Flux Linkage

Flux linkage is the total flux linked by the coil. TThe unit of the flux linkage is turn.Weber.

$$\lambda = N_{turns} \Phi$$

Faraday's Law

Kirchhoff's voltage law is just a simplified version of Faraday's Law and it is not valid when there is an external magnetic field. (Please watch: Kirchhoff's Law vs Faraday's Law)

Faraday's Law can be expressed in integral form as:

\(\oint \vec{E} dl = -\frac{\partial\Phi}{\partial t}\)

For a single turn coil, the induced voltage is:

(e = -\frac{d \Phi}{dt} = -\frac{d\oint \vec{B} d\vec{A}}{dt} )

If the number of turns is

$$ N_{turns}

$$ , then the induced voltage becomes:

$$e(t) = \frac{\mathrm{d} \lambda(t)}{\mathrm{d}t}$$$$e(t) = N_{turns} \times \frac{\mathrm{d} \Phi(t)}{\mathrm{d}t}$$ If the magnetic flux distribution is uniform:

$$e(t) = N_{turns} \times Area \times \frac{\mathrm{d} B(t)}{\mathrm{d}t}$$ Methods to generate voltage in a coil:

• Varying magnetic flux by an AC excitation
• Varying flux linkage by changing area
• Motion or rotation of the coil thorugh a static field

Inductance

There is a direct coupling between inductors in electric circuits and magnetic circuits:

Consider a basic magnetic circuit with a coil:

From the side of the electric circuit:

$$e(t) =L \frac{\mathrm{d}I}{\mathrm{d}t}$$ From Faraday's Law

$$e(t) = \frac{\mathrm{d} \lambda}{\mathrm{d}t}$$ Thus:

$$L \frac{\mathrm{d}I}{\mathrm{d}t} = \frac{\mathrm{d} \lambda}{\mathrm{d}t}$$$$L = \frac{\mathrm{d} \lambda}{\mathrm{d}I}$$ Inductance is the flux created per current.