Transmission coefficient calculator

Non-positive impedances are rejected with invalid_input because the intrinsic impedance of a lossless medium is a strictly positive real number, not zero or a negative placeholder.

How to call the API and what comes back

Values at or below about one times ten to the minus fifteen, or below the published minimum positive floor, are treated as non-positive and rejected. Send a JSON body with two required finite numbers: z1 for the incident-medium intrinsic impedance in ohms (aliases eta1, z_incident, impedance_1) and z2 for the second medium (aliases eta2, z_transmitted, impedance_2). Both must be strictly positive and stay within the published absolute limit of one trillion so the sum, product, and squared ratio remain finite in IEEE double precision on the Worker and in the browser. On success the response echoes rounded z1 and z2, sets power_transmission and T to the same power transmission coefficient as a fraction between zero and one, power_transmission_percent to one hundred times T, power_reflection and R to the complementary reflected fraction, amplitude_reflection to Γ, amplitude_transmission to τ, formula to the string T equals 4 η₁ η₂ over (η₁ plus η₂) squared, units for ohm impedances and fraction power coefficients, message to a short English summary, and steps to an ordered array of substitution lines. Failed validation—missing fields, non-finite numbers, values outside absolute limits, or non-positive impedances—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item. Prefer sending JSON numbers rather than pre-formatted strings when you control the client; numeric strings that match a strict finite pattern are still accepted so spreadsheet paste paths keep working.

Practical tips, limits, and related searches

Use the intrinsic impedances at the frequency of interest when the media are dispersive: μ and ε of real materials change across microwave bands, so a single η pair is a monochromatic approximation. Keep both impedances inside the published positive range so the denominator cannot collapse and the transmission coefficient cannot overflow into a non-finite value that the solver must reject. Prefer this capability when you already have two positive intrinsic impedances and only need the head-on power transmission coefficient; if your source is a material datasheet with εᵣ and μᵣ, first form η equals η₀ times the square root of μᵣ over εᵣ and then call this tool. When teaching, pair examples: free-space η₁ equals 377 ohms onto η₂ equals 150 ohms yields a clear mismatch with T below one; equal impedances η₁ equals η₂ yield T equals one and R equals zero; a small but still positive η₂ demonstrates a near-conductor interface with tiny transmitted power; swapping the two media changes Γ and τ but leaves T and R unchanged for lossless real η; and z1 or z2 equal to zero demonstrates the hard invalid_input path. This landing targets people searching for a transmission coefficient calculator, power transmission from intrinsic impedance, or electromagnetic interface T from η₁ and η₂; it is not an angle-dependent Fresnel curve tool, not a Brewster-angle polarizer calculator, not a critical-angle total-internal-reflection tool, and not a multilayer thin-film stack solver. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when learners deliberately send non-positive η. Round-trip checks are easy: recompute 4 η₁ η₂ over (η₁ plus η₂) squared by hand or in a calculator and confirm it equals power_transmission within the same twelve-decimal rounding, then verify that T plus R equals one.