Formula sheet

Applied Electromagnetics and Wave Propagation Formula Sheet

Applied electromagnetics formulas for fields, wave impedance, skin depth, reflections, waveguides, antennas, measurement bandwidth, limits, and validation.

Branch: Engineering Physics
Content: Formula sheet
Updated: Jul 03, 2026
Revision: v1.1.0 · reviewed

This formula sheet collects first-pass relationships used when electromagnetic fields, waves, conductors, dielectrics, antennas, waveguides, transmission lines, shields and measurement systems must be turned into engineering decisions.

Use it for screening, design review and plausibility checks. It is not a replacement for full-wave simulation, calibrated measurements, compliance testing, material characterization, antenna range testing or standard-specific link budgets. Always state the geometry, reference plane, frequency range, material properties, boundary condition and validation method before applying a formula.

How to Use This Formula Sheet

Use this sheet as a calculation control surface for electromagnetic design reviews. Start with the physical regime: compare object size, conductor length, enclosure slots, cable runs, probe dimensions and antenna aperture against wavelength or edge rise time. A calculation using lumped capacitance, inductance or average field is only meaningful while the geometry is still electrically small for the phenomenon being checked.

Then choose the reference plane. Reflections, delivered power, return loss, guide wavelength, antenna gain and receiver noise all depend on where the measurement or model boundary is placed. A connector launch, fixture, cable, probe, radome, enclosure wall or chamber antenna can change the value that the formula is meant to represent.

Use the field formulas for plausibility, safety margin and order-of-magnitude decisions. Use the transmission-line, waveguide and antenna formulas when phase, propagation, impedance or field region controls the result. Use the uncertainty and validation formulas before releasing a pass/fail statement, especially when the margin is smaller than the combined measurement uncertainty.

Scope and Constants

Symbol	Meaning	Typical unit
$c$	speed of light in vacuum	m/s
$\epsilon_0$	vacuum permittivity	F/m
$\mu_0$	vacuum permeability	H/m
$\epsilon_r$	relative permittivity	dimensionless
$\mu_r$	relative permeability	dimensionless
$\sigma$	conductivity	S/m
$f$	frequency	Hz
$\omega$	angular frequency, $2\pi f$	rad/s
$\lambda$	wavelength	m
$E$	electric field magnitude	V/m
$H$	magnetic field strength	A/m
$B$	magnetic flux density	T
$\eta$	wave impedance	ohm
$S_p$	power density or Poynting magnitude	W/m^2
$Z_0$	characteristic impedance	ohm
$\Gamma$	reflection coefficient	dimensionless
$G$	antenna gain	linear or dBi
$A_e$	effective aperture	m^2
$B_n$	noise bandwidth	Hz

Useful constants:

c\approx 3.00\times10^8\ \text{m/s}

\displaystyle \eta_0=\sqrt{\frac{\mu_0}{\epsilon_0}}\approx 377\ \Omega

Use RMS field values consistently. If peak fields are used, state that convention and convert before comparing to RMS power density or regulatory limits.

Basis and Validity Limits

Most relationships in this sheet are first-order engineering formulas. They assume idealized geometry, stable material properties and a clearly defined electromagnetic regime. They are useful because they expose scale, coupling path and reference-plane sensitivity, but they do not remove the need for measurement or simulation when geometry, materials or boundary conditions are complex.

Plane-wave relations assume far-field behavior with a stable ratio between electric and magnetic field. They are not valid for reactive near-field coupling around loops, slots, cables, transformers, motor leads, high-voltage conductors or compact antennas unless a justified near-field model is used. Transmission-line relations assume a defined characteristic impedance and a line long enough, or an edge fast enough, for propagation delay to matter.

Skin-depth and shielding estimates assume a good conductor, known permeability, known conductivity and uniform material thickness. Real shielding effectiveness can be dominated by apertures, seams, gasket pressure, cable terminations, connector leakage, contact corrosion and common-mode current. Waveguide formulas assume controlled dimensions and the intended mode. Antenna aperture formulas assume a defined pattern, polarization and efficiency rather than installed-system performance by themselves.

For design release, treat every formula result as conditional evidence. Record frequency, bandwidth, geometry, material data source, temperature range, field region, instrumentation bandwidth, calibration state and uncertainty basis. If any of those assumptions cannot be defended, the calculation should be labelled screening rather than validation.

