increase rapidly with frequency. For this reason, very thin laminations, as thin as 0.1 mm, may be used at very high frequencies (such as 400 Hz in aircraft generators or 3000 Hz in certain specialty machines). The “anomalous loss” is associated with domain wall movement and is not often accounted for in empirical expressions of the iron loss.

Characterization of core loss. Core-loss data from steel suppliers is almost always obtained from

measurements in which a sinusoidal flux waveform is applied to a sample of laminations in the form of a stack of rings or an “Epstein square” made up from strips interleaved at the corners. The loss may be characterized by the so-called Steinmetz equation with separate terms for hysteresis and eddy- current loss:

The units of P are usually W/kg or W/lb. Bpk is the peak flux-density in T, and f is the frequency in Hz.

Ch is the hysteresis loss coefficient and Ce is the eddy-current loss coefficient. The exponent n is often

assumed to be 1.6!1.8, but it varies to a certain extent with B_pk. To a first approximation we can write

n = a + bBpk. With this modification,

The flux-density in motor laminations may be far from sinusoidal, and one approximate way to deal with this is to modify the Steinmetz equation in the following way, recognizing that the eddy-current loss component is expected to vary as the square of the EMF driving the eddy-currents, and that this EMF varies in proportion to dB/dt. Thus

The hysteresis loss component is unchanged, but the eddy-current component is taken to be proportional to the mean squared value of dB/dt over one cycle of the fundamental frequency. Eqn. (1.71) can be applied in the respective sections of the magnetic circuit, after calculating the relevant flux-density waveforms.

P f ' Ch Bpk a % bB_pk % C_e f B_pk2 _(1.74) P f ' D % Ef. (1.75) D ' C_h B_pka % bBpk_{. (1.76)}

log D₁ ' log C_h % (a % bB_{pk 1}) log B_{pk 1}. _(1.77)

C_e1 ' Ce

2B2. (1.72)

The eddy-current loss coefficient Ce1 in the modified form can be derived from the sinewave coefficient

Ce if we assume that eqn. (1.71) holds with B = Bpk sin (2B f t). Then dB/dt = 2B f Bpk cos (2B f t) and

(dB/dt)2 = 4B2f2B_pk2 cos2 (2B ft), the mean value of which is [dB/dt]2 = 2B2f2B_pk2. For sinewave flux- density, equations 1.70 and 1.71 give the same result if

Extracting the core loss coefficients from test data. Two procedures are used for extracting the

coefficients Ch, Ce1, a and b from sinewave loss data. The more elaborate of these requires a complete

set of curves of core loss vs. frequency at different flux-densities. When this data is not available, a simpler procedure is used based on five parameters.

Simple procedure—It is often the case that only a single value of P is available, for example, 8 W/kg at

50 Hz, measured with B_pk = 1.5 T. There is not enough data to determine the four loss coefficients uniquely, so we use an estimate for n in eqn. (1.69); for example, n = 1.7. It is further necessary to estimate the split between hysteresis and eddy-current loss. If h is the fraction of the total loss attributable to hysteresis, then it can be shown that

Then a = n; b = 0, and Ce1 = Ce/2B2.

Procedure used with complete set of core-loss data—The core loss data is usually in the form of graphs

of P vs. f at different flux-densities, or P vs. Bpk at different frequencies. The procedure is to try to

separate the hysteresis and eddy-current components of P. First we divide eqn. (1.70) by f :

We then plot graphs of P/f vs. f for three values of Bpk, e.g. 1, 1.5 and 2T with f from 50 to the highest

frequency. The graphs should be straight lines and can be represented by

The intercept D on the vertical (P/f) axis must be equal to

The intercepts D1, D2 and D3 for the three values of Bpk are substituted into the logarithm of eqn. (1.76),

giving three simultaneous linear algebraic equations for Ch, a and b of the form

C_e ' P(1 ! h)

f2_B pk2

and C_h ' hP

10_{The ampere-conductor distribution is often loosely termed the MMF (magneto-motive force), or MMF distribution.}

These are solved for log C_h, a and b; C_h is then obtained from log C_h. Next, three values of C_e are obtained from the gradients of the three graphs of P/f vs. f , eqn. (1.74). The average or the highest value can be taken for Ce. Finally Ce1 = Ce/2B2. The loss curves may be re-plotted from the formula as a

check. Any extrapolation to higher B_pk or f should be checked carefully.

Note that Ce is approximately inversely proportional to t2 , where t is the lamination thickness. This

can be used to modify C_e (or C_e1) for different thicknesses if test data is not available.