How Does a Rogowski Coil Measure AC Current?

[togahockey15]

Homework Statement

When a wire carries an AC current with a known frequency you can use a Rogowski
coil to determine the amplitude Imax of the current without disconnecting the wire to
shunt the current in a meter. The Rogowski coil, shown in the figure, simply clips
around the wire. It consists of a toroidal conductor wrapped around a circular return
cord. The toroid has n turns per unit length and a cross-sectional area A. The current
to be measured is given by I(t) = Imax sin (ω t). (a) Show that the amplitude, E, of the
emf induced in the Rogowski coil is E = μ0 n A ω Imax. (b) Explain why the wire
carrying the unknown current need not be at the center of the Rogowski coil, and why
the coil will not respond to nearby currents that it does not enclose.

Homework Equations

Emf= -N(dI/dt) where I = magnetic flux, not current

Emf = I/R = R*(dQ/dt) where I = current

Magnetic flux = *integral* (B*dA)

Emf = *surface integral* (E*dL) = -(d*magnetic flux*/dt)

Emf= (ANu/l)*(dI/dt)

The Attempt at a Solution

I am not really sure where to start - maybe by using the Emf= (ANu/l)*(dI/dt) ?? Can anyone help get me started in the right direction? Thanks!**In the attached document, the diagram is in #6, all the way at the bottom**

Edit: The figure is simply a ring of wire that has another wire wrapped around it, with a current going through the ring. Apologies for the mix up

 

[anathema]

!

First, let's define some variables:
- n = number of turns per unit length
- A = cross-sectional area of the toroid
- Imax = maximum current in the wire (given by I(t) = Imax sin(ωt))
- ω = angular frequency of the current

a) To find the induced emf in the Rogowski coil, we can use Faraday's law of induction:
Emf = -N(dΦ/dt)
where N is the number of turns in the coil and Φ is the magnetic flux through the coil.

The magnetic flux through the coil can be found by taking the surface integral of the magnetic field, B, over the surface of the coil:
Φ = ∫B*dA

Now, we know that the magnetic field inside a toroid is given by:
B = μ0nI
where μ0 is the permeability of free space, n is the number of turns per unit length, and I is the current.

Substituting this into the integral for Φ, we get:
Φ = ∫μ0nI*dA

Since the cross-sectional area of the toroid is A, we can rewrite this as:
Φ = μ0nAI

Now, we can substitute this into our equation for the induced emf:
Emf = -N(dΦ/dt)
Emf = -N(d(μ0nAI)/dt)
Emf = -μ0nAN(dI/dt)

We know that dI/dt = ωImax cos(ωt), so:
Emf = -μ0nAN(ωImax cos(ωt)/dt)
Emf = -μ0nAωImax sin(ωt)

Since we are looking for the amplitude of the emf, we can use the maximum value of sin(ωt), which is 1, so:
Emf = μ0nAωImax

b) The wire carrying the unknown current does not need to be at the center of the Rogowski coil because the magnetic field inside the toroid is uniform, so the magnetic flux through the coil will be the same regardless of the position of the wire.

The coil will not respond to nearby currents that it does not enclose because the magnetic field outside the toroid is negligible. This means that there will be no change in magnetic flux through the coil, and

 

[nlightner]

!Sure, let's start by looking at the diagram. It seems like there is a wire carrying an AC current, and the Rogowski coil is wrapped around it. The Rogowski coil itself is a toroidal conductor wrapped around a circular return cord. This means that the wire carrying the current is not enclosed by the toroid, but rather passes through the center of it.

Now, the goal of the Rogowski coil is to measure the amplitude of the current without disconnecting the wire. This is done by measuring the emf (electromotive force) induced in the coil. We know that emf is given by the change in magnetic flux over time, so let's start by looking at the magnetic flux through the coil.

From the diagram, we can see that the magnetic field lines are passing through the circular return cord. This means that the magnetic flux through the coil is given by the magnetic field times the area of the circular return cord, which we'll call A.

So, the magnetic flux through the coil is B*A. We also know that the current in the wire is given by I(t) = Imax sin (ω t). We can use this to calculate the magnetic field at any point along the circular return cord.

Since the current is changing with time, we need to use the equation for the magnetic field due to a current-carrying wire, which is B = μ0*I/(2π*r). Here, r is the distance from the wire to the point where we want to calculate the magnetic field. In this case, we want to calculate the magnetic field at the circular return cord, so r is the distance from the wire to the center of the circular return cord.

Next, we need to consider the number of turns in the toroid, which is given by n. This tells us how many times the magnetic field lines are passing through the circular return cord. So, the total magnetic field passing through the coil is B*A*n, which is the same as the magnetic flux through the coil.

Now, we can use Faraday's law to calculate the emf induced in the coil. Faraday's law states that the emf is equal to the negative of the change in magnetic flux over time. In this case, the change in magnetic flux is B*A*n, and the change in time is just the period of the AC current, which is 2π/ω.

So, we have emf = -(B*A