Magnetism & Magnetic Circuit Formulas

A Comprehensive Guide for Electrical Engineers

Important Nomenclature and Notation

\[ \begin{aligned} \phi &= \text{magnetic flux in Wb (webers)} \\ B &= \text{magnetic flux density in T (tesla)} \\ H &= \text{magnetic field intensity in AT/m} \\ 1~\mathrm{T} &= 1~\mathrm{Wb/m^2} = 10,000~\text{gauss} \\ A &= \text{cross-sectional area in}~\mathrm{m^2} \\ \mathcal{F} &= \text{magnetomotive force (MMF) in AT (amp-turns)} \\ \mathcal{R} &= \text{reluctance of the material in AT/Wb} \\ P &= \text{permeance of the material in Wb/AT} \\ N &= \text{number of loops or turns in the coil} \\ I &= \text{current in the coil in A (amperes)} \\ l &= \text{average length of the material in m (meters)} \\ L &= \text{inductance of the coil in H (henry)} \\ \mu &= \text{permeability of the material in H/m} \\ \mu_0 &= \text{permeability of free space} = 4\pi \times 10^{-7}~\mathrm{H/m} \\ \mu_r &= \text{relative permeability (dimensionless)} \\ \chi_m &= \text{magnetic susceptibility (dimensionless)} \\ M &= \text{magnetization in A/m} \\ W_m &= \text{energy stored in magnetic field in J (joules)} \end{aligned} \]

Analogy: Electric Circuit vs Magnetic Circuit

Magnetic circuits follow similar principles to electric circuits, making analysis intuitive for electrical engineers.

Electric Circuit	Magnetic Circuit	Relationship
EMF (Electromotive force) \(\mathcal{E}\)	MMF (Magnetomotive force) \(\mathcal{F}\)	\(\mathcal{F} = NI\)
Current \(i\)	Magnetic Flux \(\phi\)	\(\phi = \frac{\mathcal{F}}{\mathcal{R}}\)
Resistance \(R\)	Reluctance \(\mathcal{R}\)	\(\mathcal{R} = \frac{l}{\mu A}\)
Conductance \(G\)	Permeance \(P\)	\(P = \frac{1}{\mathcal{R}}\)
Current Density \(J\)	Flux Density \(B\)	\(B = \frac{\phi}{A}\)
Ohm's Law: \(V = IR\)	Hopkinson's Law: \(\mathcal{F} = \phi \mathcal{R}\)	Analogous Laws

Fundamental Formulas

Magnetomotive Force (MMF)

\[ \mathcal{F} = NI \quad \text{(in ampere-turns)} \]

Definition: MMF is the driving force that produces magnetic flux in a magnetic circuit, analogous to EMF in electric circuits.

Magnetic Field Intensity

\[ H = \frac{\mathcal{F}}{l} = \frac{NI}{l} \quad \text{(in AT/m)} \]

Magnetic Flux

\[ \phi = \frac{\mathcal{F}}{\mathcal{R}} = \frac{NI}{\mathcal{R}} \]

Magnetic Flux Density

\[ \begin{aligned} B &= \frac{\phi}{A} \\ B &= \mu H = \mu_0 \mu_r H \end{aligned} \]

Permeability Relationships

\[ \begin{aligned} \mu &= \mu_0 \mu_r \\ \mu_r &= 1 + \chi_m \\ \mu_0 &= 4\pi \times 10^{-7}~\mathrm{H/m} \end{aligned} \]

Key Point: Relative permeability (\(\mu_r\)) values:

Diamagnetic materials: \(\mu_r < 1\) (e.g., copper, silver)
Paramagnetic materials: \(\mu_r > 1\) (slightly) (e.g., aluminum, platinum)
Ferromagnetic materials: \(\mu_r \gg 1\) (e.g., iron: 5,000, silicon steel: 7,000)

Reluctance

\[ \begin{aligned} \mathcal{R} &= \frac{l}{\mu_0 \mu_r A} = \frac{l}{\mu A} \\ \mathcal{R} &= \frac{\mathcal{F}}{\phi} = \frac{NI}{\phi} \end{aligned} \]

Induced EMF (Faraday's Law)

\[ e = -N \frac{d\phi}{dt} = -L \frac{di}{dt} \]

