3 hours · 3 marks
Magnetic circuits analyse flux paths using analogies to electric circuits. This short chapter also covers B–H behaviour, hysteresis and eddy losses, Faraday induction, and force on current-carrying conductors — all frequent short-answer / numerical topics.
| Magnetic | Symbol / unit | Electric analogy |
|---|---|---|
| MMF (magnetomotive force) | \(\mathcal{F}=NI\) (AT) | EMF \(E\) |
| Flux | \(\Phi\) (Wb) | Current \(I\) |
| Reluctance | \(\mathcal{R}=l/(\mu A)\) (AT/Wb) | Resistance \(R\) |
| Permeance | \(\mathcal{P}=1/\mathcal{R}\) | Conductance \(G\) |
| Flux density | \(B=\Phi/A\) (T) | — |
| Field intensity | \(H=\mathcal{F}/l=NI/l\) (AT/m) | — |
Same flux \(\Phi\) through each section (core + air gap, etc.); MMFs (reluctance drops) add:
\[ NI=\Phi(\mathcal{R}_1+\mathcal{R}_2+\cdots)=H_1\ell_1+H_2\ell_2+\cdots \]Air gaps dominate because \(\mu_r\approx 1\) ⇒ large \(\mathcal{R}_g=\ell_g/(\mu_0 A_g)\). Fringing increases effective gap area slightly; often neglected in first-order exam problems.
Same MMF across parallel branches; fluxes add:
\[ \Phi_{\text{total}}=\Phi_a+\Phi_b,\qquad NI=H_a\ell_a=H_b\ell_b \]Composite cores (window frames, multiple limbs) are solved like parallel–series resistor networks.
Plotting \(B\) vs \(H\) for a ferromagnetic core while slowly increasing then decreasing magnetizing current yields a hysteresis loop.
| Point / quantity | Meaning |
|---|---|
| Residual flux density \(B_r\) | \(B\) remaining when \(H=0\) after magnetization |
| Coercive force \(H_c\) | \(H\) needed to bring \(B\) back to zero |
| Saturation \(B_{\text{sat}}\) | maximum practical \(B\) |
| Loop area | energy loss per cycle per unit volume (DC cycling) |
For DC (or very slow) magnetization, domain alignment lags \(H\), producing the loop. Remanence and coercivity characterise soft vs hard materials.
Under AC excitation the operating point traverses the hysteresis loop every cycle. Instantaneous \(B\) and \(H\) are not single-valued; the loop area still represents energy dissipated as heat per cycle.
At higher frequency the loop may widen slightly (dynamic effects), but the classical separation still treats hysteresis loss as proportional to frequency (once per cycle) and eddy loss as proportional to \(f^2\).
Changing flux induces circulating currents in the core (Faraday). These \(I^2R\) losses are reduced by laminating the core (thin sheets insulated from each other):
\[ P_e=k_e\,B_m^{2}\,f^{2}\,t^{2}\,V \] where \(t\) is lamination thickness. Finer laminations ⇒ smaller \(P_e\).Separation of losses: measure core loss at two frequencies (same \(B_m\)) or use the \(P/f\) vs \(f\) method.
Faraday’s law: the induced EMF in a coil equals the rate of change of flux linkage:
\[ e=-N\frac{d\Phi}{dt}=-\frac{d\lambda}{dt} \]Lenz’s law: the minus sign — induced current opposes the change causing it.
Conductor / coil is stationary; flux through it changes with time (transformer action).
For sinusoidal flux \(\Phi=\Phi_m\sin\omega t\):
\[ E_{\text{rms}}=4.44\,f\,N\,\Phi_m \] (transformer EMF equation).Flux density is constant in time, but conductors move in the field (generator action):
\[ e=B\,\ell\,v\sin\theta \] where \(\theta\) is the angle between velocity and \(\mathbf{B}\) (maximum when motion is perpendicular to \(\mathbf{B}\) and to the conductor length).A conductor of length \(\ell\) carrying current \(I\) in a magnetic field \(\mathbf{B}\) experiences
\[ \mathbf{F}=I\,(\boldsymbol{\ell}\times\mathbf{B}),\qquad F=B I \ell\sin\theta \]Direction: Fleming’s left-hand rule (motors) — Forefinger (Field), Centre finger (Current), Thumb (Motion/Force).
For two long parallel conductors:
\[ F/l=\frac{\mu_0 I_1 I_2}{2\pi d} \] (attractive if currents same direction).Classic numerical
A circular magnetic ring has mean circumference (mean length of iron path) \(0.5\,\text{m}\), cross-section \(A=100\,\text{cm}^2\), air gap \(4\,\text{mm}\), flux \(\Phi=2\,\text{mWb}\), relative permeability of iron \(\mu_r=800\), turns \(N=200\). Find the MMF (and magnetizing current).
Given data (SI):
Flux density (assume same \(A\) in gap, neglect fringing):
\[ B=\frac{\Phi}{A}=\frac{0.002}{0.01}=0.2\,\text{T} \]Field in iron:
\[ H_i=\frac{B}{\mu_0\mu_r}=\frac{0.2}{4\pi\times 10^{-7}\times 800} =\frac{0.2}{1.0053\times 10^{-3}}\approx 199\,\text{AT/m} \] \[ \text{AT for iron}=H_i\ell_i\approx 199\times 0.5\approx 99.5\,\text{AT} \]Field in air gap (\(\mu_r=1\)):
\[ H_g=\frac{B}{\mu_0}=\frac{0.2}{4\pi\times 10^{-7}}\approx 1.59\times 10^{5}\,\text{AT/m} \] \[ \text{AT for gap}=H_g\ell_g\approx 1.59\times 10^{5}\times 0.004\approx 636\,\text{AT} \]Total MMF:
\[ NI=H_i\ell_i+H_g\ell_g\approx 99.5+636=735.5\,\text{AT} \] \[ I=\frac{NI}{N}=\frac{735.5}{200}\approx 3.68\,\text{A} \]Observation: the tiny air gap accounts for most of the MMF (\(\approx 86\%\)). This is why air-gap design dominates machine/magnetics calculations.
Reluctance check:
\[ \mathcal{R}_i=\frac{\ell_i}{\mu_0\mu_r A},\quad \mathcal{R}_g=\frac{\ell_g}{\mu_0 A},\quad NI=\Phi(\mathcal{R}_i+\mathcal{R}_g) \] yields the same numerical result.| Series | Parallel | |
|---|---|---|
| Flux | Same \(\Phi\) in each part | Splits among branches |
| MMF | Drops add (\(NI=\sum H\ell\)) | Same \(NI\) on each branch |
| Typical structure | Ring + one air gap | Multiple limbs / leakage paths |