Radiation Fundamentals, Retarded Potentials, and Dipole Antennas

Unit 8: Computational electromagnetics and validation

What the unit establishes

Discretize Maxwell equations, evaluate stability and mesh convergence, implement physically meaningful boundaries, and distinguish numerical agreement from validation against the real world.

Reproducible investigation

Construct a one-dimensional wave-propagation benchmark with a Yee-style electric/magnetic staggered grid. For at least three spatial steps, calculate the permitted time step under the CFL relation and predict a travel delay over the same physical distance. Compare numerical results at a Courant factor below one and one above one, remembering that selected low-frequency curves may appear plausible even when higher spatial frequencies become unstable. Add a simple dielectric boundary and verify the Fresnel reflection coefficient against the grid result. Move the computational boundary farther away and repeat the test with an absorbing termination; quantify the change. In a complementary finite-element thought experiment, explain why vector edge bases and mesh refinement near interfaces matter. Finish by proposing an inverse parameter fit and showing how different parameter sets can explain the same observed phase when only the refractive-index-thickness product is known.

Four connected lessons

1. Yee-grid finite-difference time-domain method

The starting point is the physical situation considered in the [full lesson](../lessons/U08-L01-yee-grid-finite-difference-time-domain-method.html). Its key mathematical relationship is the following, stated with the particular geometrical or material assumptions of this topic:

For a uniform Cartesian cell in a lossless 3D explicit Yee scheme, a commonly used Courant condition is Δt≤1/[v sqrt(Δx^{-2}+Δy^{-2}+Δz^{-2})] with v the maximum relevant phase speed in the modeled nondispersive medium. In one dimension this reduces to Δt≤Δx/v. Update E and H with staggered discrete curls; material discontinuities require careful placement of ε,μ,σ.

Work through its numerical example on paper before using the model. Compare the predicted result with the situation described in the lesson's engineering case; check which assumption restricts the validity of the formula. The goal is not memorization of symbols but identifying whether a simplified scalar parameter is sufficient for this geometry and frequency.

2. Finite-element weak forms and mesh quality

The starting point is the physical situation considered in the [full lesson](../lessons/U08-L02-finite-element-weak-forms-and-mesh-quality.html). Its key mathematical relationship is the following, stated with the particular geometrical or material assumptions of this topic:

Starting from frequency-domain curl-curl equation ∇×(μ^{-1}∇×E)−ω²ε E=jωJ under a chosen phasor convention, multiply by vector test function v and integrate over Ω. Integration by parts produces a weak form with boundary terms that encode specified tangential fields or natural conditions. Appropriate Nédélec edge elements enforce tangential continuity across element interfaces without forcing incorrect normal continuity of E.

Work through its numerical example on paper before using the model. Compare the predicted result with the situation described in the lesson's engineering case; check which assumption restricts the validity of the formula. The goal is not memorization of symbols but identifying whether a simplified scalar parameter is sufficient for this geometry and frequency.

3. Absorbing boundaries, PML and convergence tests

The starting point is the physical situation considered in the [full lesson](../lessons/U08-L03-absorbing-boundaries-pml-and-convergence-tests.html). Its key mathematical relationship is the following, stated with the particular geometrical or material assumptions of this topic:

At a simple one-dimensional boundary, first-order outgoing-wave behavior satisfies ∂E/∂t±v∂E/∂x≈0 for the relevant direction. A PML can be derived by complex coordinate stretching in the frequency domain or equivalent split-field/auxiliary updates. Reflection error is measured relative to a reference run or analytic solution, using a consistent source normalization and incident-field subtraction.

Work through its numerical example on paper before using the model. Compare the predicted result with the situation described in the lesson's engineering case; check which assumption restricts the validity of the formula. The goal is not memorization of symbols but identifying whether a simplified scalar parameter is sufficient for this geometry and frequency.

4. Inverse material estimation and design optimization

The starting point is the physical situation considered in the [full lesson](../lessons/U08-L04-inverse-material-estimation-and-design-optimization.html). Its key mathematical relationship is the following, stated with the particular geometrical or material assumptions of this topic:

A simple inverse objective might be J(θ)=Σ_k w_k |S21_model(f_k;θ)−S21_obs(f_k)|²+λR(θ), with θ material and geometry parameters and R regularization. Identifiability can be examined through the Jacobian ∂observable/∂θ and its conditioning. Reserve independent measurements for validation. If several parameter sets fit within observation error, report the family of plausible solutions rather than one overconfident point estimate.

