About leakage inductance in transformers

The usual technique for measuring the leakage of a two-winding transformer
is to short one winding and measure the inductance at the other winding.

Imagine a two-winding transformer with one winding (denote this the
primary) having a self inductance of L1, and a winding resistance of R1.
The other winding (the secondary) has a self inductance of L2 and a winding
resistance of R2. The mutual inductance between the two windings is m.

Develop an expression for the impedance (involving the complex frequency s)
seen at the primary if the secondary is shorted.

This impedance will have a real part and an imaginary part. A good LCR
meter can measure both parts, and will probably be able to display the
imaginary part as an inductance. Is this really the leakage inductance?
Under what conditions might it not be a good value for the leakage
inductance?

Can anybody beat Jim Thompson to the punch in developing the complex
impedance expression?

 

Insufficient data....(?) my guess

X = R1 + M.SQRT(L1/L2)R2 + sLleak

DNA

 

If I say it is and state the conditions.
Large signal/different excitation frequency conditions.

DNA

 

Assuming a steel cored transformer. I've yet to find -any- LCR meter that
will measure the inductive component.
They're good at measuring R (so is any cheap dvm) but are magnificently
inaccurate in any guesstimate of inductance. A better guess is to multiply
the offered value by 10, or even just pluck a value out of thin air.
john.

 

How do you know that the LCR meter is giving you a wrong value for the
inductance unless you have another method which you are convinced is giving
the correct value? What is that other method that you trust?

 

Essentially by comparison to direct measurement at normal working voltage
and frequency (eg 240Vac 50Hz).
For a Q meter design, I'd noticed all my steel cored test transformers and
chokes were giving weird inductance and loss value readings that varied
nearly directly with the AC test level, running upto a couple of volts. (L
up, R down). Not my usual 10-20% errors (my kind of precision!) but up in
the 1000s%.
Only semblence of reasonable values started to turn up at the 40V~50V drive
level. Move the frequency up a tad and even these readings turned to mush.
5 non-agreeing and different LCR meters (bridge and vectored) later and I
gave up.
Figured Steel/Iron takes the real meaning of the word "complex". Take the
Steel out and all suddenly becomes sweetness and light and textbook
john

 

I thought this was what you were talking about, and it's well known.

It's an interesting topic in its own, but in this thread I'm talking
about the inductance of one winding with the other winding shorted. Under
that condition the phenomenon you're referring to doesn't happen.

 

[...]

Yes. I was on a flyer there.

12Watt, 240V to 24V, 50hz, mains transformer. Shorted secondary
50Hz constant current to primary
10ma = 3.67Vac primary volts
1ma = 0.36Vac
0.1ma= 0.035Vac
i.e. constant Z

Open secondary
10ma =157Vac primary volts
1ma = 5.8Vac
0.1ma= 0.15Vac
i.e a mess.

 

What you are seeing here (the open circuit numbers) is the magnetizing
loss. It is highly non-linear and only approximately modeled by an L and
R. It is due to core losses and hysteresis and it increases dramatically
as the core approaches saturation (as your numbers indicate).

This is the primary (no pun intended) reason that leakage impedance
measurements are best taken with a shorted winding. The voltages are
lower and the contribution of the magnetizing loss is minimized.

 

An HP LCR Bridge and this method is exactly how we measured it...
ALL THE TIME. Interwinding capacitance as well.

 

OPERATOR ERROR :-]

You have to hold your tongue like Michael Jordan does. :-]

 

Would cheating do? Open Prof Wm Frazer's book,
"Telecommunications", to page 80, "Inductively-
-Coupled Circuits with a Resistive Load on the
Secondary.". Read off eqns 4.19 and 4.20......


w^2.M^2.(Rl+Rs)
Effective Rpri = Rp + --------------------
(Rs+Rl)^2 + (w.Ls)^2


( (w^2.M^2).Ls )
Effective Xpri = j.w.( Lp - -------------------- )
( (Rs+Rl)^2 + (w.Ls)^2 )

For a short circuit secondary set Rl to 0.

Interesting that the effective primary resistance
is increased as Rl reduces in value.
Slightly counter-intuitive, (to me anyway).

 

I can measure a power transformer primary inductance with three
different bridges and get wildly differing values, a 5:1 spread not
being unusual. And none of those values are predictive of actual
line-voltage magnetizing current. Silicon steel laminations are hugely
nonlinear on flux density and frequency. You see the same thing in
commercial current transformers... they're great from about 5 to 100%
of rated current, but go to hell - increasing losses and phase shift -
at low currents.

