Another Inductance winding Number question

[The Electrician]

The analysis can be extended to accommodate the resistance of the wire used to wind the inductances. R1 is in series with L1 and R2 is in series with L2:

I wound a couple of inductors on high permeability, low loss, ferrite toroids. The two windings on each core consist of 20 turns of 16 gauge wire for L1 and 10 turns of 16 gauge wire for L2:

One of them was wound with tight coupling between L1 and L2; the other has considerably looser coupling. The values of L1, L2, R1, R2, k and f (the measurement frequency) for the tightly coupled inductors can be seen in the following image, along with the calculated values of the currents. The real and imaginary parts of I1 are blue and red; the real and imaginary parts of I2 are green and orange. Just below those values is the value of I_T. Also shown is the calculated value of the apparent inductance of the two inductors in parallel, with R1 and R2 taken into account.

The formula for the apparent inductance of the parallel combination of L1 and L2 with the resistances of the winding taken into account is shown in magenta. This formula corresponds to the formula of the referenced web page. Notice the presence of the frequency variable f in the magenta formula. This was unexpected.

I measured the apparent inductance of the tightly coupled inductor pair with an LCR meter at three frequencies and compared the measured result to the calculated values:

Frequency      Calculated      Measured
   1000 Hz       2.023 uH        2.02 uH
    100 Hz       4.658 uH        4.62 uH
     20 Hz       66.92 uH        67.0 uH

The calculated results are close to the measured results, considering that it's somewhat difficult to get good results of inductance measurements on ferromagnetic cores.

 

[Baluncore]

However, the formula apparently fails when L1 = L2 and k=1 (M = sqrt{L1*L2).

Banter is considering parallel equal inductors with perfect coupling.
That is the special case of one inductor wound with two wires twisted together, that is, thick wire. Lt = L1 = L2.

But if L1 = L2 and M =sqrt(L1*L2) then (L1+L2 – 2M) is zero.
So throughout the equations he is actually dividing zero by zero.

 

[jim hardy]

Wow ! Great find there, and wonderful experiment ! (I was still trying to remember where my old CT cores are for a similar experiment.)

So, that formula only fails when L1 and L2 are equal, ie same number of turns?

I too never expected frequency to be show up in a calculation of inductance.
I'll be some time plodding through your work , plodder that i am, until it becomes clear why.

can M involve f(frequency) ?

The mutual inductance M can be defined as the proportionalitiy between the emf generated in coil 2 to the change in current in coil 1 which produced it.

THANKS, guys !

old jim

 

[The Electrician]

So, that formula only fails when L1 and L2 are equal, ie same number of turns?

It apparently fails when L1 and L2 are equal, and when the coupling coefficient is unity (M = sqrt(L1*L2) in other words).

I said apparently because when a mathematical expression degenerates to the form 0/0, oftentimes using a limiting process will give a proper result, such as I get with the limit function in my CAS software:

The result after simplifying the odd expression is simply L1, the expected result.

I too never expected frequency to be show up in a calculation of inductance.
I'll be some time plodding through your work , plodder that i am, until it becomes clear why.

can M involve f(frequency) ?

It could, for example due to skin and proximity effect, L1 and L2 could change a small amount with frequency, and so could M.

But the effect we're seeing here is very large increases in effective inductance at low frequencies due to changes in the ratio of reactance to resistance in the windings. The analysis shows that it happens, but an easy intuitive understanding doesn't leap out at me. We just have to trust the math, which experiment verifies.

 

[jim hardy]

but an easy intuitive understanding doesn't leap out at me.

Me either. It's the sort of thing one mulls over in background for days.

Great job ! Thank You !

 

[The Electrician]

This coupled inductor circuit is very sensitive to the coupling coefficient. For example, plotting the phase of the total current vs. k shows that the impedance of the parallel connection of L1 an L2 looks quite inductive until the coupling coefficient gets very close to unity:

We can't really see what's going on when k gets close to unity.

Plotting with 1-k as the independent variable rather than k, and using a log plot allows us to see the details:

It looks like the phase is zero when k=1 but it isn't. As long as there is some resistance in the windings, the overall impedance is slightly inductive even when k=1, and the total current has a reactive component when k=1, as long as the permeability of the core isn't infinite.

