Transmission line parameters are the four distributed values R, L, G and C, per unit length, together with the quantities derived from them: the propagation constant γ and the characteristic impedance Z0. They describe how much of a signal survives along the line, how fast it travels and what impedance it sees. Geometry and materials fix the four primary values, and everything else follows by calculation.
This article defines each parameter and works through a complete example. It builds on the introduction to transmission line theory, part of the RF fundamentals material.
Primary parameters: R, L, G and C
The primary constants come from the distributed model of a line.
- R (Ω/m): series resistance of both conductors. It rises with the square root of frequency once the skin effect confines current to the surface.
- L (H/m): series inductance, from magnetic energy stored around the conductors.
- G (S/m): shunt conductance, from loss in the dielectric. At RF it is usually modelled as G = ωC tan δ, so it rises with frequency.
- C (F/m): shunt capacitance, from electric energy stored between the conductors.
In a line with a uniform dielectric, L and C are linked, because LC = μ0ε. Changing geometry to raise L lowers C by the same factor. This is why velocity depends on the dielectric and impedance depends on both geometry and dielectric. The article on types of transmission lines gives the coaxial formulas as an example.
Secondary parameters: the propagation constant
The secondary quantities describe the wave, and both are complex in general.
γ = α + jβ = √((R + jωL)(G + jωC))
Attenuation constant
The real part α, in nepers per metre, sets the decay of the wave as e−αz. Multiplying by 8.686 converts it to dB/m. For a low-loss line it splits neatly into conductor and dielectric parts:
α ≈ R / (2Z0) + GZ0 / 2
The physical sources of these two terms, and what they mean for a real cable run, are covered in transmission line losses and power handling capacity.
Phase constant
The imaginary part β, in radians per metre, sets the phase change per unit length. On a low-loss line, β ≈ ω√(LC). The wavelength on the line is λ = 2π/β.
Phase velocity
Phase velocity is vp = ω/β. For a low-loss line, vp ≈ 1/√(LC) and is independent of frequency, so all components of a signal arrive together. At lower frequencies the approximation fails and the line becomes dispersive. The wave propagation article looks at the time-domain view of this.
Characteristic impedance
The characteristic impedance is:
Z0 = √((R + jωL) / (G + jωC))
Three regions follow from this expression.
- High frequency (ωL ≫ R and ωC ≫ G): Z0 ≈ √(L/C). The value is real and nearly constant. This is the figure printed on a cable.
- Very low frequency (ωL ≪ R and ωC ≪ G): Z0 ≈ √(R/G). This is also real, but it can differ greatly from the RF value.
- Between the two: Z0 is complex and varies with frequency.
Worked example: from RLGC to Z0 and γ
Take a line with R = 0.8 Ω/m, L = 250 nH/m, G = 20 μS/m and C = 100 pF/m. To keep the arithmetic clear, the values are treated as constant, although R and G rise with frequency in a real line.
Calculation at 1 GHz
With ω = 2π × 109 rad/s, the series and shunt terms are:
- R + jωL = 0.8 + j1570.8 Ω/m
- G + jωC = 0.00002 + j0.62832 S/m
Taking the square roots of the ratio and product gives:
- Z0 = 50.000 − j0.012 Ω
- γ = 0.0085 + j31.42 per metre
From these results:
- Attenuation: α = 0.0085 Np/m, or 0.0738 dB/m.
- Phase velocity: vp = 2π × 109 / 31.42 = 2.000 × 108 m/s, a velocity factor of 0.667.
- Wavelength on the line: 200 mm.
The low-loss approximation agrees closely. R/(2Z0) = 0.0080 Np/m and GZ0/2 = 0.0005 Np/m, which sum to 0.0085 Np/m. Conductor loss dominates by a factor of 16 in this line.
Calculation at 1 kHz
At 1 kHz the same line behaves differently, because R and G now dominate over ωL and ωC.
- Z0 = 199.9 − j2.9 Ω, so |Z0| is about 200 Ω.
- α = 0.0040 Np/m, or 0.0347 dB/m.
- vp = 9.41 × 107 m/s, less than half the RF value.
Figure 1 shows the change across the whole range. The transition sits around the frequencies where ωL = R and ωC = G, at 509 kHz and 31.8 kHz for these values.
Figure 2 shows the same transition in phase velocity. In the transition region, different frequency components travel at different speeds, which distorts wideband pulses but rarely troubles a narrowband RF signal.
The distortionless line
A line can keep a constant velocity and attenuation at all frequencies if its primary values satisfy the Heaviside condition:
R/L = G/C
When this holds, Z0 = √(L/C) exactly, α = √(RG), and vp = 1/√(LC) at every frequency, so a pulse shrinks in amplitude but keeps its shape. The example line does not meet the condition. Its R/L is 3.2 × 106 per second and its G/C is 2 × 105 per second, a ratio of 16, the same factor that separates its conductor and dielectric loss at RF. Meeting the condition by raising G alone would need G = RC/L = 320 μS/m, and attenuation would rise to 0.016 Np/m, or 0.139 dB/m. Practical lines never add loss in this way. Long telephone circuits once used series loading coils to raise L instead, which is an early and well-known use of the same relation.
Measuring transmission line parameters
Primary values can be extracted from measurements rather than formulas.
- Open and short measurements: a short length of line measured with its far end open gives the shunt admittance, and with its far end shorted gives the series impedance. The product of the two input impedances equals Z02.
- Time-domain reflectometry: a step shows the impedance profile along a line directly, and delay gives velocity.
- Vector network analyzer: measured S-parameters of two lengths of line can yield γ and Z0, after de-embedding the fixtures.
Field solvers do the same job at the design stage, and the microstrip calculator provides closed-form estimates for that line type.
Frequently asked questions
What are the four primary transmission line parameters?
They are series resistance R, series inductance L, shunt conductance G and shunt capacitance C, all per unit length. R and G cause loss, while L and C store energy and set the velocity and impedance.
Is the characteristic impedance of a line ever complex?
Yes. Z0 is complex whenever R or G is significant relative to ωL or ωC. On low-loss RF lines the imaginary part is tiny, as in the worked example, but it grows at low frequency and on lossy on-chip interconnect.
What is the difference between α and β?
α describes the loss of amplitude with distance and β describes the change of phase with distance. Together they form the propagation constant γ = α + jβ. On a lossless line, α is zero.
Why does velocity depend on the dielectric but not the conductor spacing?
For a uniform non-magnetic dielectric, the product LC equals μ0ε for any geometry, so spacing changes L and C in opposite directions. Velocity 1/√(LC) therefore depends only on ε. Mixed dielectrics such as microstrip are the exception, where geometry changes εeff.
Related topics
These parameters feed directly into reflection and transmission coefficients and transmission line terminations. They also explain the differences between microstrip and stripline designs.