The only sensible way to measure power transformer primary inductance
is by measuring current and phase angle with 120 volts applied. And
even then, the harmonics in the current may be significant.

John

 

Yes, you cheated! But, you could have derived them if you had to, right?

Anyway, those are correct. If you leave the j.w. off the second
expression, you will have a formula for Effective Lpri.

It's even stranger than that. If the secondary resistance (Rs+Rl, or
just Rs if you set Rl to zero) is less than w*Ls, then the Effective Rpri
will decrease with decreases of Rs. If the secondary resistance is greater
than w*Ls, then Rpri will *increase* with decreases of Rs.

Just take the derivative (with respect to Rs) of the first formula you
gave above to see this.

Now that we've got the formulas I was asking for, use these typical
values for a small 60 Hz 120 volt to 12.6 volt transformer:

L1 = 12.5 mH
R1 = 32.7 ohms
L2 = 938 uH
R2 = 1.72 ohms
m = 2.8 mH

and plot the Effective Lpri vs frequency from 60 Hz to 10 kHz.

Then answer my question:

"Is this really the leakage inductance? Under what conditions might it not
be a good value for the leakage inductance?"

 

With 120 volts applied.

John

 

With the secondary shorted, the core has essentially no effect, and the
measured value of Lpri is independent of the excitation level. See John
Jardine's latest post where his measurements confirm this.

Of course, with 120 volts applied, the resistance of the windings will be
increasing rapidly due to heating and it will be difficult to get a steady
measurement.

 

I think the leakage inductance still depends on permeability.

But he didn't apply enough excitation to substantially change
permeability.

With 120 volts applied, the permeability of the iron will be seriously
different from the low-flux value, so the leakage inductance will
indeed be different. You can of course define leakage inductance any
way you like, but if you care about things like charging caps through
bridge rectifiers, or any of the other things power transformers
usually do, leakage inductance measured at low level may not be
predictive.

John

 

I don't define it myself; I accept the definition found in the textbooks.
It is well known that the E-I iron core of a standard transformer (and not
the tricky geometry you proposed in another thread) has a small effect on
leakage inductance. It *does* have an effect, but it's small. The leakage
inductance is mostly due to flux in the air space between the windings, and
therefore scarcely affected by the core.

I cite from "Magnetic Circuits and Transformers", by members of the Staff
of the MIT Department of Electrical Engineering, a book well worth having.

They say "Since the leakage fields are not greatly affected by the core,
the effect of the iron core is sometimes entirely neglected and the leakage
inductances are computed as in Eqs. 91 and 92 from formulas giving the
self- and mutual inductances of air-core coils."

One way to convince yourself of this is to get a transformer that hasn't
been varnished so that you can remove and re-insert the laminations. If
the leakage inductance is greatly affected by the core, then it should
change substantially when you remove the core. The permeability of
ordinary transformer laminations is about 1000 with low flux, increasing to
a peak of maybe 6000 at about 8000 gauss, then decreasing again as the flux
density approaches saturation. Inserting the laminations will put material
of permeability of 1000 in the bobbin. Increasing the flux to the point of
maximum incremental permeability will only increase the permeability by
another factor of maybe 10, maximum.

Surely replacing the air in the center of the bobbin with a material of
permeability 1000 will have a greater effect than increasing the
permeability by another factor of 10.

So, just take a transformer with removeable laminations, short the
secondary and measure the inductance at the primary with the laminations in
place. Then remove the laminations and measure Lpri again. If the
measured inductance doesn't change much with a change of 1000 times in the
permeability of the space inside the bobbin, then it probably isn't going
to change much with another factor of 10 increase.

The measurements also depend on whether the winding you short is the inner
winding on the bobbin, or not. I'll post some measurements later.

If this experiment doesn't convince you, then propose one of your own.

My purpose in this thread is to raise awareness that sometimes measuring
the inductance of a winding with the other winding shorted may not
accurately reflect the leakage inductance. It would be good to know when
it does, and when it doesn't. Tim Williams is going in the right
direction.

 

It probably will, but not much, since the shorted secondary is doing its
damnedest to keep flux out of the core.

robert

 

Either you are being obtuse or you are being stupid.

Of course you might be being both.

DNA