1. Basic Electromagnetic

We start with a circuit filled with magnetic fields:

Where, \(\vec B = \mu \vec H\)

• \(\vec B\) is the flux density.
• \(\vec H\) is magnetic filed strength.
• \(\mu\) is the permeability.

1. Permeability

Permeability is different for different materials, commonly, we have:

Materials	Permeability
param materials	\(\mu \approx \mu_0 = 4\pi \cdot 10^{-7}\)
ferr materials	\(\mu = 1000 \sim 10000 \mu_0\)
diamag materials	\(\mu < \mu_0\)

For the electrical machines, we only consider param materials and ferr materials.

2. BH Curve

Within the permeability, we can draw the BH curve:

In the BH curve, the ferr material keeps linear until \(1.2 \sim 1.5 T\), then due to the material saturates, the slope will be reduced, and after saturation, the slope is similar to \(\mu_0\).

3. Faraday's Law

Faraday's law helps us to find the relations between flux density (\(B\)) and electric motive force (\(e\)):

\[ e = \frac{d\psi}{dt} \]

Where, \(\psi = N\phi = \int_S (\vec B \cdot \vec n) dS\).

• \(\psi\) is flux linkage
• \(N\) is the number of turns of the coil
• \(\phi\) is flux
• \(\vec n\) is the unit vector perpendicular to the plane
• \(\vec B\) is the magnetic field

We can use the right hand rule to define the direction of \(e\):

If the magnetic field is uniform, then the EMF is:

\[ e = \frac{d\psi}{dt} = \frac{d}{dt}(BS\cos \theta) = -BS\omega \sin(\omega t) \]

Where \(\theta\) is the angle between \(\vec B\) and \(\vec n\).

4. Ampere's Law

The current can produce a circled magnetic field,

This can be expressed by the Ampere's law:

\[ i_{tot} = \oint_l \vec H dl \]

If we have multiple turns of coils, thus we have: \(N_{coils} i = \vec H l\).

And the magnetic permeability have the following relationship with electric permeability:

\[ c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \]

• \(c\) is the speed of light.
• \(\mu_0\) is the magnetic permeability of vacuum.
• \(\varepsilon_0\) is the electric permeability of vacuum.

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