Magnetic Circuit and Induction
Magnetomotive force, reluctance, air-gap fringing, B-H loops, Steinmetz core losses, and electromagnetic induction fundamentals.
6.1 & 6.2 Magnetic Circuits vs. Electric Circuits
Concept & Fundamental Laws
A magnetic circuit is the closed path traversed by magnetic lines of force (flux). Magnetomotive force (MMF) drives magnetic flux through the reluctance of magnetic materials, strictly analogous to how EMF drives electric current through resistance:
\[\mathcal{F} = N I = \Phi \mathcal{R}\]
Where:
• \(\mathcal{F}\) = Magnetomotive Force (MMF) [Ampere-Turns, AT]
• \(\Phi\) = Magnetic flux [Webers, Wb]
• \(\mathcal{R}\) = Reluctance [AT/Wb] = \(\frac{l}{\mu A} = \frac{l}{\mu_0 \mu_r A}\)
• \(\mathcal{P}\) = Permeance [Wb/AT] = \(1/\mathcal{R}\)
Comparison Table: Magnetic vs. Electric Circuits
| Feature | Electric Circuit | Magnetic Circuit |
|---|---|---|
| Driving Potential | Electromotive Force (\(\text{EMF} = V\)) [Volts] | Magnetomotive Force (\(\text{MMF} = N I\)) [Ampere-turns] |
| Resulting Flow / Field | Electric Current (\(I\)) [Amperes] | Magnetic Flux (\(\Phi\)) [Webers] |
| Opposition | Resistance \(R = \rho \frac{l}{A} = \frac{l}{\sigma A}\) [\(\Omega\)] | Reluctance \(\mathcal{R} = \frac{l}{\mu_0 \mu_r A}\) [AT/Wb] |
| Conductivity / Permeability | Conductivity \(\sigma\) (constant with current) | Permeability \(\mu = \mu_0 \mu_r\) (varies with \(B\)) |
| Current / Flux Density | Current Density \(J = I/A\) [A/m\(^2\)] | Flux Density \(B = \Phi/A\) [Tesla] |
| Potential Gradient | Electric Field \(E = V/l\) [V/m] | Magnetic Field Intensity \(H = \mathcal{F}/l\) [AT/m] |
| Energy Consumption | Energy is continuously dissipated as heat (\(I^2 R\)). | Energy is required only to establish the field; no continuous power is consumed to maintain static flux! |
Air Gap, Leakage Flux & Fringing
• Leakage Flux: Magnetic flux that strays outside the intended iron path and closes through ambient air.
• Fringing Effect: At an air gap, magnetic flux lines repel each other and bulge outward into surrounding space. This increases the effective cross-sectional area of the air gap (\(A_g > A_{\text{core}}\)), which slightly reduces the actual flux density in the gap.
6.3 & 6.4 B-H Curve, Hysteresis & Eddy Current Losses
The Hysteresis Phenomenon
When a ferromagnetic core is magnetized by an alternating current, the flux density \(B\) lags behind the magnetizing force \(H\). This lag is known as magnetic hysteresis.
• Retentivity (Residual Magnetism): The flux density \(B_r\) remaining in the core when the magnetizing field \(H\) is reduced to zero.
• Coercivity (Coercive Force): The reverse magnetic field intensity \(H_c\) required to completely demagnetize the core (\(B = 0\)).
Core Loss Formulations
6.5 & 6.6 Electromagnetic Induction & Conductor Force
| Category | Statically Induced EMF | Dynamically Induced EMF |
|---|---|---|
| Mechanism | Conductor remains stationary; magnetic flux links it and varies with time. | Magnetic field is stationary; conductor physically moves across flux lines. |
| Governing Law | Faraday's Law: \(e = -N \frac{d\Phi}{dt} = -L \frac{di}{dt}\) | Lorentz Induction: \(e = B l v \sin\theta\) |
| Direction Rule | Lenz's Law (opposes rate of change of flux) | Fleming's Right-Hand Rule (Thumb = Motion, Forefinger = Field, Middle = Induced Current) |
| Application | Transformers, inductors, chokes. | DC Generators, AC Alternators. |
Master Worked Example: MMF with Air Gap (PYQ 2082 Bhadra Q8)
Cross-sectional area \(A = 100\text{ cm}^2 = 0.01\text{ m}^2\)
Air gap length \(l_g = 4\text{ mm} = 0.004\text{ m}\)
Required air gap flux \(\Phi = 2\text{ mWb} = 0.002\text{ Wb}\)
Coil turns \(N = 200\), Relative permeability \(\mu_r = 800\).