Electrical Length
Describing a transmission line as "10 feet long" is not very useful to an RF engineer. Ten feet means completely different things at 3.5 MHz (about 2° of electrical length) and at 432 MHz (about 265° of electrical length). A much more powerful way to describe feedline 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. Electrical length is the language of transmission line design, and it is the key to understanding stubs, matching sections, and all the other feedline tricks you will learn in the remaining lessons of this module.
- 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
- 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 10-foot length of RG-213 is electrically 90° long at about 16 MHz, 180° at 32 MHz, and 360° at 64 MHz. The same 10 feet is only about 22° 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 antenna and feedline design. 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 jigs, coax delay lines |
| 90° (quarter-wave) | ¼ λ | Impedance inversion: Zin = Z₀² / ZL | Matching transformers, stubs, phasing lines |
| 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 antenna 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 lesson M13K.
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 trap 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. The analyzer still reads the true load impedance.
More practically for antenna work: adding or subtracting a half-wavelength of cable from a matched feedline does not change the SWR at the transmitter end. This provides a useful check — if adding a half-wave of 50-ohm coax to a supposedly matched system changes the SWR, something is wrong with the matching or there is a cable fault.
You want to check the impedance of an antenna at 7.2 MHz from inside the shack, but the antenna is 30 feet above the roof and you do not want to carry a VNA up the ladder. You have 30 feet of RG-213 (VF = 0.66) connecting the antenna to the shack.
Half-wave at 7.2 MHz in RG-213 = (300/7.2) × 0.66 / 2 = 41.67 × 0.66 / 2 = 13.75 meters = 45.1 feet
Your 30-foot cable is not a half-wave, so it transforms the impedance. You would need to add about 15 feet (4.6 meters) to make the total 45.1 feet. Then your analyzer in the shack would read the antenna impedance directly.
Alternatively, add a full wavelength (90.2 feet) to the existing 30 feet for the same effect — though in practice 120 feet of coax to the antenna would be installed permanently and you would just note the correct total length for measurement purposes.
Practical Applications of Electrical Length
Electrical length is used throughout antenna and feedline design. Here are the most common situations you will encounter as an amateur radio operator:
Phasing harnesses for vertical arrays
When feeding multiple antennas (such as a two-element vertical array) with a specific phase difference between them — typically 90° or 180° — you use transmission line sections of precise electrical lengths. A 90° section introduces exactly a quarter-wave delay. Getting this wrong by even a few degrees causes the beam to point in the wrong direction and reduces the front-to-back ratio dramatically.
Stub filters and traps
A short-circuit or open-circuit stub of specific electrical length acts as a filter or trap at the design frequency. Quarter-wave stubs are used to trap unwanted harmonics or to reject interference. Stub design is covered in lesson M13J.
Quarter-wave impedance transformers
A quarter-wave section of the correct characteristic impedance can match any two real impedances. A common example is matching the 200-ohm impedance of a 2-element Yagi feedpoint to 50-ohm coax using a quarter-wave section of 100-ohm coax (or two 50-ohm cables in parallel). This is covered in lesson M13K.
Antenna design
The physical length of a half-wave dipole or quarter-wave vertical is determined by the wavelength in free space (not in a cable), because the antenna radiates in air. But the antenna's feedpoint impedance depends on the electrical length of the antenna elements, and velocity factor in the conductor material itself (which is very close to 1.0 for typical copper wire) causes the resonant length to be slightly shorter than the free-space half-wavelength. This is why the standard formula 468/f(MHz) for a half-wave dipole gives a length slightly less than 492/f(MHz) (the true free-space half-wavelength in feet).
Test Your Knowledge
Answer the questions below to check your understanding. Every answer can be found in the lesson above.