Electrical Length of an Antenna: Definition, Formula and Worked Examples

The electrical length of an antenna, cable or circuit trace is its length measured in wavelengths, or in degrees of phase, at a stated frequency. It is not the same as the physical length in millimetres. One wavelength equals 360°, so a quarter-wave element is 90° long and a half-wave dipole is 180° long. The formula is θ = 360° × L ÷ λ, where L is the physical length and λ is the wavelength of the signal on that particular structure. Because λ depends on frequency, the same piece of wire or cable has a different electrical length at every frequency.

What electrical length means

A signal does not reach the far end of a conductor instantly. It travels as a wave, and every wavelength it covers adds one full cycle (360°) of phase delay. Electrical length is a way of stating that delay relative to the signal's own period:

• In wavelengths: L ÷ λ. A 31 mm wire at 2.4 GHz is about 0.25 λ, a quarter wave.
• In degrees: θ = 360° × L ÷ λ. The same wire is about 90° long at 2.4 GHz.
• In radians: βL = 2π × L ÷ λ, the form used in transmission-line equations.

The wavelength on the structure is λ = VF × c ÷ f, where c is the speed of light in a vacuum (299,792,458 m/s), f is the frequency in hertz and VF is the velocity factor, the speed of the wave on that structure as a fraction of c. Combining the two gives the practical formula:

θ (degrees) = 360° × f × L ÷ (VF × c)

Electrical length can also be read as a time delay. The wave takes τ = L ÷ (VF × c) seconds to travel the length L, and the phase delay is 360° × f × τ. One cycle at 868 MHz lasts about 1.15 ns, so any path that delays the signal by 1.15 ns is one wavelength (360°) long at that frequency.

Why electrical length differs from physical length

Two different effects make a structure electrically longer or shorter than its size in free-space wavelengths suggests. It helps to keep them apart.

Velocity factor in cables and circuit boards

In a coaxial cable or a printed trace, the wave travels through a dielectric and moves more slowly than in air. For a cable filled with a uniform dielectric, VF = 1 ÷ √ε_r, where ε_r is the relative permittivity of the dielectric. Typical values are:

• Solid polyethylene (ε_r ≈ 2.25): VF ≈ 0.66.
• Solid PTFE (ε_r ≈ 2.1): VF ≈ 0.69 to 0.70. RG-316, for example, is often listed at about 0.695.
• Foamed polyethylene: VF of roughly 0.8 to 0.88, depending on the foam.
• Microstrip on FR-4: part of the field is in the board and part is in the air, so the effective permittivity of a 50 Ω line is roughly 3.2 to 3.4 and VF is about 0.55. The exact value depends on the laminate and the trace geometry.

A slower wave means a shorter wavelength, so a cable is electrically longer than the same length of wire in air. Take a cable's velocity factor from the manufacturer's datasheet rather than from a generic table. For a PCB line, calculate or measure the effective permittivity from the laminate data, the actual stack-up and the trace geometry; the laminate's own permittivity is not the line's velocity factor.

Resonant length of an antenna element

A radiating element is not a transmission line with a datasheet velocity factor, but the same idea applies: the length that resonates is not exactly the free-space value. A thin half-wave dipole in air resonates a few per cent short of λ/2, often at about 95 % of it, partly because of the capacitance at the open ends (the end effect) and partly because of the conductor's thickness. A thicker element resonates shorter still. Some texts call this ratio a shortening factor or end-effect factor. It depends on the geometry, so it is not the same number as a cable's velocity factor.

Dielectric material where the element's near field is strong, such as a printed circuit board, a plastic housing or a ceramic body, usually lowers the resonant frequency, so a shorter element is needed. How much depends on how close the material is and how much of the field it fills. Lossy material such as the user's hand also absorbs power. The ground plane and nearby metal shift the resonance too. This is why the "right" length of an antenna is finally set by measurement in the finished product, not by the formula alone. For how size and frequency relate in general, see the relationship between antenna size and wavelength.

Worked quarter-wave examples

The table gives the free-space wavelength and quarter-wave length at common bands, a thin-wire starting point at about 95 % of the free-space quarter wave, and the length of a quarter wave inside two common coaxial dielectrics. All values are calculated with c = 299,792,458 m/s.