The case where the core has infinite permeability means that the inductors have infinite inductance. If the windings have turns wound with a finite ratio, say 2 to 1, but the core has infinite permeability, the inductances would be infinite, but mathematically this case can still be treated--infinite inductances for the windings due to the infinite permeability, but with inductances having a finite ratio. This case doesn't occur with real wire and real cores.

Another interesting thing about this circuit is the fact that the currents in L1 are apparently flowing in the opposite direction to those in L2 when k is unity; this is the case with the early analyses in this thread. It turns out that this property is also affected by the coupling coefficient and the resistance of the windings. Here's a plot showing the ratio of the real part of the current in L1 to the real part of the current in L2 (blue curve). The red curve shows the ratio of the imaginary part of the currents in L1 to L2.

Notice that the sign of the ratios is negative when k is approaching unity, which means that the currents are flowing in opposite directions. As k decreases from unity to zero, the sign of the ratios becomes positive for some k. This is not unexpected. Obviously, if the inductors are not coupled at all (wound on separate cores), but connected in parallel, the currents should be in the same direction with respect to the energized terminals:

It almost looks like there is a discontinuity at k=1. Adding 1 ohm to the resistance of each winding and then plotting these ratios shows what's happening; for k large enough, the imaginary parts of the currents are again flowing in the same direction. However, for the special case where k=1 but the resistances are zero, all bets are off; the permeability of the core comes into play. The coupling in my real inductors isn't close enough to unity, and the resistances aren't large enough, to see the current directions become the same, as these plots suggest would be the case.

 

[Baluncore]

This all makes a lot of sense. My simple transformer circuit model with series winding resistance can be changed to move part of the 100% coupled transformer inductance out of coupling into isolated series with the winding resistance. The division of voltage between the winding resistance and the isolated part of the inductance is then a frequency dependent function. I have seen that clearly in the spice model. Changing frequency or M changes the phase shift of the voltage to current and therefore the effective inductance.

 

[tim9000]

Yeah...wow, everytime I've come back to this thread it's grown. It's going to take me days to digest all that's been said and shown here for me to sumarise it, atm I have little no no idea lol

 

[Baluncore]

It's going to take me days to digest all that's been said and shown here for me to sumarise it,

There are some things electronics engineers spend their time trying to avoid. One of those things is “shorted turns” in wound magnetic components such as chokes and transformers. Your suggestion that inductors with 100% coupling, (wound on the same core, but with different numbers of turns), might be useful, has led to this little explored field of analysis. It is little explored because in real circuits it appears to always result in higher losses and inefficiency. We have yet to identify a justifiable use for such an engineered situation. I would class this field as fault analysis.

Science involves the reduction of complex situations into simple relationships. Some of us are sufficiently annoyed that we were not able to understand and explain this field using our simple rules, that we have taken your thread and torn our analysis apart in the search for a consistent interpretation.

The summary at present is that you should avoid having windings with different numbers of turns in parallel on the same magnetic core. We have a two dimensional map of coupling from 0 to 100% plotted against turns ratio, n, from 1 to infinity. The point on that map where coupling=100% and n=1 is the very special case of using one thicker wire instead of two separate wires. All other points with significant coupling appear to result in an unnecessary circulating current, higher losses and a waste of investment in winding and core materials.

 

[The Electrician]

Jim mentioned in post #33: "There exist cores with a hole drilled through so that the last turn can encircle part of the flux, so as to approximate a fractional-turn for non-integer turns ratio."

One place where I've seen that is on a 10 kW variac. Here's a picture of a smaller variac:

http://en.wikipedia.org/wiki/Autotransformer#/media/File:Variable_Transformer_01.jpg

The sliding brush shorts out several adjacent turns on the toroidal core. This represents some power loss, and how much depends on how many turns per volt the core has been wound for. The 10 kW variac we used actually had two windings side by side in a single layer in the manner seen in the picture. But one of the windings was threaded at the end of the winding through a hole drilled through the core. Thus that winding was enclosing just a little less mmf; another way to look at it is the winding had a half turn. This meant that the voltage between adjacent turns (that were shorted by the sliding brush) was less, and the loss due to heating of the sliding brush was reduced.

