I read in sci.electronics.design that Tony Williams
You can also obtain the result by simple differentiation. Since it's a
text-book result, I don't propose to post the mathematics here.
Cutting and pasting from your previous post of the mathematics:
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Well, it's an example of a general principle, 'fixed loss' = 'variable
loss' is the condition for highest efficiency. For a fixed core size
and induction, the 'iron' loss is fixed, and the copper loss is
variable.
You can do it the other way round, but you get the same answer. You
can find it in Section 121 of Fundamentals of Electrical Engineering,
by E Hughes, Longmans, London 1955, but I'm sure it's also in many
better-known textbooks.
In Hughes' treatment, the only assumption is that the copper loss is
small compared with the load power.
The output power is Vs*Is*cos[phi].
The input power is Vs*Is*cos[phi] + P(iron) + Is^2*Res.
phi = phase angle of load (it doesn't affect the result),
Res = Rs + Rp*n^2,
n = turns ratio.
So the efficiency is:
Vs*cos[phi]/{Vs*cos[phi] + P(iron)/Is + IsRes},
which is a maximum when the denominator is a minimum, if Vs is nearly
constant, i.e. copper loss is small).
d/dIs(denom) = 0 if P(iron)/Is^2 = Res,
i.e. P(iron) = Is^2*Res.
Q.E.D.
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and I replied:
Thanks, John. I think that one additional assumption is required to
extend the observation that a transformers efficiency is maximized by
the above equations when core and copper losses are equal, to the
statement that a transformer should be designed for equal losses: that
the 'cost' of copper efficiency is equal to the 'cost' of core
efficiency. Where 'cost' is whatever combination of volume, weight,
price etc. is important to your application.
Since the 'cost' of copper and core efficiency is generally not equal,
given any set of total cost limits, the efficiency/cost optimized
design having generally unequal losses will be more efficient than an
equal loss design.
An example of true optimization is presented in Smith, "Magnetic
Components" for the case of volume optimization (lowest volume for a
given efficiency):
Expressing core loss as C1 * B^X,
C1 = constant (actually a function of frequency and B, but constant
under design conditions)
B = Flux Density
X = Flux Density Exponent, typically around 2.5
(where C1 and X are derived from core mfgr's data sheet)
Then for various values of Flux Density Exponent, the optimum % core
loss (from the minimum volume optimization equations) is:
X core loss
3 40%
2.5 44% (most typical)
2 50%
1.5 57%
1 67%
Optimizing for something other than efficiency per unit volume would
no doubt result in different results.
But I will conceed that the equal loss rule is a reasonable first
order approximation, even though Smith correctly points out that it is
not in any way useful for the design of truly optimized transformers.