Wavelength, Velocity and Electrical Size

Free-space wavelength:

\displaystyle \lambda_0=\frac{c}{f}

Phase velocity in a low-loss material:

\displaystyle v_p=\frac{c}{\sqrt{\epsilon_r\mu_r}}

Wavelength in that material:

\displaystyle \lambda=\frac{v_p}{f}=\frac{\lambda_0}{\sqrt{\epsilon_r\mu_r}}

For a PCB trace with effective relative permittivity:

\displaystyle v_p\approx\frac{c}{\sqrt{\epsilon_{eff}}}

Propagation delay:

\displaystyle t_d=\frac{l}{v_p}

A conservative regime check is:

\displaystyle l<\frac{\lambda}{10}

for an electrically short structure at a single frequency. Fast digital edges should also be checked against rise time:

\displaystyle l_{crit}\approx\frac{v_p t_r}{10}

Common Mistakes

Checking only clock frequency while ignoring edge-rate spectral content.
Using free-space wavelength for a wave guided by a dielectric or trace.
Treating a cable, PCB trace, enclosure slot or sensor lead as lumped after delay and phase matter.

Electric and Magnetic Field Screening

Average electric field across a gap:

\displaystyle E\approx\frac{V}{d}

Magnetic flux density:

B=\mu H

Magnetic flux through an area:

\displaystyle \Phi=\int B\cdot dA

For approximately uniform normal flux:

\Phi\approx BA

Faraday induction:

\displaystyle v=-N\frac{d\Phi}{dt}

These are screening relationships. Sharp edges, partial discharge, contamination, humidity, altitude, insulation aging, fringing fields, nearby conductors and material saturation can dominate real field limits.

Plane Wave Relations

Intrinsic wave impedance:

\displaystyle \eta=\sqrt{\frac{\mu}{\epsilon}}

For a uniform plane wave:

\displaystyle H=\frac{E}{\eta}

For RMS field values:

\displaystyle S_p=\frac{E_{rms}^2}{\eta}=\eta H_{rms}^2

For peak field values:

\displaystyle S_{p,avg}=\frac{E_{pk}^2}{2\eta}=\frac{\eta H_{pk}^2}{2}

Free-space field from power density:

E_{rms}=\sqrt{S_p\eta_0}

Plane-wave formulas apply in the far field and away from strong reflections. Near-field electric and magnetic coupling can have very different field ratios.

Radiated Power Density and Far Field

Power density from an antenna with transmit power $P_t$ and gain $G_t$ at range $R$ :

\displaystyle S_p=\frac{P_tG_t}{4\pi R^2}

Far-field distance for an antenna or aperture with largest dimension $D$ :

\displaystyle R_{FF}\approx\frac{2D^2}{\lambda}

This far-field rule is a screening rule. Electrically small antennas, near-field probes, chamber limits, reactive fields, ground planes and nearby structures may require more careful methods.

Transmission Lines and Reflections

Propagation delay:

\displaystyle t_d=\frac{l}{v_p}

For a lossless line, load voltage reflection coefficient:

\displaystyle \Gamma_L=\frac{Z_L-Z_0}{Z_L+Z_0}

Reflected power fraction:

\displaystyle \frac{P_r}{P_i}=|\Gamma_L|^2

Voltage standing wave ratio:

\displaystyle VSWR=\frac{1+|\Gamma|}{1-|\Gamma|}

Return loss:

RL=-20\log_{10}|\Gamma|

Mismatch loss:

ML=-10\log_{10}(1-|\Gamma|^2)

Transmission-line calculations must state the reference plane. A connector, via, fixture adapter, cable, probe or board launch can move the practical reference plane away from the schematic node.

Skin Depth and Conductor Loss

Skin depth in a good conductor:

\displaystyle \delta=\sqrt{\frac{2}{\omega\mu\sigma}}

Surface resistance:

\displaystyle R_s=\frac{1}{\sigma\delta}=\sqrt{\frac{\pi f\mu}{\sigma}}

Thickness in skin depths:

\displaystyle n_\delta=\frac{t}{\delta}

Approximate absorption loss through a conductor thickness $t$ :

\displaystyle A_{dB}\approx 8.686\frac{t}{\delta}

This absorption estimate is not the whole shielding problem. Apertures, seams, cable shields, gasket compression, contact resistance, low-frequency magnetic fields and common-mode currents can dominate shielding performance.

Capacitance, Inductance and Coupling Checks

Capacitive reactance:

\displaystyle X_C=\frac{1}{\omega C}

Inductive reactance:

X_L=\omega L

Capacitive displacement current estimate:

\displaystyle I_C=C\frac{dV}{dt}

Inductive voltage estimate:

\displaystyle V_L=L\frac{dI}{dt}

Mutual inductive coupling:

\displaystyle V_M=M\frac{dI}{dt}

These formulas explain many EMC failures. The relevant capacitance or inductance may be parasitic geometry rather than a named component.

