Log-Periodic Antenna Calculator (LPDA)

Step 1

Design

Pick the band and either a gain target or the scale factor τ and spacing σ. The calculator follows Carrel's procedure for the element count, lengths and spacings, works out the feeder impedance that gives your input resistance and the boom spacing that makes that feeder, then solves the whole array across the band.

Display units

Band & structure

Band preset

Lowest frequency

MHz

Highest frequency

MHz

Design by

Gain target

dBi

Spacing σ

Low-end margin

Elements, booms & feed

Element material

Element diameter

Diameter along the array

Boom conductors

Boom width

Element mounting

Input resistance wanted

Feeder impedance

Rear termination

Balun

System impedance

Solver accuracy

Design changes need a new run. The display units apply immediately.

Layout top view, to scale; beam to the right

Element halves on boom AElement halves on boom BBooms (the feeder)Feed point and terminationApex lines, 2α

Carrel design parameters from τ, σ and the band

Feeder & booms the crossed line that feeds every element

Cut list

Step 2

Analyze

Each frequency point is a full method-of-moments solve of every element with the crossed feeder and termination. SWR is at the coax, after the balun. The shaded region is your design band; dashed horizontal lines are your goals.

Sweep span

Points

SWR at the coax referenced to the system impedance

Input impedance at the feed point, Ω

Gain against frequency forward, dBi

Front-to-back and front-to-rear dB

Beamwidth against frequency −3 dB, degrees

Smith chart at the coax, design band in accent

Active region element current against frequency, each column scaled to its strongest element

Each column is one frequency of the sweep. The bright band is the active region: the few elements near half a wavelength that carry most of the current and do the radiating. It should glide steadily from the rear (long elements) to the front as the frequency rises, and stay clear of both ends of the array inside your band. The dashed line marks where each element is a half wave long.

At one frequency

Radiation pattern and currents

The E-plane contains the elements and the boom; for a horizontally mounted array it is the azimuth pattern. The H-plane is at right angles to the elements, the elevation pattern of a horizontal array. Free space, no ground reflection.

Pattern frequency

MHz

3D radiation pattern

Radiation pattern gain, dBi; forward is up

Element currents centre current relative to the strongest element

Bars show the magnitude; the line shows the phase relative to the feed element, unwrapped from the front. Inside the active region the phase advances by roughly 180° plus the line delay per element: the backward wave that sends the beam toward the short end.

Pattern cuts

Touchstone export

One-port S11 at the coax over the sweep in Touchstone 1.1 format (MHz, magnitude/angle, referenced to the system impedance). Paste into a .s1p file for a circuit simulator or scikit-rf.

Model. Galerkin method of moments with piecewise-sinusoidal basis and test functions on each thin wire (Richmond), segmented at the chosen density per wavelength at each frequency. All elements are solved together for their full port admittance matrix; the booms are a lossless two-wire line crossed between every pair of elements, the rear termination sits on the longest element and a 1 A source drives the front element. Conductor loss comes from the skin-effect surface impedance of the tubing. Gain is the IEEE gain (radiation over accepted power), so it includes conductor loss and the power dissipated in a termination resistor; realized gain also subtracts the mismatch at the coax. The two booms are treated as a pure transmission line: their small vertical offset of the element halves and any radiation from the booms are not modelled.

About this log-periodic antenna calculator

This free calculator from NextAntennas designs a log-periodic dipole array (LPDA) from the frequency range and either a gain target or the scale factor τ and relative spacing σ, then analyzes and optimizes it in the browser. It follows Carrel's design procedure for the number of elements, their lengths and spacings and the boom length, works out the feeder impedance for your input resistance and the spacing of the two booms that gives it, and then solves every element together with the crossed feeder using a method-of-moments wire solver, the same class of solver as NEC. The result is a cut list, an NEC deck for 4nec2, and the gain, front-to-back ratio, beamwidth and SWR you can expect across the whole band.

How to use it

Design: choose a band such as HF 20–10 m, 2 m and 70 cm, UHF TV, cellular or Wi-Fi, a gain target or τ and σ, the element tubing and the booms. The calculator returns the element lengths and spacings, the apex angle, the feeder impedance, the boom spacing and a cut list with half lengths for each boom.
Analyze: SWR and input impedance at the coax, forward gain in dBi and dBd, front-to-back and front-to-rear ratio, E- and H-plane beamwidth across the band, the active-region map, polar and 3D radiation patterns and the element currents at any frequency.
Optimize: set goals for the minimum gain, gain flatness, front-to-back, SWR, boom length and element count, and let Nelder–Mead or differential evolution adjust τ, σ, the low-end margin, the feeder impedance and the stub.

