Transmission line theory describes how voltage and current travel as waves along a pair of conductors when the conductors are long compared with the signal’s wavelength. Instead of treating a connection as a node with one voltage, it treats the line as a distributed circuit with a characteristic impedance, a propagation delay and possible reflections. As a working rule, any interconnect longer than about one tenth of a wavelength should be analyzed as a transmission line.
This article is the starting point for the transmission line section of the site and sits under RF fundamentals. It explains why ordinary circuit theory stops working, introduces the core equations, and points to a more detailed article on each part of the subject.
Why circuit theory stops working at RF
Kirchhoff’s laws assume that a change at one end of a wire appears at the other end at once. That is a good approximation when the signal period is long compared with the time the signal takes to cross the wire. At radio frequencies the two times can be similar. The voltage at the far end then belongs to an earlier moment of the waveform than the voltage at the near end, and the two can even have opposite signs.
Two consequences follow. A conductor no longer has a single voltage along its length, so the phase of the signal depends on position. The impedance seen at the input also stops being a simple property of the load, because the line transforms it. Neither effect appears in a lumped schematic, and both are large enough to detune a matching network or ruin a measurement.
Electrical length
The useful measure is not physical length but electrical length, the length expressed in wavelengths or degrees of phase:
θ = βl = 360° × l / λ, with λ = vp / f
Here vp is the phase velocity, c / √εeff for a non-magnetic line with effective permittivity εeff. Figure 1 plots the one-tenth-wavelength length for three common media. A trace that is harmless at VHF sits well above the threshold in the 2.4 GHz band.
Worked example: is this PCB trace a transmission line?
Take a 75 mm microstrip trace on FR-4 with εeff ≈ 3.3. The phase velocity is vp = c / √3.3 = 1.65 × 108 m/s, or 0.55c. The one-way delay is 0.075 / (1.65 × 108) = 454 ps.
- At 10 MHz the wavelength is 16.5 m. The trace is 0.0045 wavelengths long, a phase shift of 1.6°. It behaves as a wire with some capacitance. The λ/10 limit here is 1.65 m.
- At 2.4 GHz the wavelength is 68.8 mm. The same trace is 1.09 wavelengths long, a phase shift of 393°. The λ/10 limit is 6.9 mm, about one eleventh of the trace length, so width, reference plane and termination all matter.
The copper has not changed. Only the ratio of length to wavelength has.
The distributed model of a line
Transmission line theory divides the line into sections of length Δz, each far shorter than a wavelength, so that each one can be treated as a lumped circuit. Every section has series resistance and inductance and shunt conductance and capacitance, as Figure 2 shows. R, L, G and C are quantities per unit length, set by the geometry and materials.
Applying Kirchhoff’s laws to one section and letting Δz approach zero gives the telegrapher’s equations:
∂V/∂z = −(R + jωL) I
∂I/∂z = −(G + jωC) V
Combining them yields a wave equation whose solutions are one wave travelling forward and one travelling backward:
V(z) = V+e−γz + V−e+γz
Two quantities describe the result. The propagation constant γ = α + jβ = √((R + jωL)(G + jωC)) sets the attenuation α and the phase constant β. The characteristic impedance Z0 = √((R + jωL)/(G + jωC)) is the ratio of voltage to current in a single travelling wave. For a low-loss line at RF these reduce to Z0 ≈ √(L/C) and vp ≈ 1/√(LC). The article on transmission line parameters works through these relations with numbers, including the low-frequency region where the approximations fail.
Characteristic impedance is not a resistance
A 50 Ω line does not dissipate power the way a 50 Ω resistor does. Z0 is the ratio a travelling wave imposes between voltage and current. An ideal lossless line absorbs nothing, yet a source connected to a long line sees Z0 until the first reflection returns, because energy is being stored in the inductance and capacitance of each new section the wave reaches. That energy keeps moving, which is why the line looks resistive to the source.
Z0 depends only on cross-section and materials. Length does not enter into it, so cutting a cable shorter leaves Z0 unchanged.
Reflections and input impedance
When a wave reaches a load that differs from Z0, the voltage and current cannot keep the ratio the wave requires, so part of the wave returns. The size of that returned wave is the reflection coefficient:
Γ = (ZL − Z0) / (ZL + Z0)
The forward and reflected waves add to form a standing wave. Their combination changes the impedance seen along the line, and on a lossless line of length l the input impedance is:
Zin = Z0 (ZL + jZ0 tan βl) / (Z0 + jZL tan βl)
As a short illustration, a 100 Ω resistor at the end of a 50 Ω line one eighth of a wavelength long presents 40 − j30 Ω at the input. A pure resistance has become a capacitive load through line length alone. Half a wavelength reproduces the load exactly, and a quarter wavelength inverts it to Z02/ZL. The Smith chart exists largely to make this transformation easy to see and use.
Transmission line topics in this section
The six companion articles and the related pages below cover each area in depth.
Theory and behaviour
- Characteristics and Types of Transmission Lines: how coax, parallel wire, planar lines and waveguide compare.
- Transmission Line Parameters: RLGC values, the propagation constant and complex Z0.
- Wave Propagation in Transmission Lines: phase and group velocity, step response and standing waves.
- Reflection and Transmission Coefficients: Γ, τ, complex loads and junctions between lines.
- Transmission Line Terminations: matched, open, short and reactive loads, and stubs.
- Transmission Line Losses and Power Handling Capacity: attenuation, heating and voltage breakdown.
Line types and tools
- Microstrip calculator: width and impedance of the most common PCB line.
- Stripline: the buried PCB line with both reference planes.
- Coplanar waveguide: signal and grounds on the same layer.
- Waveguides: rectangular guide, cutoff and modes.
- VSWR, return loss and mismatch loss: the measures of reflection, with a calculator.
- Smith charts for RF engineers: impedance transformation on one graph.
Frequently asked questions
When does a wire need to be treated as a transmission line?
A common rule is that transmission line analysis is needed once the length exceeds about one tenth of a wavelength in the medium. The boundary is not sharp. Shorter connections can still matter in precise matching work, and for digital signals the rise time sets the relevant bandwidth rather than the clock rate.
What does characteristic impedance actually mean?
It is the ratio of voltage to current in one wave travelling along the line in one direction. It is fixed by the conductor geometry and dielectric, not by length. A line terminated in its characteristic impedance produces no reflection, so the input looks like Z0 at any length.
Why is 50 Ω used so widely in RF systems?
It is a compromise between the coaxial impedance for lowest loss, near 77 Ω in air, and the impedance for highest peak power, near 30 Ω. The article on line losses and power handling plots that trade-off. Video and cable television systems often use 75 Ω because low loss matters more to them than power.
Does a lossless line consume power?
No. An ideal lossless line stores energy and passes it along, and all power entering the line reaches the load or returns to the source. Real lines add loss through conductor resistance and dielectric absorption, with both effects growing with frequency.
Related topics
Readers working on real boards should continue with the FR-4 versus Rogers comparison, and those measuring lines will need S-parameters. Matching a load to a line is covered by the impedance matching calculator.