How a dipole radiates

Connect a transmitter to two straight wires, one on each side of a small gap. Leave the far ends open. A receiver across the room can pick up the signal. How does energy get from those wires to the receiver?

A dipole is one of the simplest antennas to build. Yet changing its length can turn a useful antenna into a difficult load, or move its strongest radiation away from the receiver. To understand why, start with what happens in the wires.

Two wires, an alternating current

The two wires are the dipole’s arms. The transmitter connects across the gap between them, at the feed, and applies an alternating voltage. It moves charge from one arm to the other, then back again.

The open tips do not stop this back-and-forth current. Charge can accumulate on an arm without leaving its tip. When current flows into the upper arm, that arm becomes more positive and the lower arm more negative. Here the arrows follow conventional current: the direction positive charge would move. After the current reverses, the accumulated charge starts to decrease. It takes time to remove it, so the upper arm remains positive for a while.

Watch that reversal below. The charge signs change after the current arrows reverse, and the change in the electric field takes time to reach the probe.

From wire current to a radio wave

Set the slider just past ¼ cycle. Is the upper arm still positive?

Highlighted field line
DipoleField probe

Slow-motion visualization. Large arrows show current; small arrowheads show field direction. The wire and nearby field are enlarged. The probe is 0.75 wavelengths from the feed.

Current reverses at ¼ cycle. The upper arm stays positive until ½ cycle.

The electrons move back and forth within the metal; they do not fly across the room. The transmitter makes them speed up, slow down and reverse direction. This acceleration produces a radiating field that travels outward at the speed of light, carrying energy from the transmitter. At the receiving antenna, the wave’s electric field moves charge and produces a voltage.

Why start with a short dipole?

“Short” means short compared with the wavelength, the distance a radio wave travels in one cycle. If the whole dipole is much shorter than that distance, current rises and falls at almost the same time along both arms. We can understand its radiation before dealing with different parts of a longer wire carrying current in opposite directions.

The current is not equally strong everywhere. It is largest at the feed and falls to zero at the open tips, where charge cannot continue along the wire. For a short, thin dipole, this current distribution is nearly triangular. A total length of about one tenth of a wavelength or less is a common working range for the approximation; there is no sudden physical boundary at one tenth.

We will start with that simple case, then lengthen the wire. First, where does its radiation go?

Where does the signal go?

A vertical short dipole radiates most strongly to its sides and has a radiation null along its axis, above and below the tips. Its three-dimensional pattern looks like a ring around the wire. The ring shows relative strength by direction, not the shape of a wave moving through space.

Imagine walking around the antenna at the height of its feed, always the same distance away. In open space, the signal stays the same. This is what omnidirectional means here: equal radiation around the wire’s axis. It does not mean equal radiation in every direction.

Now move the receiver over a tip while keeping its distance fixed. You leave the strongest direction, called broadside, and approach the null. The radiated electric field is transverse: it points across the direction of travel. Only the component of wire current perpendicular to your line of sight contributes to this far field. Seen end-on, that component is zero.

Try both paths below. The receiving antenna stays aligned with the local electric field so you can isolate the effect of position.

Walk around the short dipole

Move the receiver while keeping its wire aligned with the local electric field. The transmitter current stays fixed.

0°180°90°
Electric field100.0%
Power density100.0%
The copper dot marks the receiver. Broadside at 3 m, both readings are 100%. The wire is drawn larger so you can see it.

Reading the pattern

Measure the angle θ from the upward wire axis: 0° is above the tip and 90° is broadside. For the short dipole, field amplitude varies as sin θ and power density as sin² θ. Power density is power passing through a unit area, measured in W/m².

At 45° and 135°, power density is half its broadside value. Those directions are 90° apart, giving a half-power beamwidth of 90°. Broadside power density is 1.5 times what an antenna radiating the same total power equally in all directions would produce. That ratio is the short dipole’s peak directivity: 1.5, or 1.76 dBi. Directivity describes concentration by direction, not transmitter power or antenna efficiency.

Position and orientation are different

Stay broadside and rotate the receiving wire. The field is still strong, but the received signal falls because the wire responds to the electric field along its own length. A vertical dipole produces a vertically oriented field broadside. The field reverses along that same line, so its polarization is linear.

Tilt a small receiving dipole by an angle ψ away from the field and its voltage falls by cos ψ. With the same load, power falls by cos² ψ. A 60° tilt halves the voltage and leaves one quarter of the power; a 90° tilt gives an ideal null. This polarization loss is separate from the transmitting antenna’s directional null.

Why distance weakens the signal

Double the distance without changing direction. The radiated power spreads over four times the area, so power density falls to one quarter. Electric and magnetic field amplitudes each fall to one half. These are far-field rules: far enough away, the pattern no longer changes shape with distance, and the electric and magnetic fields rise and fall together.

The short-dipole equations collect those relationships. Eθ is the electric field in the plane containing the wire and observer; Hφ is the perpendicular magnetic field, directed around the wire axis. Their far-field ratio in free space is η₀ ≈ 377 Ω.

Eθ≃jη0kIfeedℓeff4πre−jkrsin⁡θE_\theta\simeq j\eta_0\frac{k I_{\mathrm{feed}}\ell_{\mathrm{eff}}}{4\pi r}e^{-jkr}\sin\thetaHϕ≃Eθη0H_\phi\simeq\frac{E_\theta}{\eta_0}

Here r is distance, Ifeed is peak feed current and k = 2π/λ. The effective length, ℓeff, is about L/2 for this triangular current distribution: the average current amplitude along the wire is half the feed amplitude. Thus a longer short dipole produces a stronger field at the same feed current. The sin θ and 1/r factors describe direction and distance. The factors j and exp(−jkr) specify phase; they do not change the amplitude.