Log-periodic design rules of thumb

Gain follows τ: about 7.5 dBi at τ = 0.86, 9 dBi at τ = 0.90, 11 dBi at τ = 0.95 and 13 dBi at τ = 0.97 on the optimum-σ line, and it stays within a dB or so across the band. Each step closer to τ = 1 costs elements and boom: a 2:1 band needs 11 elements on a 0.95 λ_max boom at τ = 0.86, and 25 elements on a 2.8 λ_max boom at τ = 0.95. Below τ ≈ 0.85 the gain ripples by 3 dB or more from one log period to the next. Carrel makes the longest element half a wavelength at the lowest frequency; making it about 10% longer keeps the gain and front-to-back up at the bottom edge. The shortest is somewhat shorter than half a wavelength at the highest frequency, because the active region needs a few elements beyond the band edge. Thicker elements broaden the active region, and a feeder impedance of 50–100 Ω usually gives an input resistance close to 50 Ω.

Related antenna and RF calculators

Log-periodic antenna FAQ

How do you design a log-periodic dipole antenna?

Choose the scale factor τ and the relative spacing σ; Carrel's optimum is σ = 0.258τ − 0.066. The apex half-angle is α = arctan((1 − τ)/4σ), the active-region bandwidth is Bar = 1.1 + 7.7(1 − τ)² cot α and the structure bandwidth is Bs = B × Bar, where B = fmax/fmin. The longest element is half a wavelength at the lowest frequency, each following element is τ times shorter, each spacing is 2σ times the longer element's length, and the number of elements is N = 1 + ln(Bs)/ln(1/τ), rounded up. This calculator also makes the longest element about 10% longer and keeps Bar at 1.5 or more, which gives flatter gain at both band edges.

How much gain does a log-periodic antenna have?

Gain is set mainly by the scale factor τ. On the optimum-σ line a log-periodic dipole array gives about 7.5 dBi at τ = 0.86, 9 dBi at τ = 0.90, 11 dBi at τ = 0.95 and 13 dBi at τ = 0.97, and the gain stays within about 1 dB across the band. Higher τ needs more elements and a longer boom. Carrel's original design charts overstated gain by 1 to 2 dB.

How many elements does a log-periodic antenna need?

It depends on τ and the frequency ratio. For a 2:1 band this calculator gives 11 elements at τ = 0.86, 14 at τ = 0.90 and 25 at τ = 0.95. A 9 dBi array for 140–450 MHz, a 3.2:1 ratio, needs 18 elements on a 3.2 m boom.

How is a log-periodic antenna fed?

All elements connect to a two-wire feeder that is crossed between every pair, so adjacent elements are fed in antiphase, and the feed point is at the short (front) end. In practice the two booms are the feeder: the coax runs inside one boom from the rear to the front, where its shield connects to that boom and its centre conductor to the other. This arrangement is called an infinite balun.

Should a log-periodic antenna use a termination stub or a resistor?

A resistor equal to the feeder impedance across the booms at the longest element absorbs the power that passes the active region. It gives the smoothest SWR and front-to-back ratio and costs 0.2 to 0.8 dB of gain, most of it in the lower half of the band. A short-circuited stub about λmax/8 long wastes no power but can cause narrow SWR spikes and front-to-back dips inside the band, so check the sweep before choosing a stub.

Should I build a log-periodic or a Yagi?

A Yagi on the same boom length has 3 to 6 dB more gain, but only over a few percent of bandwidth. A log-periodic keeps its gain, pattern and SWR over a frequency ratio of two, three or ten to one, so it suits TV reception, EMC testing, spectrum monitoring and multiband HF beams.

Why are the elements of a log-periodic antenna crossed?

Adjacent elements are fed 180° apart plus the delay along the feeder. In the active region this makes the currents form a wave travelling toward the short end, so the beam points toward the short elements and away from the long ones.

How accurate is this log-periodic antenna calculator?

Its method-of-moments wire solver agrees with NEC-2 to within about 0.2 dB in gain and a few ohms in impedance on the arrays we checked. Build accuracy matters more: keep the boom spacing constant, run the coax inside one boom and put each element half on the correct boom. The model is in free space and treats the booms as an ideal transmission line.

Next step — the rest of the RF chain Free Antenna & RF Calculators Yagis, dipoles and monopoles, patch antennas, transmission lines, filters, RF exposure and more — instantly, free, no sign-up.

Engineering Theory

Theory Behind the Equations

How the calculator works: Carrel's frequency-independent geometry for the elements and the feeder, then a method-of-moments solve of all the wires joined by the crossed two-wire line for the currents, the impedance, the far field and the gain.

1. What Is a Log-Periodic Dipole Array?

A log-periodic dipole array is a row of half-wave dipoles whose lengths and spacings shrink by the same scale factor τ from one element to the next. All of them are fed from a common two-wire line that is crossed between every pair. Because the geometry repeats itself when every dimension is multiplied by τ, the electrical behaviour repeats when the frequency is multiplied by 1/τ: the impedance and pattern vary periodically with the logarithm of frequency, and with a small enough τ the variation within each period is negligible. The antenna is then usefully frequency-independent between the frequencies where its longest and shortest elements are half a wavelength.

