Electromagnetics for Engineers
1. At a glance
Electromagnetics is the physics underlying every transformer, motor, antenna, transmission line, optical fiber, and printed-circuit-board trace operating into the GHz. In engineering practice, very few people solve Maxwell’s equations directly. Instead, the field theory is reduced to specialised abstractions and rules of thumb that handle each operating regime: magnetic-circuit theory for transformer and motor cores; lumped-element AC analysis for circuits whose physical dimensions are small compared to a wavelength; transmission-line theory for interconnects, cables, and PCB traces that aren’t electrically short; antenna theory for the radiating boundary case; and electromagnetic-compatibility (EMC) practice for managing unwanted coupling between everything else. Each of those abstractions is Maxwell’s equations specialised to one regime — and every one of them stops working at the regime boundary, which is where most engineering blunders happen (running a “circuit” at 5 GHz on a 0.5 m trace; treating a transformer core as linear past saturation; trusting a “ground plane” as zero impedance in a switching converter).
This note provides the engineering-grade Maxwell foundation and the most common application slices. It is not a physics derivation; it is a working reference for the practising electrical / electronic / RF engineer who has to size an inductor, route a 5 Gb/s differential pair, design a shielded enclosure for FCC Part 15 compliance, or specify a Wi-Fi link budget. The DC and AC circuit foundations live at [[Engineering/circuit-analysis]] and [[Engineering/ac-analysis-three-phase]]; the machinery built on top lives at [[Engineering/electric-motors]], [[Engineering/transformers-power-systems]], and [[Engineering/power-electronics]].
2. First principles
Maxwell’s equations
The four field equations, in differential and integral form (free-space + matter, SI units):
Gauss’s law for electricity:
- Differential: ∇·E = ρ/ε₀
- Integral: ∮ E·dA = Q_enclosed / ε₀
- Engineering reading: free electric charge is the source of the electric field. Sets capacitance (Q on the plates determines E between them).
Gauss’s law for magnetism:
- Differential: ∇·B = 0
- Integral: ∮ B·dA = 0
- Engineering reading: no magnetic monopoles exist; magnetic field lines always close on themselves. Therefore magnetic flux Φ entering any closed surface equals flux leaving — the basis of the magnetic-circuit conservation law.
Faraday’s law of induction:
- Differential: ∇×E = −∂B/∂t
- Integral: ∮ E·dℓ = −dΦ/dt (where Φ = ∫B·dA is magnetic flux through the loop)
- Engineering reading: a time-varying magnetic flux through a loop induces an EMF around the loop. The physical foundation of every transformer, generator, inductor, current-transformer probe, and inductive sensor.
Ampère–Maxwell law:
- Differential: ∇×B = µ₀ ( J + ε₀ ∂E/∂t )
- Integral: ∮ B·dℓ = µ₀ ( I_enc + ε₀ dΦ_E/dt )
- Engineering reading: current and time-varying electric flux both create a magnetic field. The ε₀ ∂E/∂t term is Maxwell’s displacement current — what makes a capacitor “conduct” AC, and what propagates EM waves through free space.
Auxiliary quantities
When matter is involved, two auxiliary field vectors simplify the bookkeeping:
- D = ε₀E + P = εE = ε₀ ε_r E — electric displacement, units C/m². The polarisation P absorbs the response of bound charges; ε_r is the relative permittivity (dielectric constant) of the material.
- H = B/µ₀ − M = B/µ = B/(µ₀ µ_r) — magnetic field intensity, units A/m. The magnetisation M absorbs the response of bound currents; µ_r is the relative permeability.
In terms of D and H, Gauss and Ampère re-write to use only free charges and free currents:
- ∇·D = ρ_free
- ∇×H = J_free + ∂D/∂t
This is the form used in engineering electromagnetics, because the engineer specifies the material (ε_r, µ_r) and the free sources (the wire carrying current I); the bound response of the material is hidden inside ε and µ.
Lorentz force
The force on a charge q moving at velocity v through fields:
F = q ( E + v × B )
For a wire of length ℓ carrying current I in a field B, integrating over the charges gives:
F = I ℓ × B
This is the basis of every motor, loudspeaker voice-coil actuator, galvanometer, MEMS magnetometer, and the steering of every cathode-ray tube and electron microscope. See [[Engineering/electric-motors]].
Continuity equation
Charge conservation: ∇·J = −∂ρ/∂t. Combined with Ampère–Maxwell, it forces the displacement-current term to exist (otherwise charge would not be conserved during transient capacitor charging).
The wave equation
Take the curl of Faraday, substitute Ampère–Maxwell, assume free space (J = 0, ρ = 0):
∇²E = µ₀ ε₀ ∂²E/∂t²
This is the classical wave equation with propagation speed c = 1/√(µ₀ε₀) ≈ 2.998 × 10⁸ m/s. The fact that this matched the measured speed of light (Maxwell, 1865) is the historical moment when “electromagnetism” and “optics” became one subject.
In a linear medium with permittivity ε = ε₀ε_r and permeability µ = µ₀µ_r the wave speed drops to v = c/√(ε_r µ_r) — the basis of refractive index n = √(ε_r µ_r) ≈ √ε_r for most non-magnetic dielectrics.
Energy density
EM energy is stored in the fields themselves:
- Electric: u_E = ½ ε₀ E² = ½ ε E² (in matter) — J/m³
- Magnetic: u_B = (1/2µ₀) B² = (1/2µ) B² (in matter) — J/m³
Integrated over the volume of a capacitor this gives ½ C V²; over the volume of an inductor this gives ½ L I². Field-domain and circuit-domain energy expressions are the same physics.
