Calculation of inductance of rectangle coil

Question

I'm currently working on a project about inductance. I want to calculate the inductance of a rectangle-shaped coil with a pitch different from zero. I tried this by applying the law of Biot-Savart. If i do this for one wire in a cartesians axis system i get a formula of the shape \$B=\int{\frac{(y-y_0)}{\sqrt{((x-x_0)^2+(y-y_0)^2)}^3}}\$ from 0 to \$x_0\$ or \$y_0\$. Which i can't integrate by hand or by numerical integration in matlab (singularity at \$y_0\$). If i write it out in polar coordinate system, i get \$B=constant*(\sin(\theta_2)-\sin(\theta_1))\$. But i can't figure out how to calculate the magnetic flux (phi=int{B dA}). And numerical integration has again some problems with the singularity close to the wire.

Has someone a better idea to calculate the inductance instead of using Biot-Savart? Or someone who knows how to integrate those integrands. Thanks a lot!

Answer 1

There is a web-calculator for this. And here, you can find an excellent paper about methods of calculation for different shapes of coils.

Answer 2

Consider a line in length of \$\dfrac{y_0}{2}\$ carrying DC current \$I\$. We wish to find the magnetic field at the point \$P\left(\dfrac{x_0}{2},0,0\right)\$.

The Biot Savart formula:

$$ \mathbf{B} = \dfrac{\mu I}{4 \pi} \int \limits_C \dfrac{d\mathbf{\ell} \times \mathbf{a_R}}{R^2} = \dfrac{\mu I}{4 \pi} \int \limits_C \dfrac{d\mathbf{\ell} \times \mathbf{R}}{R^3} $$

Substitute the quantities:

$$ \mathbf{\ell} = y \; \mathbf{a_y}, \quad \mathbf{R} = \dfrac{x_0}{2}\mathbf{a_x} - y \; \mathbf{a_y}, \quad d\mathbf{\ell} \times \mathbf{R} = \dfrac{x_0}{2}(-\mathbf{a_z}) $$

Considering that the path \$C\$ is from the bottom to top of the wire, it becomes:

$$ \mathbf{B} = \dfrac{\mu I}{4 \pi} \int \limits_C \dfrac{d\mathbf{\ell} \times \mathbf{R}}{R^3} = \dfrac{\mu I}{4 \pi} \int \limits_C \dfrac{\dfrac{x_0}{2}(-\mathbf{a_z})}{R^3} = -\dfrac{\mu I}{8 \pi} \int \limits_C \dfrac{x_0\mathbf{a_z}}{R^3} = -\dfrac{2 \mu I y_0}{\pi x_0 \sqrt{x_0^2 + y_0^2}} \mathbf{a_z} $$

If the entire structure is a rectangular of \$x_0\$ by \$y_0\$, just sum all four line segments due to the superposition theorem:

$$ \mathbf{B} = -2\dfrac{2 \mu I y_0}{\pi x_0 \sqrt{x_0^2 + y_0^2}} \mathbf{a_z} - 2\dfrac{2 \mu I x_0}{\pi y_0 \sqrt{x_0^2 + y_0^2}} \mathbf{a_z} = -\dfrac{4 \mu I}{\pi x_0 y_0 \sqrt{x_0^2 + y_0^2}} \left[ x_0^2 + y_0^2 \right] \mathbf{a_z} = \dfrac{4 \mu I}{\pi x_0 y_0 \sqrt{x_0^2 + y_0^2}} \left[ x_0^2 + y_0^2 \right] (-\mathbf{a_z}) = \dfrac{4 \mu I}{\pi x_0 y_0 \sqrt{x_0^2 + y_0^2}} \left[ x_0^2 + y_0^2 \right] \mathbf{a_{-z}} $$