At any frequency only a few elements near half a wavelength, the active region, carry much current. Longer elements behind them act as weak reflectors and shorter elements in front as weak directors; most of the power is radiated before it reaches them. As the frequency rises the active region moves toward the short end.

2. Carrel's Design Equations

The geometry is fixed by τ and the relative spacing σ, the distance between two elements divided by twice the length of the longer one. Together they set the half-angle α of the wedge that the element tips lie on:

Scale factor, relative spacing and apex angle

τ = \frac{L_{n + 1}}{L_{n}} = \frac{R_{n + 1}}{R_{n}}

σ = \frac{d_{n}}{2 L_{n}}

α = \arctan \frac{1 - τ}{4 σ}

L_n is the length of element n, d_n its distance to the next shorter element and R_n its distance from the virtual apex. Carrel's optimum spacing for the most directivity at a given τ is σ_opt ≈ 0.258τ − 0.066.

The active region spans a band of its own, which Carrel found empirically. The structure must cover the operating band B = f_max/f_min multiplied by that active-region bandwidth, and that fixes the number of elements:

Active-region bandwidth, structure bandwidth and element count

B_{ar} = 1.1 + 7.7 {(1 - τ)}^{2} \cot α

B_{s} = B B_{ar}

N = 1 + \frac{\ln B_{s}}{\ln (1 / τ)}

N is rounded up. The longest element is L₁ = λ_max/2 with λ_max = c/f_min; each following element is τ times shorter and spaced d_n = 2σL_n in front of the previous one.

Checking Carrel's procedure with the solver shows two weak spots, and the calculator corrects both. The longest element only just resonates at the lowest frequency, so nothing behind it reflects and the gain and front-to-back sag at the bottom edge. And above τ ≈ 0.93 the empirical active-region bandwidth is too small, so the array runs out of short elements before the top of the band. With a low-end margin k (default 1.10):

Corrections used by this calculator

L_{1} = k \frac{λ_{\max}}{2}

B_{ar}^{'} = \max (B_{ar}, 1.5)

N = 1 + \frac{\ln (k B B_{ar}^{'})}{\ln (1 / τ)}

The margin lengthens element 1 and adds elements so the shortest element stays where Carrel puts it. On a 140–450 MHz design at τ = 0.893, the 10% margin raises the worst front-to-back in the band from 16 to 26 dB and the lowest gain from 6.9 to 8.2 dBi. At τ = 0.97 the 1.5 floor raises the gain at 450 MHz from 10.5 to 12.2 dBi.

Boom length

L_{boom} = \sum_{n = 1}^{N - 1} 2 σ L_{n} \approx \frac{k λ_{\max}}{4} (1 - \frac{1}{k B_{s}}) \cot α

3. Feeder Impedance and Boom Spacing

The elements load the feeder periodically and lower its impedance. Carrel's relation gives the characteristic impedance Z₀ of the bare feeder that produces a wanted input resistance R₀, using the mean characteristic impedance Z_a of the elements:

Feeder impedance

Z_{0} = \frac{R_{0}^{2}}{8 σ^{'} Z_{a}} + R_{0} \sqrt{{(\frac{R_{0}}{8 σ^{'} Z_{a}})}^{2} + 1}

σ^{'} = \frac{σ}{\sqrt{τ}}, Z_{a} = 120 [\ln \frac{L_{n}}{d_{n}} - 2.25]

L_n/d_n is the length-to-diameter ratio of the elements; for a constant tube size the calculator uses the geometric mean element.

The two booms are the feeder. Two parallel round conductors of diameter d at a centre spacing s form an air line of impedance

Two-conductor line

Z_{0} = 120 arccosh \frac{s}{d} \Rightarrow s = d \cosh \frac{Z_{0}}{120}

A square tube of width w behaves like a round tube of diameter about 1.18 w. A 50 Ω array usually needs a feeder of 55–75 Ω, which means booms only a few millimetres apart; insulating spacers keep the gap constant.

4. Feeding: the Infinite Balun

The array is fed at the short end. The usual construction runs the coax inside one boom from the rear to the front, where its centre conductor crosses to the other boom. The outside of the coax shield is then just part of that boom and carries no common-mode current, so the booms themselves form a balun, the so-called infinite balun. A 4:1 balun is an alternative when the array is designed for about 200 Ω.