Boundary conditions at material interfaces
At any sharp interface between two media:
- E_tangential is continuous (unless an idealised surface EMF is present)
- D_normal jumps by the free surface-charge density σ_s: D₁ₙ − D₂ₙ = σ_s
- B_normal is continuous (always)
- H_tangential jumps by the surface current density K_s: H₁ₜ − H₂ₜ = K_s
These are the rules that make refraction work (Snell’s law follows from continuity of tangential E and H across a dielectric interface) and that make magnetic-circuit problems tractable (B_normal continuous → flux conserved at an air gap).
3. Practical math / design equations
Magnetic circuit analogy
A closed loop of magnetic flux in iron + air gap behaves like an electric DC circuit, with the following correspondences:
| Electric | Magnetic |
|---|---|
| EMF, V (volts) | MMF, ℱ = N·I (ampere-turns) |
| Current, I (A) | Flux, Φ (Wb) |
| Resistance, R (Ω) | Reluctance, ℛ (A-turns/Wb) |
| Conductivity σ | Permeability µ |
| Ohm’s law V = I·R | Hopkinson’s law ℱ = Φ·ℛ |
For a uniform-section path of length ℓ, area A, permeability µ:
ℛ = ℓ / (µ·A)
Reluctances in series add; in parallel they combine like resistors. For a magnetic circuit with an air gap the gap reluctance ℛ_gap = ℓ_gap / (µ₀·A) almost always dominates the total because µ_r of iron is 1000× to 10 000× larger than µ_r of air — this is why all practical inductors are gapped to set L and prevent saturation.
Inductance
For a coil of N turns wrapped on a magnetic path of total reluctance ℛ:
L = N² / ℛ
For an ideal toroid with mean path length ℓ, cross-section A, relative permeability µ_r:
L = µ_r µ₀ N² A / ℓ
Energy stored in inductors and capacitors
W_L = ½ L I² W_C = ½ C V²
Mutual inductance
Between two coupled coils with self-inductances L₁ and L₂:
M = k · √(L₁ · L₂) 0 ≤ k ≤ 1
k = 1 is the ideal-transformer limit (every flux line that links coil 1 also links coil 2). Real transformers achieve k ≈ 0.95–0.999; loosely coupled inductors and wireless-power coils run at k ≈ 0.1–0.5.
Transformer turns relation
For an ideal transformer (k = 1, no losses): V₁/V₂ = N₁/N₂ and I₁/I₂ = N₂/N₁. See [[Engineering/transformers-power-systems]].
Wave impedance of free space
η₀ = √(µ₀/ε₀) = 376.73 Ω ≈ 377 Ω
The ratio |E|/|H| for a plane wave in vacuum. In matter η = η₀ · √(µ_r/ε_r) ≈ η₀/√ε_r for non-magnetic dielectrics.
Transmission-line characteristic impedance
For a lossless transmission line with distributed inductance L’ (H/m) and capacitance C’ (F/m):
Z₀ = √(L’/C’) v_p = 1/√(L’·C’)
Common engineering values:
- 50 Ω: RF / lab equipment / GHz PCB.
- 75 Ω: video, CATV (lower loss for a given attenuation at a chosen size).
- 100 Ω differential: USB, LVDS, HDMI, DDR.
Reflection coefficient and VSWR
A transmission line of impedance Z₀ terminated in load Z_L sees a reflected wave:
Γ = (Z_L − Z₀) / (Z_L + Z₀)
with |Γ| = 0 for a matched load and |Γ| = 1 for an open or short. The voltage standing-wave ratio:
VSWR = (1 + |Γ|) / (1 − |Γ|)
A “good” RF match is VSWR < 1.5 (|Γ| < 0.2, return loss > 14 dB); a “great” match is VSWR < 1.2 (return loss > 21 dB).
Skin depth
In a good conductor (σ ≫ ωε), AC current crowds into a surface layer of depth:
δ = √( 2 / (ω µ σ) )
For copper at 60 Hz: δ ≈ 8.5 mm. At 1 GHz: δ ≈ 2 µm. This drives the use of Litz wire in switching-converter windings, the plating of waveguide and PCB copper, and the choice of dielectric-loss-tangent specs at GHz frequencies. See [[Engineering/ac-analysis-three-phase]] §4 for the same table.
Antenna gain and effective area
A reciprocal antenna has gain G and effective aperture A_eff related by:
G = 4π A_eff / λ² A_eff = G λ² / (4π)
A small antenna (dimension < λ) has G of order unity (dipole G = 1.64 ≈ 2.15 dBi); a large dish at λ = 3 cm has G of 10⁴ to 10⁶.
Friis transmission formula
The received power for a free-space line-of-sight link:
P_r = P_t · G_t · G_r · ( λ / (4π·d) )²
In dB:
P_r[dBm] = P_t[dBm] + G_t[dBi] + G_r[dBi] − 20·log₁₀(d) − 20·log₁₀(f) − 32.44
with d in km and f in MHz (the constant absorbs the 4π/c units). The 20·log(d) + 20·log(f) term is the free-space path loss (FSPL) — universally tabulated and the centrepiece of every link budget.
Worked example 1 — toroidal inductor
A ferrite toroid has mean magnetic path length ℓ = 100 mm = 0.1 m, cross-section A = 100 mm² = 1 × 10⁻⁴ m², relative permeability µ_r = 1500. Wound with N = 100 turns. Find reluctance, inductance, and stored energy at I = 1 A.
- Reluctance ℛ = ℓ / (µ_r µ₀ A) = 0.1 / (1500 · 4π × 10⁻⁷ · 1 × 10⁻⁴) = 0.1 / (1.885 × 10⁻⁷) = 5.305 × 10⁵ A-turns/Wb.
- Inductance L = N² / ℛ = 10⁴ / 5.305 × 10⁵ = 0.01885 H = 18.85 mH.
- Cross-check with the direct formula: L = µ_r µ₀ N² A / ℓ = 1500 · 4π × 10⁻⁷ · 10⁴ · 10⁻⁴ / 0.1 = 1.885 × 10⁻² H ✓.
