Electrical calculator

Coax Impedance Calculator

Compute coaxial characteristic impedance Z0 from shield inner diameter D, center-conductor diameter d, and dielectric εr. Ideal TEM, μr≈1. Runs locally in your browser.

Instant result

Result

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Enter values to calculate.

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Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance

Input interpretation: Enter values to calculate.
Result: —
Assurance: Engineering
Declared partition coverage: PASS · 2/2 declared partitions (z0, invalid-domain) · Matrix
Known limitations: Ideal TEM coax; homogeneous, lossless, non-magnetic dielectric (μr≈1); conductor loss and dielectric dispersion neglected. Omitted εr defaults to 1.0.
Core CVP does not include live graph, viewport, or pointer interaction.
Model: Coaxial Z0 and VF.
Scope: Ideal TEM coaxial transmission line
Verification: Engine tested · Source checked · v1.0.1 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance· View Manifest · CVP overview · Specification
Versions: Calculation 1.0.1 · CVP protocol 1.0.0-proposed
CVP identity: 5/5 property · digest eb5f53204f54
Legacy regression: 21/21 tests · Production surface contract 4/4
Trust layers: Verification VERIFIED · Production CURRENT · overall VERIFIED
Reference: O1 model · O2 expected_values · O2 numerical_behavior
Interfaces: PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review: Internal · Pass · rf-engineering
Named expert review: Not performed
CVP suite: 1/1 golden · 9/9 CVP boundary · 18/18 invalid · 5/5 property · 2/2 metamorphic · 3/3 cross-interface · 4/4 CVP contract · Manifest
Sources: David M. Pozar, Microwave Engineering — 4th ed., Ch. 2 Transmission Line Theory — coaxial line
Robert E. Collin, Foundations for Microwave Engineering — 2nd ed., coaxial transmission line
Sources
Evidence: 4 legacy golden · 8 legacy boundary · legacy regression suite · 1/1 oracle-backed golden · 18/18 invalid · Artifact integrity PASS · CalculatorX physics/RF review
Semantic contract: PASS

Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Z0Z0 = (η0/(2π√εr)) · ln(D/d)

VFVF = 1/√εr

iη0 ≈ 376.73 Ω. D = shield (outer conductor) inner diameter. d = center conductor outer diameter. Homogeneous lossless dielectric with μr≈1.

How to use

Enter shield inner diameter D

Inner diameter of the outer conductor / shield — not the cable jacket OD. D and d must use the same unit; D > d > 0.

Enter center conductor diameter d

Outer diameter of the inner conductor. Typical coax uses millimetres or inches.

Enter relative permittivity εr

Dielectric filling the coax (≈2.25 solid PE, ≈2.1 PTFE). Omit in the API to default to 1.0 (air/vacuum approximation).

Read Z0 and velocity factor

Z0 = (η0/(2π√εr)) · ln(D/d). VF = 1/√εr under the same non-magnetic assumption.

Example calculations

Common configurations with formula and result.

50 Ω solid PE

D/d≈3.493 · εr=2.25

(η0/(2π√2.25))ln(3.493)

50 Ω

75 Ω solid PE

D/d≈6.529 · εr=2.25

(η0/(2π√2.25))ln(6.529)

75 Ω

Solid PE example

D=6.6 mm · d=1.6 mm · εr=2.25

(η0/(2π√2.25))ln(4.125)

≈56.643 Ω

Coax Impedance calculator specification

Version 1.0.1 · Engine tested · Supplemental domain review · Internal · 2026-09-12

Calculation status

Engine tested 21/21 tests · Production surface contract 4/4
Supplemental domain review Internal · Pass · rf-engineering · 2026-09-12
Named expert review Not performed
Calculation version 1.0.1

Review policy · Evidence · Reviewed by CalculatorX physics/RF review (rf-engineering)

