Electrical Length
Describing a cable or a circuit-board trace as "3 meters long" tells you very little about how it will treat a signal. Three meters is a negligible fraction of a wavelength for a 1 MHz sine wave (about 5° of electrical length in typical coax) but about one and a half wavelengths at 100 MHz (about 545°). A much more powerful way to describe line length is in electrical degrees — a unit that captures the phase relationship between the signal entering one end of the line and the signal emerging at the other — or, for digital signals, as a delay compared with the signal’s rise time. Electrical length is the language of transmission line design, and it tells you when a connection must be treated as a transmission line at all.
- What Electrical Length Means
- Converting Between Physical and Electrical Length
- Calculator: Physical Length to Electrical Degrees
- Calculator: Electrical Degrees to Physical Length
- Special Electrical Lengths and Their Properties
- The Quarter-Wave Section in Detail
- The Half-Wave Section: Electrical Transparency
- When Is a Line Electrically Long?
- Practical Applications of Electrical Length
Electrical length divides the transmission line into phase increments of a traveling wave. A line that introduces exactly 90° of phase shift is a quarter-wave section; 180° is a half-wave section. The physical length depends on both the frequency and the cable's velocity factor.
View LargerWhat Electrical Length Means
Electrical length is the phase shift introduced by a transmission line — the number of degrees by which a traveling wave is delayed as it passes from one end of the line to the other. Because a complete cycle of a sine wave spans 360°, a line that is exactly one wavelength long at the operating frequency introduces exactly 360° of phase shift. A half-wave line introduces 180°, and a quarter-wave line introduces 90°.
The concept is closely related to what you already know about phase angles in AC circuits, but applied to traveling waves rather than lumped-element circuits. When a signal enters one end of a transmission line, it takes a finite time to reach the other end. During that time, the phase of the driving signal has advanced. The electrical length in degrees is simply that phase advance.
Because wavelength shortens inside a cable (due to velocity factor), the electrical length of a given physical length of cable depends on three things:
- The physical length of the cable
- The velocity factor of the cable
- The frequency of the signal
A 3-meter length of RG-58 (VF 0.66) is electrically 90° long at about 16.5 MHz, 180° at 33 MHz, and 360° at 66 MHz. The same 3 meters is only about 19° long at 3.5 MHz. Physical length alone tells you nothing about electrical behavior — you need all three parameters.
Converting Between Physical and Electrical Length
The fundamental formulas for converting between physical length and electrical degrees:
θ (degrees) = 360 × (physical length / λcable)
where λcable = (c × VF) / f
Simplified:
θ (degrees) = 360 × f × physical_length / (c × VF)
- f = frequency (Hz)
- physical_length = cable length (meters)
- c = 3 × 10⁸ m/s
- VF = velocity factor
In practical units with f in MHz and length in feet:
θ (degrees) = physical_length (feet) × f (MHz) × 1.0667 / VFElectrical degrees to physical length:
physical_length (meters) = θ × c × VF / (360 × f)
physical_length (feet) = θ × VF × 984 / (360 × fMHz)
Calculator: Physical Length to Electrical Degrees
Physical Length → Electrical Degrees
Enter the cable's physical length, its velocity factor, and the frequency. The calculator returns the electrical length in degrees.
Calculator: Electrical Degrees to Physical Length
Electrical Degrees → Physical Length
Enter the desired electrical length in degrees (e.g. 90 for a quarter-wave section), the cable's velocity factor, and the frequency. The calculator returns the required physical length in both feet and meters.
Special Electrical Lengths and Their Properties
Certain electrical lengths have special properties that make them extremely useful in RF circuits, test fixtures and matching networks. The most important are the quarter-wave (90°) and half-wave (180°) sections.
| Electrical Length | Fraction of λ | Key Property | Applications |
|---|---|---|---|
| 0° (or 360°, 720°...) | 0, 1, 2... λ | Load impedance reproduced exactly at input | Test fixtures, coax delay lines |
| 90° (quarter-wave) | ¼ λ | Impedance inversion: Zin = Z₀² / ZL | Matching transformers, stubs, notch filters |
| 180° (half-wave) | ½ λ | Load impedance reproduced exactly at input | Remote impedance measurement, identical to 0° |
| 270° (three-quarter-wave) | ¾ λ | Impedance inversion (same as 90°) | Alternative to quarter-wave section |
The pattern repeats every 180°: any section that is a multiple of 180° (0°, 180°, 360°...) reproduces the load impedance at the input when terminated with a matched load. Any section that is an odd multiple of 90° (90°, 270°, 450°...) performs an impedance inversion.
The Quarter-Wave Section in Detail
The quarter-wave section is arguably the single most useful tool in the RF engineer's toolbox. A transmission line of exactly 90° electrical length has the remarkable property of inverting the load impedance relative to Z₀²:
Zin = Z₀² / ZL
- Zin = impedance seen at the input of the quarter-wave section
- Z₀ = characteristic impedance of the section
- ZL = load impedance at the far end
This impedance inversion has three crucial consequences:
1. Open circuit becomes short circuit: If you open-circuit the far end of a quarter-wave line (ZL = ∞), the input looks like a short circuit (Zin = Z₀²/∞ = 0). This is the basis of open-circuit stubs acting as short circuits.