Example 1: one wire, two frequencies. A 31 mm wire is about a quarter wave (roughly 90°) at 2.4 GHz. At 868 MHz the free-space wavelength is 345.4 mm, so the same wire is 31 ÷ 345.4 ≈ 0.09 λ, or about 32°. Used as a monopole at 868 MHz it is far too short to resonate on its own. A length specification without a frequency says very little.

Example 2: a short cable at two frequencies. A 100 mm RG-316 jumper with VF 0.695 has an internal wavelength of 85.0 mm at 2.45 GHz. Its electrical length is 100 ÷ 85.0 ≈ 1.18 λ, or about 423°, which is one full cycle plus 63°. At 868 MHz the internal wavelength is 240.0 mm and the same jumper is about 0.42 λ, or 150°. Each extra millimetre of this cable adds about 4.2° of phase at 2.45 GHz.

Example 3: a printed feed line. On FR-4 with VF ≈ 0.55, a 10 mm microstrip at 2.45 GHz is about 0.15 λ, or roughly 53°. That is enough to move the impedance a long way around the Smith chart, which is why matching components are placed close to the antenna feed and why measurements are referred to the feed point.

Resonance and impedance: what happens when the electrical length is wrong

An antenna is resonant at the frequency where the reactive (imaginary) part of its input impedance is zero. Near its first resonance, an element that is electrically too short looks capacitive (negative reactance), and one that is too long looks inductive (positive reactance).

• Half-wave dipole. An idealised thin dipole cut to exactly λ/2 has an input impedance of about 73 + j42 Ω. Shortening it by a few per cent removes the reactance and leaves a resistance of roughly 65 to 70 Ω. More background: half-wavelength antennas.
• Quarter-wave monopole. Over a large, flat ground plane the impedance is about half that of the dipole, roughly 36 + j21 Ω at exactly λ/4, and a little over 30 Ω once trimmed to resonance. It is therefore not exactly 50 Ω. Ground radials sloped downward, a matching network or the shape of a device's ground plane bring it closer.
• Electrically short monopole. For the 31 mm wire at 868 MHz from Example 1, standing on an ideal ground plane, the short-monopole approximation R ≈ 395 × (h/λ)² gives a radiation resistance of only about 3 Ω. Its capacitive reactance is large; the exact value depends on the wire diameter, the feed and the surroundings. Connected directly to a 50 Ω radio, it reflects most of the power.

The mismatch between the antenna impedance Z and the line impedance Z₀ is described by the reflection coefficient Γ = (Z − Z₀) ÷ (Z + Z₀). The fraction of power reflected is |Γ|², and VSWR = (1 + |Γ|) ÷ (1 − |Γ|).

A mismatch means less power gets into the antenna; the rest is reflected back toward the radio. Whatever caused the detuning can also change the antenna's radiation efficiency and pattern. A hand or a metal part nearby, for example, both detunes the antenna and absorbs or redirects power. Restoring the match therefore does not always restore the radiated performance. Efficiency is also the main concern with electrically short and heavily loaded antennas, because their small radiation resistance competes with the loss resistance of conductors and loading parts. For practical matching, see antenna impedance matching and VSWR optimization for IoT antennas.

Resonance is a single frequency, but radio bands have width. As a fraction of the centre frequency, the EU 863–870 MHz band spans about 0.8 %, the US 902–928 MHz band about 2.8 % and the 2400–2483.5 MHz band about 3.4 %. The impedance bandwidth of an antenna is the range where its VSWR stays within the design target, often 2:1. The usable bandwidth must also meet the application's efficiency, gain and pattern requirements across the whole band.

Making an antenna electrically longer or shorter

When there is no room for a full quarter wave, reactive loading makes a short element behave as if it were longer. The basic rule is simple: series inductance makes an element electrically longer, and series capacitance makes it electrically shorter.

