Diagnostics/Magnetic/Theory/Rogowski_coil/index

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title: Rogowski Coil
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Introduction
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Rogowski coil is an electrical device used for contactless measurement of flowing current. It is a solenoidal coil whose ends are brought around together to form a torus. The whole assembly is then wrapped around a straight conductor. Its voltage output is proportional to the rate of change of current in the conductor.

The advantage of a Rogowski coil over other types of induction coils is that it can be made open-ended and flexible, allowing it to be wrapped round a live conductor being measured without disturbing it.

<table>
<caption align="bottom"><em>Schmatic diagram of the Rogowski coil</em></caption>
<tr><td><img src="Rog_coil.png"></td></tr>
</table>

Theory
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Consider a coil of the uniform cross-sectional area $A$, with constant turns per unit length - $n$. Provided the magnetic field varies little over one turn spacing. That is, if

$$|\nabla B|/B \ll n ,$$

the total flux linkage by the coil $\Phi$ can be written as an integral rather than a sum over individual turns:

$$\Phi = n\oint_l dl \oint_A dA\ B$$

where $dl$  is the line element along the solenoidal axis as illustrated in this figure

<table>
<caption align="bottom"><em>Geometry for the integral of magnetic flux through a Rogowski coil</em></caption>
<tr><td><img src="Rog_scheme.png"></td></tr>
</table>

Note that it is important to have the return wire back down the coil as shown in figure. Otherwise, eqution for $\Phi$ also includes a term arising from the flux passing through the torus center.

Using Amper’s law :

$$\oint_l B\ dl = \mu I$$

where $I$ is the total current encircled by $l$ and $\mu$ is the magnetic permeability of the medium in the solenoid, the magnetic flux through the Rogowski coil can be written as:

$$\Phi = nA\mu I$$

The output voltage from the Roqowski coil is

$$V = \dot{\Phi} = nA\mu\dot{I}$$

It is seen, that the output voltage from the Rogowski coil is directly proportional to the time derivative of total current flowing through its cross-section. Note that it is independent of the distribution of the flowing current.

Credit: Excerpt from

* I. Ďuran: *<a href="/Library/IvanDuranPhdThesis.pdf">Fluctuations of magnetic field in the CASTOR tokamak</a>*, Dissertation Thesis, 2003.
* J. Sentkerestiová: *Systematic measurements of plasma position on CASTOR tokamak using magnetic coils and Hall sensors*, Diploma Thesis, 2006