Phase notation and a field calculation

The equations use complex amplitudes for sinusoidal fields. The imaginary unit j represents a quarter-cycle phase shift; exp(−jkr) represents propagation delay. Multiply by exp(jωt) and take the real part to recover the instantaneous field, with ω = 2πf.

For a numerical example, choose 2.45 GHz. The free-space wavelength is c/f ≈ 122.4 mm, where c is the speed of light. A wire 0.1λ long has an effective length of about 6.12 mm. With 10 mA peak feed current, the peak electric field three meters broadside is 377 × 51.35 × 0.010 × 0.00612 / (4π × 3) ≈ 31.4 mV/m. The magnetic field is E/377 ≈ 83.3 µA/m. Both amplitudes halve at six meters.

Close to the wire, some energy comes back

Near the antenna, energy also moves into and out of the surrounding fields during each cycle. The electric and magnetic fields can reverse at different times, so their instantaneous power flow can briefly point toward the antenna even while the average is outward.

The following comparison uses a separate, very small ideal current element: 1 mm at 300 MHz. It isolates the effect of distance. Watch the power trace at 0.05λ cross below zero, then compare it with the trace at 1λ.

Energy flow near and far from the wire

Slow-motion visualization · Two probes broadside to the same source

Eη₀HAverage outward power density

H direction: ⊙ out of the page · ⊗ into the page.

Near 0.05λ from the source

E and η₀H / local E peakRadial power / peak00EHInwardone source cycle

Cycle average: 189 µW/m² outward

Farther 1λ from the source

E and η₀H / local E peakRadial power / peak00EHOutwardone source cycle

Cycle average: 0.472 µW/m² outward

One source clock · travel delay included
Watch when E and H cross zero. Near the source they reverse at different times. Farther away the crossings nearly line up. The average flow is outward at both probes.

Compare timing across the two rows. Each row has its own scale. To compare average power density, use the values in µW/m² below the plots.

What the plots measure

The source is a 1 mm ideal current element at 300 MHz with 0.1 A peak current. Both probes are far from the element compared with its length. The average outward power density is 400 times as large at the near probe.

Multiplying H by η₀ puts it in the same units as E. Each row uses the peak E at that probe as the scale for both fields. The power trace uses its own positive peak. The dotted line marks the average over one cycle, and the horizontal arrow shows the direction of flow.

The plots show the fields and power flow at two points. Even at 1λ, the exact solution shows a brief interval of inward flow. The change from near field to far field is gradual.

For a field measurement near the wire, you need both E and H and their relative timing. The outward component of their cross product, E × H, gives instantaneous outward power density. Average it over a cycle. In the far field, their known relationship lets you use the simpler ERMS²/377 Ω instead. A large electric-field reading close to an antenna is therefore not, by itself, evidence of strong radiation.

What changes when the wire gets longer?

So far, the current has had one simple shape. As the wire becomes a larger fraction of a wavelength, propagation along the arms and reflection from the open ends become important. They produce a standing-wave current distribution: its shape stays in place while the current oscillates. The tips remain current zeros, but longer wires can have additional zeros between them.

To put dimensions on the comparison, keep the frequency at 2.45 GHz. One wavelength in free space is 122.4 mm. A half-wave dipole is therefore 61.2 mm from tip to tip, with two quarter-wave arms. All lengths below refer to the whole dipole, not one arm.

Compare 0.25λ with 0.5λ in the explorer. Current is strongest at the feed in both. Both radiate most strongly broadside, but the half-wave beam is narrower. Next try 1λ and 1.5λ: the current peaks move, and the pattern changes much more.

Wire current and radiation pattern

Move the slider to change the wire length. The copper dot marks the feed, and the copper curve shows the current along the wire.

Wire current

+1λ+½λFeed−½λ−1λ−10+11 = half-wave feed current
Current at feed +1.00

Distances along the wire start at the feed.

Power cut

0°90°radius = power / peak
Thin-line 3D power surface
Selected power cutFixed 0.5λ power cut
Peak directivity2.15 dBi
Peak direction from wire90.0°
The current scale stays fixed as the wire changes length. Parts of the curve on opposite sides of the wire show current flowing in opposite directions at the same instant. Each power pattern uses its own peak as 1. The wire in the 3D view is drawn on a separate scale.

The model sets the current along the wire. In a real dipole, the feed current depends on the source and the input impedance.

Why the beam narrows—and then splits

Every region of the wire contributes to the field at a distant receiver. The result depends on both the current in that region and the time its field takes to reach the receiver.

On a full-wave dipole, current has a peak midway along each arm. The two peaks are half a wavelength apart and rise and fall together. Broadside, their fields travel essentially equal distances and reinforce each other. Move away from broadside and one field has farther to travel. Its extra delay makes the fields add less strongly.

With the strong-current regions farther apart, a smaller change in angle produces a significant difference in arrival phase. That is why the full-wave beam is narrower: its half-power points are about 24° either side of broadside, compared with 39° for a half-wave dipole.

At 1.5λ, current near the center flows opposite to current farther out. Broadside, their fields partly cancel. In other directions, different travel delays reduce that cancellation, so the fields can add more strongly. The strongest radiation moves to about 43° and 137° from the wire axis. Compare the two 1.5λ presets below and follow the bold total field.

