Transmission Line Impedance Calculator Formula

Understand the math behind the transmission line impedance calculator. Each variable explained with a worked example.

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Formulas Used

Characteristic Impedance

z0 = 138 / sqrt(dielectric_constant) * log10(outer_diameter_mm / inner_diameter_mm)

Velocity Factor

velocity_factor = 1 / sqrt(dielectric_constant)

Capacitance per Meter

capacitance_pf_m = 24.13 * dielectric_constant / log10(outer_diameter_mm / inner_diameter_mm)

Variables

VariableDescriptionDefault
outer_diameter_mmOuter Conductor Inner Diameter(mm)7
inner_diameter_mmInner Conductor Outer Diameter(mm)1.63
dielectric_constantDielectric Constant (Er)2.3

How It Works

Coaxial Cable Characteristic Impedance

The characteristic impedance of a coaxial transmission line depends on the ratio of conductor diameters and the dielectric material.

Formula

Z_0 = (138 / sqrt(Er)) x log10(D / d)

Where D is the outer conductor inner diameter, d is the inner conductor outer diameter, and Er is the relative dielectric constant.

Common Values

  • 50 ohm: Standard for RF and instrumentation
  • 75 ohm: Video and cable TV
  • 93 ohm: Early computer networks

    • The velocity factor is 1/sqrt(Er), which determines signal propagation speed.

    Worked Example

    Coaxial cable with D=7.0 mm, d=1.63 mm, polyethylene dielectric (Er=2.3).

    outer_diameter_mm = 7inner_diameter_mm = 1.63dielectric_constant = 2.3
    1. 01Diameter ratio: 7.0 / 1.63 = 4.294
    2. 02log10(4.294) = 0.633
    3. 03Z0 = 138 / sqrt(2.3) x 0.633 = 138 / 1.517 x 0.633 = 57.6 ohms
    4. 04Velocity factor: 1 / sqrt(2.3) = 0.659

    Frequently Asked Questions

    Why are 50 and 75 ohms standard?

    50 ohms balances power handling and attenuation. 75 ohms minimizes attenuation and matches the impedance of dipole antennas.

    How does dielectric constant affect impedance?

    Higher dielectric constant lowers impedance for the same physical dimensions. It also reduces velocity factor, making the cable electrically longer.

    Can I use this for other transmission line types?

    This formula is for coaxial lines. Microstrip, stripline, and waveguide have different impedance formulas based on their geometry.

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