Inductance

\[ \begin{aligned} L &= N \frac{d\phi}{di} = \frac{N\phi}{I} \\ L &= \frac{\mu_0 \mu_r N^2 A}{l} = \frac{N^2}{\mathcal{R}} \\ L &= \frac{N^2 P}{\text{where } P = \frac{1}{\mathcal{R}} \text{ (permeance)}} \end{aligned} \]

Magnetic Materials Classification

1. Diamagnetic Materials

Property: \(\mu_r < 1\), \(\chi_m < 0\)
Behavior: Weakly repelled by magnetic fields
Examples: Copper, gold, silver, bismuth, water
Applications: Magnetic levitation, shielding

2. Paramagnetic Materials

Property: \(\mu_r > 1\) (slightly), \(\chi_m > 0\) (small)
Behavior: Weakly attracted to magnetic fields
Examples: Aluminum, platinum, manganese, oxygen
Applications: MRI contrast agents, oxygen sensors

3. Ferromagnetic Materials

Property: \(\mu_r \gg 1\), \(\chi_m \gg 0\)
Behavior: Strong attraction, exhibits hysteresis
Examples: Iron, cobalt, nickel, and their alloys
Curie Temperature: Above this temperature, ferromagnetic materials become paramagnetic
Applications: Transformers, motors, generators, magnetic storage

4. Ferrimagnetic Materials

Property: Similar to ferromagnetic but with lower magnetization
Examples: Ferrites (Fe₃O₄), garnets
Applications: High-frequency transformers, inductors, microwave devices

B-H Curve and Magnetic Hysteresis

B-H Curve Characteristics

\[ B = \mu_0 \mu_r H = \mu_0 (H + M) \]

Important Points on B-H Curve:

Origin (O): Unmagnetized state
Saturation Point: Maximum flux density achieved
Remanence (Br): Residual flux density when H = 0
Coercivity (Hc): Reverse field intensity needed to reduce B to zero
Hysteresis Loop: Area represents energy loss per cycle

Hysteresis Loss

\[ \begin{aligned} W_h &= \eta B_{max}^{1.6} \cdot \text{Volume} \quad \text{(Steinmetz equation)} \\ P_h &= \eta f B_{max}^{1.6} \cdot \text{Volume} \quad \text{(power loss)} \end{aligned} \]

where \(\eta\) is Steinmetz hysteresis coefficient and \(f\) is frequency.

Types of Magnetic Materials by Hysteresis

Soft Magnetic Materials: Narrow hysteresis loop, low coercivity, easy to magnetize/demagnetize (e.g., silicon steel, soft iron) - Used in transformers and electromagnets
Hard Magnetic Materials: Wide hysteresis loop, high coercivity, retain magnetization (e.g., Alnico, NdFeB) - Used in permanent magnets

Magnetic Circuits with Air Gap

Series Magnetic Circuit with Air Gap

\[ \begin{aligned} NI &= H_c l_c + H_g l_g \\ NI &= \frac{B_c}{\mu_c} l_c + \frac{B_g}{\mu_0} l_g \\ \mathcal{R}_{total} &= \mathcal{R}_c + \mathcal{R}_g = \frac{l_c}{\mu_c A_c} + \frac{l_g}{\mu_0 A_g} \end{aligned} \]

Important Note: Even a small air gap significantly increases the total reluctance because \(\mu_r\) of air = 1, while \(\mu_r\) of magnetic materials can be thousands.

Fringing Effect

In air gaps, magnetic flux spreads out beyond the core cross-section. This is called fringing. To account for this: \[ A_g^{effective} = (w + l_g)(d + l_g) \] where \(w\) and \(d\) are core dimensions and \(l_g\) is air gap length.

Series and Parallel Magnetic Circuits

Reluctances in Series

\[ \begin{aligned} \mathcal{R}_{total} &= \mathcal{R}_1 + \mathcal{R}_2 + \mathcal{R}_3 + \cdots + \mathcal{R}_N \\ \phi_{total} &= \phi_1 = \phi_2 = \phi_3 = \cdots = \phi_N \\ \mathcal{F}_{total} &= \mathcal{F}_1 + \mathcal{F}_2 + \mathcal{F}_3 + \cdots + \mathcal{F}_N \end{aligned} \]

Reluctances in Parallel

\[ \begin{aligned} \mathcal{R}_{total} &= \frac{1}{\frac{1}{\mathcal{R}_1} + \frac{1}{\mathcal{R}_2} + \frac{1}{\mathcal{R}_3} + \cdots + \frac{1}{\mathcal{R}_N}} \\ \phi_{total} &= \phi_1 + \phi_2 + \phi_3 + \cdots + \phi_N \\ \mathcal{F}_{total} &= \mathcal{F}_1 = \mathcal{F}_2 = \mathcal{F}_3 = \cdots = \mathcal{F}_N \end{aligned} \]