How fields from the wire combine

Three sections of wire · one receiver direction · slow motion

Instantaneous currentLCUwire−10+1Relative pathsLCUEqual paths at 90°Common reference plane
Relative electric field00one cycle
Lower (L)Center (C)Upper (U)Total field
Each length uses its own peak current as the scale.
Diagram B. At the cursor, add the three field readings with their signs to get the bold trace. The traces from the two outer sections overlap here. Count both. Even when current is zero at the feed, current elsewhere on the wire can still radiate.

The sketch shows paths from three points on the wire. Each field trace adds up the field from its whole section. The plot subtracts the travel time from the wire center so you can compare the delays from different parts of the wire.

How the model adds the fields

For each section, the model adds the field from every point using the current there, the angle to the receiver and the travel time. The three fields and their sum share one scale at each length. That scale is proportional to the wire length in wavelengths. A negative current means it flows in the opposite direction along the same straight wire. The parallel paths end at a shared plane to show the difference in travel distance to a far-away receiver.

Longer does not always mean higher peak directivity. In this model it rises from 2.15 dBi at 0.5λ to 5.16 dBi at 1.25λ, then falls to 3.48 dBi at 1.5λ as radiation is redistributed among lobes. The ideal straight wire remains omnidirectional around its axis and linearly polarized wherever its field is nonzero.

Each displayed power pattern is scaled to its own peak. It tells you where the antenna radiates most strongly, not how many watts it radiates. That second question needs the feed impedance and losses.

The model behind the curves

For a thin straight wire, the following sinusoidal current is a useful approximation. L is total length, z is distance along the wire from the feed, and Is sets the current scale. With k = 2π/λ, the expression gives zero current at both tips. For a very short wire it approaches the triangular shape we started with.

I(z)=Issin⁡ ⁣[k(L2−∣z∣)]I(z)=I_s\sin\!\left[k\left(\frac{L}{2}-|z|\right)\right]−L2≤z≤L2-\frac{L}{2}\leq z\leq\frac{L}{2}

Adding the fields from the whole wire, including their travel delays, gives the finite-dipole field:

Eθ≃jη0Ise−jkr2πrF(θ)E_\theta\simeq\frac{j\eta_0 I_s e^{-jkr}}{2\pi r}F(\theta)F(θ)=cos⁡[(kL/2)cos⁡θ]−cos⁡(kL/2)sin⁡θF(\theta)=\frac{\cos[(kL/2)\cos\theta]-\cos(kL/2)}{\sin\theta}

F(θ) describes variation with direction; |F|² gives the power-pattern shape. Eθ, r and η₀ have the same meanings as before, and Hφ = Eθ/η₀ in the far field. Is equals feed current only at particular lengths, including 0.5λ. The model prescribes current; it does not establish how much a real transmitter can drive into the antenna.

Check the half-wave field at broadside

For L = λ/2 and θ = 90°, F = [cos 0 − cos(π/2)]/1 = 1. With 10 mA peak feed current and r = 3 m, the peak electric field is 377 × 0.010 / (2π × 3) ≈ 0.200 V/m. This is larger than the short-wire result at the same feed current. It is not a comparison at equal transmitter power.

What does the transmitter have to drive?

The radiation pattern assumes a current distribution. The transmitter has to produce that current. How much voltage is needed, and does the current rise at the same time as the voltage? The antenna’s input impedance, Z = R + jX, answers both questions at the feed.

The resistance R accounts for average power leaving the circuit as radiation or heat. Radiation resistance represents the radiated part; it is not a physical resistor heating the wire. Loss resistance represents heat. Reactance X describes the effect of energy stored in the fields and returned during the cycle.

Why a short wire looks capacitive

Remember the charge building up on the two arms. More separated charge means more voltage across the gap. Current adds charge fastest before the charge—and therefore the voltage—reaches its peak. This is capacitor-like behavior, giving negative reactance. A very short dipole can require a high feed voltage for a small current.

Lengthen the wire toward half a wavelength and its reactance moves toward zero. Where X = 0, voltage and current are in phase: the antenna is resonant. Fields extend beyond the wire tips, making the antenna electrically a little longer than its physical length suggests. The first resonance is consequently slightly below a geometric half wavelength. Wire diameter, feed geometry and nearby materials affect the exact value. End effects.

Resonance is not the same as a match

A 50 Ω feed line reflects no power only when it sees 50 + j0 Ω. Removing the reactance is only part of that requirement. A resonant antenna with a different resistance still reflects some power.

The voltage standing-wave ratio (VSWR) describes the voltage maxima and minima formed by forward and reflected waves on the line. A value of 1:1 means no reflection; larger values mean more reflection. Compare First resonance with Exactly 0.5λ below. This reference calculation uses a lossless wire with a fixed radius of 0.0001λ to isolate the effect of length.

Resonance and a 50 Ω match

Thin-wire calculation · wire radius 0.0001λ · 50 Ω line

Full Length Range

Resistance (Ω · logarithmic)0.110100050 ΩReactance (Ω · linear)-4000040000.05λ0.8λ

Near First Resonance · Enlarged

Resistance (Ω · linear)5075Reactance (Ω · linear)-1000100120.45λ0.52λ
1 · Resonance · 0.4846λ2 · Half-wave · 0.5000λ

Outlined points follow your selection. A 50 Ω match needs R = 50 Ω and X = 0 together.

Input impedance73.1 + j42.5 Ω
VSWR on 50 Ω2.182:1
Incident power accepted86.2%
Positive reactance means the dipole looks inductive at the feed. The model keeps the radius fixed as a fraction of a wavelength. In a frequency sweep on a real wire, the wavelength changes while the radius stays the same.

