Electromagnetic Solver Calculator

This solver uses the standard closed-form plane-wave solution in a homogeneous, isotropic, linear medium.

• ω = 2πf, ε = ε0εr, μ = μ0μr
• Define p = σ/(ωε) and R = √(1+p²)
• Propagation constant: γ = α + jβ
• α = ω √(με/2) · √(R − 1)
• β = ω √(με/2) · √(R + 1)
• Wavelength and phase velocity: λ = 2π/β, v_p = ω/β
• Envelope decay: |E(z)| = E0 e^−αz
• Intrinsic impedance: η = √(jωμ/(σ + jωε))

For σ → 0, the medium becomes (approximately) lossless and α approaches zero.

1. Enter the wave frequency and select its unit.
2. Provide material properties: εr, μr, and conductivity σ.
3. Set distance z to evaluate how much the field decays.
4. Set E0 to scale the field magnitude at z = 0.
5. Click Solve to view results above the form.
6. Use Download CSV or Download PDF for reports.

1) What this calculator models

This calculator solves the uniform plane‑wave problem in a homogeneous material using frequency, relative permittivity, relative permeability, and conductivity. It reports propagation constants (α, β), intrinsic impedance magnitude and phase, wavelength, phase velocity, and the exponential field envelope at a chosen distance.

2) Inputs that matter most

Frequency sets the angular frequency ω, which directly scales both attenuation and phase terms. The pair (εr, μr) determines the baseline wave speed and impedance, while conductivity σ introduces loss. A convenient diagnostic is the loss ratio σ/(ωε): when it is much less than 1, the medium behaves nearly lossless.

3) Typical material data points

Common engineering ranges help with realistic scenarios. Air is near εr ≈ 1 and μr ≈ 1 with negligible σ. Many plastics fall around εr ≈ 2–3. FR‑4 laminates are often near εr ≈ 4 with loss that increases with frequency. Good conductors can have very large σ, producing strong attenuation and shallow penetration.

4) Interpreting attenuation α

Attenuation α is given in nepers per meter (Np/m). A practical conversion is dB/m ≈ 8.686·α, which the calculator also provides. Higher σ or higher ω generally increases α, meaning the field magnitude drops rapidly with distance. Use the distance input z to evaluate the envelope |E(z)|.

5) Phase constant β and wavelength

The phase constant β (rad/m) sets how fast the wave oscillates in space. Wavelength follows λ = 2π/β, so larger β means shorter λ. Short wavelengths in high‑εr media can impact antenna spacing, PCB trace behavior, and electromagnetic shielding design.

6) Impedance and E–H relationship

The intrinsic impedance η links electric and magnetic fields: |H| ≈ |E|/|η| for the magnitude. In lossy media, η is complex, so the calculator also outputs an impedance angle. For low‑loss cases, η approaches √(μ/ε), which is useful for estimating reflections and power flow.

7) Skin depth as a design indicator

Skin depth δ is reported as approximately 1/α in this model. It gives a quick estimate of how deep fields penetrate before decaying significantly. When δ is small compared with the structure thickness, currents concentrate near surfaces and shielding becomes more effective, but losses can rise.

8) Reporting and verification workflow

Start by entering catalog or measured material properties, then sweep frequency and σ to see sensitivity. Compare λ against physical dimensions to anticipate resonance or interference. Export CSV for spreadsheets and PDF for documentation. For critical designs, validate inputs against material datasheets and measurement conditions.

1) What does σ/(ωε) tell me?

It compares conduction current to displacement current. Values ≪ 1 indicate a near‑lossless dielectric, while values ≫ 1 indicate conductor‑like behavior with strong attenuation and shallow penetration.

2) Why is α sometimes close to zero?

If conductivity is very small or frequency is high enough that σ/(ωε) is tiny, the medium is effectively lossless. In that limit, attenuation becomes negligible and the wave mainly accumulates phase.

3) How do I use z and E0?

E0 sets the field magnitude at z = 0. The calculator applies |E(z)| = E0·e^{−αz} to estimate the envelope at distance z. This helps evaluate decay through materials or along paths.

4) Is the impedance output useful for reflections?

Yes. The intrinsic impedance of each medium is a core input for reflection and transmission estimates. In lossy materials, the angle indicates a phase shift between E and H that can affect power flow.

5) What units should I use for conductivity?

Use S/m for most material data. The mS/m and uS/m options help when dealing with weakly conductive dielectrics. Converting correctly is important because σ strongly influences attenuation.

6) Does this replace full Maxwell solvers?

No. This is a fast, closed‑form plane‑wave model for homogeneous media. It is excellent for estimates and learning, but complex geometries, boundaries, and sources require numerical solvers.

7) Why might my results differ from a datasheet?

Material properties depend on frequency, temperature, moisture, and test method. Datasheets often specify εr and loss under particular conditions. Use the closest available parameters, and consider measurement uncertainty for precise work.