Work through its numerical example on paper before using the model. Compare the predicted result with the situation described in the lesson's engineering case; check which assumption restricts the validity of the formula. The goal is not memorization of symbols but identifying whether a simplified scalar parameter is sufficient for this geometry and frequency.

Unit test of understanding

Use the [eight solved questions](../assessments/unit-08.html) as a closed-book audit. Then complete the [unit investigation](../projects/unit-08.html) and compare its computational trend with the [unit interactive model](../simulations/model-08.html). Write down any mismatch between the ideal analytical limit and the plotted values; if an output appears to violate conservation or dimensional consistency, diagnose the implementation before adjusting a physical parameter to force agreement.

Evidence standard

A completed study record contains the source equations, declared time and orientation convention, parameter units, at least two numerical cases, a model-limitation statement and a reproducible test. Nothing in this unit requires live mains or high-voltage apparatus; synthetic numerical work is adequate for the computational project. A credible claim about real equipment would additionally require calibration and qualified verification appropriate to that application.

U08-L01: Yee-grid finite-difference time-domain method

Book 22 · Unit 8 — Computational electromagnetics and validation Learning objective: Reconstruct the physical model, calculate one representative case and explain where its assumptions cease to apply.

Physical principles

The Yee scheme staggers electric and magnetic field components in both space and time, turning curl equations into explicit updates that naturally encode circulation around discrete cells. This placement helps preserve discrete conservation properties when implemented correctly. A simulation domain requires grid spacing small enough to resolve both the shortest wavelength and geometric features. Time-step selection must satisfy stability limits, but stability alone does not guarantee accuracy. FDTD is appealing for broadband transients because one run can produce time records across many frequencies, provided material dispersion and boundary absorption are modeled.

Derivation and governing equations

For a uniform Cartesian cell in a lossless 3D explicit Yee scheme, a commonly used Courant condition is Δt≤1/[v sqrt(Δx^{-2}+Δy^{-2}+Δz^{-2})] with v the maximum relevant phase speed in the modeled nondispersive medium. In one dimension this reduces to Δt≤Δx/v. Update E and H with staggered discrete curls; material discontinuities require careful placement of ε,μ,σ.

Fully specified worked example

For a vacuum 1D grid with Δx=1 mm, Δt≤3.336 ps. With Courant factor 0.9 choose about 3.00 ps. Simulating 1 ns then requires roughly 334 time steps. If Δx halves but temporal Courant factor stays fixed, the time step also halves and the number of time steps doubles.

Independent numerical verification. Reconstruct the arithmetic in the preceding illustration using SI base units, then change one input by a factor of two and predict the consequence from the actual governing expression. Specify which inputs remain fixed. When the result involves a signed voltage or complex amplitude, report the chosen polarity or phase reference; a magnitude alone cannot settle the sign. Compare the result against an analytically known limit rather than blindly trusting significant figures from a calculator.

Engineering decision scenario

A pulse incident on a dielectric slab can be used to verify arrival time and reflection coefficient from analytic formulas. Run at two grid resolutions, compare delays and amplitudes, and ensure waves leave the computational region without artificial reflection. A pretty field animation is not enough evidence that the numerical phase velocity is accurate.

Scope, failure modes and falsification

A method stable in free space may require different treatment for dispersive materials, subcell features or highly conductive regions. Numerical dispersion shifts wave phase even when the update is stable. Poor boundary conditions and discontinuity staircasing can dominate error.

What evidence would change this conclusion? Propose one measurement or full-wave simulation that would expose the limitation described above, name the raw observable, and determine whether it distinguishes an omitted physical effect from a code implementation error. Numerical convergence and physical validation are different claims. No idealized wave or electromagnetic enclosure model in this chapter constitutes a hardware safety approval.

Three thinking problems

1. Compute and check: Recalculate the central numerical result above with a clearly stated changed parameter. Show unit conversion and the primary scaling law, not just a final decimal.

2. Derive and compare: Starting with the main field law above, reconstruct the next relation using the assumptions explicitly stated in the lesson. Say which boundary condition or sign convention is indispensable.

3. Design critique: Apply the limitation above to the engineering scenario. Specify the experiment, frequency range, geometry or calibration that would need to be checked before an engineer could accept the simple calculation.

Continue: [Unit 8 overview](../units/unit-08.html) · [Assessment and answer key](../assessments/unit-08.html) · [Interactive model](../simulations/model-08.html).

U08-L02: Finite-element weak forms and mesh quality

Book 22 · Unit 8 — Computational electromagnetics and validation Learning objective: Reconstruct the physical model, calculate one representative case and explain where its assumptions cease to apply.