- Energy at I = 1 A: W = ½ L I² = 0.5 · 0.01885 · 1² = 9.43 mJ.
- Flux density check: B = µ_r µ₀ N I / ℓ = 1500 · 4π × 10⁻⁷ · 100 · 1 / 0.1 = 1.885 T — above the typical ferrite saturation flux density of ~0.4 T. In practice this core would saturate hard at less than ~200 mA. Either reduce turns, increase area, gap the core, or pick a different material. Saturation is a first-order design constraint; the linear inductance formula above is valid only while B is below the knee of the B-H curve.
Worked example 2 — PCB microstrip 50 Ω trace
A microstrip trace on FR-4 (ε_r ≈ 4.4) with dielectric thickness H = 0.2 mm and trace width W. Find W for Z₀ = 50 Ω, and discuss when the trace must be treated as a transmission line versus a lumped element.
The standard Wheeler / Hammerstad approximation for microstrip (W/H > 1):
Z₀ ≈ (120π / √ε_eff) / ( W/H + 1.393 + 0.667·ln(W/H + 1.444) )
with effective permittivity (some field lines run in air, some in dielectric):
ε_eff ≈ (ε_r + 1)/2 + ((ε_r − 1)/2) · (1 + 12 H/W)^(−1/2)
Iterating: try W = 0.36 mm → W/H = 1.8 → ε_eff ≈ 3.33 → denominator ≈ 1.8 + 1.393 + 0.667·ln(3.244) = 3.193 + 0.785 = 3.978 → Z₀ ≈ (120π / √3.33) / 3.978 = 206.5 / 3.978 ≈ 51.9 Ω — close enough to 50 Ω that an EM solver or a curve-fitter (Saturn PCB Toolkit, KiCad’s calculator, Polar Si9000) is needed for production accuracy.
Critical length. Phase velocity v_p = c / √ε_eff = 3 × 10⁸ / √3.33 ≈ 1.64 × 10⁸ m/s. Wavelength λ = v_p / f.
| Frequency | λ on this microstrip | λ/10 (transmission-line threshold) |
|---|---|---|
| 100 MHz | 1.64 m | 164 mm |
| 1 GHz | 164 mm | 16 mm |
| 5 GHz | 33 mm | 3.3 mm |
A standard ~100 mm trace acts as a lumped wire at 100 MHz (well under λ/10) but is decidedly distributed at 1 GHz (about 6·λ/10) — controlled impedance, length matching, and termination matter. At 5 GHz even a 10 mm trace is electrically long.
Worked example 3 — 2.4 GHz Wi-Fi link budget
Transmit power P_t = 20 dBm (100 mW), G_t = G_r = 5 dBi, free-space distance d = 30 m, receiver sensitivity S = −80 dBm. Find received power and link margin.
- λ = c/f = 3 × 10⁸ / 2.4 × 10⁹ = 0.125 m.
- FSPL = 20·log₁₀(4π·d/λ) = 20·log₁₀(4π · 30 / 0.125) = 20·log₁₀(3015) = 69.6 dB.
- P_r = P_t + G_t + G_r − FSPL = 20 + 5 + 5 − 69.6 = −39.6 dBm.
- Link margin = P_r − S = −39.6 − (−80) = 40.4 dB.
40 dB of margin is generous, which is realistic for 30 m line-of-sight — most of it is consumed by indoor multipath, wall attenuation (drywall ~3 dB, brick 10–15 dB per wall, reinforced concrete 20–30 dB), interference, and fading. Real-world Wi-Fi at 30 m through two interior walls might see only 5–10 dB margin, which is why repeaters and mesh systems exist.
4. Reference data
Permittivity (relative) of common dielectrics
| Material | ε_r | Frequency dependence | Engineering use |
|---|---|---|---|
| Vacuum | 1 (exact, ε₀ = 8.854 × 10⁻¹² F/m) | — | reference |
| Air (1 atm) | 1.0006 | flat to ~100 GHz | almost always treated as 1 |
| PTFE (Teflon) | 2.1 | very flat, low loss | high-frequency coax, RF connectors |
| Polyethylene | 2.25 | flat | coax dielectric |
| Polystyrene | 2.5 | flat | low-loss caps |
| Rogers RO4003C | 3.38 | flat to 20 GHz | high-frequency PCB |
| Rogers RO4350B | 3.66 | flat to 20 GHz | high-frequency PCB |
| FR-4 (typical) | 4.0–4.7 (≈ 4.4 nominal) | drops slightly with f, tan δ ≈ 0.02 | the workhorse PCB material below ~6 GHz |
| Glass | 4–7 | flat | windows in waveguides |
| Alumina (Al₂O₃ 99.5%) | 9.8 | flat to 30 GHz | hybrid microwave substrates |
| Silicon (intrinsic) | 11.7 | flat | semiconductor substrate |
| GaAs | 12.9 | flat | MMIC substrate |
| Water (20 °C) | 80 | drops to ~6 at 30 GHz | absorbs microwaves (microwave oven physics) |
| BaTiO₃ (barium titanate) | 1 000–10 000 | strongly T- and V-dependent | Class-II / Class-III MLCC capacitors (X7R, Y5V) |
For FR-4 the variation of ε_r with frequency is significant: ε_r ≈ 4.7 at 1 kHz, 4.4 at 1 MHz, 4.1 at 5 GHz. The loss tangent tan δ grows from ~0.01 at 1 MHz to ~0.03 at 10 GHz — which is why high-frequency PCBs migrate to Rogers or Megtron above ~6 GHz.
