About

Impedance mismatch in transmission lines causes signal reflections quantified by the voltage standing wave ratio (VSWR). A mismatch of just 5Ω on a 50Ω system produces a return loss of roughly −26dB, degrading signal integrity in high-frequency circuits. This calculator computes the characteristic impedance Z₀ for four common geometries: coaxial, twin-lead (parallel wire), microstrip, and twisted pair. It derives secondary parameters - velocity factor VF, propagation delay t_pd, capacitance per meter C′, and inductance per meter L′ - from closed-form electromagnetic equations using the relative permittivity ε_r of the dielectric. Results assume lossless TEM-mode propagation and homogeneous dielectric fill; for microstrip, the Hammerstad-Jensen effective permittivity approximation is applied.

Accuracy degrades above frequencies where higher-order modes propagate or where skin-effect losses dominate. For coaxial lines this threshold is approximately f_c ≈ c₀ ÷ (π ⋅ (D + d) ⋅ 12). For PCB microstrip designs, always verify against a 2D field solver when W/h falls outside the range 0.1 - 10.

Formulas

The characteristic impedance Z₀ depends on cable geometry. All formulas assume lossless, TEM-mode propagation in a homogeneous dielectric with relative permittivity ε_r.

Coaxial Cable

Z₀ = 138√ε_r ⋅ log₁₀ (Dd)

Twin-Lead (Parallel Wire)

Z₀ = 276√ε_r ⋅ log₁₀ (2Sd)

Twisted Pair

Z₀ = 120√ε_r ⋅ ln (2Sd)

Microstrip (Hammerstad-Jensen)

ε_eff = ε_r + 12 + ε_r − 12 ⋅ 1√1 + 12h/W

For W/h ≤ 1:

Z₀ = 60√ε_eff ⋅ ln (8hW + W4h)

For W/h > 1:

Z₀ = 120π√ε_eff ⋅ [W/h + 1.393 + 0.667 ln(W/h + 1.444)]

Derived Parameters (all types):

VF = 1√ε_eff v_p = c₀ ⋅ VF t_pd = 1v_p L′ = Z₀v_p C′ = 1Z₀ ⋅ v_p

Where D = outer conductor inner diameter, d = inner conductor outer diameter (or wire diameter), S = center-to-center spacing, W = trace width, h = substrate height, ε_r = relative permittivity, ε_eff = effective permittivity (microstrip), c₀ = 299792458 m/s.

Reference Data

Material / Cable Type	ε_r	Typical Z₀ (Ω)	Velocity Factor	Application
Air (vacuum reference)	1.00	-	1.00	Open-wire, ladder line
PTFE (Teflon)	2.10	50 - 75	0.69	Premium coax (RG-142)
Polyethylene (solid)	2.25	50 - 75	0.66	RG-58, RG-59, RG-6
Foam Polyethylene	1.50	50 - 75	0.82	Low-loss coax (LMR-400)
Polypropylene	2.20	50 - 100	0.67	Twin-lead, some coax
FR-4 (PCB)	4.40	50 - 120	0.48	Microstrip, stripline
Rogers RO4003C	3.38	50	0.54	RF PCB, microwave
Rogers RO3010	10.20	50	0.31	Antenna substrates
Alumina (Al₂O₃)	9.80	50	0.32	Hybrid microwave ICs
PVC	3.40	75 - 120	0.54	Low-freq twisted pair
RG-58/U (Coaxial)	2.25	50	0.66	General RF, test leads
RG-59/U (Coaxial)	2.25	75	0.66	Video, CATV
RG-213/U (Coaxial)	2.25	50	0.66	High-power HF/VHF
LMR-400 (Coaxial)	1.52	50	0.85	Base station feeds
Cat5e (Twisted Pair)	2.00	100	0.64	Ethernet 1 Gbps
Cat6 (Twisted Pair)	2.00	100	0.65	Ethernet 10 Gbps
300 Ω TV Twin-Lead	1.00	300	0.82	VHF/UHF antenna feed
450 Ω Window Line	1.00	450	0.91	Ham radio balanced feed
Sapphire	11.50	-	0.29	mmWave substrates
Quartz (fused silica)	3.78	-	0.51	Precision RF delay lines

Frequently Asked Questions

Characteristic impedance is inversely proportional to the square root of the relative permittivity ε_r. Doubling ε_r from 2.25 to 4.50 reduces Z₀ by approximately 29%. This is why foam dielectrics (lower ε_r) are used in high-impedance, low-loss cables.

In microstrip geometry, electric field lines travel partly through the substrate and partly through the air above the trace. The effective permittivity ε_eff is a weighted average that accounts for this mixed dielectric environment. It always satisfies 1 < ε_eff < ε_r. Wider traces (higher W/h) concentrate more field in the substrate, pushing ε_eff closer to ε_r.

These closed-form equations assume TEM or quasi-TEM mode propagation with lossless conductors and homogeneous dielectric. For coaxial cable, the first higher-order mode (TE11) appears at f_c ≈ c₀ ÷ (π(D + d) ÷ 2). For microstrip, accuracy degrades above roughly 10GHz on FR-4 due to dispersion and dielectric loss tangent. Use full-wave EM simulators (HFSS, CST) for mmWave designs.

Surface roughness increases effective conductor resistance at high frequencies due to extended current path length along the rough surface. This does not change Z₀ directly (which is a geometric/dielectric property), but it increases attenuation (α) and introduces a small reactive component. The Hammerstad model for roughness correction scales loss by a factor of 1 + 2π arctan(1.4(Δ/δ)²), where Δ is RMS roughness and δ is skin depth.

50Ω cable minimizes the sum of attenuation and power-handling losses in air-dielectric coax (theoretical optimum is 30Ω for power, 77Ω for minimum loss; 50Ω is the geometric mean). It is standard for RF transmitters, test equipment, and cellular infrastructure. 75Ω cable is closer to the minimum-loss optimum and is standard for video, CATV, and receive-only antenna feeds where power handling is not critical.

This calculator computes single-ended microstrip impedance. Differential impedance Z_diff depends on the coupling between two adjacent traces (gap spacing s). A common approximation is Z_diff ≈ 2 ⋅ Z₀ ⋅ (1 − 0.48 ⋅ exp(−0.96 ⋅ s/h)). For tightly coupled traces (s < h), Z_diff drops well below 2Z₀.