Physical principles

Finite-element electromagnetics partitions a domain into elements and approximates unknown fields with basis functions. The weak formulation enforces a weighted integral residual, making complex geometry and material heterogeneity tractable. Vector electromagnetic problems benefit from curl-conforming edge elements, since naïve nodal vector bases can introduce spurious electromagnetic modes. Mesh quality must be assessed where fields vary sharply, at material interfaces and near resonant features. A solver residual says the algebraic system was solved, but mesh refinement and a physical benchmark are still necessary to establish a useful engineering result.

Derivation and governing equations

Starting from frequency-domain curl-curl equation ∇×(μ^{-1}∇×E)−ω²ε E=jωJ under a chosen phasor convention, multiply by vector test function v and integrate over Ω. Integration by parts produces a weak form with boundary terms that encode specified tangential fields or natural conditions. Appropriate Nédélec edge elements enforce tangential continuity across element interfaces without forcing incorrect normal continuity of E.

Fully specified worked example

Consider a mesh of a 1 cm cavity at 10 GHz where λ0≈3 cm. A 3 mm nominal element size gives only about ten elements per free-space wavelength; whether this is sufficient depends on basis order, dielectric wavelength and required phase accuracy. Doubling mesh resolution in each three-dimensional direction increases element count by roughly eight before accounting for local refinement.

Independent numerical verification. Reconstruct the arithmetic in the preceding illustration using SI base units, then change one input by a factor of two and predict the consequence from the actual governing expression. Specify which inputs remain fixed. When the result involves a signed voltage or complex amplitude, report the chosen polarity or phase reference; a magnitude alone cannot settle the sign. Compare the result against an analytically known limit rather than blindly trusting significant figures from a calculator.

Engineering decision scenario

An RF connector model may resolve the feed pin and dielectric discontinuity locally while using a coarser mesh in a distant homogeneous region. The refinement criterion should focus on changes in S11, stored energy and peak field in defined regions, not merely on the total number of tetrahedra.

Scope, failure modes and falsification

Artificial eigenmodes may arise from inappropriate elements or incomplete constraints. An extremely fine mesh of an incorrect geometry can still yield a precise but wrong result. Sharp perfect-conductor corners create singular fields for which pointwise peak values may not converge without a physically rounded geometry.

What evidence would change this conclusion? Propose one measurement or full-wave simulation that would expose the limitation described above, name the raw observable, and determine whether it distinguishes an omitted physical effect from a code implementation error. Numerical convergence and physical validation are different claims. No idealized wave or electromagnetic enclosure model in this chapter constitutes a hardware safety approval.

Three thinking problems

1. Compute and check: Recalculate the central numerical result above with a clearly stated changed parameter. Show unit conversion and the primary scaling law, not just a final decimal.

2. Derive and compare: Starting with the main field law above, reconstruct the next relation using the assumptions explicitly stated in the lesson. Say which boundary condition or sign convention is indispensable.

3. Design critique: Apply the limitation above to the engineering scenario. Specify the experiment, frequency range, geometry or calibration that would need to be checked before an engineer could accept the simple calculation.

Continue: [Unit 8 overview](../units/unit-08.html) · [Assessment and answer key](../assessments/unit-08.html) · [Interactive model](../simulations/model-08.html).

U08-L03: Absorbing boundaries, PML and convergence tests

Book 22 · Unit 8 — Computational electromagnetics and validation Learning objective: Reconstruct the physical model, calculate one representative case and explain where its assumptions cease to apply.

Physical principles

A computational region cannot represent infinite free space directly, so an outgoing-wave condition or absorbing layer is usually required. A perfectly matched layer is an artificial region designed to attenuate outgoing waves with minimal reflection over its intended parameter range. Poor layer grading, insufficient thickness, grazing incidence or incorrect implementation can create artifacts that look like physical reflections. Verification compares against known solutions and changes both mesh and computational-domain size. A result that changes when the absorbing boundary moves is not yet reliable as a free-space prediction.

Derivation and governing equations

At a simple one-dimensional boundary, first-order outgoing-wave behavior satisfies ∂E/∂t±v∂E/∂x≈0 for the relevant direction. A PML can be derived by complex coordinate stretching in the frequency domain or equivalent split-field/auxiliary updates. Reflection error is measured relative to a reference run or analytic solution, using a consistent source normalization and incident-field subtraction.

Fully specified worked example

Suppose a test pulse's incident peak is 1.0 V/m and a numerical boundary sends back 0.01 V/m into the observation plane. Its returned field amplitude is −40 dB relative to incident amplitude, corresponding to 0.01% of incident power in the matched plane-wave idealization. This one figure does not guarantee equal performance across frequencies and incidence angles.