Permeability (relative) of magnetic materials
| Material | µ_r | B_sat (T) | Use |
|---|---|---|---|
| Vacuum / air | 1 (exact, µ₀ = 4π × 10⁻⁷ H/m exact) | — | reference |
| Aluminium, copper, gold | ≈ 1 | — | non-magnetic |
| Pure iron (annealed) | 200 000 | 2.2 | reference for “soft” iron |
| Low-carbon steel | 100–4 000 | 1.6–2.0 | construction, motor frames |
| Silicon steel (3 % Si, grain-oriented) | 4 000–10 000 | 1.8–2.0 | transformer + motor laminations |
| 50 % Ni–Fe (permalloy) | 50 000–100 000 | 1.5 | sensitive instruments |
| 80 % Ni–Fe (mu-metal) | 100 000 (low-field) | 0.7 | low-frequency magnetic shielding |
| MnZn ferrite (e.g. N87, 3C90) | 1 500–6 000 | 0.4 | switching converters 10 kHz–1 MHz |
| NiZn ferrite | 100–1 500 | 0.3 | RF chokes, EMI suppression, 1 MHz–GHz |
| Powdered iron (Micrometals) | 10–100 | 1.4 (distributed gap) | DC choke, less prone to sudden saturation |
| Amorphous (Metglas) | 10 000–50 000 | 1.5 | low-loss distribution transformers |
| Nanocrystalline (Vitroperm) | 50 000–150 000 | 1.2 | common-mode chokes, current sensors |
| NdFeB rare-earth (magnetised) | 1.05 (recoil) | 1.4 (remanence) | PMSM rotors, voice coils |
Note that µ_r of a soft magnetic material is operating-point dependent — pulled from a B-H curve at a given DC bias, AC swing, frequency, and temperature. Catalog values are typically maximum or initial; design must use the value at the actual operating B.
Conductivity (σ, S/m at 20 °C)
| Material | σ (S/m) | Use |
|---|---|---|
| Silver | 6.30 × 10⁷ | plated for highest-Q resonators |
| Copper (annealed, IACS reference) | 5.96 × 10⁷ | universal wire and PCB |
| Gold | 4.10 × 10⁷ | corrosion-resistant connector plating |
| Aluminium | 3.50 × 10⁷ | overhead transmission, motor windings |
| Brass | 1.5 × 10⁷ | connector bodies |
| Lead | 4.6 × 10⁶ | shielding (X-ray; not RF) |
| Stainless 304 | 1.4 × 10⁶ | enclosures (mediocre conductor, decent shield by absorption) |
| Carbon (graphite) | 1 × 10⁵ | brushes, EMI gaskets |
| Seawater | 4–5 | marine antennas, ground returns |
| FR-4 (volume) | ~10⁻¹⁴ | insulator |
Skin depth in copper
| Frequency | δ (Cu, 20 °C) | Engineering implication |
|---|---|---|
| 50 Hz | 9.3 mm | uniform conduction through solid bars |
| 60 Hz | 8.5 mm | uniform conduction through solid bars |
| 1 kHz | 2.1 mm | start of Litz benefit in transformers |
| 100 kHz | 0.21 mm | switching converters use Litz / foil |
| 1 MHz | 66 µm | RF conductors plated, not bulk |
| 10 MHz | 21 µm | tubular conductors as efficient as solid |
| 1 GHz | 2.1 µm | PCB-copper surface roughness dominates loss |
| 10 GHz | 0.66 µm | ”rolled” copper > “electrodeposited” for low loss |
| 30 GHz | 0.4 µm | mm-wave substrates: smooth-Cu Rogers / PTFE |
Free-space wavelength
| Frequency | λ in vacuum | Note |
|---|---|---|
| 50 Hz | 6 000 km | power-line frequency |
| 60 Hz | 5 000 km | power-line frequency |
| 1 MHz | 300 m | AM broadcast |
| 100 MHz | 3.0 m | FM broadcast |
| 1 GHz | 30 cm | cellular, GPS L1 (1.575 GHz, λ ≈ 19 cm) |
| 2.4 GHz | 12.5 cm | Wi-Fi 2.4, Bluetooth |
| 5 GHz | 6.0 cm | Wi-Fi 5/6 |
| 28 GHz | 1.07 cm | 5G mm-wave |
| 100 GHz | 3.0 mm | automotive radar, W-band |
| 1 THz | 0.30 mm | submillimetre / terahertz imaging |
Common transmission-line impedances
| System | Z₀ | Where |
|---|---|---|
| Lab/RF coax (RG-58, RG-213, SMA, BNC, N-type) | 50 Ω | universal RF test, cellular, radar, GHz PCB |
| Video / CATV coax (RG-6, RG-59, F-type) | 75 Ω | video, satellite, cable TV |
| PCB single-ended high-speed | 50 Ω | DDR address, ECL, SerDes single-ended |
| PCB differential pair | 100 Ω (USB, LVDS, HDMI), 90 Ω (USB 3, PCIe) | high-speed digital |
| Cat-5e/6/6A Ethernet differential | 100 Ω | structured cabling |
| Telephone twisted pair | 600 Ω (audio), 100 Ω (DSL) | varies by service |
| Antenna feed (older HF amateur, TV) | 300 Ω (ribbon), 450 Ω (ladder line) | low-loss balanced lines |
| Free-space “antenna” (wave impedance) | 377 Ω | matching target for antennas |
5p. Theory
Vector calculus identities. Two identities are needed constantly:
- ∇×(∇×A) = ∇(∇·A) − ∇²A — used to derive the wave equation from Faraday + Ampère.
- ∇·(∇×A) = 0 — automatically satisfies Gauss’s law for magnetism if B = ∇×A (the magnetic vector potential).
The scalar potential V and vector potential A unify electrostatics and magnetostatics:
- E = −∇V − ∂A/∂t
- B = ∇×A
Gauge freedom (Lorenz gauge ∇·A + (1/c²)∂V/∂t = 0) decouples the potential equations into the familiar inhomogeneous wave equations driven by ρ and J. This is the natural language for radiation problems.
