■ FIGURE 3
■ FIGURE 4
permeability is a
like iron. Unlike
permeability is not
constant; it changes
with flux. Actually,
it changes with flux
Flux density is
B = Φ/Ac and
magnetizing force is H = mmf/lF. Since μ changes with B,
the B-H graph is non-linear as in Figure 3.
Permeability is the slope of the B-H curve at a
specified value of B: μ= ΔB/ΔH. As H increases beyond a
certain point, the curve flattens out. That's saturation: the
maximum flux density a given material can have.
In 1831, Michael Faraday showed that moving a coil
through a magnetic field (or moving a magnetic field
through a coil) induces voltage in the coil: V= N(ΔΦ/Δt)
material is more complicated than Figure 3. Apply a
magnetizing force to a bar of steel. When you remove
the mmf, the bar is magnetized with residual flux. To
demagnetize the bar, you have to apply reverse mmf. If
the battery in Figure 2 is replaced by an AC source, you
get the B-H curve of Figure 4: a hysteresis loop. Every
time the core flux cycles around the loop, some electrical
energy gets converted into heat. It's a form of core loss.
A changing magnetic field will induce voltage into
any conductor it passes through, including the core itself.
Voltage in the core causes eddy currents to flow
producing I R core loss. I R in the coils is copper loss.
Ferrite vs. Steel
V is the induced voltage.
N is the number of turns of wire in the coil.
ΔΦ is the amount of magnetic flux the coil "cuts."
t is the amount of time the coils cuts
through the flux.
In Figure 2, closing the switch
applies voltage across the coil. As coil
current increases, mmf increases,
increases, and a voltage is induced in
the coil that is equal but opposite to the
applied voltage. The induced voltage
lasts until the core saturates. Then, the
coil becomes a short-circuit across the
battery. So, the maximum Δt in the
above equation is the time it takes for
the flux to go from zero to saturation
(Φsat). Saturation causes current spikes
so the maximum flux used (Φm) must
be less than Φsat.
Currents, and Power Loss
Both hysteresis and eddy current losses increase
rapidly with frequency. At audio frequencies, transformer
cores are made from insulated steel
laminations. Steel has very high
permeability, but a lot of loss at
switch-mode frequencies where ferrite is
Ferrite is a ceramic containing
iron and other elements. It starts as
clay-like material that can be formed into
many shapes, then baked hard. It's
brittle and can crack or break from too
much mechanical stress. It has less
permeability than steel, but much less
loss at high frequency. Ferrite is
conductive and abrasive, so often it's
coated with insulation. Figure 5 shows
a toroid core like the one we will use.
A toroid is the ideal shape for a
Voltage and Flux
■ FIGURE 5
The B-H curve for magnetic
In Figure 6, the switches alternately
March 2009 47