Calculate Impedance of Copper Coil w/ Iron Core

[levi]

I have to calculate the impedance (=R+j*w*L) of an infinite copper coil (par meter), with an iron core. Given data are the magnetic permeability, conductivity and radius of the copper wire, inside and outside radius of the coil, radius of the core and permeability and conductivity of the iron.

I don't have to use some formula like pouillet's law or something like that, but start from maxwell's equations or EQS or MQS approximations. The result should be a function of the frequency. Hence the symmetry of the problem I think I should use integral equations in cilindrical coordinates, but i have no idea where to start.
Can anyone help me?
thanks

 

[anathema]

Thank you for your question. I am happy to assist you with calculating the impedance of an infinite copper coil with an iron core.

First, let's review some important equations that will help us solve this problem. Maxwell's equations describe the relationship between electric and magnetic fields, and they are the basis for understanding electromagnetic phenomena. The equations are:

1. Gauss's law for electric fields: ∇ · E = ρ/ε0
2. Gauss's law for magnetic fields: ∇ · B = 0
3. Faraday's law: ∇ × E = -∂B/∂t
4. Ampere's law: ∇ × B = μ0(J + ε0∂E/∂t)

In these equations, ε0 is the permittivity of free space, μ0 is the permeability of free space, ρ is the electric charge density, J is the current density, and ∂/∂t denotes the partial derivative with respect to time.

To solve this problem, we will use the equations for the magnetic field in cylindrical coordinates, which are:

∇ · B = 0
∇ × B = μ0(J + ε0∂E/∂t)

We will also use the equation for the electric field in cylindrical coordinates:

∇ · E = ρ/ε0

Now, let's consider the geometry of the problem. We have an infinite copper coil with an iron core. We can assume that the coil is wound tightly, so we can consider the coil to be a solenoid. The iron core is cylindrical in shape, and it is located inside the coil.

To start, we will consider the magnetic field inside the solenoid. Using the first equation above, we know that the divergence of the magnetic field is zero. This means that the magnetic field lines are closed loops, and there is no net flow of magnetic field through any closed surface. This is true for both the inside and outside of the solenoid.

Next, we will use the second equation to find the magnetic field inside the solenoid. Since the electric field is zero inside the solenoid, we can simplify the equation to:

∇ × B = μ0J

We know that the current density, J, is related to the current, I, and the cross-sectional area of the solenoid, A,

 

[quicknote]

To calculate the impedance of an infinite copper coil with an iron core, we can start by using Maxwell's equations in cylindrical coordinates. These equations describe the behavior of electromagnetic fields in a given material. In this case, we are interested in the behavior of the electric and magnetic fields within the copper coil and iron core.

First, we can use Ampere's law to calculate the magnetic field within the coil. This law states that the line integral of the magnetic field around a closed loop is equal to the current passing through the loop multiplied by the permeability of the material. In this case, we can use the magnetic permeability of copper to calculate the magnetic field within the coil.

Next, we can use Faraday's law to calculate the induced electric field within the coil. This law states that the line integral of the electric field around a closed loop is equal to the rate of change of the magnetic flux passing through the loop. In this case, we can use the magnetic field calculated from Ampere's law and the radius of the coil to calculate the induced electric field.

Now, we can use Ohm's law to calculate the current within the coil. This law states that the current passing through a material is equal to the electric field divided by the material's conductivity. In this case, we can use the induced electric field and the conductivity of copper to calculate the current within the coil.

Finally, we can use the impedance formula, which is equal to the resistance (R) plus the reactance (j*w*L), to calculate the impedance of the coil. The resistance can be calculated from the current and the radius of the coil, while the reactance can be calculated from the current and the inductance (L) of the coil, which can be found using the magnetic field and the radius of the coil.

To take into account the iron core, we can use the magnetic permeability and conductivity of iron in the calculations for the magnetic field and current. This will affect the overall impedance of the coil and must be taken into consideration in the calculations.

In summary, to calculate the impedance of an infinite copper coil with an iron core, we can use Maxwell's equations, Ampere's law, Faraday's law, Ohm's law, and the impedance formula. With these equations and the given data, we can find a function of frequency that describes the impedance of the coil.