Poynting vector and power flow. Energy flow density in an EM field:
S = E × H (W/m²)
Time-averaged for sinusoids: ⟨S⟩ = ½ Re{ E × H* }. Integrating ⟨S⟩ over a closed surface gives the average power crossing that surface. This is the physical mechanism by which a wire carries power: the energy travels in the dielectric around the conductors, not through the metal — a useful mental model for understanding PCB return-current paths and the fields outside coax.
Near-field vs far-field. Close to an antenna of largest dimension D, fields are dominated by reactive (storage) components that fall as 1/r² and 1/r³. The boundary into the radiating far-field is conventionally:
d_ff ≈ 2 D² / λ
For a 0.1 m antenna at 2.4 GHz (λ = 0.125 m), d_ff ≈ 0.16 m — almost any practical measurement is far-field. For a 30 m dish at 1 GHz, d_ff ≈ 6 km — antenna ranges and OTA test facilities must be built accordingly, or compact-range / near-field-to-far-field transformation must be used.
Guided waves in waveguides and optical fibers. A hollow conductor (rectangular or circular waveguide) supports TE and TM modes above a cutoff frequency f_c (set by the largest cross-sectional dimension). Below f_c the wave is evanescent; above, it propagates. Dominant mode in rectangular WR-90 (X-band, 8.2–12.4 GHz) is TE₁₀ with f_c = c/(2a) where a is the broad-wall width.
Optical fibers guide by total internal reflection (single-mode SMF-28 has core/cladding indices that give numerical aperture NA ≈ 0.14 and V parameter V < 2.405 for f at 1310/1550 nm). The physics is the same wave equation with different boundary conditions; the engineering details live at [[Engineering/photonics]].
Quasi-static regimes. When all dimensions are ≪ λ, ∂D/∂t and ∂B/∂t are small relative to J, and:
- Magneto-quasi-static (MQS): keep ∂B/∂t (Faraday’s law for inductors, transformers, eddy currents). Drop ∂D/∂t.
- Electro-quasi-static (EQS): keep ∂D/∂t (capacitors). Drop ∂B/∂t.
These are the regimes inside which lumped-element circuit theory is exact. The transition out of quasi-static is precisely where transmission-line theory takes over. Knowing which regime you are in is the most useful single piece of engineering EM judgement.
6p. Application
Magnetic-circuit design
Used for transformers, motors, relays, solenoids, magnetic recording heads, MRI gradient coils. Workflow:
- Compute required flux Φ from voltage and time (Faraday: V = N dΦ/dt → Φ_peak ≈ V/(4.44 N f) for sinusoidal RMS V).
- Pick a core area A so B_peak = Φ/A stays below saturation (silicon-steel transformers run at B_peak ≈ 1.4–1.7 T; ferrites at 0.2–0.3 T; powdered iron at ~1 T).
- Compute reluctance of the path (gap reluctance dominates if you have a gap).
- Solve for N from L = N²/ℛ or V/N = (4.44 f Φ).
- Size the wire for the current (current density typically 2–6 A/mm² for naturally air-cooled magnetics; 4 A/mm² is a good first cut).
- Check the window-fill factor (copper area / available window) — typically 0.3–0.5 for round magnet wire.
- Compute core loss (Steinmetz: P_v = k · f^α · B^β, with α ≈ 1.5, β ≈ 2.5 for ferrites; catalogue data preferred).
Inductor topology selection
- Air-core (no magnetic material). No saturation, no hysteresis loss, very linear, very stable with temperature. Low inductance per turn (large turn counts needed). Stray field radiates strongly — needs to be far from sensitive circuits or enclosed. Used in RF tuned circuits, anti-aliasing filters, current-viewing resistors / Rogowski coils, MRI gradient coils.
- Ferrite-core (MnZn or NiZn). High µ_r → high L per turn. Saturation at ~0.3–0.4 T limits DC bias; gaps avoid hard saturation. Low conductivity (ferrite is a ceramic) → low eddy-current loss; usable to 1 MHz (MnZn) or 1 GHz (NiZn). The standard for switching-converter transformers and chokes.
- Powdered iron (distributed gap). Iron particles bonded with insulating binder; inherent distributed gap gives soft saturation. Higher loss than ferrite above ~200 kHz but tolerates higher B. Common in PFC chokes and output filters.
- Silicon-steel laminations. 0.2–0.5 mm laminations of grain-oriented Fe-Si separated by varnish; very low loss at 50/60 Hz, high B_sat ≈ 1.8 T. The standard for line-frequency transformers and motors. Lamination thickness < skin depth keeps eddy currents bounded — see
[[Engineering/materials-steel]]. - Toroidal geometry. Closed magnetic path → no air gap → low stray field → low EMI radiation. Per ampere-turn the toroid is the most efficient and electromagnetically the cleanest geometry. Awkward to wind by hand; cores are wound on automatic toroid winders or designed for split assembly.
EMI / EMC — the three-element model
Every interference problem has three elements; address any one to break the chain:
- Source. Switching-edge dV/dt and dI/dt are the universal culprits — every nanosecond-fast edge in a modern SMPS, motor drive, or digital line generates a continuous spectrum extending to several hundred MHz (rule of thumb: −20 dB/dec roll-off above f_BW = 1/(π·t_rise)). Common-mode source impedance (charge on heatsinks, leakage capacitance across switching transformer windings) couples the noise into wires and chassis.
- Coupling path. Conducted (through cables, both differential-mode and common-mode); radiated (electric coupling at high impedance, magnetic coupling at low impedance, plane-wave coupling in the far field).
- Victim. Sensitive analog inputs, RF receivers, control logic with marginal noise margin.
Mitigation hierarchy, in priority order:
- Source-side: soft switching, snubbers, spread-spectrum clocking, slew-rate control on digital outputs, common-mode chokes on motor leads, careful PCB layout around switching nodes (small dV/dt loops, small dI/dt loops).
- Coupling-path: shielded cables, twisted pairs (cancel low-frequency H-field pickup), differential signalling (rejects common-mode), gaskets at enclosure seams, filter capacitors at I/O penetrations, ferrite beads on cables.
