Why are 50 ohms and 75 ohms the standard coaxial impedances?

The choice of 50 and 75 ohms comes from optimizing different performance parameters. The impedance that minimizes attenuation (conductor loss) in an air-dielectric coaxial line is 77 ohms (at the b/a ratio of 3.59 where the product of current and resistance is minimized). The impedance that maximizes power handling capacity is 30 ohms (at b/a of 1.65 where the peak electric field at the inner conductor surface is minimized). The geometric mean of these two optima is sqrt(77*30) = 48 ohms, which was rounded to 50 ohms as a practical compromise for general-purpose RF systems. The 75-ohm standard evolved from the video industry, where it is close to the minimum-loss impedance (77 ohms) and matches the impedance of folded dipole antennas used for television reception. With PTFE dielectric (epsilon_r = 2.1), the minimum-loss impedance shifts to 77/sqrt(2.1) = 53 ohms, further justifying the 50-ohm choice for solid-dielectric cables.

What causes attenuation in coaxial lines?

Three mechanisms cause signal loss. Conductor loss is the dominant mechanism for most cables, arising from the finite conductivity of the center and outer conductors. At RF frequencies, current flows only within the skin depth (delta = sqrt(2/(omega*mu*sigma))), which is 2.1 micrometers in copper at 1 GHz. Conductor loss scales as sqrt(f) because skin depth decreases with frequency, concentrating current in a thinner layer with higher effective resistance. Dielectric loss is caused by the dissipation factor (tan delta) of the insulating material between conductors. It scales linearly with frequency and dominates in low-loss cables at high frequencies. PTFE has tan delta of 0.0002 (very low), foam polyethylene is 0.0001, while standard polyethylene is 0.0004. Radiation loss occurs when shield coverage is imperfect (braided shields have 85 to 95% coverage) and increases with frequency. Solid outer conductors (semi-rigid cables, corrugated hardline) have essentially zero radiation loss. For RG-58 (50 ohm, solid PE) at 1 GHz: total attenuation is approximately 0.5 dB/m, of which 85% is conductor loss and 15% is dielectric loss.

What is the upper frequency limit of a coaxial line?

The upper frequency limit is set by the cutoff of the first higher-order mode (TE11), above which the coaxial line supports multiple propagation modes that cause signal distortion and connector mismatch. The TE11 cutoff frequency is approximately fc = c/(pi*(a+b)*sqrt(epsilon_r)), where the mean circumference equals one wavelength. For standard connectors: SMA (3.5 mm outer) has fc of approximately 26 GHz, 2.92 mm connectors reach 40 GHz, 2.4 mm reaches 50 GHz, 1.85 mm reaches 67 GHz, and 1.0 mm reaches 110 GHz. Below cutoff, the TEM mode provides dispersionless propagation (all frequencies travel at the same velocity), which is why coaxial lines are used for broadband signal transmission. Precision air-dielectric coaxial lines (used in metrology) have the best performance because air has epsilon_r = 1 (maximum cutoff frequency) and zero dielectric loss. The 7 mm precision airline (APC-7 connector) is the primary impedance standard for RF calibration from DC to 18 GHz.

Transmission Lines

Coaxial Line

/koh-ak-see-ul line/

A coaxial line propagates the TEM mode from DC to the TE11 cutoff frequency, with electric field radial and magnetic field circumferential between concentric conductors. Characteristic impedance Z₀ = (60/√ε_r) ln(b/a). Standard: 50 Ω (RF/microwave) and 75 Ω (video/CATV). Shielding of 60 to 120 dB. Connector standards from SMA (26 GHz) through 1.0 mm (110 GHz). The foundational transmission line of RF engineering.

Category: Transmission Lines

Mode: TEM (dispersionless)

Impedance: 50 or 75 Ω

Understanding Coaxial Lines

The coaxial transmission line is the most ubiquitous RF interconnect, used in everything from cable television distribution to particle accelerator RF feeds, from cell phone base station jumpers to satellite ground terminal connections. Its enduring dominance comes from a unique combination of properties: broadband operation from DC with no low-frequency cutoff, excellent shielding from the complete enclosure of the field by the outer conductor, mechanical flexibility (in cable form), and standardized connector interfaces that enable interoperability across manufacturers.

The TEM mode in a coaxial line is particularly valuable because it is dispersionless: all frequencies propagate at the same phase velocity v = c/√ε_r regardless of frequency. This means a coaxial cable transmits broadband signals without distortion from dispersion, in contrast to waveguides where the phase velocity varies with frequency. The only signal degradation comes from attenuation (which does vary with frequency but does not cause pulse spreading), making coax the preferred medium for analog signal distribution and broadband digital communications. The TEM mode exists from DC up to the first higher-order mode cutoff, above which mode conversion causes unpredictable behavior. This cutoff frequency, determined by the coax dimensions and dielectric, establishes the maximum usable frequency for each connector and cable type.

Coaxial Line Equations

Characteristic Impedance:
Z₀ = (60/√ε_r) × ln(b/a) (Ω)

TE11 Cutoff Frequency:
f_c ≈ c / (π(a+b)√ε_r)

Conductor Attenuation:
α_c = R_s(1/a + 1/b) / (4πZ₀) (Np/m)

Where b = outer conductor inner radius, a = inner conductor outer radius, R_s = √(πfμ₀/σ) = surface resistance. For 50 Ω air-filled coax: ln(b/a) = 2.30, b/a = 2.30. Optimal for min loss: b/a = 3.59 (Z₀ = 77 Ω). Optimal for max power: b/a = 1.65 (Z₀ = 30 Ω).

Coaxial Connector Standards

Connector	Outer ø	Max Frequency	Impedance	Application
N-type	7.0 mm	18 GHz	50 Ω	Test, base station, lab
SMA	3.5 mm	18 to 26 GHz	50 Ω	General RF, PCB
2.92 mm (K)	2.92 mm	40 GHz	50 Ω	Ka-band, test cables
1.85 mm (V)	1.85 mm	67 GHz	50 Ω	V-band, 5G mmWave
1.0 mm (W)	1.0 mm	110 GHz	50 Ω	W-band, automotive radar

Common Questions

Frequently Asked Questions

Why 50 and 75 ohms?

Minimum-loss impedance (air): 77 Ω (b/a = 3.59). Maximum-power impedance: 30 Ω (b/a = 1.65). Geometric mean: √(77×30) = 48 Ω, rounded to 50 Ω as compromise. 75 Ω adopted for video/CATV (close to min-loss, matches folded dipole). With PTFE (ε_r=2.1), min-loss shifts to 53 Ω, further justifying 50 Ω.

What causes coax attenuation?

Conductor loss (dominant, scales as √f due to skin depth, 85% of total for RG-58 at 1 GHz). Dielectric loss (linear with f, depends on tanδ: PTFE 0.0002, foam PE 0.0001). Radiation loss (braided shields: 85 to 95% coverage; solid shields: zero). RG-58 at 1 GHz: ~0.5 dB/m total.

What is the upper frequency limit?

TE11 cutoff: f_c ≈ c/(π(a+b)√ε_r). SMA: ~26 GHz, 2.92 mm: 40 GHz, 1.85 mm: 67 GHz, 1.0 mm: 110 GHz. Below cutoff, TEM mode is dispersionless (all frequencies travel at v = c/√ε_r). Precision air-dielectric lines (APC-7) are primary calibration standards: DC to 18 GHz.

Related Terms

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