Key Components:

• Magnetic core (iron, steel, ferrite)

• Coil/winding carrying current

• Air gaps (if any)

Applications:

• Transformers

• Electric motors & generators

• Relays & solenoids

• Inductors

Where:

• \(\Phi\) = Magnetic flux (Wb)

• \(B\) = Flux density (T or Wb/m ² )

• \(A\) = Cross-sectional area (m ² )

• \(\theta\) = Angle between \(\vec{B}\) and normal

Unit: Weber (Wb)

Physical Meaning:

• Measures “concentration” of flux

• Higher \(B\) \(\Rightarrow\) stronger field in region

• Also called magnetic induction

Units:

• SI: Tesla (T) = Wb/m ²

• CGS: Gauss (G), where \(1\,\text{T} = 10^4\,\text{G}\)

Where:

• \(H\) = Field intensity (A/m or At/m)

• \(N\) = Number of turns

• \(I\) = Current (A)

• \(l\) = Mean path length (m)

Analogy with Electric Circuits:

Unit: Ampere-turns (At)

Where:

• \(l\) = Length of magnetic path (m)

• \(A\) = Cross-sectional area (m ² )

• \(\mu_0 = 4\pi \times 10^{-7}\) H/m

• \(\mu_r\) = Relative permeability

Unit: At/Wb or H ⁻¹

Analogy:

Higher \(\mu_r\) \(\Rightarrow\) Lower \(\mathcal{R}\)

Types:

• \(\mu_0\) = Permeability of free space
  \(= 4\pi \times 10^{-7}\) H/m

• \(\mu_r\) = Relative permeability (dimensionless)

• \(\mu = \mu_0 \mu_r\) = Absolute permeability

Key Points:

• Same flux \(\Phi\) through all sections

• MMF drops add up

Key Points:

• Flux divides: \(\Phi = \Phi_1 + \Phi_2\)

• Same MMF across parallel paths

Why Air Gap Matters:

• \(\mu_r\) of air \(= 1\) (vs. iron \(\approx\) 5000)

• Air gap has very high reluctance

• Small gap \(\Rightarrow\) large MMF drop

Fringing Effect:

• Flux spreads out at air gap

• Effective area increases

• Often neglected for small gaps

Fringing Factor ( \(k_f\) ):

• Flux spreads beyond core area

• Effective gap area \(>\) core area

Corrected Gap Reluctance: \[\mathcal{R}_g = \frac{l_g}{\mu_0 A_{eff}} = \frac{l_g}{\mu_0 k_f A}\]

Total Reluctance: \[\mathcal{R}_{total} = \mathcal{R}_{core} + \mathcal{R}_{gap}\] \[\mathcal{R}_{total} = \frac{l_c}{\mu_0 \mu_r A_c} + \frac{l_g}{\mu_0 A_g}\]

Total MMF Required: \[NI = \Phi \cdot \mathcal{R}_{total}\] \[NI = H_c l_c + H_g l_g\]

Since \(B_g = \mu_0 H_g\) : \[H_g = \frac{B_g}{\mu_0} = \frac{B}{\mu_0}\]

(Assuming no fringing: \(B_g = B_c = B\) )

Definitions:

• Useful flux ( \(\Phi_u\) ): Flux linking both coils or passing through intended path

• Leakage flux ( \(\Phi_l\) ): Flux that takes path through air

Typical values: \(\lambda = 1.1\) to \(1.25\)

Why Leakage Matters:

• Reduces useful flux available

• Lowers transformer/machine efficiency

• Causes voltage drops in transformers

• Affects coupling between windings

Minimizing Leakage:

1. Use high-permeability core

2. Minimize air gaps

3. Place windings close together

4. Use interleaved windings

5. Proper core geometry design

Machines designed to operate at knee for efficiency.

Three Regions:

1. Linear Region:

• \(B \propto H\) (constant \(\mu\) )

• Domains align gradually

2. Knee Region:

• Transition zone

• \(\mu\) starts decreasing

3. Saturation Region:

• All domains aligned

• \(B\) increases very slowly

• \(\mu \rightarrow \mu_0\)

Key Parameters:

Retentivity ( \(B_r\) ):

• Residual flux when \(H = 0\)

• Material “remembers” magnetization

Coercivity ( \(H_c\) ):

• \(H\) needed to demagnetize

• Reverse field to make \(B = 0\)

Loop Area = Energy Loss

• Dissipated as heat per cycle

• Hysteresis loss \(\propto\) frequency

Energy lost per cycle = Area of hysteresis loop

Where:

• \(P_h\) = Hysteresis loss (W)

• \(\eta\) = Steinmetz coefficient

• \(B_{max}\) = Maximum flux density (T)

• \(n\) = Steinmetz exponent ( \(\approx 1.6\) – \(2.0\) )

• \(f\) = Frequency (Hz)

• \(V\) = Volume of core (m ³ )

What are Eddy Currents?

• Circulating currents induced in conductor

• Caused by changing magnetic flux

• Follow Lenz’s Law (oppose change)

Effects:

• Power loss ( \(I^2R\) heating)

• Useful in induction heating

• Useful in electromagnetic braking

Laminations oriented parallel to flux direction.

Why Laminations Work:

• Core divided into thin sheets

• Each sheet insulated (varnish/oxide)

• Eddy current paths broken

• Smaller loops \(\Rightarrow\) higher resistance

Key Points:

• \(P_e \propto t^2\) (thickness squared)

• Thinner laminations = less loss

• Typical: 0.35–0.5 mm thick

• Used in transformers, motors

Separation Method: \[\frac{P_{core}}{f} = k_h B_{max}^n + k_e B_{max}^2 f \cdot t^2\] Plot \(\frac{P_{core}}{f}\) vs \(f\) \(\Rightarrow\) straight line

Key Observations:

• At low frequency : \(P_h\) dominates

• At high frequency : \(P_e\) dominates

• Crossover depends on lamination thickness

To Reduce Hysteresis Loss:

• Use soft magnetic materials

• Choose materials with narrow B-H loop

• Silicon steel (3–4% Si) preferred

• Grain-oriented steel for transformers

To Reduce Eddy Current Loss:

• Use laminated cores

• Thinner laminations for higher frequency

• Add silicon to increase resistivity

• Use ferrites at very high frequencies

Typical Lamination Thickness

Thinner = less eddy loss but higher cost