At first resonance, the model gives 66.7 + j0 Ω. At exactly half a wavelength it gives 73.1 + j42.5 Ω: already inductive. These are different conditions, and neither is a perfect 50 Ω match. The half-wave wire accepts about 86.2% of incident power.

The lowest VSWR occurs very slightly below resonance in this calculation. The difference is small—1.332:1 versus 1.335:1—but the distinction matters: find resonance using zero reactance, and find the best match using minimum reflection.

Calculate reflection, accepted power and VSWR

For Z = R + jX connected to a 50 Ω line, Γ is reflected voltage divided by forward voltage. Its squared magnitude is the reflected power fraction, so the remaining fraction enters the antenna:

∣Γ∣2=(R−50 Ω)2+X2(R+50 Ω)2+X2,Pacc=Pinc(1−∣Γ∣2)|\Gamma|^2=\frac{(R-50\,\Omega)^2+X^2}{(R+50\,\Omega)^2+X^2},\qquad P_{\mathrm{acc}}=P_{\mathrm{inc}}(1-|\Gamma|^2)VSWR=1+∣Γ∣1−∣Γ∣\mathrm{VSWR}=\frac{1+|\Gamma|}{1-|\Gamma|}

For 73.1 + j42.5 Ω, |Γ| ≈ 0.371, giving VSWR ≈ 2.18:1 and accepted power ≈ 86.2% of incident power. These quantities describe the same connection; none tells you how much accepted power becomes heat.

Why a full-wave dipole is hard to feed at the center

At 1λ, the current plot has strong peaks along the arms but a minimum at the feed. A small feed current can therefore coexist with strong current elsewhere. Since input impedance is feed voltage divided by feed current, this center connection has high impedance and is a poor direct match to 50 Ω.

The ideal sinusoidal model gives exactly zero feed current at 1λ and 2λ. Real antennas have high, finite impedance peaks near these lengths; wire thickness and the feed gap prevent the ideal infinity. A narrower beam alone is therefore not a reason to choose a full-wave dipole. You also need a practical way to feed it. Full-wave measurements.

From transmitter power to a working antenna

An antenna can be efficient with the power it accepts and still radiate very little of the power arriving from the transmitter. Our short dipole makes that distinction especially clear.

At 2.45 GHz, take a copper dipole 12.2 mm tip to tip (0.1λ), made from 0.25 mm diameter wire. Connect it directly to a lossless 50 Ω line with 1 W incident at the feed. The short-wire model predicts that almost all of that watt reflects. Only 0.334 mW enters the antenna; 0.323 mW radiates and 0.011 mW heats the wire.

Its radiation efficiency is nevertheless about 97%: radiated power divided by accepted power. Efficiency excludes the power that never entered the antenna. This is why matching and efficiency must be checked separately.

What makes a short antenna difficult?

Radiation resistance falls roughly with the square of electrical length. Halve a short dipole’s length at the same frequency and feed current, and it radiates about one quarter as much power. The copper loss resistance falls only in proportion to length at fixed diameter. As the wire gets shorter, a greater share of accepted power therefore becomes heat.

For this wire, radiation resistance is about 1.97 Ω, copper loss resistance 0.067 Ω, and reactance −1104 Ω. The copper loss is small. The large negative reactance is the immediate problem: the feed impedance, 2.04 − j1104 Ω, is very far from 50 Ω.

Calculate radiation resistance, copper loss and reactance

Use total length L = 0.1λ = 12.236 mm, radius a = 0.125 mm and copper conductivity σ = 5.8 × 10⁷ S/m. For the triangular current distribution, radiation resistance referred to the feed is

Rrad≃η0π6(Lλ)2≈1.97 Ω.R_{\mathrm{rad}}\simeq\frac{\eta_0\pi}{6}\left(\frac{L}{\lambda}\right)^2\approx1.97\,\Omega.

At RF, current is concentrated near the copper surface. With free-space permeability μ₀ ≈ 4π × 10⁻⁷ H/m, skin depth δ and surface resistance Rs are

δ=1πfμ0σ≈1.34 μm,Rs=1σδ≈0.0129 Ω.\delta=\frac{1}{\sqrt{\pi f\mu_0\sigma}}\approx1.34\,\mu\mathrm{m},\qquad R_s=\frac{1}{\sigma\delta}\approx0.0129\,\Omega.

Integrating the heat loss along the triangular current profile gives

Rwire≃RsL6πa≈0.067 Ω,ηrad=RradRrad+Rwire≈96.7%.R_{\mathrm{wire}}\simeq\frac{R_sL}{6\pi a}\approx0.067\,\Omega,\qquad \eta_{\mathrm{rad}}=\frac{R_{\mathrm{rad}}}{R_{\mathrm{rad}}+R_{\mathrm{wire}}}\approx96.7\%.

For a thin, straight, short wire with a small feed gap in free space, the approximate capacitive reactance is

X≃−120 ΩπL/λ[ln⁡(L2a)−1]≈−1104 Ω.X\simeq-\frac{120\,\Omega}{\pi L/\lambda}\left[\ln\left(\frac{L}{2a}\right)-1\right]\approx-1104\,\Omega.

Here ln is the natural logarithm. Reactance formula and assumptions. Use Z = Rrad + Rwire + jX in the reflection calculation: the accepted fraction is about 0.000334. Multiplying 1 W by that fraction and by 0.967 gives 0.323 mW radiated. These are estimates for the stated geometry, not measured hardware performance.

