Transmission lines
A quarter-wave length of cable turns 100 Ω into 25 Ω without a single component. At those frequencies, the length of the cable is part of the circuit.
01When a cable stops being a cable
In circuit theory, a cable is a node: the voltage is the same at both of its ends at the same instant. That holds as long as the signal takes much less time to travel along it than it takes to change. When the length of the cable approaches an appreciable fraction of the wavelength —the rule of thumb is one tenth— or when the rise time of a pulse is comparable to the travel time, the cable becomes a transmission line: the signal travels along it as a wave, and what happens at one end depends on what is at the other. The page on lumped models in Circuit Theory I marked that limit.
02The distributed model
A short section of line is modeled with four parameters per unit length: the resistance \( R \) and the inductance \( L \) of the conductors, and the conductance \( G \) and the capacitance \( C \) between them. From these come the telegrapher's equations, which describe a wave traveling in both directions, with two parameters that characterize the line:
| Line | \( Z_0 \) | Velocity |
|---|---|---|
| RG-58 coaxial (polyethylene) | 50 Ω | 0.66 c |
| RG-6 coaxial (television) | 75 Ω | 0.82 c (foam) |
| Data twisted pair | 100 Ω | Around 0.65 c |
| 1.9 mm microstrip on 1 mm FR-4 | About 50 Ω | Around 0.55 c |
In a coaxial cable, \( Z_0 = \dfrac{60}{\sqrt{\varepsilon_r}} \ln \dfrac{b}{a} \), with \( a \) the radius of the inner conductor and \( b \) that of the shield: the impedance depends on the geometry and the dielectric, not on the length.
03Reflection and standing waves
If the line ends in a load \( Z_L \) different from \( Z_0 \), part of the wave is reflected, just like the plane wave at the boundary between two media:
The incident and the reflected waves add with phases that depend on position, and the voltage along the line has fixed maxima and minima separated by half a wavelength. For the same reason, the impedance seen from the input depends on the length:
Solving this expression by hand is tedious, and that is why the Smith chart exists, which turns it into a rotation on a graph.
A lossless line of impedance \( Z_0 \) ends in a load \( R_L + jX_L \). The graph shows the peak value of the voltage along the line, from the load (on the right) toward the generator, for an incident wave of amplitude 1. The dot marks the position where the input impedance is calculated.
04Pulses and reflectometry
In the time domain, a step entering the line travels to the load and comes back reflected with an amplitude \( \Gamma_L \) times the incident one, after a time equal to twice the delay of the section. If the source is not matched either, the reflection bounces back again, and the voltage settles in a staircase of echoes. Time-domain reflectometry (TDR) uses this principle to locate faults: the time of the echo gives the distance and its shape, the type of fault. This is how a break in a buried cable or a discontinuity in a board trace is found.
05Where it appears in electronics
| Case | What matters about the line |
|---|---|
| High-speed bus on a board | Controlled-impedance traces and terminations to avoid reflections |
| Antenna cable of a transmitter | Low SWR so that the power reaches the antenna |
| Adapter between 50 and 75 Ω | A quarter-wave section of intermediate impedance |
| Ethernet network | 100 Ω twisted pairs terminated at each end |
| Locating faults in cables | Time-domain reflectometry |
06In the lab
With a fast pulse generator, 20 m of RG-58 and an oscilloscope at the input, observe the echo with the far end open, shorted and terminated in 50 Ω. Measure the delay and calculate the propagation velocity.
With a network analyzer, measure the input impedance of a section of coaxial cable terminated in 100 Ω and find the frequency at which it equals 25 Ω. Relate it to the length of the section.
Measure the SWR of an amateur radio antenna with an SWR meter across its band and locate the frequency of best match.
07Common mistakes
- Treating as a node a cable that is long compared with the wavelength. The voltage changes along the cable.
- Believing that \( Z_0 \) depends on the length. It depends only on the cross-section and the dielectric.
- Measuring with a 1 MΩ probe at the end of an unterminated 50 Ω line. The pulse is reflected and the waveform lies.
- Forgetting the velocity of the dielectric when calculating a quarter wave. In RG-58, the physical length is 66% of the electrical one.
08Self-assessment
What SWR does a 25 Ω load produce on a 50 Ω line?
\( \Gamma = -1/3 \), SWR = 2.
What impedance is seen a quarter wave away from a 100 Ω load on a 50 Ω line?
\( 50^2 / 100 = 25 \) Ω.
How long is a quarter wave of RG-58 at 100 MHz?
\( 0.66 \times 3\text{ m} / 4 \approx 0.50 \) m.
What is seen on a TDR if the cable is cut?
An echo of the same polarity as the pulse (open circuit, \( \Gamma = +1 \)) at the time that corresponds to the distance of the break.
09Further reading
- David M. Pozar. Microwave Engineering. 4th ed., Wiley, 2011. Its chapter on transmission line theory is the reference, with the Smith chart and matching.
- Howard Johnson and Martin Graham. High-Speed Digital Design: A Handbook of Black Magic. Prentice Hall, 1993. Lines seen from digital design: reflections, terminations and crosstalk.
- Fawwaz T. Ulaby and Umberto Ravaioli. Fundamentals of Applied Electromagnetics. 7th ed., Pearson, 2015. Lines in the sinusoidal and transient regimes, with many figures.