ELECTROMAGNETISM
& ELECTROMECHANICS
Principles · Waves · Diagrams · Formulas · Applications
01 — Foundation
Core Electromagnetic Principles
⚡ Coulomb’s Law
F = k·|q₁q₂| / r²
k = 8.99×10⁹ N·m²/C²
ε₀ = 8.85×10⁻¹² F/m
k = 8.99×10⁹ N·m²/C²
ε₀ = 8.85×10⁻¹² F/m
Electric force between point charges is proportional to their product and inversely proportional to distance squared.
🧲 Magnetic Field — Biot-Savart
dB = (μ₀/4π)·(I·dl×r̂)/r²
B = μ₀I / 2πr (long wire)
μ₀ = 4π×10⁻⁷ T·m/A
B = μ₀I / 2πr (long wire)
μ₀ = 4π×10⁻⁷ T·m/A
Magnetic field forms concentric circles around a current-carrying conductor.
⚡ Faraday’s Induction Law
ε = −dΦ_B / dt
Φ_B = ∫ B · dA
N turns: ε = −N·dΦ/dt
Φ_B = ∫ B · dA
N turns: ε = −N·dΦ/dt
A changing magnetic flux through a loop induces an EMF. The negative sign is Lenz’s law.
💫 Lorentz Force
F = q(E + v×B)
|F_mag| = qvB·sinθ
F⊥v → circular motion
|F_mag| = qvB·sinθ
F⊥v → circular motion
Force on a charged particle in electric and magnetic fields. Basis of all electric motors.
🔄 Ampère’s Law
∮ B·dl = μ₀·I_enc
B = μ₀nI (solenoid)
n = turns/meter
B = μ₀nI (solenoid)
n = turns/meter
The line integral of B around a closed loop equals μ₀ times the enclosed current.
⚡ Energy & Power
U_E = ½CV² = Q²/2C
U_B = ½LI²
P = VI = I²R = V²/R
U_B = ½LI²
P = VI = I²R = V²/R
Energy storage in electric (capacitor) and magnetic (inductor) fields.
02 — Unified Theory
Maxwell’s Equations — The Complete Framework
📐 Differential Form (Modern)
∇·E = ρ/ε₀ ← Gauss (Electric)
∇·B = 0 ← Gauss (Magnetic)
∇×E = −∂B/∂t ← Faraday
∇×B = μ₀J + μ₀ε₀∂E/∂t ← Ampère-Maxwell
∇·B = 0 ← Gauss (Magnetic)
∇×E = −∂B/∂t ← Faraday
∇×B = μ₀J + μ₀ε₀∂E/∂t ← Ampère-Maxwell
Insight: These 4 equations completely describe all classical electromagnetic phenomena including light, radio waves, and electric/magnetic forces.
📊 Integral Form
∮ E·dA = Q_enc/ε₀ ← Total charge enclosed
∮ B·dA = 0 ← No magnetic monopoles
∮ E·dl = −dΦ_B/dt ← Induction EMF
∮ B·dl = μ₀(I+ε₀dΦ_E/dt) ← Full Ampère
∮ B·dA = 0 ← No magnetic monopoles
∮ E·dl = −dΦ_B/dt ← Induction EMF
∮ B·dl = μ₀(I+ε₀dΦ_E/dt) ← Full Ampère
Wave Prediction: From ∇×(∇×E): ∇²E = μ₀ε₀ ∂²E/∂t²
Speed: c = 1/√(μ₀ε₀) ≈ 3×10⁸ m/s
Speed: c = 1/√(μ₀ε₀) ≈ 3×10⁸ m/s
03 — Wave Theory
Electromagnetic Waves — Spectrum & Waveforms
〰 Wave Structure (3D View)
E(z,t) = E₀·cos(kz−ωt)·ŷ
B(z,t) = B₀·cos(kz−ωt)·x̂
E₀/B₀ = c, k=2π/λ, ω=2πf
c = fλ = 3×10⁸ m/s
B(z,t) = B₀·cos(kz−ωt)·x̂
E₀/B₀ = c, k=2π/λ, ω=2πf
c = fλ = 3×10⁸ m/s
📊 EM Spectrum
c = fλ = 3×10⁸ m/s
E_photon = hf = hc/λ
h = 6.626×10⁻³⁴ J·s (Planck)
E_photon = hf = hc/λ
h = 6.626×10⁻³⁴ J·s (Planck)
〰 Waveform Visualizer
E(t) = E₀·sin(ωt) — Pure Sinusoidal Electromagnetic Wave
04 — Energy Conversion
Electromechanics — Motors, Generators & Actuators
🔄 DC Motor — Cross Section
τ = NIAB·sinθ (torque)
ε_back = BLv (back EMF)
P_mech = τ·ω = Tω
η = P_out/P_in × 100%
ε_back = BLv (back EMF)
P_mech = τ·ω = Tω
η = P_out/P_in × 100%
⚡ AC Generator — Operation