Matching helps—but the components can waste power

A series coil can supply positive reactance to cancel the antenna’s negative reactance. For this teaching model, a 71.7 nH coil does so at 2.45 GHz. Give it a quality factor Q = 100: Q is reactance divided by series loss resistance at this frequency, so the coil adds about 11 Ω of loss. That is much larger than the antenna’s radiation resistance.

Adding the coil lets more power enter, raising radiation to about 99 mW. An ideal transformer can then match the remaining resistance to 50 Ω. The complete assembly accepts the whole watt and radiates about 151 mW. These are circuit-model results; a real coil’s parasitic capacitance and self-resonance matter at 2.45 GHz. Follow the three connections below; most of the newly accepted power heats the coil.

Power at three feed connections

Each bar starts with 1 W arriving at the connection shown. The 50 Ω line has no loss.

Bare wireAt the wire terminals
Radiated
0.323 mW
Wire heat
0.011 mW
Coil heat
0 mW
Reflected
999.7 mW
Wire and series coilAt the coil and wire input
Radiated
99.14 mW
Wire heat
3.371 mW
Coil heat
555 mW
Reflected
342.5 mW
Wire, coil and transformerAt the transformer input
Radiated
150.8 mW
Wire heat
5.127 mW
Coil heat
844.1 mW
Reflected
0 mW
Compare the last two bars. The transformer helps the antenna and coil accept more power. In both cases, 15.1% of the accepted power radiates.
Size the tuning coil and include its loss

The coil must supply Xcoil = −Xantenna. Its inductance and loss are

Lcoil=Xcoil2πf,Rcoil=XcoilQ.L_{\mathrm{coil}}=\frac{X_{\mathrm{coil}}}{2\pi f},\qquad R_{\mathrm{coil}}=\frac{X_{\mathrm{coil}}}{Q}.

Using 1104 Ω at 2.45 GHz gives 71.7 nH and, with Q = 100, about 11 Ω loss. Include that loss in the efficiency: Rrad/(Rrad + Rwire + Rcoil) ≈ 15.1%. Even a lossless tuning coil would leave about 2.04 Ω of resistance, so canceling reactance alone would still not give a 50 Ω match.

The coil and transformer are circuit models. At 2.45 GHz, a real 71.7 nH component may be limited by its self-resonance and parasitic capacitance. Check component data or measured impedance; the example is not a component recommendation.

A match at one frequency is not a bandwidth

Keep the network fixed and change frequency. Below the tuned frequency, the coil no longer cancels all the capacitive reactance; above it, the combination becomes inductive. Reflection grows on either side. A short antenna stores substantial field energy relative to the power it radiates, which makes a low-loss match sensitive to frequency. This sweep holds inductance, Q and the ideal transformer ratio fixed.

Define the usable match bandwidth here as the range with VSWR ≤ 2:1. At that limit, one ninth of incident power reflects. Sweep the frequency to find the two edges.

VSWR at the transformer input

Change the frequency without retuning the coil or transformer. The model uses Q = 100 across the sweep.

VSWR 1.00:1
The two dots mark VSWR = 2. The shaded band runs from 2.4398 to 2.4603 GHz, a width of 20.5 MHz or 0.84% of 2.45 GHz.

This model gives about 20.5 MHz, or 0.84% of 2.45 GHz. That is a match bandwidth, not a guarantee of useful radiation throughout it. Loss can broaden a good-looking match by absorbing power. Check radiation efficiency and directional performance as well as VSWR.

If there is room, start near half a wavelength

For a first 2.45 GHz prototype, a straight dipole near half a wavelength is easier to feed than our very short wire. Half a wavelength is 61.2 mm total. Allowing for end effects gives a slightly shorter resonant length. For wire diameter d, this estimate gives a useful first cut:

Lres≃λ2[1−0.2257ln⁡(λ/d)−1]L_{\mathrm{res}}\simeq\frac{\lambda}{2}\left[1-\frac{0.2257}{\ln(\lambda/d)-1}\right]

With 0.5 mm diameter wire, the estimate is 58.1 mm tip to tip, about 29 mm per arm with a small feed gap. Start slightly long and trim both arms equally. Shortening the wire raises its first resonant frequency. The estimate assumes a straight dipole in free space; supports, an enclosure and nearby metal or ground can change both resonance and pattern. First-resonance estimate.

Measure the antenna, not just a good-looking match

Measure with the final feed and mounting in place. A vector network analyzer (VNA) measures reflection; its calibration determines where that reflection is reported. Calibrate at the cable connector that will attach to the antenna. Anything beyond that connector—including an adapter or balun—remains part of the measurement.

Cable loss can be deceptive. Suppose the antenna has 10 dB return loss, meaning 10% of incident power reflects. Add a matched cable with 3 dB one-way loss, but leave the calibration at the VNA port. The return loss now reads about 16 dB: the signal loses 3 dB going out and another 3 dB coming back. The antenna has not improved. Less power reaches it, and less reflection returns to the instrument.

The feed cable can become part of the antenna

Inside the coax, signal current flows on the center conductor and the inner surface of the shield in opposite directions. At the dipole, the center conductor connects to one arm and the shield to the other. That connection can also drive common-mode current on the shield’s outer surface. The cable then radiates too, changing the pattern you intended to build.

Follow the two paths below. A current balun or choke at the feed can reduce the outside current.