• Loading coils. A coil cancels the capacitive reactance of a short element. The reactance of an ideal inductor is X = 2πfL, so cancelling 300 Ω of capacitive reactance at 868 MHz takes about 55 nH. A real coil or chip inductor must be chosen by its impedance, Q and self-resonant frequency at the operating frequency, then verified by measurement. A coil at the base is the simplest. A coil placed higher up, at the centre for example, leaves more of the element carrying a large current, which raises the radiation resistance, but it needs more inductance for the same tuning.
• Losses in the loading. In a short antenna, every ohm of loss matters. If a 31 mm element at 868 MHz has about 3 Ω of radiation resistance, and the coil and conductors add 2 Ω of loss resistance, only about 60 % of the power accepted by the antenna is radiated. High-Q coils and good conductors are part of the antenna's efficiency, not just its tuning.
• Capacitive top loading. A disc, spokes or a horizontal wire at the top of a short vertical element (a "top hat") keeps the current from falling to zero at the tip. The current along the vertical part becomes more uniform. In the ideal case of uniform current, the radiation resistance is up to four times that of the same height without top loading, and less inductance is needed to reach resonance.
• Helical (spring) elements. Winding the radiator into a helix spreads the inductance along its length. The wire inside a helical whip can be longer than a quarter wave while the antenna itself is much shorter, and its resonance depends on the pitch, diameter and number of turns, not on the wire length alone. See helical antenna functions.
• Meanders and folded shapes. PCB antennas fold or meander the radiator to fit the available space. The added inductance and capacitance between the folds change the electrical length, so the trace length alone does not predict the resonant frequency.
• Dielectric loading. A high-permittivity material shrinks the resonant size by roughly the square root of the effective permittivity. At GPS L1 a half wave in air is 95.1 mm. In a ceramic with ε_r around 20 it is about 21 mm before fringing corrections, which is why ceramic GPS patches are only a few centimetres across.

Shortening comes with trade-offs. The smaller an antenna is in wavelengths, the narrower its bandwidth, the more sensitive it is to its surroundings, and the more its efficiency depends on losses. For bandwidth there is a physical bound: the Chu limit sets a minimum radiation Q for a passive antenna that fits inside a sphere of a given electrical size, which limits the bandwidth it can reach at a given efficiency and matching target. Adding loss can widen the matched bandwidth, but only by turning more of the power into heat. Efficiency and sensitivity also depend on the materials, the loading and the whole radiating structure, including the device's ground. See also physical dimensions of antennas.

Multi-band antennas, such as one that serves cellular bands as well as 2.4 GHz Wi-Fi and Bluetooth, need the right electrical length in each band. Common methods are traps (parallel LC circuits that isolate the outer part of an element at the higher band), separate branches for each band, stepped-impedance sections, matching networks with several resonances, and switched or tunable matching. A quarter-wave monopole also resonates again near three times its design frequency, but that second resonance rarely falls on a band that is wanted.

Electrical length of coaxial cables and phasing lines

For a cable, the same formula applies with the velocity factor from the datasheet: θ = 360° × f × L ÷ (VF × c). The delay is about 5.05 ns per metre at VF 0.66 and about 4.80 ns per metre at VF 0.695. Electrical length matters in cables for several reasons.

• Quarter-wave transformers. A λ/4 line of impedance Z₀ transforms a load Z_L to Z_in = Z₀² ÷ Z_L. Matching a 100 Ω load to 50 Ω needs Z₀ = √(50 × 100) ≈ 70.7 Ω, and 75 Ω cable comes close. At 868 MHz, a quarter wave of solid-polyethylene cable (VF 0.66) is about 57 mm long.
• Half-wave lines. A lossless line repeats the load impedance at its input whenever its electrical length is a whole number of half waves (180°, 360° and so on). Real low-loss lines come close to this, but only near those frequencies.
• Stubs. A quarter-wave stub shorted at the far end looks like an open circuit at its design frequency. Connected across a feed line, it provides a DC path to ground while barely affecting the signal near that frequency. It drains static charge, but it is not lightning protection on its own: that needs a properly rated and grounded protection system.
• Phasing lines. In an array, the phase of each element's feed sets the pattern. With equal-phase input signals and matched feeds, two lines that differ by 90° of electrical length excite the elements in quadrature. Crossed dipoles fed this way radiate circular polarisation in the directions where their two orthogonal fields are also equal in amplitude. For in-phase feeding, the feed paths must have the same transmission phase at the design frequency, and lines of equal electrical length keep that phase relationship best across a band. Two cables of different types with the same physical length can differ by many degrees. See antenna phase coherence.

Phase shift is proportional to frequency, so a phasing line is exact at only one frequency. A line that is 90° long at 868 MHz is about 94.9° long at 915 MHz. Across 863–870 MHz it varies only from about 89.5° to 90.2°, but across a wide band the error grows.

