## An electric anomaly

20.03.2010 19:26

Last week I got an email from Lucy Clarke asking me if I could explain how the following machine works and the readings she gets on the multimeter:

The coil is bifilar, meaning it has two winds of wire on it, one (heavier gauge) for taking power from DC and one (lighter gauge) going out to the multi-meter.
The coil also has a ferrite rod core so that, when the reed switch is open and there is no power, the magnets are attracted to the iron core, but when the reed-switch is closed and the power flows, it acts like an electromagnet and pushes away the magnet, so there is no 'sticky point'.
The peak reading on the multi-meter is 600V at 10.0A (...) When the rotor spins fast enough the voltage and the amps readings on the multimeter are off it's scale.

First, this is a variant of a brushless direct-current electric motor where the commutation is performed magnetically via the reed relay. A very inefficient one actually. For instance, it's only drawing power half of the time - ordinary two-pole DC motors will invert the direction of the current in the coil while it passes over the opposite magnetic pole, while this one merely switches the current off. There's also a huge air gap between the north and south poles of the permanent magnets reducing the strength of the magnetic field upon which the current in the coil can act.

If we forget about the moving parts for a moment, this device is very similar to an induction coil. Here's what an equivalent circuit looks like:

On the primary side is a constant voltage source that gets periodically connected and disconnected to and from the primary side of a transformer. On the secondary side is a resistor that represents a model of the multimeter.

Here's how primary and secondary side voltage and current look like versus time for such a circuit in somewhat idealized conditions. The switch is closed in A and opened in B time interval:

What is happening here? In time interval A the voltage of the source U1 appears on the secondary side and runs a current U1/R through the resistance of the multimeter. This current is also reflected on the primary side (the dotted line in i1(t) graph). However because of the self-inductance of the primary coil, the primary side current also has another component that is linearly increasing with time. So while the magnetic flux in the core of the coil due to U1/R on the primary side is compensated by the same current on the secondary side, the flux due to the self-inductance of the primary side is not (current difference I0 on the graph). This means that the magnetic flux in the core is steadily rising during interval A.

Magnetic field cannot collapse in an instance, so when the relay is switched off and primary side current is interrupted the only way it can be be sustained is by current on the secondary side. This effect causes the secondary side current to jump by I0, which then exponentially decays towards 0 as energy is lost in the resistance. This is also the principle of flyback converters.

Now we can explain the high voltage reading on the multimeter: when the multimeter is in voltage measurement mode, it has a high resistance between its probes (say 100 kΩ, but probably much higher). This means high R and a high R I0 voltage on the secondary side, since I0 current is forced through it by the collapsing magnetic field of the coil.

Since U1/R is negligible, the whole 0.5 A current budget appears as I0. 0.5 A times 100 kΩ gives a pretty respectable theoretical peak voltage. Of course, in practice this is much lower, but still high enough to affect the well-being of a multimeter.

On the other hand the high current readings are harder to explain. The theoretical peak current on the secondary side can't be higher than the peak on the primary side with this model. If the current limit on the power supply was working correctly, I can only offer some hand-waving explanations. One is that at higher speeds the machine gets in resonance with a tank capacitor in the power supply. So while the average current stayed below 0.5 A, peak current could have been well above that. The other is that some auto-ranging digital multimeters are very bad at measuring quickly changing values and may show completely wrong readings before they settle.

This analysis also ignores the effect of the permanent magnets moving in front of the coil. The effect of those is hard to judge, but my guess would be it is pretty insignificant due to the large air gap. For reference, I've included an idealized graph of the magnetic flux Φm through the core that is contributed by the magnets. If anything, the effect of this flux counteracts the self-induction of the coil and only causes the inducted voltages and currents in the circuit to be lower (which is logical, since the wheel is taking some energy from the system).

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