Magnetic Circuit Mastery: Core Concepts & Calculations

Demonstrative Video

Magnetic Circuit and its Analysis

Closed path followed by magnetic flux is called magnetic circuit
Usually consists of magnetic materials having high permeability (e.g., iron, soft steel, etc.).

\[\begin{aligned} \text{Flux density }B & =\dfrac{\phi}{a}~\text{Wb/m\textsuperscript{2}}\\ \text{Magnetising force }H & =\dfrac{B}{\mu_{0}\mu_{r}}\\ & =\dfrac{\phi}{a\mu_{0}\mu_{r}}~\text{AT/m} \end{aligned}\]
magnetic flux starts from a point and finishes at the same point after completing its path

\[\begin{aligned} & \rightarrow Hl=NI\\ \Rightarrow & \dfrac{\phi}{a\mu_{0}\mu_{r}}\times l=NI\\ \Rightarrow & \phi=\dfrac{NI}{\left(l/a\mu_{0}\mu_{r}\right)}~\text{Wb} \end{aligned}\]

\[\begin{aligned} & \boxed{\phi =\dfrac{NI}{\left(l/a\mu_{0}\mu_{r}\right)}~\text{Wb}} \end{aligned}\]

\[\Rightarrow \phi ~\text{increases if either of the two increases and vice-versa}\]
magnetomotive force (mmf)\(NI\)\(I\)\(\propto N\)
\[\begin{aligned} \Rightarrow &\text{Reluctance: opposition offered to}~ \phi~ \text{by the magnetic path} \\ \Rightarrow &\text{lower reluctance higher will be} ~\phi ~\text{and vice-versa} \end{aligned}\]
of the magnetic path called inversely
\[\phi = \dfrac{\text{mmf}}{\text{reluctance}}\]
Therefore,
Note: expression strong resemblance to Ohm’s Law for electric current (I=emf/resistance)
referred to Ohm’s law of magnetic circuits

Important terms in magnetism

Magnetic field: The region around a magnet where its poles exhibit a force of attraction or repulsion
Magnetic flux (\(\phi\)):
- The amount of magnetic lines of force set-up in a magnetic circuit
- unit is weber (Wb).
- analogous to electric current \(I\) in electric circuit
\[B=\frac{\phi}{A} \mathrm{~Wb} / \mathrm{m}^{2} \quad \text { or } \quad \mathrm{T}\left(1 \mathrm{~Wb} / \mathrm{m}^{2}=1 \times 10^{4} \mathrm{~Wb} / \mathrm{cm}^{2}\right)\]
Magnetic flux density (\(B\)):

Permeability (\(\mu\)): The ability of a material to conduct magnetic lines of force through it
- greater the \(\mu\) of a material, the greater is its conductivity for the magnetic lines of force and vice-versa
- air or vacuum is the poorest represented \(\mu_0=4\pi\times10^{-7}\) H/m
- Relative permeability (\(\mu_r\)): The absolute (or actual) permeability of a magnetic material is much greater than \(\mu_0\).
- \[\mu_{r}=\frac{\mu}{\mu_{0}} \quad \text { or } \quad \mu=\mu_{0} \mu_{r}\]
  \(\mu_r\)
- Obviously, \(\mu_r\) of air would be 1.
- \(\mu_r\) of all the non-magnetic materials is also 1.
- However, its value is as high as 8000 for soft iron, whereas, its value for mumetal (iron 22% and nickel 78%) is as high as 1,20,000.

\[H=\frac{\mathrm{m} \cdot \mathrm{m} \cdot \mathrm{f}}{\text { length of magnetic path }}=\frac{N I}{l} \mathrm{AT} / \mathrm{m}\]
Magnetic field intensity (\(H\)):
Magneto-motive force (mmf): The magnetic pressure which sets-up or tends to set-up magnetic flux in a magnetic circuit
- \[\text{mmf} = NI ~\text{ampere-turns (AT)}\]
  As per work law it may be defined as: The work done in moving a unit magnetic pole (1 Wb) once round the magnetic circuit
- analogous to emf in an electric circuit

\[\text { Reluctance, } S=\frac{l}{a \mu_{0} \mu_{r}}\]
Reluctance (\(S\)):
- analogous to resistance in an electric circuit
Permeance: It is a measure of the ease with which flux can be set-up in the material.
- \[\text { Permeance }=\frac{1}{\text { reluctance }}=\frac{a \mu_{0} \mu_{r}}{l} \text { Wb/AT or } \mathrm{H}\]
  It is just reciprocal of reluctance of the material
- analogous to conductance in an electric circuit
Reluctivity: It is specific reluctance and analogous to resistivity in electric circuit