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I have two coils covered in a box and I don't know the physical arrangement of the two coils and mode of each winding in a magnetically couple DC circuit. How can I make an experiment to determine the dot convention of the two coils?

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    \$\begingroup\$ Send a sine wave in one side and check the polarity (phase) of the output. \$\endgroup\$ – George Herold Sep 5 '14 at 12:33
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You can use an inductance meter and connect the two windings in series. The configuration with the higher inductance has the windings connected dot-to-no-dot.

The actual mutual inductance is the half the difference between the sum of the two inductances (measured separately with the other winding open) and the total inductance with the coils in series and phased as above. In other words if the inductance of one winding is \$L_1\$ and the inductance of the second winding is \$L_2\$ and the total inductance measured with the two in series is \$L_X\$ then the mutual inductance is M = \$\dfrac{L_X - L_1 - L_2}{2}\$.

By shorting one of the coils and measuring the inductance of the other (preferably with an instrument that gives you an L+R measurement) you can get a measurement of the leakage inductance.

Of course the placement of the dot is (usually) arbitrary- if you reverse the dots on all the windings it is exactly the same thing for a typical inductor or transformer (there are some types of inductors that are magnetically pre-biased so the are not symmetrical).

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Applying a sinewave to one coil and measuring the output amplitude on the other might be seen as a good start. If it were a perfect transformer (i.e. 100% coupled coils) then the ratio of the output coil voltage to the input coil voltage tells you the relative turn ratio BUT I guess there won't be 100% coupling between coils so it's a little harder.

To understand the dot convention is easy - if the voltage applied on one coil is in-phase with the voltage seen on the other coil then, the dots can be drawn on the signal-in wire and the o-scope signal-out wire. If you wish to go further read-on...

I'd definitely start by understanding each coil in turn - use a signal generator, o-scope and tuning capacitor to measure the inductance of each coil whilst the other coil is open circuited and playing no role. This is OK providing the coils each have a high self-resonant frequency where one can assume that the self resonant frequency of the "open-circuit" coil is not influencing readings on the coil being measured. So, to avoid this try making the coil resonate at the lowest practical frequency.

Once you have a reasonable value of inductance for each coil you then have to delve into the mutual coupling properties. For perfectly coupled coils with no flux leakage: -

\$M_{perfect} = \sqrt{L_1 L_2}\$ henries

But, for coils that aren't 100% coupled the formula becomes: -

\$M_{real} = k\sqrt{L_1 L_2}\$ henries where k is the coupling coefficient and has a value between 0 and 1.

Next is understanding what M is in a bit more detail. This site shows a good picture: -

enter image description here enter image description here

Note the formula - it says that the induced voltage in a 2nd coil is: -

Induced voltage = \$-M\dfrac{\Delta I_1}{\Delta t}\$

This allows you to determine M by injecting a sinewave into the 1st coil and integrating the voltage sinewave produced on the 2nd coil.

Once you have actual M you can calculate the coupling coefficient, k using the \$M_{real}\$ formula given higher up in my answer.

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You must connect a generator with a known signal (square, sine) at the input and use a two-channel oscilloscope. With a channel you see the input signal and the other channel to see the output. The phase relationship will tell you what the homopolar terminals.

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