2. Short circuit becomes open circuit: If you short-circuit the far end (ZL = 0), the input looks like an open circuit (Zin = Z₀²/0 = ∞). This is the basis of short-circuit stubs acting as open circuits.
3. Impedance transformation: A quarter-wave section of appropriate Z₀ can match any two real impedances. This is the quarter-wave transformer principle, covered in detail in the Quarter-Wave Transformers lesson.
A 50-ohm coaxial stub, quarter-wave long, is open-circuited at its far end. What impedance does it present at its input at exactly the design frequency?
Zin = Z₀²/ZL = 50² / ∞ = 0 ohms
The stub looks like a short circuit — zero impedance — at the design frequency. If this stub is connected in parallel (shunt) with a signal line, it will short the signal to ground at that frequency, acting as a notch filter. This is exactly how a stub notch filter works.
At frequencies other than the design frequency, the stub is not exactly 90° long, so its input impedance is not exactly zero — it rises on either side of the design frequency, making the trap increasingly transparent.
The Half-Wave Section: Electrical Transparency
A half-wave (180°) transmission line section has a property that initially seems magical: it reproduces the load impedance exactly at its input, regardless of the cable's characteristic impedance. This is called electrical transparency.
More precisely, for a line of characteristic impedance Z₀ terminated in a load ZL:
- At 0° and 360° and every multiple of 360°: Zin = ZL
- At 180° and every odd multiple of 180°: Zin = ZL
This means that if you know the impedance of a load, you can measure it remotely by inserting a half-wave section of cable of any Z₀ between the load and your analyzer. At that frequency, the analyzer still reads the true load impedance.
More practically: adding or subtracting a half-wavelength of cable from a line that is already matched does not change anything at the source end. This provides a useful check — if adding a half-wave of 50-ohm coax to a supposedly matched system changes the reading on your network analyzer, something is wrong with the matching or there is a cable fault.
You want to measure the input impedance of a 50 MHz amplifier module that is mounted inside an enclosure, and the only access is through 2.5 meters of RG-58 (VF = 0.66) to a connector on the outside.
Half-wave at 50 MHz in RG-58 = (300/50) × 0.66 / 2 = 6 × 0.66 / 2 = 1.98 meters
Your 2.5-meter cable is not a half-wave, so it transforms the impedance. A full wavelength in this cable is 3.96 meters, so adding about 1.46 meters (to make 3.96 m) gives a length of exactly one wavelength. Then your analyzer at the outside connector would read the module's input impedance directly at 50 MHz.
Most modern network analyzers can do this mathematically instead — a feature called port extension or de-embedding subtracts the effect of a known cable — but the half-wave trick is the principle behind it, and it works with any instrument.
When Is a Line Electrically Long?
For sine waves, the usual rule of thumb is that a line shorter than about one-tenth of a wavelength (36°) can be treated as a simple wire, while anything longer must be treated as a transmission line.
For digital signals the frequency of the clock is not the right measure. What matters is how fast each edge rises. An edge with rise time tr contains significant energy up to a frequency of roughly
and a connection must be treated as a transmission line when its one-way delay td is more than about one-sixth of the rise time:
Critical length ≈ tr / (6 × delay per unit length)Some designers use one-quarter or one-half of the rise time instead of one-sixth; the exact fraction depends on how much overshoot you can tolerate. The principle is the same: if the reflection from the far end returns while the edge is still rising, it blends into the edge harmlessly; if it returns later, it appears as a separate step or ringing.
A microcontroller's clock output has a rise time of 1 ns. The trace runs on the outer layer of an FR-4 board (about 6.1 ps/mm).
fknee ≈ 0.35 / 1 ns = 350 MHz — even if the clock itself is only 8 MHz.
Critical length ≈ 1000 ps / (6 × 6.1 ps/mm) = 1000 / 36.6 = 27 mm
A trace shorter than about 27 mm can be routed without termination. A longer trace — or any trace that leaves the board through a connector and cable — needs a controlled impedance and a termination, covered in the Termination lesson.
Practical Applications of Electrical Length
Electrical length is used throughout high-speed and RF design. Here are the most common situations you will encounter:
Matching delays
When two signals must arrive together — clock and data on a fast bus, the two halves of a differential pair, or two oscilloscope channels you are comparing — their electrical lengths must match. On an oscilloscope, a few centimetres difference in probe or cable length shows up as a fraction of a nanosecond of skew between channels, which is why scopes include a channel deskew adjustment.
Stub filters and notches
A short-circuit or open-circuit stub of specific electrical length acts as a filter at the design frequency. A quarter-wave open stub connected across a line shorts that frequency to ground, providing a simple notch filter for an unwanted signal or harmonic in an RF circuit or test setup.
Quarter-wave impedance transformers
A quarter-wave section of the correct characteristic impedance can match any two real impedances. A common example is matching a 100-ohm load to a 50-ohm system using a quarter-wave section of about 70-ohm line. This is covered in the Quarter-Wave Transformers lesson.
Cable fault location
A time-domain reflectometer measures the delay to a fault and converts it into distance using the cable's velocity factor. Getting the velocity factor wrong by 10% puts the fault 10% further away or closer than it really is — an expensive mistake when the cable is buried or inside a wall.
Test Your Knowledge
Answer the questions below to check your understanding of this lesson. Every answer can be found in the lesson above.