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The Rogowski Coil is a very simple device for measuring fast, strong electric currents. This article will describe the theory of operation of a Rogowski coil with the basic math showing the simple operation.

Let’s start by following a closed path around a current carrying wire or some other conductive path. According to Ampere’s law, the line integral of the magnetic field is proportional to the current piercing the surface defined by our closed path:

If that path coincides with the axis of a solenoid (with the turns small compared to the path so that the magnetic field , is essentially constant over the surface of any individual turn in the solenoid), then is parallel to and the magnetic flux linking the turns of the solenoid is proportional to the same integral:

In this equation is the area of each turn of the solenoid and is the number of turns per unit length. By using Faraday’s law of induction, we can find the induced in our solenoid:

Now we have an induced in a solenoid which is proportional to the change in current threading the surface defined by the axis of our solenoid. This solenoid is a Rogowski loop. By letting the terminal end of the solenoid trace a path back down the axis of the solenoid, we avoid and produced by changing magnetic fields normal to the surface defined by our loop.

We also have a circuit consisting of a an inductance (the solenoid), a resistance and an induced . Describing this circuit is the differential equation:

This is a linear equation in which the right hand side can be turned into an exact differential by an integrating factor

If the right hand side is to be an exact differential, then there exists a function, , such that which is valid if and . If this is true then which leaves:

The solution to which is:

Taking to be 1, carrying out the integration in the exponent and putting the integrating factor back into the equation above we get the following:

Integrating gives:

And, now we can write this as:

We can solve for the constant of integration by using the current:

Then

Let’s look at an example, such as a square wave current that you might expect from a charged particle beam with a finite length:

Then

The current in the coil then becomes:

Carrying out the integration and scanning the time through the different regions we get:

This gives us an output waveform that looks like the following:

We can see that if is small, then in the central region the output has a spike at zero after which the exponential causes a very rapid decay to zero. When the time comes that the pulse ends, we get a negative spike which again decays to zero. This is exactly what we expect for a differentiating coil with output proportional to the derivative of the current.

We could also try a current described by a ramp function:

In this case the output is zero until time equals zero and any time thereafter it can be described by:

Rewriting this in the voltage output regime:

If is small, then the output quickly rises from zero to a fixed asymptotic value which indicates the derivative of the function representing the current to be measured. If is large, however, this function representing the output does not exactly look like the output of a self integrating coil, the asymptotic value is larger and the exponential is more spread out, but we still don’t see a ramp function. Here we have reached a limit; the pulse width is infinite which makes a self integrating coil impossible since there’s no way to make larger than the pulse width.

For further examples we can look at sinusoidal functions describing the current to be measured.

At times greater than zero:

Then the coil current, and thus the output, for times greater than zero is:

Carrying out an integration by parts yields:

And combining terms:

In this solution the term that dominates is determined by the time compared to the period of the current being measured. If the term is large compared tot eh period, then we have a self integrating coil and the dominant term is proportional to the current being measured, while a smaller leaves the dominant term proportional to the derivative of the current being measured. Note that in the self integrating case the magnitude of the output voltage is not significantly dependant on the frequency of the input signal while in the derivative proportionality case the magnitude of the output voltage definitely does depend on the frequency.

By keeping as small as possible, we can be certain that we are operating our coil in the self integrating mode for a rather high frequency current that we most often want to measure.

-Troy Stark

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