- Victim-side: input filtering, common-mode rejection in differential amplifiers, signal-integrity techniques on high-speed lines.
Compliance standards:
- CISPR 22 / CISPR 32 / EN 55032 — emissions limits for ITE / multimedia at the cable port (conducted, 150 kHz–30 MHz) and via OATS / SAC measurement (radiated, 30 MHz–6 GHz). Class A for industrial, Class B (stricter) for residential.
- FCC Part 15 Subpart B — US equivalent for unintentional radiators; Class A vs Class B same split.
- CISPR 11 / EN 55011 — ISM (industrial/scientific/medical) equipment.
- IEC 61000-4-x — immunity test methods (ESD 61000-4-2, RF 4-3, EFT 4-4, surge 4-5, conducted-RF 4-6, magnetic field 4-8, voltage dips 4-11).
- MIL-STD-461 — US military emissions and immunity, much stricter envelope, RE102/CE102 are the headline tests.
Shielding
A shield reduces field penetration through three mechanisms (in dB):
SE = R + A + B
- Reflection loss R dominates for thin, highly-conductive shields in the far field. R ≈ 168 + 10·log₁₀(σ_r / (µ_r · f)) dB for plane waves (σ_r relative to Cu).
- Absorption loss A dominates for thick or magnetically permeable shields. A ≈ 131·t·√(σ_r·µ_r·f) dB with t in metres. Both σ and µ help.
- Multiple-reflection correction B ≈ small for A > 10 dB; matters only for thin shields at low frequency.
Practical consequences:
- High-frequency E-field shielding: any decent conductor works; aluminium foil is plenty above ~1 MHz.
- Low-frequency H-field shielding: need µ_r → use mu-metal or co-netic enclosures, and anneal after forming (working mu-metal destroys µ_r).
- Joints and apertures dominate real shield performance — a perfect Faraday cage with a 50 mm slot leaks horribly above f = c/(2·50 mm) = 3 GHz. Conductive gaskets, finger-stock, screw spacing all matter.
IEEE 299 standardises shielding-effectiveness measurement for enclosures from 9 kHz to 18 GHz.
Antennas — practical types
- Half-wave dipole. Resonant at f₀ where physical length ≈ 0.48·λ (5% velocity-factor correction). Z_in ≈ 73 Ω, omnidirectional in azimuth, G = 2.15 dBi. The reference antenna.
- Quarter-wave monopole over ground plane. Length ≈ λ/4, Z_in ≈ 36 Ω, image-theory gain G = 5.2 dBi (3 dB over dipole because radiation is into half-space). Whip antennas, car AM/FM, GPS patches over ground.
- Patch (microstrip) antenna. A copper rectangle ~λ/2 on a thin dielectric over a ground plane. Bandwidth narrow (~1–5%), G = 6–9 dBi, very PCB-friendly. The default for 2.4 / 5 GHz Wi-Fi modules, GPS, RFID.
- Helical (axial mode). Several turns of wire pitched ~λ/4 around a ground plane. Circularly polarised, G = 10–15 dBi, wide bandwidth. Standard for satellite uplinks (Iridium, Inmarsat hand-helds).
- Yagi-Uda. Driven dipole plus reflector and director(s). G = 7–15 dBi, narrow bandwidth, linearly polarised. The classic rooftop TV antenna and amateur HF/VHF beam.
- Parabolic dish. G ≈ η·(4πA/λ²) with η ≈ 0.55–0.70 aperture efficiency. G = 30–50 dBi for satellite TV and microwave point-to-point links.
- Phased array. N elements with independent phase/amplitude control. Electronic beam steering, multi-beam, MIMO. Now dominant at mm-wave (5G, automotive radar) where the array fits in a thumbnail.
Polarisation matching matters: a cross-polarised mismatch is ~20 dB of loss. Circular-polarised antennas (helix, crossed dipoles, sequentially-rotated patch arrays) tolerate any linear orientation with a fixed 3 dB loss.
Eddy currents — loss and braking
A time-varying flux in a conductor induces circulating currents (Faraday’s law). For a lamination of thickness t and resistivity ρ, with uniform AC flux density B_peak at frequency f:
P_eddy ∝ ( B_peak · f · t )² / ρ W/m³
Mitigation:
- Laminate the iron — every doubled lamination count reduces eddy loss by 4×.
- Use silicon steel (3 % Si) — silicon raises ρ by ~4× over plain iron.
- Above ~1 kHz, switch to ferrite (σ ≈ 1 S/m, eight orders of magnitude less than iron).
- For very high f, use thin-film magnetics (sputtered NiFe on silicon) or stay non-magnetic.
Eddy currents also do useful work in eddy-current brakes (no contact, no wear, used in trains and roller-coasters), induction cooktops (high-frequency field heats the ferromagnetic pan), and induction heating for through-hardening. Skin effect in AC conductors is a closely-related eddy-current phenomenon — the current self-shields toward the surface.
PCB layout at high speed
The transition from low-speed PCB (where every trace is “just a wire”) to high-speed (where every trace is a transmission line) is when t_rise of any source ≤ ~2 × (trace propagation delay). For an FR-4 microstrip at v_p ≈ 1.6 × 10⁸ m/s, a 100 mm trace has a one-way delay of ~600 ps; any signal with t_rise < ~1.2 ns must be treated as distributed.
Rules of practice:
- Controlled impedance — define Z₀ (typically 50 Ω single-ended, 100 Ω differential) and stack-up to match.
- Ground / return planes — every signal trace needs a return path right below it; splits in the plane force return current to detour, radiating EMI.
- Length matching for parallel buses (DDR, parallel ADC) — typically within λ/10 of the highest harmonic of interest.
- Termination — series at the source for point-to-point CMOS; parallel at the receiver for terminated lines (ECL, LVDS); differential 100 Ω at the receiver for USB/LVDS/HDMI.