Rectangular Waveguide

Dominant TE10 cutoff frequency for a rectangular waveguide with broad dimension $a$ :

\displaystyle f_c=\frac{c}{2a}

Free-space wavelength:

\displaystyle \lambda_0=\frac{c}{f}

Guide wavelength above cutoff:

\displaystyle \lambda_g=\frac{\lambda_0}{\sqrt{1-(f_c/f)^2}}

Phase velocity:

\displaystyle v_p=\frac{c}{\sqrt{1-(f_c/f)^2}}

Group velocity:

v_g=c\sqrt{1-(f_c/f)^2}

Waveguide formulas assume the intended mode, suitable dimensions, good conducting walls, controlled flanges, acceptable bends and no unintended mode conversion.

Antenna Aperture and Gain

Effective aperture and gain:

\displaystyle A_e=\frac{G\lambda^2}{4\pi}

Gain from physical aperture area $A$ and aperture efficiency $\eta_a$ :

\displaystyle G=\eta_a\frac{4\pi A}{\lambda^2}

Gain in dBi:

G_{dBi}=10\log_{10}(G)

Polarization mismatch loss:

L_{pol}=|\hat{p}_t\cdot \hat{p}_r|^2

For a polarization angle error $\theta$ :

L_{pol}=\cos^2\theta

High gain narrows the useful direction. Antenna validation should state mounting, ground plane, nearby structures, cable route, polarization, pointing tolerance and whether the reported result is gain, pattern, return loss or service performance.

Noise Bandwidth and Receiver Screening

Thermal noise power:

N=kTB_n

Noise power in dBm at approximately room temperature:

N_{dBm}\approx -174+10\log_{10}(B_n)

Including receiver noise figure:

N_{dBm}\approx -174+10\log_{10}(B_n)+NF_{dB}

Signal-to-noise ratio:

SNR_{dB}=P_{signal,dBm}-N_{dBm}

Use this only after bandwidth is defined. Occupied bandwidth, filter 3 dB bandwidth, equivalent noise bandwidth and measurement resolution bandwidth are not interchangeable.

Sampling, Jitter and Measurement Bandwidth

Nyquist frequency:

\displaystyle f_N=\frac{f_s}{2}

Sampling period:

\displaystyle T_s=\frac{1}{f_s}

Jitter-limited SNR for a sinusoid:

SNR_{j,dB}\approx -20\log_{10}(2\pi f_{in}\sigma_t)

where $\sigma_t$ is RMS sampling jitter. Practical electromagnetic measurements require anti-alias filtering, calibrated probes, controlled cable routing, adequate dynamic range and a documented reference plane.

Uncertainty and Validation

For independent standard uncertainty components:

u_c=\sqrt{u_1^2+u_2^2+\cdots+u_n^2}

Expanded uncertainty:

U=ku_c

Validation should include:

reference plane and calibration method;
frequency range, bandwidth and resolution bandwidth;
probe loading and fixture de-embedding where relevant;
cable routing, shield terminations and grounding;
chamber, open-area, near-field or installed test configuration;
environmental conditions, nearby structures and operating modes;
pass/fail limits and engineering consequence of measurement uncertainty.

Common Formula Mistakes

The most common applied electromagnetics error is mixing regimes. A free-space wavelength check does not validate a dielectric-loaded trace, a near-field probe, a waveguide, an enclosure slot or a cable harness routed against a metal structure. Always repeat the check with the effective dielectric constant, propagation path and installed geometry that actually control phase.

Do not mix peak, RMS and average quantities. Field strength, power density, conducted power, oscilloscope voltage, spectrum-analyzer power and receiver noise all carry implicit conventions. A single missing factor of two can turn a conservative EMC calculation into a misleading release statement.

Do not treat impedance as a schematic-only property. A nominal 50 ohm system can be dominated by launches, vias, adapters, probe loading, fixture de-embedding, connector repeatability or cable bending. State the calibration plane and do not compare values taken at different planes as if they were the same measurement.

Do not use shielding absorption as total shielding effectiveness. High-frequency copper absorption can look excellent while an enclosure still fails through apertures, pigtail shield terminations, poor gasket compression or common-mode cable radiation. For low-frequency magnetic fields, permeability, saturation and geometry may matter more than conductivity.

