Basic Concepts of Magnetic Circuits

When a current flowing in a wire, it creates a magnetic field around it. And, if the current carrying wire is a coil, the created magnetic field induces a voltage in it. Furthermore, when the current carrying conductor facing another magnetic field, it induces a force. Moreover, when the current carrying wire, moving in a magnetic field, it induces a voltage.

Magnetic field intensity around a closed path

Magnetic field intensity H is the ability of current to create a magnetic field in a circuit. To calculate the magnetic field intensity, caused by a current, we can use the below equation, Ampere’s law.

$latex \displaystyle \oint H~dl~=~{{I}_{{net}}}$

H, Magnetic field intensity produced by the current

$latex \displaystyle dI\text{ }\!\!~\!\!\text{ }$, Differential element of length along the path of intergration

$latex \displaystyle {{I}_{{net}}}$, The current passing along the path of integration

If there is a ferromagnetic metal inside a coil, the magnetic field created by the coil is absorbed into the core.

$latex \displaystyle {{T}_{{net}}}=Ni$

N is the number of loops

i is the amount of current flow in a loop

$latex \displaystyle {{l}_{C}}\text{ }\!\!~\!\!\text{ }$is the magnetic path length

So, the Ampere’s law

$latex \displaystyle H{{I}_{c}}=Ni$

H is the magnitude of the magnetic field intensity vector,

$latex \displaystyle H=\frac{{Ni}}{{{{l}_{c}}}}$

Magnetic field intensity and the core

If there is a ferromagnetic metal inside the coil, the magnetic field created by the coil is absorbed into the core. However, the strength of the magnetic field, depends on the core.

$latex \displaystyle B=\text{ }\!\!\mu\!\!\text{ }H$

B is the magnetic flux density.

µ is the permeability of the material.

We can also write this as,


$latex \displaystyle B=\text{ }\!\!\mu\!\!\text{ }\frac{{Ni}}{{{{l}_{c}}}}$

Total flux in a given area ϕ

$latex \displaystyle \phi $ = $latex \displaystyle \mathop{\oint }_{A}^{0}B\cdot dA$

dA is the differential unit of area.

If the flux density is a constant and has a $latex \displaystyle \theta $ angle to a plane of area A, we can write the equation as,

$latex \displaystyle \phi \text{ }\!\!~\!\!\text{ }=BA\cos \left( \theta  \right)$

If the flux density is perpendicular to the area and a constant, we can rewrite the above equation as,

$latex \displaystyle \phi $ = $latex \displaystyle BA$

So, to get to know the total magnetic flux generated by the current, we can substitute,

B=µH or $latex \displaystyle B=\text{ }\!\!\mu\!\!\text{ }\frac{{Ni}}{{{{l}_{c}}}}$

Into the equation, therefore the magnetic flux density,

$latex \displaystyle \phi $ = $latex \displaystyle \text{ }\!\!\mu\!\!\text{ }HA$

$latex \displaystyle \phi $ = $latex \displaystyle \text{ }\!\!\mu\!\!\text{ }\frac{{Ni}}{{{{l}_{c}}}}A$

Force that drives the current flow

In a normal electrical circuit, the voltage or the EMF (electromotive force) drives the current flow. So it can be described using the Ohm’s law,

$latex \displaystyle V=IR$

But in magnetic circuits, the force that drives the current flow is MMF (magnetomotive force).  And it’s represented by script letter $latex \displaystyle \mathcal{F},~$ and measured by Ampere turns. The equation for magnetomotive force is,

$latex \displaystyle \mathcal{F}=\text{Ni}$

So, we can write the magnetic flux as,

$latex \displaystyle \phi $ = $latex \displaystyle \mathcal{F}\left( {\frac{\text{ }\!\!\mu\!\!\text{ }}{{{{l}_{c}}}}A} \right)$

MMF, flux and reluctance

Magnetomotive force $latex \displaystyle \mathcal{F}$ (MMF) is the cause of the generation of the magnetic flux ∅. Magnetic reluctance is a similar concept to the electrical resistance. And the units of reluctance s ampere-turns per weber. Just like in the Ohm’s law, we can build a relationship as follows,

$latex \displaystyle \mathcal{F}=\varnothing \mathcal{R}$

So, we can write the magnetic flux as,

$latex \displaystyle \frac{\mathcal{F}}{\mathcal{R}}$ = $latex \displaystyle \mathcal{F}\left( {\frac{\text{ }\!\!\mu\!\!\text{}}{{{{l}_{c}}}}A} \right)$

$latex \displaystyle \mathcal{R}=\frac{{{{\text{I}}_{\text{c}}}}}{{\text{ }\!\!\mu\!\!\text{ A}}}$

Reluctance in series

$latex \displaystyle {{R}_{{eq}}}={{R}_{1}}+{{R}_{2}}+{{R}_{3}}+{{R}_{4}}+\ldots $

Reluctance in parallel

$latex \displaystyle{{R}_{{eq}}}=\frac{1}{{{{R}_{1}}}}+\frac{1}{{{{R}_{2}}}}+\frac{1}{{{{R}_{3}}}}+\frac{1}{{{{R}_{4}}}}+\ldots $

Permeance

As there is reluctance for the resistance, what is there for the conductance. There is permeance for the resistance.

$latex \displaystyle \mathcal{P}=\frac{1}{\mathcal{R}}$