A common myth: cutting coax to "fix" VSWR. On a low-loss line, the VSWR is the same everywhere along the line. Changing the cable length changes the impedance the radio sees, but not the size of the mismatch. Cable loss actually makes the VSWR measured at the radio look better than it is at the antenna, because the reflected signal is attenuated twice. The return loss seen at the radio is roughly the antenna's return loss plus twice the one-way cable loss. An antenna with a 2:1 VSWR (9.5 dB return loss) behind a cable with 3 dB of loss looks like 15.5 dB, or about 1.4:1, at the radio end. If the reading still changes when the cable is moved or re-routed, after allowing for cable loss, calibration and connector condition, current is probably flowing on the outside of the coax, which makes the cable part of the antenna. That is a feed problem to fix with a choke, a balun or a better ground plane, not with a particular cable length. More on where reflections arise: positions of reflection in cable and antenna systems.

How electrical length is measured

A vector network analyser (VNA) measures phase directly, so it is the standard tool.

• Antenna resonance. Calibrate the VNA as close to the antenna feed as possible. Port extension can move the reference plane further along a well-matched, characterised feed cable (with loss compensation where the instrument supports it), but it does not remove the effect of a poorly matched fixture; that needs de-embedding. Sweep S11 and look at the Smith chart: resonance is where the trace crosses the real axis (zero reactance). The best match, where S11 is lowest and the return loss is highest, is often close to resonance but does not have to coincide with it. Measure with the antenna in its final housing and on its real ground plane.
• Trimming a simple element. For a plain wire, the resonant frequency is roughly inversely proportional to length. A whip measured at 900 MHz that should resonate at 868 MHz is electrically too short; scaling it by 900 ÷ 868 takes an 80 mm element to about 83 mm. Re-measure afterwards, because surroundings and feed details make the relationship approximate. For a loaded antenna, the radiator length, the loading and the matching network are all tuning options; choose using impedance and radiation measurements in the final assembly, because a matching network alone can hide a poorly radiating element.
• Cable phase and delay. With a calibrated two-port measurement of a well-matched cable, the transmission phase (S21) gives the electrical length. A single reading repeats every 360°, and a positive delay shows as a negative phase, so sweep enough frequency points to unwrap the phase and use the approximate length to resolve whole cycles. For a cable whose velocity factor does not vary with frequency, the one-way delay τ gives θ = 360° × f × τ and L = VF × c × τ. Phase-matched cable assemblies are usually specified as a phase tolerance in degrees at a stated frequency.
• Velocity factor from a quarter-wave test. Leave the far end of a low-loss sample open and calibrate at its input. The lowest frequency at which its input impedance passes through a minimum (the trace comes closest to the short-circuit point of the Smith chart) is approximately where the sample is λ/4 long, so VF ≈ 4 × L × f ÷ c. A 200 mm sample whose first impedance minimum is at 247 MHz has VF ≈ 0.66. Connector delay and the capacitance at the open end shift the result, so longer samples give more accurate values.
• Time-domain measurements. Time-domain reflectometry, or the distance-to-fault mode of a cable analyser, shows the delay to each reflection. That delay Δt is a round trip, so the distance is d = VF × c × Δt ÷ 2, and with a known length VF = 2d ÷ (c × Δt). Check whether the instrument already displays the one-way distance.

Checklist for engineers and buyers

1. Always state the frequency. An electrical length, whether "quarter wave", "90°" or "half wave", only means something at a stated frequency or band.
2. Ask for measured VSWR or return loss across the whole band, with the test set-up stated: ground plane size, cable, and whether the antenna was inside the housing.
3. Ask for efficiency as well as VSWR for small, loaded or embedded antennas. A lossy antenna can show a good match.
4. Specify phasing and matched cables by electrical length: the phase in degrees at the operating frequency, the tolerance and the cable type, not only the physical length.
5. Plan to retune in the final product. Housings, batteries, cables and hands shift the resonance.
6. Do not cut coax to tune an antenna. Fix the match at the antenna, and fix common-mode current on the cable where it starts.

Related reading

• Relationship between antenna size and wavelength
• Half-wavelength antennas
• Antenna impedance matching
• Helical antenna functions
• Antenna phase coherence

Talk to VLG about your antenna

Need an antenna or cable assembly where electrical length matters, such as a short loaded whip, an embedded antenna or a set of phase-matched cables? Contact VLG with your frequency bands, available space, cable lengths and mounting, and we can go through suitable options with you.