Independent numerical verification. Reconstruct the arithmetic in the preceding illustration using SI base units, then change one input by a factor of two and predict the consequence from the actual governing expression. Specify which inputs remain fixed. When the result involves a signed voltage or complex amplitude, report the chosen polarity or phase reference; a magnitude alone cannot settle the sign. Compare the result against an analytically known limit rather than blindly trusting significant figures from a calculator.

Engineering decision scenario

When modeling antenna radiation, repeat the simulation with PML boundaries pushed farther from the feed while keeping a meaningful mesh refinement. If the observed pattern changes, either near-field coupling to the boundary or discretization error is significant. Report convergence of the target metric rather than only convergence of the solver iteration count.

Scope, failure modes and falsification

No finite PML is perfectly transparent at every numerical frequency and angle. Strong evanescent content, anisotropy, nonlinearity and complex dispersive media can require specialized formulations. A low reported reflection at one test frequency does not establish correct open-region behavior for a broadband pulse.

What evidence would change this conclusion? Propose one measurement or full-wave simulation that would expose the limitation described above, name the raw observable, and determine whether it distinguishes an omitted physical effect from a code implementation error. Numerical convergence and physical validation are different claims. No idealized wave or electromagnetic enclosure model in this chapter constitutes a hardware safety approval.

Three thinking problems

1. Compute and check: Recalculate the central numerical result above with a clearly stated changed parameter. Show unit conversion and the primary scaling law, not just a final decimal.

2. Derive and compare: Starting with the main field law above, reconstruct the next relation using the assumptions explicitly stated in the lesson. Say which boundary condition or sign convention is indispensable.

3. Design critique: Apply the limitation above to the engineering scenario. Specify the experiment, frequency range, geometry or calibration that would need to be checked before an engineer could accept the simple calculation.

Continue: [Unit 8 overview](../units/unit-08.html) · [Assessment and answer key](../assessments/unit-08.html) · [Interactive model](../simulations/model-08.html).

U08-L04: Inverse material estimation and design optimization

Book 22 · Unit 8 — Computational electromagnetics and validation Learning objective: Reconstruct the physical model, calculate one representative case and explain where its assumptions cease to apply.

Physical principles

Forward electromagnetics predicts field response from known geometry and material parameters. An inverse problem asks which parameters could explain measured or simulated responses; it is often nonunique. Two different combinations of dielectric constant, sample thickness and conductor loss can produce similar S-parameters over a narrow band. Optimization can reduce a chosen numerical objective without discovering the true physical specimen. Defensible inference therefore includes calibration, sensitivity, parameter bounds, independent validation frequencies and uncertainty intervals. A black-box minimum is not enough to establish a unique or causal material explanation.

Derivation and governing equations

A simple inverse objective might be J(θ)=Σ_k w_k |S21_model(f_k;θ)−S21_obs(f_k)|²+λR(θ), with θ material and geometry parameters and R regularization. Identifiability can be examined through the Jacobian ∂observable/∂θ and its conditioning. Reserve independent measurements for validation. If several parameter sets fit within observation error, report the family of plausible solutions rather than one overconfident point estimate.

Fully specified worked example

A lossless slab at normal incidence accumulates phase approximately φ=2π f n d/c. If the measured phase constrains only the product nd, doubling d and halving n leaves that phase unchanged, showing a simple ambiguity. A second measurement across frequency or independent thickness measurement can help resolve this ambiguity.

Independent numerical verification. Reconstruct the arithmetic in the preceding illustration using SI base units, then change one input by a factor of two and predict the consequence from the actual governing expression. Specify which inputs remain fixed. When the result involves a signed voltage or complex amplitude, report the chosen polarity or phase reference; a magnitude alone cannot settle the sign. Compare the result against an analytically known limit rather than blindly trusting significant figures from a calculator.

Engineering decision scenario

For an RF sensing product, material retrieval from a measured network response should include instrument calibration and uncertainty. Measure a known reference sample, estimate parasitic fixture parameters, and test whether the inferred dielectric property predicts a second specimen or frequency range. Report whether the fitted value is stable when initialization changes.

Scope, failure modes and falsification

Models can be structurally wrong even when residual loss is small. Measurement drift, correlated noise, fixture mismatch and unmodeled modes bias parameter estimates. Overfitting on a dense frequency sweep can give the illusion of abundant independent evidence when the measurements are strongly correlated.

What evidence would change this conclusion? Propose one measurement or full-wave simulation that would expose the limitation described above, name the raw observable, and determine whether it distinguishes an omitted physical effect from a code implementation error. Numerical convergence and physical validation are different claims. No idealized wave or electromagnetic enclosure model in this chapter constitutes a hardware safety approval.