Current inside and outside the coax

Interactive connection schematic · arrows show one illustrative instant

Internal pairExteriorshield pathTwo dipole armsCoax · spacing enlarged for clarity
Current can flow on the outside of the shield while the coax carries the signal inside. A choke at the feed can reduce that outside current. Measure it at the frequency you plan to use.

If moving the cable changes the VNA trace, investigate the feed arrangement. To test a choke, use an RF current probe around the whole coax and compare readings with and without it. Keep the frequency, drive level, probe position and cable routing fixed. The internal currents cancel in that measurement; the probe detects the outside current. A change in reflection alone does not tell you whether the choke worked.

Finally, check radiation toward the intended receiver. Realized gain combines the pattern, radiation efficiency and input mismatch: it describes directional radiation relative to the power incident at the antenna’s input. Length helps establish the current distribution; the pattern tells you where radiation goes; impedance determines what the feed line delivers; losses determine what becomes heat. A useful antenna has to satisfy all of them in its actual installation.

Try it yourself

Use these optional questions to check your reasoning. Each has feedback and a worked solution; your written notes are for your own comparison.

Checkpoint 1 / Reason it throughHow much charge accumulates now?

Current can flow into the upper arm at the feed. The tip is open, so no current leaves there. The feed current tells you how fast the total charge on that arm changes. For a sine wave,

dQupperdt=Ifeed,Qpk=Ipkω\frac{dQ_{\mathrm{upper}}}{dt}=I_{\mathrm{feed}},\qquad Q_{\mathrm{pk}}=\frac{I_{\mathrm{pk}}}{\omega}

Here Q is total arm charge in coulombs, and ω = 2πf. Faster reversal leaves less time to accumulate charge at the same peak current.

Double the frequency and raise the peak feed current from 40 mA to 60 mA. The same wire is electrically short at both frequencies, and the current is sinusoidal. What fraction of the original peak charge builds up on the arm now?

Choose an answer

How does the higher current affect the charge? How does the shorter half-cycle affect it?

Notes stay on this page while it’s open.
Use the worked solution to check your reasoning.
Show the worked solution

Peak charge falls by 25%. A larger current transfers charge faster. Doubling the frequency halves the time for charge to build up. You can compare the two peaks using the ratios of current and frequency.

QnewQold=InewIoldfoldfnew=6040×12=0.75\frac{Q_{\mathrm{new}}}{Q_{\mathrm{old}}}=\frac{I_{\mathrm{new}}}{I_{\mathrm{old}}}\frac{f_{\mathrm{old}}}{f_{\mathrm{new}}}=\frac{60}{40}\times\frac{1}{2}=0.75

Revisit current and charge, then return to your explanation.

Checkpoint 2 / Reason it throughWhere would you place the receiver?

In the thin-wire model, set the total length to 1.8λ. Would a distant receiver get the strongest signal broadside or at an angle to broadside? Keep the distance fixed and the receiving wire aligned with the electric field. Make a prediction before checking the power cut.

Choose an answer

In the far-field model, fields from all parts of the wire have the same travel time broadside. What happens when some parts carry current in opposite directions? How do the travel times change when you move away from broadside?

Notes stay on this page while it’s open.
Use the worked solution to check your reasoning.
Show the worked solution
  1. Read the current plot. The current near the center flows one way while the current farther out flows the other way.
  2. Compare directions. Broadside, the travel times are the same. The areas on opposite sides of zero in the current plot are nearly equal, so their fields nearly cancel. Away from broadside, the different travel times can reduce that cancellation.
  3. Check the power cut. The model peaks near 52.4° and 127.6° from the upward wire axis. Broadside power density is about 0.74% of the peak.

Revisit the length and pattern explorer, then return to your explanation.

Checkpoint 3 / Reason it throughWhich load is resonant? Which accepts more power?

Connect each load to the same 50 Ω line. At the connection, load A is 80 + j0 Ω and load B is 50 + j20 Ω. The same incident power reaches each load. Which load is resonant? Which accepts more power? Use the reflection calculation if you need it.

Choose an answer

Find the reactance of each load. Then calculate the fraction of incident power each one reflects.

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Use the worked solution to check your reasoning.
Show the worked solution

A is resonant because its reactance is zero. B accepts more power because it reflects less. Use both resistance and reactance to calculate the reflected fraction, |Γ|². Subtract that fraction from 1 to get the fraction the load accepts.

∣Γ∣2=(R−50 Ω)2+X2(R+50 Ω)2+X2,PaccPinc=1−∣Γ∣2|\Gamma|^2=\frac{(R-50\,\Omega)^2+X^2}{(R+50\,\Omega)^2+X^2},\qquad \frac{P_{\mathrm{acc}}}{P_{\mathrm{inc}}}=1-|\Gamma|^2

For A, use R = 80 Ω and X = 0:

∣ΓA∣2=(30130)2≈0.05325,Pacc,APinc≈94.67%|\Gamma_A|^2=\left(\frac{30}{130}\right)^2\approx0.05325,\qquad \frac{P_{\mathrm{acc},A}}{P_{\mathrm{inc}}}\approx94.67\%

For B, use R = 50 Ω and X = 20 Ω:

∣ΓB∣2=2021002+202≈0.03846,Pacc,BPinc≈96.15%|\Gamma_B|^2=\frac{20^2}{100^2+20^2}\approx0.03846,\qquad \frac{P_{\mathrm{acc},B}}{P_{\mathrm{inc}}}\approx96.15\%

If these loads are antennas, some of the accepted power may become heat. To compare how much they radiate, you also need their radiation efficiencies.

Revisit matching and the power balance, then return to your explanation.