- Vias introduce inductance and impedance discontinuities; minimise count on high-speed nets and back-drill stubs > λ/20.
- Decoupling networks — multiple capacitor values (100 nF + 10 nF + 1 nF + small reservoir) across the relevant frequency range. The PCB plane itself has inductance and resonance modes — see PI (power integrity) tooling.
7p. Edge cases & assumptions
Non-linear magnetic materials. Permeability µ = dB/dH is not constant — the B-H curve has an initial low-field slope (small-signal permeability), rises to a peak (incremental permeability), and falls through the knee into saturation where dB/dH → µ₀. Beyond saturation an inductor stops being an inductor — current rises rapidly with no further inductive impedance. Add to that hysteresis — the loop encloses an energy area that is dissipated as heat each cycle. Saturation flux density B_sat and the loop area (loss density) at the design B_peak, f, and temperature are the design-driving parameters. Always look at the manufacturer’s B-H curve at the operating point, not the catalog “µ_r” headline.
Frequency-dependent ε_r and tan δ. FR-4 is a glass-reinforced epoxy composite, not a uniform dielectric; ε_r varies with glass weave orientation (the “fibre-weave effect” causes per-trace impedance variation in DDR systems), and tan δ rises with frequency. For anything above ~6 GHz the variation is enough that controlled-impedance tolerances cannot be met without a TDR-verified stack-up and tighter substrate spec (Rogers RO4350B, Megtron 6, Isola I-Tera). Pre-preg vs core dielectric properties also differ slightly and matter for stripline.
Skin and proximity effects in AC conductors. Skin effect crowds current to the surface of a single conductor; proximity effect further crowds current toward (or away from) adjacent conductors carrying parallel current. In a multi-layer transformer winding, proximity effect can multiply the AC resistance by 5–10× over the DC value. Mitigation: interleave primary and secondary layers, use Litz wire, run at lower current density, design with Dowell’s analytical model or 2D FEM (FEMM, Ansys Maxwell 2D).
Near-field vs far-field confusion. Friis transmission and free-space path loss are far-field formulas; they fail inside d_ff. Cell-phone SAR testing, RFID, wireless power, and any reactive coupling are near-field problems requiring different analysis (the dipole-near-field formulas, full-wave simulation, or impedance/coupling matrices).
Non-ideal grounds. A “ground plane” has impedance — finite skin depth, finite spreading resistance, induced loop currents from displacement and conduction currents elsewhere. At GHz frequencies even a solid copper plane shows mV-level “ground bounce” between widely-separated points, which is enough to disturb sensitive analog or trigger flip-flops. The ground concept becomes a network analysis problem; PDN (power-distribution-network) impedance vs frequency is a design output, not an assumption.
EMI is statistical. Worst-case compliance is set by a fast resonance / antenna coincidence on a specific harmonic of a specific clock. The job of the EMC engineer is to push every plausible resonance below the limit; the job of the test lab is to find the worst-case orientation, cable layout, and frequency. Margin of 6 dB over the limit is a typical design target.
PCB crosstalk. Adjacent traces couple capacitively (low-impedance victim) and inductively (high-impedance victim), with the two contributions partly cancelling on the forward signal and adding on the reverse (near-end crosstalk, NEXT). Differential signalling, increased trace spacing (the “3W rule” — centre-to-centre ≥ 3× trace width), and a solid reference plane between layers are standard mitigations.
Saturation makes magnetic shields fail. A mu-metal shield around a sensitive instrument is wonderful until an external DC field saturates it; then µ_r collapses and shielding effectiveness drops by 60 dB or more. Always check the shield does not saturate at the worst-case external field — particularly relevant for instruments operated near MRI magnets, current-carrying buses, or ferromagnetic walls.
8p. Tools & software
Full-wave 3D EM solvers (the heavy artillery for antennas, packages, complex resonators):
- ANSYS HFSS — frequency-domain FEM, the industry default for antennas, RF packaging, signal integrity.
- CST Studio Suite (Dassault) — time-domain FIT (close cousin of FDTD) plus frequency-domain solver; strong for broadband and transient problems (ESD, EMI).
- Keysight ADS with Momentum — planar-3D for RF / microwave PCB.
- COMSOL Multiphysics with RF Module — coupled EM + thermal + mechanical; general-purpose.
Specialised PCB SI / PI / EMC tools:
- Cadence Sigrity / Allegro — power integrity, signal integrity, channel analysis.
- ANSYS SIwave — planar resonance, PDN impedance, near-field EMI.
- Polar Si9000 / Si8000 — controlled-impedance stack-up calculator (the de facto reference).
- Hyperlynx (Siemens) — channel simulation, IBIS-AMI.
2D field solvers (cross-section problems like inductor windings, microstrip impedance):
- FEMM (free) — magnetostatic / electrostatic 2D FEM. Goes a long way for transformer and inductor design.
- ANSYS Maxwell 2D — production-grade 2D motor / transformer.
- QucsStudio — RF schematic + planar simulation.
Circuit-level tools that include EM models:
- LTspice — free, fast, with inductor / transformer / coupling models adequate for most converter design.
- PSpice / Cadence OrCAD — commercial, broader device library.
- PLECS — power electronics specialist with averaged + switched models.
Antenna-specific tools:
- NEC2 / NEC4 — method-of-moments wire-antenna code, FOSS lineage from Lawrence Livermore. NEC4 export-controlled but readily licensed.
- 4NEC2 / EZNEC — graphical front-ends for NEC.
- xnec2c — Linux NEC front-end.
- openEMS — open-source FDTD, capable for moderate-scale problems with no licence cost.
EMC test instruments and predictive tools:
- Spectrum analyser with EMC pre-amp — pre-compliance scans in the lab.
- LISN (line impedance stabilisation network) — defines source impedance for conducted-emission tests.