Flux Division in Parallel Paths

\[ \frac{\phi_1}{\phi_2} = \frac{\mathcal{R}_2}{\mathcal{R}_1} = \frac{l_1 \mu_2 A_2}{l_2 \mu_1 A_1} \]

Energy Storage in Magnetic Circuits

Energy Stored

\[ \begin{aligned} W_m &= \frac{1}{2} L I^2 \\ W_m &= \frac{1}{2} \mathcal{R} \phi^2 \\ W_m &= \frac{1}{2} \frac{\phi^2}{\mu A} l \\ W_m &= \frac{1}{2} \frac{B^2}{\mu} \cdot \text{Volume} \\ W_m &= \frac{1}{2} B H \cdot \text{Volume} \end{aligned} \]

Energy Density

\[ \begin{aligned} w_m &= \frac{W_m}{\text{Volume}} = \frac{1}{2} \frac{B^2}{\mu} = \frac{1}{2} \mu H^2 \\ w_m &= \frac{1}{2} B H \quad \text{(in J/m}^3\text{)} \end{aligned} \]

Application: Energy density is crucial in designing electromagnets, inductors, and magnetic actuators where energy storage capacity determines performance.

Force and Torque in Magnetic Systems

Magnetic Force on Current-Carrying Conductor

\[ \begin{aligned} \vec{F} &= I \vec{l} \times \vec{B} \\ F &= BIl \sin\theta \quad \text{(scalar form)} \\ F &= BIl \quad \text{(when perpendicular)} \end{aligned} \]

Force on Moving Charge

\[ \vec{F} = q(\vec{v} \times \vec{B}) \]

Force in Magnetic Circuit (Electromagnet)

\[ \begin{aligned} F &= \frac{B^2 A}{2\mu_0} = \frac{\phi^2}{2\mu_0 A} \\ F &= \frac{1}{2} \frac{d\mathcal{R}}{dx} \phi^2 = \frac{1}{2} I^2 \frac{dL}{dx} \end{aligned} \]

Torque on a Coil

\[ \begin{aligned} \tau &= NIBA \sin\theta \\ \tau &= \vec{m} \times \vec{B} \quad \text{where } \vec{m} = NIA\hat{n} \end{aligned} \]

Ampere's Circuital Law

\[ \begin{aligned} \oint \vec{H} \cdot d\vec{l} &= I_{enclosed} = NI \\ \oint \vec{B} \cdot d\vec{l} &= \mu_0 I_{enclosed} \end{aligned} \]

Applications:

Calculating magnetic field in solenoids: \(H = \frac{NI}{l}\)
Calculating magnetic field in toroids: \(H = \frac{NI}{2\pi r}\)
Analyzing magnetic circuits with multiple sections

Magnetic Flux Leakage

Leakage Factor (λ): Ratio of total flux produced to useful flux linking the circuit. \[ \lambda = \frac{\phi_{total}}{\phi_{useful}} = \frac{\phi_m + \phi_l}{\phi_m} \] where \(\phi_l\) is leakage flux.

Typical leakage factors: 1.15 to 1.25 for well-designed magnetic circuits.

Core Losses in Magnetic Materials

Total Core Loss

\[ P_{core} = P_h + P_e \]

Hysteresis Loss

\[ P_h = \eta f B_{max}^{1.6} \cdot V \quad \text{(watts)} \]

Eddy Current Loss

\[ P_e = K_e f^2 B_{max}^2 t^2 \cdot V \quad \text{(watts)} \]

where \(t\) is lamination thickness, \(V\) is volume, \(K_e\) is eddy current coefficient.
Reduction: Use thin laminations, high-resistivity materials, or ferrites at high frequencies.

