Measuring Currents with Rogowski Coils

[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)

The Attempt at a Solution

I am not really sure where to start - maybe by using the Emf= I/R ?? Can anyone help get me started in the right direction? Thanks!

 

[anathema]

Sure, let's start by looking at the circuit. The Rogowski coil is essentially a loop of wire with a circular return cord. When an AC current flows through the wire, it creates a changing magnetic field around the coil. This changing magnetic field will induce an emf in the coil, according to Faraday's law of induction.

In this case, we know that the current in the wire is given by I(t) = Imax sin(ωt). This means that the current is changing sinusoidally with time, which means that the magnetic field around the coil is also changing sinusoidally with time.

Now, let's think about the magnetic flux through the coil. Remember, magnetic flux is defined as the integral of the magnetic field over a surface. In this case, the surface we are interested in is the cross-sectional area of the coil, since that's where the magnetic field is passing through.

So, we can write the magnetic flux through the coil as Φ = ∫B*dA, where B is the magnetic field and dA is an infinitesimal element of area on the cross-section of the coil.

Next, let's think about the emf induced in the coil. According to Faraday's law, the emf is equal to the negative of the rate of change of magnetic flux. So, we can write the emf as E = -dΦ/dt.

Now, let's put these two equations together. We can substitute our expression for magnetic flux into the equation for emf, giving us E = -d(∫B*dA)/dt. Since the magnetic field is changing sinusoidally with time, we can pull it out of the integral, giving us E = -∫(dB/dt)*dA.

Now, let's think about what dB/dt represents. Since we are dealing with an AC current, dB/dt is equal to μ0nωImax, where μ0 is the permeability of free space, n is the number of turns per unit length, ω is the angular frequency of the current, and Imax is the maximum current.

Substituting this into our equation for emf, we get E = -∫μ0nωImax*dA. Now, remember that the integral is over the cross-sectional area of the coil. We can pull the constants μ0, n, and ω out of the integral, giving us E

 

[nlightner]

I can provide some guidance on how to approach this problem. First, it is important to understand the basic principles behind Rogowski coils and how they work. Rogowski coils are designed to measure the rate of change of magnetic flux, rather than the current itself. This is because the induced emf in the coil is directly proportional to the rate of change of the magnetic flux, as shown in the equations above.

To solve part (a) of the problem, you can start by using the formula for the induced emf in the Rogowski coil, which is given by E = -N(dI/dt), where N is the number of turns in the coil and dI/dt is the rate of change of current. In this case, we know that the current is given by I(t) = Imax sin (ωt), so we can calculate the rate of change of current as dI/dt = Imaxωcos(ωt). Substituting this into the equation for emf, we get E = -N*Imaxωcos(ωt).

Next, we can use the formula for magnetic flux, which is given by *integral* (B*dA), where B is the magnetic field and dA is the area of the coil. Since the magnetic field is constant and perpendicular to the area of the coil, we can simplify this to B*A. Substituting this into our equation for emf, we get E = -N*B*A*ωcos(ωt).

Finally, we can use the relationship between magnetic field and current, which is given by B = μ0*n*I, 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 equation for emf, we get E = -N*μ0*n*A*ωImax*cos(ωt).

Since we are only interested in the amplitude of the emf, we can ignore the cosine term and take the absolute value of the equation, which gives us E = μ0*n*A*ω*Imax. This is the desired result.

For part (b) of the problem, it is important to note that the wire carrying the unknown current does not have to be at the center of the Rogowski coil because the induced emf is proportional to the rate of change of magnetic flux, which depends on the area of the