Do not interpret antenna gain, aperture gain or far-field distance outside the measurement setup. Installed gain can change with ground plane, radome, nearby structures, cable routing, polarization, pointing error and multipath. If the result feeds a link, radar or exposure budget, it should be traceable to a validated model or measurement.

Worked Example 1: Field Strength and Power Density

An immunity test exposes equipment to an approximately uniform far-field RMS electric field:

E_{rms}=10\ \text{V/m}

Estimate magnetic field strength and power density using free-space wave impedance.

Magnetic field:

\displaystyle H_{rms}=\frac{E_{rms}}{\eta_0}=\frac{10}{377}=0.0265\ \text{A/m}

Power density:

\displaystyle S_p=\frac{E_{rms}^2}{\eta_0}=\frac{10^2}{377}=0.265\ \text{W/m}^2

Engineering Comment

The calculation is valid for a plane-wave-like far-field condition. It does not describe near-field magnetic coupling from a current loop or near-field electric coupling from a high-voltage conductor. The validation record should state antenna distance, polarization, chamber condition, dwell time, modulation and cable configuration.

Worked Example 2: PCB Trace Delay and Edge-Rate Regime

A PCB trace is:

l=150\ \text{mm}

with effective relative permittivity:

\epsilon_{eff}=3.6

The signal rise time is:

t_r=1.0\ \text{ns}

Estimate propagation delay and decide whether delay is important.

Propagation velocity:

\displaystyle v_p=\frac{c}{\sqrt{3.6}}=\frac{3.0\times10^8}{1.897}=1.58\times10^8\ \text{m/s}

Delay:

\displaystyle t_d=\frac{0.150}{1.58\times10^8}=9.49\times10^{-10}\ \text{s}

t_d=0.949\ \text{ns}

The delay is nearly equal to the rise time.

Engineering Comment

This trace must be reviewed as a transmission line for that edge. The key design evidence is not only trace length; it is characteristic impedance, return path continuity, driver impedance, receiver threshold margin, via and connector discontinuities, and waveform measurement at the correct reference plane.

Worked Example 3: Copper Skin Depth and Shield Absorption

Estimate copper skin depth at:

f=10\ \text{MHz}

Use:

\sigma=5.8\times10^7\ \text{S/m}

\mu\approx\mu_0

For copper thickness:

t=35\ \mu\text{m}

estimate thickness in skin depths and absorption loss.

Skin depth:

\displaystyle \delta=\sqrt{\frac{2}{2\pi f\mu_0\sigma}}

\delta=20.9\ \mu\text{m}

Thickness in skin depths:

\displaystyle n_\delta=\frac{35}{20.9}=1.67

Absorption estimate:

A_{dB}=8.686(1.67)=14.5\ \text{dB}

Engineering Comment

The absorption term is only one part of shielding. A thin copper layer may provide useful high-frequency electric-field shielding, yet an enclosure can still fail through slots, seams, cable shields, connector leakage or low-frequency magnetic fields. Validate the installed enclosure, not just coupon material.

Worked Example 4: Reflection, VSWR and Return Loss

A transmission line has:

Z_0=50\ \Omega

and is terminated by:

Z_L=75\ \Omega

Find reflection coefficient magnitude, reflected power fraction, VSWR and return loss.

Reflection coefficient:

\displaystyle \Gamma=\frac{75-50}{75+50}=0.20

Reflected power fraction:

|\Gamma|^2=0.20^2=0.040

So 4.0 percent of incident power is reflected at that discontinuity.

VSWR:

\displaystyle VSWR=\frac{1+0.20}{1-0.20}=1.50

Return loss:

RL=-20\log_{10}(0.20)=14.0\ \text{dB}

Engineering Comment

This mismatch may be acceptable for some broadband lab measurements and unacceptable for precision RF power transfer, clock distribution, radar paths or high-speed serial links. The decision depends on tolerance for ringing, delivered power loss, multiple discontinuities and calibration-plane uncertainty.

Worked Example 5: Rectangular Waveguide Cutoff

A rectangular waveguide has broad dimension:

a=22.86\ \text{mm}

Estimate TE10 cutoff frequency and guide wavelength at:

f=10\ \text{GHz}

Cutoff:

\displaystyle f_c=\frac{c}{2a}=\frac{3.00\times10^8}{2(0.02286)}=6.56\ \text{GHz}

Free-space wavelength at 10 GHz:

\displaystyle \lambda_0=\frac{3.00\times10^8}{10\times10^9}=30.0\ \text{mm}

Guide wavelength:

\displaystyle \lambda_g=\frac{\lambda_0}{\sqrt{1-(f_c/f)^2}}

\displaystyle \lambda_g=\frac{30.0}{\sqrt{1-(6.56/10)^2}}=39.7\ \text{mm}

Engineering Comment

The operating frequency is above TE10 cutoff, but that alone does not validate the RF path. Flange alignment, bends, surface condition, moisture, gasket condition, mode conversion and calibration reference plane can still control measured loss and match.