ε(t) = NBAω·sin(ωt)
ε_max = NBAω
V_rms = V_max/√2
f = ω/2π (Hz)
ε_max = NBAω
V_rms = V_max/√2
f = ω/2π (Hz)
🔌 Transformer — Core Principle & Equivalent Circuit
V₁/V₂ = N₁/N₂ = a (turns ratio)
I₁/I₂ = N₂/N₁ = 1/a
V₁I₁ = V₂I₂ (ideal: power conserved)
Z_in = a²·Z_L (impedance transform)
η = P_out/P_in = 1 − P_loss/P_in
I₁/I₂ = N₂/N₁ = 1/a
V₁I₁ = V₂I₂ (ideal: power conserved)
Z_in = a²·Z_L (impedance transform)
η = P_out/P_in = 1 − P_loss/P_in
Step-Up: N₂ > N₁ → V₂ > V₁, I₂ < I₁
Step-Down: N₂ < N₁ → V₂ < V₁, I₂ > I₁
Step-Down: N₂ < N₁ → V₂ < V₁, I₂ > I₁
Losses: Copper (I²R), Core (eddy currents, hysteresis), Flux leakage
05 — Signal Analysis
Graphs, Waveforms & Frequency Analysis
📊 Phasor & AC Circuit Analysis
Z = R + j(ωL − 1/ωC)
|Z| = √(R² + (X_L−X_C)²)
φ = arctan((X_L−X_C)/R)
Resonance: ω₀=1/√LC, Z=R
|Z| = √(R² + (X_L−X_C)²)
φ = arctan((X_L−X_C)/R)
Resonance: ω₀=1/√LC, Z=R
📈 Frequency Response (Bode Plot)
H(jω) = 1/(1+jωRC)
|H| = 1/√(1+(ωRC)²)
ω_c = 1/RC (cutoff freq)
−3dB at ω_c, −20dB/dec roll-off
|H| = 1/√(1+(ωRC)²)
ω_c = 1/RC (cutoff freq)
−3dB at ω_c, −20dB/dec roll-off
⚡ B-H Hysteresis Loop
B = μ₀(H + M) = μH
μᵣ = μ/μ₀ (relative permeability)
Area = energy/cycle loss
B_r = remanence, H_c = coercivity
μᵣ = μ/μ₀ (relative permeability)
Area = energy/cycle loss
B_r = remanence, H_c = coercivity
🌀 RLC Transient Response
L·d²i/dt² + R·di/dt + i/C = V/L
α = R/2L, ω₀ = 1/√LC
Underdamped: ω_d = √(ω₀²−α²)
Q = ω₀L/R (quality factor)
α = R/2L, ω₀ = 1/√LC
Underdamped: ω_d = √(ω₀²−α²)
Q = ω₀L/R (quality factor)
📋 Master Equation Reference
| Law / Principle | Formula | Variables | Application |
|---|---|---|---|
| Coulomb | F = kq₁q₂/r² | k=8.99×10⁹, q=charge, r=distance | Electrostatics, capacitor design |
| Ohm’s Law | V = IR | V=voltage, I=current, R=resistance | All circuit analysis |
| Faraday | ε = −N·dΦ/dt | N=turns, Φ=flux (Wb) | Generators, transformers |
| Lorentz | F = q(E + v×B) | v=velocity, B=magnetic field | Motors, particle accelerators |
| Maxwell Wave | c = 1/√(μ₀ε₀) | μ₀=4π×10⁻⁷, ε₀=8.85×10⁻¹² | EM wave propagation, optics |
| Biot-Savart | B = μ₀I/2πr | I=current, r=distance from wire | Inductor design, EMC |
| Transformer | V₁/V₂ = N₁/N₂ | N=turns, V=voltage | Power transmission, isolation |
| Skin Effect | δ = √(2ρ/ωμ) | δ=skin depth, ρ=resistivity | HF conductors, RF design |
06 — Machine Types
Electromechanical Machines — Types & Characteristics
⚡ Induction Motor
n_s = 120f/P (sync speed)
s = (n_s−n)/n_s
T ∝ sV²/(R₂²+s²X₂²)
P=poles, s=slip (0-1)
s = (n_s−n)/n_s
T ∝ sV²/(R₂²+s²X₂²)
P=poles, s=slip (0-1)
Most common industrial motor. Rotating magnetic field induces rotor currents via transformer action.
🔋 Linear Actuator
F = BIL = NI·(dL/dx)
F = μ₀N²I²A/(2g²)
g=air gap, A=pole area
x(t) = F/(m)·t²/2
F = μ₀N²I²A/(2g²)
g=air gap, A=pole area
x(t) = F/(m)·t²/2
Converts electrical energy directly to linear motion. Used in solenoid valves, speakers, and precision stages.