Checkpoint 4 / Reason it throughCan an electric-field reading tell you how much power flows outward?

Compare the two probes. Suppose the probe near the wire gives you only the RMS electric-field strength. Can you use ERMS²/η₀ there? If power briefly flows inward, can it still flow outward on average?

Choose an answer

What else do you need to know about the fields? How would you count inward and outward flow over one cycle?

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Use the worked solution to check your reasoning.
Show the worked solution
  1. Use both fields. You need the strength and direction of E and H at the same instant.
  2. Find the flow at the probe. Calculate E × H and take its outward component. Average that value over one cycle.
  3. Keep the sign. Count outward flow as positive and inward flow as negative. The average is outward if more energy flows out than comes back.

In a sinusoidal wave traveling through free space, E and H are perpendicular and in phase, and E/H = η₀. Under those conditions you can use the shortcut below.

⟨S⟩=ERMS2η0=Epk22η0\langle S\rangle=\frac{E_{\mathrm{RMS}}^2}{\eta_0}=\frac{E_{\mathrm{pk}}^2}{2\eta_0}

Revisit the near/far field comparison, then return to your explanation.

Checkpoint 5 / Reason it throughDid the choke reduce current on the cable?

Moving the coax changes the reflection you see on the VNA (S11). When you add a choke, the reading hardly changes. What can you conclude? How would you use an RF current probe to check current on the outside of the shield?

Choose an answer

What does the VNA reading tell you? Where would you put the current probe, and what would you keep fixed?

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Use the worked solution to check your reasoning.
Show the worked solution
  1. Start with what you saw. Moving the cable changed the reflection. Outside current, also called common-mode current here, could explain it. Moving the cable can also change how it interacts with nearby objects.
  2. Measure with and without the choke. Clamp the probe around the whole coax. Keep the frequency, source level, probe position, cable routing, calibration plane and surroundings fixed.
  3. Check more than one spot. A repeatable drop shows that the choke reduced the current at that spot and frequency. Check other positions and frequencies you plan to use. A small reading could be a current minimum or below the probe’s sensitivity.

Revisit feedline current paths and diagnostics, then return to your explanation.

Checkpoint 6 / Reason it throughWhich antenna would you prototype for a 2.45 GHz link?

You need an antenna for a short-range link from 2.44 to 2.46 GHz. The receiver is broadside and uses vertical polarization. Compare two options at 2.45 GHz.

Straight dipole. A vertical, center-fed dipole made from 1 mm diameter wire will fit. Use the worked design to estimate its length in free space. For the match calculation, assume a separate measurement at the feed gives 65 + j18 Ω on a 50 Ω line.

Compact antenna. Use a model with 2.5 Ω of radiation resistance, 0.15 Ω of wire loss resistance and −180 Ω of reactance. Tune it with a series coil that has Q = 80 at 2.45 GHz.

  1. How long would you cut the straight dipole and each arm? What could shift resonance once you install it?
  2. Is the straight dipole resonant? Find its VSWR and the fraction of power it accepts. Can you tell how much it radiates?
  3. Find the inductance, coil loss resistance and radiation efficiency for the compact antenna and coil. Does tuning it give a 50 Ω match?
  4. Which would you build first? What measurement could change your decision?

Calculation help: reflection and VSWR · coil inductance and loss.

Show your calculations and units. Say where each power figure is measured or calculated. Explain what you would build first and what you would measure next.

Notes stay on this page while it’s open.
Show the worked solution

1. Estimate the wire length

λ=299 792 458 m/s2.45×109 Hz≈122.36 mm\lambda=\frac{299\,792\,458\ \mathrm{m/s}}{2.45\times10^9\ \mathrm{Hz}}\approx122.36\ \mathrm{mm}

Use conductor diameter d = 1 mm in the first-resonance estimate.

Lres≃122.36 mm2[1−0.2257ln⁡(122.36/1)−1]≈57.6 mmL_{\mathrm{res}}\simeq\frac{122.36\ \mathrm{mm}}{2}\left[1-\frac{0.2257}{\ln(122.36/1)-1}\right]\approx57.6\ \mathrm{mm}

Allow about 28.8 mm per arm before accounting for the feed gap. Leave room to trim and check resonance with the actual feed and supports in place.

2. Check resonance and match at the feed

The measurement gives +18 Ω of reactance, so the dipole is inductive at 2.45 GHz. Resonance occurs where reactance reaches zero. For the match calculation, use 50 Ω as the line impedance.

∣Γ∣=(65−50)2+182(65+50)2+182=54913549≈0.2013|\Gamma|=\sqrt{\frac{(65-50)^2+18^2}{(65+50)^2+18^2}}=\sqrt{\frac{549}{13549}}\approx0.2013VSWR=1+∣Γ∣1−∣Γ∣≈1.504:1,PaccPinc=1−∣Γ∣2≈95.95%\mathrm{VSWR}=\frac{1+|\Gamma|}{1-|\Gamma|}\approx1.504:1,\qquad \frac{P_{\mathrm{acc}}}{P_{\mathrm{inc}}}=1-|\Gamma|^2\approx95.95\%

The dipole accepts 95.95% of the power that reaches its feed. To find how much of that power radiates, you also need to know how much becomes heat.

3. Tune the compact antenna and include coil loss

A series coil must supply +180 Ω at 2.45 GHz to cancel the antenna’s reactance.