Worked Example 6: Antenna Far-Field Distance

An antenna or aperture has largest dimension:

D=0.60\ \text{m}

It operates at:

f=5.8\ \text{GHz}

Estimate the far-field distance.

Wavelength:

\displaystyle \lambda=\frac{3.00\times10^8}{5.8\times10^9}=0.0517\ \text{m}

Far-field distance:

\displaystyle R_{FF}=\frac{2D^2}{\lambda}=\frac{2(0.60)^2}{0.0517}=13.9\ \text{m}

Engineering Comment

A measurement at 3 m would be a near-field measurement for this aperture by the simple criterion. It may still be useful, but it should not be interpreted as a far-field gain or pattern result without an appropriate method, transformation or justified standard procedure.

Worked Example 7: Aperture Gain Estimate

A microwave aperture has physical area:

A=0.12\ \text{m}^2

Aperture efficiency is estimated as:

\eta_a=0.55

Frequency is:

f=10\ \text{GHz}

Estimate gain.

Wavelength:

\displaystyle \lambda=\frac{3.00\times10^8}{10\times10^9}=0.0300\ \text{m}

Gain:

\displaystyle G=\eta_a\frac{4\pi A}{\lambda^2}

\displaystyle G=0.55\frac{4\pi(0.12)}{0.0300^2}=923

Gain in dBi:

G_{dBi}=10\log_{10}(923)=29.7\ \text{dBi}

Engineering Comment

This is an aperture estimate, not a full antenna qualification. Real gain depends on illumination, spillover, blockage, surface error, feed match, polarization, radome, mounting and nearby structure. A gain value used in a link or radar budget should be traceable to measurement or a validated model.

Validation Evidence Package

Before accepting an applied electromagnetics calculation, assemble the evidence that proves the formula was used in the correct regime. The package should let another engineer reproduce the assumptions, units, reference plane and pass/fail decision without guessing the missing context.

Include:

the physical size was compared with wavelength or rise time;
RMS and peak quantities were not mixed;
reference planes are stated for reflection, loss and power transfer;
material parameters are valid at the operating frequency and temperature;
cable, connector, enclosure and fixture effects are included or bounded;
antenna measurements are in the correct field region;
bandwidth and resolution bandwidth match the noise and SNR calculation;
uncertainty is included when the result supports pass/fail release.

Also include calibration certificates or traceability statements for field probes, network analyzers, power sensors, antennas, spectrum analyzers, oscilloscopes and current probes when those instruments drive acceptance. Record fixture de-embedding, chamber configuration, cable routing, operating mode, dwell time, modulation, environmental conditions and any deviations from the intended standard or internal test method.

The recurring failure is not using the wrong equation in isolation. It is using a correct equation outside its electromagnetic boundary, or using a screening calculation as if it were release evidence.

REF

Disciplines

Applied Electromagnetics and Wave Propagation Formula Sheet

How to Use This Formula Sheet

Scope and Constants

Basis and Validity Limits

Wavelength, Velocity and Electrical Size

Common Mistakes

Electric and Magnetic Field Screening

Plane Wave Relations

Radiated Power Density and Far Field

Transmission Lines and Reflections

Skin Depth and Conductor Loss

Capacitance, Inductance and Coupling Checks

Rectangular Waveguide

Antenna Aperture and Gain

Noise Bandwidth and Receiver Screening

Sampling, Jitter and Measurement Bandwidth

Uncertainty and Validation

Common Formula Mistakes

Worked Example 1: Field Strength and Power Density

Engineering Comment

Worked Example 2: PCB Trace Delay and Edge-Rate Regime

Engineering Comment

Worked Example 3: Copper Skin Depth and Shield Absorption

Engineering Comment

Worked Example 4: Reflection, VSWR and Return Loss

Engineering Comment

Worked Example 5: Rectangular Waveguide Cutoff

Engineering Comment

Worked Example 6: Antenna Far-Field Distance

Engineering Comment

Worked Example 7: Aperture Gain Estimate

Engineering Comment

Validation Evidence Package

See also