🔄 Stepper Motor
θ_s = 360°/(N_r·m)
Steps/rev = 360°/θ_s
N_r=rotor teeth, m=phases
T_hold = k·I (holding torque)
Steps/rev = 360°/θ_s
N_r=rotor teeth, m=phases
T_hold = k·I (holding torque)
Open-loop precision positioning. Each pulse = exact angular step. Used in 3D printers, CNC, robotics.
07 — Real World
Applications, Challenges & Engineering Solutions
🌐 Key Applications
POWER SYSTEMS
• AC generators (turbines)
• Power transformers (HV transmission)
• Smart grid control
• UPS & power conditioning
• Power transformers (HV transmission)
• Smart grid control
• UPS & power conditioning
COMMUNICATIONS
• Antenna design (dipole, patch)
• RF/microwave circuits
• Wireless power transfer
• Fiber-optic EM guides
• RF/microwave circuits
• Wireless power transfer
• Fiber-optic EM guides
MEDICAL
• MRI (gradient coils, RF coils)
• ECG/EEG signal acquisition
• Pacemaker induction charging
• Electrosurgical units
• ECG/EEG signal acquisition
• Pacemaker induction charging
• Electrosurgical units
TRANSPORT
• EV traction motors (PMSM)
• Maglev levitation & drive
• Regenerative braking
• Inductive lane charging
• Maglev levitation & drive
• Regenerative braking
• Inductive lane charging
⚠ Challenges & Solutions
Challenge: Core Losses
Eddy currents and hysteresis heat up magnetic cores, reducing efficiency.
✓ Solution: Laminated cores (silicon steel), amorphous metals, high-frequency ferrites
Eddy currents and hysteresis heat up magnetic cores, reducing efficiency.
✓ Solution: Laminated cores (silicon steel), amorphous metals, high-frequency ferrites
Challenge: EMI / EMC
Electromagnetic interference disrupts nearby circuits and violates regulations.
✓ Solution: Shielding (Faraday cage), filtering (LC), twisted-pair wiring, PCB layout rules
Electromagnetic interference disrupts nearby circuits and violates regulations.
✓ Solution: Shielding (Faraday cage), filtering (LC), twisted-pair wiring, PCB layout rules
Challenge: Skin Effect at HF
High-frequency currents concentrate at surface, increasing effective resistance.
✓ Solution: Litz wire, hollow conductors, surface plating (silver/gold)
High-frequency currents concentrate at surface, increasing effective resistance.
✓ Solution: Litz wire, hollow conductors, surface plating (silver/gold)
Challenge: Motor Thermal Management
I²R losses cause winding overtemperature and insulation degradation.
✓ Solution: Thermal design margins, liquid cooling, Class H insulation, thermal sensors
I²R losses cause winding overtemperature and insulation degradation.
✓ Solution: Thermal design margins, liquid cooling, Class H insulation, thermal sensors
🔧 Block Diagram — Modern Motor Drive System
🔢 Units
E → V/m (N/C)
B → Tesla (T)
H → A/m
Φ → Weber (Wb)
ε → Farad (F)
L → Henry (H)
ω → rad/s
Z → Ohm (Ω)
B → Tesla (T)
H → A/m
Φ → Weber (Wb)
ε → Farad (F)
L → Henry (H)
ω → rad/s
Z → Ohm (Ω)
⚡ Material Constants
μ₀ = 4π×10⁻⁷ H/m
ε₀ = 8.85×10⁻¹² F/m
c = 3×10⁸ m/s
k_e = 8.99×10⁹
σ_Cu = 5.8×10⁷ S/m
μᵣ(iron) ≈ 5000
B_sat(Fe) ≈ 2.0 T
ε₀ = 8.85×10⁻¹² F/m
c = 3×10⁸ m/s
k_e = 8.99×10⁹
σ_Cu = 5.8×10⁷ S/m
μᵣ(iron) ≈ 5000
B_sat(Fe) ≈ 2.0 T
🔄 AC Relationships
V_rms = V_pk/√2
P = V·I·cosφ
Q = V·I·sinφ
S = V·I (VA)
PF = P/S = cosφ
X_L = ωL
X_C = 1/ωC
P = V·I·cosφ
Q = V·I·sinφ
S = V·I (VA)
PF = P/S = cosφ
X_L = ωL
X_C = 1/ωC
📐 Wave Parameters
λ = c/f
k = 2π/λ
ω = 2πf
η = E/H (Ω)
η₀ = 377 Ω
S = E×H (W/m²)
dB = 20·log(V₂/V₁)
k = 2π/λ
ω = 2πf
η = E/H (Ω)
η₀ = 377 Ω
S = E×H (W/m²)
dB = 20·log(V₂/V₁)