Lcoil=180 Ω2π(2.45×109 Hz)≈11.7 nH,Rcoil=180 Ω80=2.25 ΩL_{\mathrm{coil}}=\frac{180\ \Omega}{2\pi(2.45\times10^9\ \mathrm{Hz})}\approx11.7\ \mathrm{nH},\qquad R_{\mathrm{coil}}=\frac{180\ \Omega}{80}=2.25\ \OmegaRtotal=(2.5+0.15+2.25) Ω=4.90 Ω,ηrad=2.54.90≈51.0%R_{\mathrm{total}}=(2.5+0.15+2.25)\ \Omega=4.90\ \Omega,\qquad \eta_{\mathrm{rad}}=\frac{2.5}{4.90}\approx51.0\%

The antenna and coil radiate about 51.0% of the power they accept. The coil brings the reactance to zero, so the combination is resonant. Its resistance is still 4.90 Ω, so it reflects much of the power arriving on a 50 Ω line.

4. Pick a prototype and check it on the bench

Start with the straight dipole. It fits and is close to a 50 Ω match at 2.45 GHz. If space becomes tighter, the compact antenna is worth testing with its coil and any matching components in place.

Before choosing either design, sweep its impedance from 2.44 to 2.46 GHz in its intended mounting. Measure realized gain toward the receiver with vertical polarization. If you test the compact antenna, check the mounted coil’s reactance and loss across the band.

Check your explanation

  • Did you distinguish diameter from radius, and total length from arm length?
  • Which values did you calculate, and which measurement did the exercise give you?
  • Did you include coil loss and keep accepted power separate from radiated power?
  • What measurement could change your choice?

Revisit the worked design and measurement sequence, then return to your explanation.

References and source notes

The sources for the original white paper were checked on 1 November 2026. The notes below identify the models and measurement procedures used here. Page numbers refer to the printed pages. The university lecture links may point to newer editions over time.

  1. Natalia K. Nikolova, McMaster University, Lecture 9. Linear Wire Antennas: Dipoles and Monopoles.

    Sections 1–3 cover dipole currents and patterns. Sections 4–7 cover image theory and antennas over a conducting plane, including the quarter-wave monopole on pp. 23–24.

  2. Natalia K. Nikolova, McMaster University, Lecture 10. Reciprocity. Cylindrical Antennas: Analytical Models.

    Induced-EMF method, its limitations and complex input impedance, especially pp. 8 and 14-16, Eqs. 10.40-10.41.

  3. Sean V. Hum, University of Toronto, ECE422 notes. Ideal (Hertzian) Dipole.

    Full electric and magnetic fields, peak-phasor power and current-element interpretation, especially pp. 3-4 and 7-8.

  4. Norris S. Nahman, Motohisa Kanda, Ezra B. Larsen and Myron L. Crawford, IEEE Transactions on Instrumentation and Measurement, IM-34(4), 1985, pp. 490-503. Methodology for Standard Electromagnetic Field Measurements.

    Official NIST/NBS paper. Standard dipoles, effective length, first-resonance length and thickness, calibration and measurement uncertainty. Eq. 10 and the table are on printed p. 494.

  5. Hermann A. Haus and James R. Melcher, MIT. Electromagnetic Fields and Energy, Chapter 12.

    Section 12.2 develops electrodynamic source fields; Sections 12.3 and 12.4 develop retarded superposition and antenna radiation. Original university-hosted text.

  6. Keysight Technologies, application note 5965-7707. Understanding the Fundamental Principles of Vector Network Analysis.

    Reflection, transmission-line and network-analysis conventions, complex measurement and calibration context.

  7. Richard Wallace, Texas Instruments, AN058 / SWRA161B. Antenna Selection Guide.

    Cited narrowly for practical measurement and tuning in Sections 6.2.1-6.2.5. Its introductory radiation analogies are not the basis of this paper's field explanation.

  8. Keysight Technologies, official instrument help. Removing Unwanted Effects from the Measurement.

    Calibration, port extension and fixture de-embedding are different operations. Supports the stated measurement-boundary distinctions.

  9. Steven R. Best, MITRE, author report. Optimization of the Bandwidth of Electrically Small Planar Antennas.

    Enclosing-sphere size, radiation efficiency, Q and bandwidth limitations. The lossless single-electric-mode bound used here follows the report's Eq. 1 with efficiency one.

  10. Sean V. Hum, University of Toronto, ECE422 notes. Receiving Antennas.

    Reciprocity, terminal conventions, vector effective length and the half-wave result λ/π, especially pp. 3-6.

  11. Natalia K. Nikolova, McMaster University, Lecture 11. Practical Dipole/Monopole Geometries and Matching Techniques.

    Section 1 derives the folded dipole's impedance ratio (p. 3, Eq. 11.10). Pages 19–22 discuss current on the outside of the coaxial shield and the role of a balun.

  12. Larry Lamano, QST, March 2024, pp. 33-36. Common-Mode Current and Common-Mode Chokes.

    Author's practical treatment and measurements of exterior current, common-mode chokes and frequency-dependent behavior.

  13. Sean V. Hum, University of Toronto, ECE422 notes. Free Space Radio Links.

    Gain, conjugate matching and effective aperture, especially pp. 2-6.

  14. David H. Staelin, MIT OpenCourseWare, 6.661 Receivers, Antennas, and Signals. Antennas, Chapter 3.

    Independent university treatment of receive power, conjugate matching and effective aperture, printed pp. 122-126.

  15. Steven W. Ellingson, Virginia Tech Publishing, 2020. Electromagnetics, Volume 2.

    Section 10.4, p. 172, Eq. 10.22 gives the short-wire reactance estimate used in this article and in the reference paper’s 30 cm copper-dipole example.