Practical Design Considerations

Transformer Core Design

Choose high \(\mu_r\) material (silicon steel: \(\mu_r\) ≈ 5000-7000)
Lamination thickness: 0.35-0.5 mm for 50/60 Hz
Minimize air gaps to reduce reluctance
Account for stacking factor (0.9-0.95)

Inductor Design

For high-frequency applications: Use ferrite cores (\(\mu_r\) = 1000-3000)
For DC bias: Consider air gap to prevent saturation
Powder cores for distributed gap applications
Window area must accommodate wire with insulation

Electromagnet Design

Force proportional to \(B^2A\): maximize flux density and pole area
Minimize air gap for maximum force
Consider copper losses: \(P_{cu} = I^2R\)
Thermal management critical for continuous operation

Maxwell's Equations for Magnetism

Gauss's Law for Magnetism

\[ \oint \vec{B} \cdot d\vec{A} = 0 \]

Interpretation: No magnetic monopoles exist; magnetic field lines form closed loops.

Faraday's Law of Induction

\[ \oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt} = -\frac{d}{dt}\int \vec{B} \cdot d\vec{A} \]

Ampere-Maxwell Law

\[ \oint \vec{H} \cdot d\vec{l} = I + \frac{d\Phi_D}{dt} \]

Magnetic Field of Common Geometries

Infinite Straight Wire

\[ \begin{aligned} B &= \frac{\mu_0 I}{2\pi r} \\ H &= \frac{I}{2\pi r} \end{aligned} \]

Solenoid (Long)

\[ \begin{aligned} B &= \mu_0 \mu_r n I = \mu_0 \mu_r \frac{NI}{l} \\ H &= \frac{NI}{l} = nI \end{aligned} \]

Toroid

\[ \begin{aligned} B &= \frac{\mu_0 \mu_r NI}{2\pi r} \\ H &= \frac{NI}{2\pi r} \end{aligned} \]

Circular Loop at Center

\[ B = \frac{\mu_0 I}{2R} \]

Helmholtz Coils

\[ B = \frac{8\mu_0 NI}{5\sqrt{5}R} = 0.716\frac{\mu_0 NI}{R} \]

Two identical coils separated by distance equal to their radius, producing uniform field in the central region.

Magnetic Coupling and Mutual Inductance

Mutual Inductance

\[ \begin{aligned} M &= k\sqrt{L_1 L_2} \\ M &= \frac{N_1 \phi_{12}}{I_2} = \frac{N_2 \phi_{21}}{I_1} \\ e_1 &= M\frac{dI_2}{dt}, \quad e_2 = M\frac{dI_1}{dt} \end{aligned} \]

Coupling Coefficient

\[ k = \frac{M}{\sqrt{L_1 L_2}}, \quad 0 \leq k \leq 1 \]

Coupling Types:

\(k = 1\): Perfect coupling (ideal transformer)
\(0.95 < k < 1\): Tight coupling (power transformers)
\(0.5 < k < 0.95\): Loose coupling (RF transformers)
\(k < 0.5\): Weak coupling

Dot Convention

Current entering the dotted terminal of one coil produces positive flux linkage in the other coil at its dotted terminal.

Series-Aiding and Series-Opposing

\[ \begin{aligned} L_{series-aiding} &= L_1 + L_2 + 2M \\ L_{series-opposing} &= L_1 + L_2 - 2M \\ L_{parallel} &= \frac{L_1L_2 - M^2}{L_1 + L_2 - 2M} \end{aligned} \]

Magnetic Saturation and Nonlinearity

Saturation Characteristics

Linear Region: \(B \propto H\), constant \(\mu_r\)
Knee Region: \(\mu_r\) starts decreasing
Saturation Region: \(B \approx B_{sat}\), \(\mu_r \approx 1\)

Typical Saturation Flux Densities

Silicon steel: 1.8 - 2.0 T
Pure iron: 2.15 T
Ferrites: 0.3 - 0.5 T
Permalloy: 0.8 - 1.0 T
Supermalloy: 0.7 - 0.8 T

Effects of Saturation

Inductance decreases significantly
Current increases rapidly for small voltage increase
Harmonic distortion in AC applications
Increased core losses and heating
Design rule: Operate at 70-80% of \(B_{sat}\) for safety margin

Temperature Effects on Magnetic Properties

Curie Temperature

Temperature at which ferromagnetic materials lose their ferromagnetic properties and become paramagnetic.

Curie Temperatures of Common Materials

Iron: 770°C (1043 K)
Cobalt: 1115°C (1388 K)
Nickel: 358°C (631 K)
Gadolinium: 20°C (293 K)

Temperature Coefficient

\[ \mu_r(T) = \mu_r(T_0)[1 + \alpha_T(T - T_0)] \]

where \(\alpha_T\) is temperature coefficient of permeability (typically negative for ferromagnetic materials).