So this is a situation where two windings of different numbers of turns (one half turn difference) were connected in parallel (by the sliding brush). This is the old engineering trade-off. You determine if the benefit outweighs the cost, and apparently it did in this case.

I hope the OP isn't annoyed by the extremes to which we've taken his thread, but it's all related to his question, so finally I'm going to post a picture of another mystery. Here's an interesting toroidal core I found at Boeing surplus years ago. Look at how it's wound. There are holes drilled around the periphery and the wire is threaded through all those holes. Why would one want to wind a core in a manner like this, not enclosing all the iron? I don't know the answer:

 

[jim hardy]

Indeed it really bugs me when i can't figure something out.

You guys's wonderful work confirms what intuition says - with zero induction current divides between the coils
but at some value of induction the greater coil overwhelms the lesser one...

What happens if one makes a CT from such an inductor, adding a one turn primary ,
then adds external resistance so as to indicate current by voltage across say a 1 ohm resistor,
meaning the sum of amp-turns must become larger to produce enough flux to impress voltage across that resistor?

I think i'll take a step in that direction by tinkering with @tim9000 's original inductor, 30 and 60 turns on an unknown core.

This is just a thought experiment:

If Electrician's core were one of these
magnetics inc 01540354R
with path length 12cm
core area 0.726 square cm
permeability let's assume 5000

so inductance is
μ₀ μ_relative N² X A_rea / l_ength

4Π X 1e-7 X 5E3 X N^2 X 0.726E-4 / 12E-2

= N^2 X 3.8e-6 , 10 turns giving 3.8E-4 or 0.38 millihenry and 20 turns giving 1.52 millihenry

so i guess his core has reluctance about a quarter (0.26) of the one i guessed at.

Maximum flux of my core per Magnetics Inc is around 6 kilogauss (0.6 Tesla)
https://www.google.com/?gws_rd=ssl#q=magnetics+bulletin+sr-4

so if flux were 0.6cos(120 pi t ) X area of .0007264 m^2 = 4.358E-4cos(120pi t) weber , dΦ/dt would be -120 pi X 4.358E-4 sin wt, = 0.164 weber/sec
yielding 0.164 volts/turn peak sinewave voltage. That's all the core can make.

How many amp turns would that take?

Well, Φ = μ₀ μ_relative N X I X A_rea / l_ength
N X I = Φ / ( μ₀ μ_relative X A_rea / l_ength _
NI = 4.358E-4 / (4Π X 1e-7 X 5E3 X 0.726E-4 / 12E-2 )
NI = 4.358E-4 / (3.803E-6) = 114 amp-turns

wow seems a lot for just 164 milllivolts per turn
clearly it's going to be hard to make a 50::5 pass thru transformer with this core
(probably i made a mistake, haven't done this since mid 1960's, and then in gausses oersteds and inches...
but I've checked, so if you find one - bravo)

so let me continue

this core produces 0.164 / 114 = 1.4386E-3 volts per turn per amp-turn of MMF.
Let's round that to 1.44 millivolts per turn per amp-turn.Whew.

Now we've characterized the core.

My hypothesis two pages back was

so
i hypothesize that for OP's 2::1 ratio of turns and one unit of current applied
A = 0.666 + jΦ/3R , the current in the 30 turn winding
and B = 1 - A = 0.333 - jΦ/3R , current in 60 turn winding
where Φ is a property of the core, volts-per-turn/amp-turn
and R is a property of the wire, ohms per turn.

[Now it's unfortunate i used Φ in that sense, but i really needed a simple picture for my worn out, obsolete brain.
Volts per turn is flux, because of derivative property of sinusoids, in hindsight i should have picked a name for characterizing the core that's closer to reluctance. My bad. Apologies if it distracts anybody. ]
My core characterization constant volts per turn per ampturn , Φ, has value 1.44E-3. I'll make it red to distinguish it from flux.

and i'll guess at ohms per turn...
My hypothetical core has area of 0.7264 cm^2 which cross section might be a circle of diameter 0.4808 cm. Let's round that to 0.5cm because it's probably wrapped in something. So a turn of wire is at least pi/2 cm long.

