ch4: how current becomes a wave

date: May 17 2026

this chapter: what are the fundamentals of converting current into EM waves?

1. Only accelerating charges radiate

Every electromagnetic wave traces back to a single fact: a charge whose velocity is changing produces radiation. A charge at rest, or moving at constant velocity, does not. Three cases:

Static charge. The field is the familiar radial Coulomb field. It carries energy stored in space, but the configuration is time-independent. No power flows outward.
Constant velocity. The field is a Lorentz-contracted version of the static field, still rigidly attached to the charge. By switching to the charge’s rest frame, the configuration becomes manifestly static. No radiation in any frame.
Acceleration. Now the field cannot rigidly follow the charge. Information about the charge’s new state cannot propagate faster than $c$ , so the field at large distances still points at where the charge used to be. The field lines must kink to connect the old far-field configuration to the new near-field one. That kink propagates outward at $c$ and carries energy. That kink is the electromagnetic wave.

The Larmor formula quantifies it. The total power radiated by a non-relativistic point charge with acceleration $a$ is

P \;=\; \frac{q^2 a^2}{6\pi\,\epsilon_0\,c^3}

Power scales as acceleration squared, not velocity. A charge moving at constant velocity along a curved path (cyclotron motion, for instance) still has nonzero acceleration and so still radiates. A charge moving in a straight line at constant speed does not.

Everything an antenna does follows from this. The wire geometry, the impedance of the feed, the resonance condition are all just different knobs for shaping how the charges accelerate, which is what shapes the radiated wave.

2. From current to wave

A current is a flow of charges. A changing current is charges that are accelerating. The connection between current on a wire and radiation is therefore direct: $dI/dt$ is the acceleration term that drives the Larmor formula.

DC does not radiate

A wire carrying a constant DC current has charges drifting at constant velocity. Each charge contributes zero radiated power. The wire produces a static magnetic field (Biot-Savart) but no propagating wave. A 12 V battery wired to a resistor through perfect conductors radiates exactly nothing.

Time-varying current radiates regardless of waveform

Sinusoidal current radiates. So does a square wave. So does a pulse, a chirp, a random noise current, any time-varying $I(t)$ . The antenna does not care about the shape of the waveform; it converts whatever time-domain current you feed it into the corresponding time-domain field in space, subject to its frequency response (Section 3).

The field-line picture

For an oscillating dipole (current flowing back and forth along a short wire), the electric field lines form closed loops near the wire. As the current reverses, those loops detach from the source and propagate outward at $c$ . The far-field pattern is the well-known toroidal “donut”: peak radiation broadside to the wire (perpendicular to its axis), zero along the wire’s own axis.

The mathematical statement is the radiated $E_\theta$ in the far field of a short dipole of length $L$ carrying current $I_0 \cos(\omega t)$ :

E_\theta(r, \theta, t) \;=\; \frac{\eta_0\,I_0\,L\,k}{4\pi}\,\frac{\sin\theta}{r}\,\sin\!\bigl(\omega t - k r\bigr), \qquad k = \frac{2\pi}{\lambda}, \quad \eta_0 \approx 377\,\Omega

The $\sin\theta/r$ angular dependence is the donut. The $\sin(\omega t - kr)$ is the wave propagating outward at $c$ . The factor $1/r$ (not $1/r^2$ ) is the signature of radiation, distinguishing the far field from the near field where the static dipole’s $1/r^3$ Coulomb-like falloff dominates.

Field of an oscillating dipole

$E_\theta$ in the plane of the dipole, animated as the source oscillates. The dipole is the small vertical bar at the centre. Teal and warm colors are opposite signs of the field. The wavefronts visibly propagate outward, and the field is zero along the dipole’s own axis (top and bottom).

Frequency (relative) 1.5

The $1/r$ falloff in the far field is hard to see at this scale; what is obvious is the alternating-sign concentric arcs (wavefronts) and the angular pattern (no radiation along the dipole axis). Closer to the source, the $1/r^3$ and $1/r^2$ terms of the near field dominate and the field looks more like an oscillating static dipole than a wave.

3. The antenna as a linear filter

An antenna is a linear, time-invariant system. Whatever current waveform $i(t)$ is fed in, the radiated field is the convolution of that current with the antenna’s impulse response $h(t)$ . Equivalently, in the frequency domain:

E(f) \;=\; H(f)\,I(f)

The transfer function $H(f)$ encodes everything about how the antenna couples each spectral component to free space: which frequencies are radiated efficiently, which are reflected back, which are distorted in phase. For a resonant antenna $H(f)$ is a bandpass shape centred at the resonance frequency with width set by the antenna’s $Q$ .