Magnetic Shielding

Shielding Factor

\[ S = \frac{B_{without\,shield}}{B_{with\,shield}} = 1 + \frac{\mu_r t}{R} \]

where \(t\) is shield thickness, \(R\) is shield radius, \(\mu_r\) is relative permeability.

Shielding Materials

Low frequency (DC to kHz): High \(\mu_r\) materials (mu-metal, permalloy)
High frequency (MHz to GHz): High conductivity materials (copper, aluminum)
Multi-layer shields: Combination for broadband protection

Magnetic Measurement Techniques

Hall Effect Sensor

\[ V_H = \frac{R_H I B}{t} = K_H I B \]

where \(R_H\) is Hall coefficient, \(t\) is sensor thickness, \(K_H\) is Hall sensitivity.

Search Coil Method

\[ \phi = \frac{1}{N}\int e\,dt \]

Vibrating Sample Magnetometer (VSM)

Measures magnetic moment of materials by detecting voltage induced in pickup coils due to sample vibration.

Applications in Electrical Engineering

1. Power Transformers

Core material: Grain-oriented silicon steel (GOSS)
Operating flux density: 1.5-1.7 T
Efficiency: > 99% for large units
Key design: Minimize core and copper losses

2. Electric Machines (Motors/Generators)

Core material: Non-oriented silicon steel
Air gap: Critical for electromechanical energy conversion
Key parameters: Torque density, efficiency, power factor

3. Magnetic Sensors

Hall effect sensors: Current measurement, position sensing
Magnetoresistive sensors: High sensitivity applications
Fluxgate magnetometers: Navigation, geophysical surveys

4. Magnetic Data Storage

Hard disk drives: Recording density > 1 Tb/in²
Magnetic tape: Backup and archival storage
Materials: CoFe alloys, rare earth magnets

5. Inductors and Chokes

Filter inductors: Smooth DC, suppress harmonics
RF chokes: Block high-frequency signals
Core selection: Frequency-dependent (ferrite for HF)

6. Magnetic Levitation (Maglev)

Electromagnetic suspension (EMS)
Electrodynamic suspension (EDS)
Applications: High-speed trains, bearings

Important Constants and Conversions

\[ \begin{aligned} \mu_0 &= 4\pi \times 10^{-7}~\mathrm{H/m} = 1.257 \times 10^{-6}~\mathrm{H/m} \\ 1~\mathrm{T} &= 1~\mathrm{Wb/m^2} = 10^4~\text{gauss} = 10^4~\text{G} \\ 1~\mathrm{Wb} &= 10^8~\text{maxwell} = 10^8~\text{lines} \\ 1~\mathrm{AT/m} &= 4\pi \times 10^{-3}~\text{Oe (oersted)} \\ 1~\mathrm{H} &= 1~\mathrm{Wb/A} = 1~\mathrm{V \cdot s/A} \end{aligned} \]

Problem-Solving Strategy

Step-by-Step Approach for Magnetic Circuit Analysis

Identify the circuit: Draw equivalent magnetic circuit
Define unknowns: Flux, MMF, field intensity, etc.
Calculate reluctances: For each section including air gaps
Apply Hopkinson's Law: \(\mathcal{F} = \phi\mathcal{R}\)
Use KCL for flux: Sum of fluxes at junction = 0
Use KVL for MMF: Sum of MMF drops = Total MMF
Check B-H curve: Verify operation in linear region
Account for fringing: Especially for air gaps
Calculate energy/force: If required
Verify reasonableness: Check against typical values

Quick Reference Summary

Most Important Equations

\[ \begin{aligned} &\text{MMF:} & \mathcal{F} &= NI \\ &\text{Field Intensity:} & H &= \frac{NI}{l} \\ &\text{Flux Density:} & B &= \mu_0\mu_r H = \frac{\phi}{A} \\ &\text{Reluctance:} & \mathcal{R} &= \frac{l}{\mu A} \\ &\text{Hopkinson's Law:} & \mathcal{F} &= \phi\mathcal{R} \\ &\text{Inductance:} & L &= \frac{\mu N^2 A}{l} = \frac{N^2}{\mathcal{R}} \\ &\text{Energy:} & W_m &= \frac{1}{2}LI^2 = \frac{1}{2}\mathcal{R}\phi^2 \\ &\text{Force:} & F &= \frac{B^2A}{2\mu_0} \\ &\text{Induced EMF:} & e &= -N\frac{d\phi}{dt} \end{aligned} \]