Definition: Ideal TEM coax with non-magnetic dielectric (μr≈1): Z0 = (η0/(2π√εr))·ln(D/d) and VF = 1/√εr. D is the shield inner diameter; d is the center-conductor outer diameter.
What it calculates: Coaxial Z0 and VF.
Inputs: D_m
d_m
er?
Outputs: Z0_ohm
VF
er
Formula: Z0=(η0/(2π√εr))ln(D/d)
Assumptions: Ideal TEM coaxial transmission line
Homogeneous dielectric
Lossless / low-loss approximation
Non-magnetic dielectric, μr ≈ 1
Omitted εr defaults to 1.0 (air/vacuum approximation)
Units: m → Ω
Boundary conditions: D≤d → VALUE_OUT_OF_RANGE
non-positive sizes or εr → VALUE_MUST_BE_POSITIVE
Example: D=6.6mm d=1.6mm εr=2.25 → ≈56.643 Ω
Validation cases: 4 published on this page · 21/21 tests · Production surface contract 4/4 · View evidence
D=0.0066 d=0.0016 er=2.25 → Z0≈56.643 Ω
preset-50ohm-pe D/d≈3.493364655 er=2.25 → Z0≈50 Ω
preset-75ohm-pe D/d≈6.529288864 er=2.25 → Z0≈75 Ω
D=0.0066 d=0.0016 er omitted → er=1.0 · Z0≈84.965 Ω
Sources: David M. Pozar, Microwave Engineering — 4th ed., Ch. 2 Transmission Line Theory — coaxial line · 2012
Supports: Ideal TEM coax Z0 = (η0/(2π√εr)) ln(D/d) with μr≈1
Robert E. Collin, Foundations for Microwave Engineering — 2nd ed., coaxial transmission line · 2001
Supports: Z0 ∝ ln(D/d)/√εr for homogeneous non-magnetic dielectric
Last reviewed: 2026-09-12
Reviewed by: CalculatorX physics/RF review (rf-engineering)
Calculation version: 1.0.1

Background

Interpretation and common distinctions.

Status: Internally reviewed — RF engineering · pin calculation_version: 1.0.1.

v1.0.1 changelog: Review label aligned to L2 (reviewKind: internal). Human inputs now name shield inner diameter / center-conductor diameter, with millimetre-first units. Result provenance includes D and d. Model scope states μr≈1. er omitted defaults to 1.0.

What is characteristic impedance?

Characteristic impedance Z₀ is the ratio of voltage to current for a travelling TEM wave. It is not the DC loop resistance of the cable.

Why 50 Ω and 75 Ω?

50 Ω is the common RF / test-system compromise. 75 Ω is common in video and CATV, closer to the loss minimum of typical PE dielectrics. This calculator does not pick a standard for you — it reports Z₀ from geometry and varepsilonᵣ.

What this ideal model ignores

Ideal TEM; homogeneous lossless dielectric with μᵣ ≈ 1. Conductor loss, dielectric loss, dispersion, imperfect shields, connectors, and higher-order modes are out of scope.

Supported and not supported

Supported — Z0 + VF from D, d, εr · API electrical.rf.coax · omitted er defaults to 1.0 (air/vacuum approximation)

Not supported — Attenuation, skin-effect R, magnetic dielectrics (μᵣ ≠ 1), higher-order modes

Agent / API notes

Capability id: electrical.rf.coax · tool id: coax-impedance · pin 1.0.1.

Canonical inputs are SI metres (D_m, d_m). er is optional; omitted εr defaults to 1.0 (air/vacuum approximation). Human UI may enter mm / cm / in; share URLs stay SI.

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Frequently asked questions

Key distinctions behind the calculation.

Where do I measure D and d?

D is the inner diameter of the shield / outer conductor. d is the outer diameter of the center conductor. Do not use the cable jacket outer diameter for D.

What if D ≤ d?

The API returns VALUE_OUT_OF_RANGE — shield inner diameter must exceed the center conductor.

What does omitted εr mean?

εr is optional. If omitted, the engine uses 1.0 (air/vacuum approximation). Non-positive εr returns VALUE_MUST_BE_POSITIVE.

Does this include magnetic dielectrics?

No. The published formula assumes μr≈1. A magnetic filling would use Z0 = (η0/2π)·√(μr/εr)·ln(D/d).