The small power that passes the active region reaches the rear. A resistor equal to the feeder impedance across the booms at the longest element absorbs it, which gives the smoothest SWR and front-to-back; it costs 0.2 to 0.8 dB of gain, most of it in the lower half of the band, where more power gets past the active region. The classic alternative is a short-circuited stub about λ_max/8 behind the longest element. It wastes no power, but the reflected wave can set up resonances between the stub and the active region that appear as narrow spikes in SWR and dips in front-to-back, often inside the band. The solver predicts them, and NEC-2 shows them at the same frequencies, so check the sweep before choosing a stub.

5. Method of Moments with a Feeder Network

The elements are thin parallel wires solved with the Galerkin method of moments, using overlapping piecewise-sinusoidal basis and test functions:

Piecewise-sinusoidal basis and moment equations

f_{n} (z) = \frac{\sin (k (Δ - | z - z_{n} |))}{\sin (k Δ)}

\sum_{n} Z_{mn} I_{n} = V_{m}

Each element is segmented at the chosen density per wavelength at every frequency, so long rear elements at the top of the band get enough segments. The near field of each basis has an exact closed form (Schelkunoff), so every matrix entry is a one-dimensional integral.

Every element is centre-fed, so each one is a port. The matrix is factored once per frequency and solved with 1 V at each port in turn, which gives the full admittance matrix Y_a of the wires including all mutual coupling. The feeder is then added as a nodal network: each section between elements n and n+1 is a line of impedance Z₀ and length d_n, and the crossing flips the sign of the element voltage on every other element:

Feeder sections and the crossed connection

Y_{line} = \frac{1}{Z_{0}} [\begin{matrix} \coth γ d & - csch γ d \\ - csch γ d & \coth γ d \end{matrix}]

v_{n} = {(- 1)}^{n} V_{n}, (S Y_{a} S + Y_{feeder}) V = J

S is the diagonal matrix of ±1, V the line voltages at the elements and J a 1 A source at the front element. The rear node also carries the termination: coth(γℓ)/Z₀ for a shorted stub of length ℓ, or 1/Z₀ for a matched resistor. The input impedance is the front voltage.

The solved line voltages give the voltage at every element and hence every basis current, from which the far field, the gain and the element currents follow. This is the same model NEC builds with TL cards and a negative (crossed) characteristic impedance.

6. Gain, Front-to-Back and Beamwidth

Each basis radiates like a short sinusoidal dipole, so the far field is a closed-form sum over all bases. The gain is the radiation intensity over the accepted power, and realized gain also includes the mismatch at the coax:

Gain, realized gain and front-to-back

G = \frac{4 π U}{P_{in}}, G_{r} = G (1 - {| Γ |}^{2})

F / B = G_{dB} (0 °) - G_{dB} (180 °)

P_in = ½Re(Z_in)|I|² includes conductor loss and any power in a termination resistor, so both lower the gain. Front-to-rear compares the forward gain with the strongest lobe anywhere in the rear half of the two principal cuts.

The E-plane beamwidth of an LPDA is narrower than the H-plane beamwidth, typically 50–65° against 60–100°, because the dipole elements themselves have no directivity in the H-plane. Both stay nearly constant across the band, which is the main attraction of the design.

7. Validity and Limitations

The thin-wire method of moments is accurate for element diameters below about 0.02 λ and segments longer than a few diameters. The model is in free space and treats the booms as an ideal transmission line:

No ground: over ground the elevation pattern gains reflection lobes and the low-angle gain rises by up to 6 dB.
The element halves on the two booms are offset by the boom spacing; the model puts each element on one line. The effect is small for spacings well under 0.05 λ.
Feeder loss, boom radiation, the coax inside the boom and mounting hardware are not modelled.
Carrel's gain charts overstate gain by 1–2 dB; the gain target here uses solver results, but always read the solved figures.

Treat the results as a strong starting point, and check the built antenna with an antenna analyzer across the band before final assembly.

References

Carrel, R.L., "Analysis and Design of the Log-Periodic Dipole Antenna", University of Illinois Antenna Laboratory, Tech. Report 52, 1961; and IRE International Convention Record, 1961.
Isbell, D.E., "Log Periodic Dipole Arrays", IRE Transactions on Antennas and Propagation, AP-8, 1960.
Balanis, C.A., Antenna Theory: Analysis and Design, 4th ed., Wiley, 2016, ch. 11.
Butson, P.C. and Thompson, G.T., "A Note on the Calculation of the Gain of Log-Periodic Dipole Antennas", IEEE Transactions on Antennas and Propagation, AP-24, 1976.
Stutzman, W.L. and Thiele, G.A., Antenna Theory and Design, 3rd ed., Wiley, 2012.
Richmond, J.H., "Radiation and scattering by thin-wire structures in the complex frequency domain", NASA CR-2396, 1974.
Burke, G.J. and Poggio, A.J., Numerical Electromagnetics Code (NEC) — Method of Moments, LLNL, 1981.
The ARRL Antenna Book, chapter on log-periodic arrays.