- Near-field probes (Beehive Electronics, Aaronia) — locating noise sources on a PCB or cable bundle.
- Tekbox / EMC PreCompliance kits — affordable LISN + probes + antennas for in-house pre-compliance.
Free / open-source numerical EM:
- openEMS (FDTD), FEniCS (general FEM), FreeFEM++ (general FEM), Elmer (multiphysics FEM), gprMax (FDTD for ground-penetrating-radar style problems), scikit-rf (Python network-parameter analysis), MEEP (FDTD, from MIT, optics-flavoured).
11. Cross-references
[[Engineering/circuit-analysis]]— the lumped-element DC and AC circuit foundation; quasi-static EM is what makes that foundation valid.[[Engineering/ac-analysis-three-phase]]— phasor and frequency-domain machinery used throughout this note.[[Engineering/electric-motors]]— Lorentz force, magnetic-circuit reluctance, and rotating-field theory applied to actuators.[[Engineering/transformers-power-systems]]— Faraday’s law turned into a power-delivery component; per-unit, leakage inductance, magnetising current.[[Engineering/semiconductor-devices]]— drift, diffusion, and depletion-region physics live inside the same Gauss + continuity equations.[[Engineering/op-amps]]— frequency response and stability margins are H(jω) descriptions of a small-signal EM problem.[[Engineering/pcb-design]]— controlled impedance, stack-up, return paths, decoupling networks.[[Engineering/power-electronics]]— switching transients, snubbers, common-mode chokes, EMC of converters.[[Engineering/rf-design]]— distributed-parameter analysis, Smith chart, S-parameters.[[Engineering/materials-steel]]— silicon-steel laminations, grain orientation, hysteresis loops.[[Robotics/sensors-pose-motion]]— Hall-effect, magnetometer, current-sensor, and inductive proximity-sensor physics.[[Robotics/power-systems]]— battery → inverter → motor signal chain; EMC challenges in mobile robotics.[[Languages/Tier3/network-protocol-dsls]]— the physical-layer EM constraints (eye diagrams, jitter, BER) that protocol stacks rest on.[[Languages/Tier3/energy-power]]— utility-scale EM where transmission lines become hundreds of km long and earth-return currents matter.
12. Citations
- Griffiths, D. J. (2017). Introduction to Electrodynamics (5th ed.). Cambridge University Press. The standard undergraduate text; clean derivations of Maxwell, waves, and radiation.
- Cheng, D. K. (1989). Field and Wave Electromagnetics (2nd ed.). Addison-Wesley. Engineering-flavoured, with explicit emphasis on guided waves and antennas.
- Hayt, W. H. & Buck, J. A. (2018). Engineering Electromagnetics (9th ed.). McGraw-Hill. The most widely-used engineering EM textbook in the US.
- Pozar, D. M. (2011). Microwave Engineering (4th ed.). Wiley. The reference for transmission lines, S-parameters, matching networks, and microwave circuits.
- Balanis, C. A. (2016). Antenna Theory: Analysis and Design (4th ed.). Wiley. The standard antenna text.
- Stutzman, W. L. & Thiele, G. A. (2012). Antenna Theory and Design (3rd ed.). Wiley. Companion / alternative to Balanis with stronger practical emphasis.
- Paul, C. R. (2006). Introduction to Electromagnetic Compatibility (2nd ed.). Wiley. The standard EMC textbook; chapter on conducted vs radiated emissions is the canonical engineering reference.
- Ott, H. W. (2009). Electromagnetic Compatibility Engineering. Wiley. The practitioner’s bible — grounding, shielding, filtering, PCB layout from the EMC perspective.
- Johnson, H. & Graham, M. (1993). High-Speed Digital Design: A Handbook of Black Magic. Prentice-Hall. Required reading for anyone routing digital traces above 100 MHz; chapters on transmission lines, ground, terminations, vias.
- Mohan, N., Undeland, T. & Robbins, W. (2003). Power Electronics: Converters, Applications, and Design (3rd ed.). Wiley. Switching-edge spectra and converter-EMC chapters.
- Maxwell, J. C. (1873). A Treatise on Electricity and Magnetism (Vols. I, II). Clarendon Press. The original — readable in places, archaic in others, historically essential.
- IEEE Std 299-2006 (R2012). IEEE Standard Method for Measuring the Effectiveness of Electromagnetic Shielding Enclosures. Measurement methodology for enclosures from 9 kHz to 18 GHz.
- IEC 61000 series. Electromagnetic compatibility (EMC). The international EMC framework; 61000-3-x (emissions), 61000-4-x (immunity test methods), 61000-6-x (generic standards).
- CISPR 22:2008 / CISPR 32:2015+A2:2023. Information technology equipment / Multimedia equipment — Radio disturbance characteristics — Limits and methods of measurement. The headline emissions standards for consumer and ITE equipment.
- CISPR 11:2024. Industrial, scientific and medical equipment — Radio-frequency disturbance characteristics. ISM equipment emissions.
- MIL-STD-461G (2015). Requirements for the Control of Electromagnetic Interference Characteristics of Subsystems and Equipment. US military EMC; CE101/CE102 conducted, RE101/RE102 radiated, CS/RS susceptibility.
- FCC 47 CFR Part 15 — Subpart B (unintentional radiators), Subpart C (intentional radiators). US emissions and licensing for low-power devices.
- Steinmetz, C. P. (1892). “On the Law of Hysteresis.” Transactions of the AIEE. Original empirical formula for core loss still used in modern magnetic design.
- Wheeler, H. A. (1965). “Transmission-Line Properties of Parallel Strips Separated by a Dielectric Sheet.” IEEE Trans. MTT 13(2): 172–185. Foundational microstrip-impedance paper.
- Hammerstad, E. & Jensen, Ø. (1980). “Accurate Models for Microstrip Computer-Aided Design.” IEEE MTT-S Digest 80: 407–409. Refined microstrip formulas used in most CAD tools.