#22 is probably the smallest one would use for 5 amps?
#22 is about 53 ohms/km, or 5.3E-4 ohms/cm
for our pi/2 long turns that's 8.325 milliohms/turn
Round up to 8.5 milliohms per turn because each turn is really a little longer than pi/2 , because with each turn it progresses partway around the 12 cm core length? And we have to add half the diameter of #22 to our radius?

Okay, now i have both Φ and ohms/turn.
A = 0.666 + jΦ/3R = 2/3 +j1.44E-3/8.5E-3, = 2/3 +j0.169 , which = 0.688 angle 14.2 degrees
B = 1 - A = 0.333 - jΦ/3R = 1/3 -j1.44E-3/8.5E-3 = 1/3 -j0.169 = 0.374 angle 26.9 degrees

So now let's sum mmf's.
twopages back,
A was the 30 turn winding
and B was the 60.

So we have
in A: 30 X (2/3 +j 0.169) amp turns
in B: 60 X (1/3 -j0.169) amp turns

sum is 40-j5.07 , which is 40.323 angle -7.22 degrees

so with an amp applied we'd have induction of 40.323 X 1.44E-3 = 10.4 millivolts per turn (peak)
which is well below the core's capability of 164 millivolts per turn.

Whew - I'm worn out.

Maybe you guys would do sanity checks on my arithmetic. Your simulations should make it a snap.

How much voltage would it take to impress an amp through that 30::60 turn inductor? Should be ohm's law...

If my arithmetic has been good, i might now have an approach for figuring out that 50::5 CT with reported dual secondaries of nine and ten turns. Could it work or not? I'll apply a fixed 50 amp turn primary and see what are currents and terminal voltage. But not right now...

Thanks guys for indulging an old fellow. I'm not done yet, but am sure done for today.

G'nite all.

Criticism welcome. Hope y'all enjoyed this.

old jim

 

[jim hardy]

PS @The Electrician

Have you the particulars on your core ?

 

[jim hardy]

Why would one want to wind a core in a manner like this, not enclosing all the iron? I don't know the answer:

oh my gosh...

 

[Baluncore]

Sorry about the OP thread diversion.

Why would one want to wind a core in a manner like this, not enclosing all the iron?

An interesting one.
It would be very difficult to wind such a core with the wrong number of turns or with poor spacing.

That must be a Current Transformer core with a 24 turn secondary for control or instrumentation.
Could the holes drop the ratio back to maybe 1 : 20 ?

There is no scale, OD, ID, stack or lamination thickness shown. It looks like it is wound with a flexible wire, plastic insulated, through a stack of laminations. If laminations are standard thickness it is for mains frequency use. Thinner than normal laminations would be for 400Hz aircraft power systems. Can you count laminations, measure stack height to compute lamination thickness ?

Were the laminations pre-punched or drilled after stacking? There would be no eddy current loop problems with lack of insulation between laminations if drill holes passed through all the laminations after stacking. You might be able to tell by; if the holes run perpendicular to the face it was drilled, if each hole staggered slightly through part of the turn pitch then it was pre-punched.

Note that one of the lead wires is NOT wound back to cancel the induced turn in order to reduce external interference, so it is probably being used as a current transformer rather than an inductor.

 

[The Electrician]

If Electrician's core were one of these
magnetics inc 01540354R
with path length 12cm
core area 0.726 square cm
permeability let's assume 5000

My core is a Magnetics inc W6113 as seen here: http://www.mag-inc.com/products/ferrite-cores/ferrite-toroids

Here's a current transformer design guide that mentions drilled holes in the core for fractional turn corrections: http://permag.net/documents/CurrentTransformerDesignGuide.pdf

We should start a new thread to continue this discussion.

 

[jim hardy]

Thanks !

nH/T² = 13690 ± 30% that'll be the proper way of expressing my Φ term...

nice core, μ_relative is 10,000

Sorry about the OP thread diversion.

Sorry for what ? You broke ground.
I learned a lot, lots more to go.