Three thinking problems

1. Compute and check: Recalculate the central numerical result above with a clearly stated changed parameter. Show unit conversion and the primary scaling law, not just a final decimal.

2. Derive and compare: Starting with the main field law above, reconstruct the next relation using the assumptions explicitly stated in the lesson. Say which boundary condition or sign convention is indispensable.

3. Design critique: Apply the limitation above to the engineering scenario. Specify the experiment, frequency range, geometry or calibration that would need to be checked before an engineer could accept the simple calculation.

Continue: [Unit 8 overview](../units/unit-08.html) · [Assessment and answer key](../assessments/unit-08.html) · [Interactive model](../simulations/model-08.html).

Unit 8 engineering project: Numerical wave solver verification campaign

Type: Computational desk investigation using synthetic data only. No mains wiring or high-voltage hardware work.

The problem and experimental model

Implement or inspect a one-dimensional Yee-grid wave model at three spatial resolutions and two Courant factors. Compare arrival time, phase and interface reflection with an analytic benchmark. Introduce an absorbing boundary and determine whether the observed reflected field changes when domain size is increased.

Required evidence

Submit mesh/time-step tables, conservation or energy trend, boundary-reflection comparison, evidence of convergence in the chosen quantity and a written distinction between verification and validation.

Independent benchmark and failure test

Before generating graphs, derive a baseline from the four unit lessons. Establish a numerical prediction from known dimensions, wave speed or material law. Run a second numerical calculation that uses a different formulation or a limiting case. Inspect whether the model conserves charge or power where the assumptions require it; use the proper amplitude or power reference. Compare input perturbations one at a time so the direction of the observed effect can be explained physically.

Analysis requirements

Every plot must label input units, dependent quantities, coordinate reference and whether the curve is synthetic. A statement such as 'the signal became better' is incomplete without specifying the chosen metric and the control variables. Include a short error budget separating numerical resolution, material uncertainty, geometrical approximation and measurement effects that would matter in a real experiment. An unexplained mismatch must remain visible in the report rather than being edited out.

Deliverables and grading

A concise technical report should include: purpose and assumptions (15%), derivation and units (20%), numerical verification (25%), parameter sensitivity and omitted physics (25%), and evidence-based engineering recommendation (15%). Name one specific datum whose collection would change the recommendation. Do not claim that a synthetic dataset confirms real equipment performance.

References inside the book: [Unit guide](../units/unit-08.html) · [Analytical model](../simulations/model-08.html) · [Assessment](../assessments/unit-08.html).

Unit 8 assessment: Computational electromagnetics and validation

Eight subject-specific questions, with explanatory answers. Attempt all questions before reading the key. These are educational examples, not external exam past papers.

Question 1

For Δx=2 mm in vacuum 1D FDTD, calculate maximum Courant time step.

Answer and rationale: Δt≤Δx/c≈6.67 ps.

Question 2

Does meeting CFL guarantee exact wave phase?

Answer and rationale: No, a stable discretization can still produce numerical dispersion and inaccurate geometry.

Question 3

Why are curl-conforming edge elements preferred for many vector EM finite-element problems?

Answer and rationale: They preserve appropriate tangential continuity and help suppress nonphysical spurious modes.

Question 4

What is the physical consequence of an unresolved sharp conductor corner?

Answer and rationale: A mathematically singular ideal peak may change with mesh refinement, so no trustworthy pointwise physical maximum is established.

Question 5

A reflected field is 0.001 of incident amplitude. What is amplitude return in dB?

Answer and rationale: 20log10(0.001)=−60 dB, corresponding to a power fraction of 10^-6 in a matched plane-wave comparison.

Question 6

Why should PML distance and thickness be varied in verification?

Answer and rationale: A numerical boundary may reflect or interact with near fields; output dependence on its placement signals an unresolved numerical artifact.

Question 7

Why can a single measured slab phase fail to identify both refractive index and thickness?

Answer and rationale: At a fixed frequency phase approximately depends on their product nd; multiple parameter pairs fit.

Question 8

What would you do when two very different parameter sets fit S21 equally well?

Answer and rationale: Report nonidentifiability and seek independent thickness, frequency, reflection or calibration information rather than asserting one unique material.

Beyond the answer key

For at least two numerical problems, write a second route to the result using a limiting case or power/charge conservation argument. For conceptual questions, identify a counterexample to an overgeneralized version of the answer. A problem is not fully understood until the applicable physical assumptions are stated alongside the formula.