What this means for waveform shape

Pure sine at $f_0$ . The spectrum is a single line. If $H(f_0)$ is large (you fed the antenna at its resonance), the line passes through; the radiated field is the same sine. Efficient.
Square wave at $f_0$ . The spectrum has lines at $f_0, 3f_0, 5f_0, \ldots$ (odd harmonics). The fundamental passes through the bandpass, the higher harmonics are attenuated. The radiated waveform looks like a rounded version of the input square; the harmonics that gave it sharp edges have been filtered off.
Short pulse. The spectrum is broadband. Only the slice within the antenna’s bandwidth radiates. The radiated waveform is the input pulse convolved with $h(t)$ , which for a narrow-band antenna is a damped sinusoid at the resonant frequency. The antenna rings.
Chirp. A linear sweep of frequency. If the sweep stays within the antenna bandwidth, it passes through with a modest phase distortion. If it exceeds the bandwidth, the chirp edges get attenuated and the FMCW range-FFT picks up the spectral edges as ghost targets.

The antenna is not a frequency analyzer; it is a frequency-selective transducer. Frequency discrimination at the receiver happens later, in the mixer and the digital signal processor. The antenna’s job is to deliver whatever falls inside $H(f)$ into free space, faithfully in phase and amplitude.

Waveform through the antenna's bandpass

Pick an input waveform. The four panels show: input in time, input spectrum, the antenna's |H(f)| overlaid on that spectrum, and the radiated waveform after H(f) filtering. The antenna's centre frequency and Q are adjustable.

Antenna f₀ (relative) 1.00 Antenna Q 6

Top row: input waveform (time) and its spectrum. Bottom row: antenna |H(f)| shown as a shaded region behind the input spectrum, and the resulting output waveform. Narrowing the antenna (large Q) rejects more of the input spectrum; mis-tuning f₀ shifts which part of the input survives.

4. Near field vs far field

Close to an antenna, the field is dominated by Coulomb-like static terms that fall off as $1/r^3$ (reactive near field) and induction-like terms that fall off as $1/r^2$ . Far from the antenna, only the $1/r$ radiation term survives; everything else has died off much faster.

E_\text{near}(r) \;\sim\; \frac{1}{r^3} + \frac{1}{r^2} + \frac{1}{r}, \qquad E_\text{far}(r) \;\sim\; \frac{1}{r}

The boundary

For a small antenna (much smaller than $\lambda$ ), the transition from near to far field happens at roughly $r \sim \lambda/(2\pi)$ , where the three terms become comparable. For a large antenna of largest dimension $D$ , the relevant boundary is the Fraunhofer distance,

r_\text{far} \;\approx\; \frac{2 D^2}{\lambda},

beyond which the wavefronts from different parts of the antenna are close enough to plane wavefronts that the standard far-field formulas apply. For a 1 m dish at 10 GHz, $r_\text{far} \approx 67$ m. Inside that range, the “antenna pattern” you might measure on a workbench will not match the pattern in deployment.

Energy flow

In the reactive near field, the net outward power flow averaged over a cycle is essentially zero. Energy stored in the field oscillates between the antenna and the surrounding space without leaving. This is reactive (imaginary-valued) impedance at the feedpoint.

In the far field, the net outward power flow is radial and nonzero. Energy is flowing outward and will not return. This is the real-valued radiation resistance at the feedpoint (Section 5).

Antenna gain and pattern are far-field quantities, so measuring them inside the Fraunhofer distance gives wrong answers. For small electrically-short antennas this is rarely a problem (the boundary is centimeters). For dish antennas or large arrays at high frequency the test range needs to be tens or hundreds of meters, or you need compact-range optics.

5. The antenna feels its own radiation

Radiated power has to come from somewhere. The source driving the antenna has to deliver, on average, the power that ends up in the far field. From the source’s perspective, the antenna looks like a resistor at the feedpoint — even though no actual resistor is present.

This apparent resistance is the radiation resistance $R_\text{rad}$ . It is not dissipation in the wire (that is the separate ohmic loss resistance); it is the impedance you would measure at the feedpoint that accounts for energy leaving as radiation. For a short dipole of length $L \ll \lambda$ :

R_\text{rad} \;\approx\; 80\pi^2 \left(\frac{L}{\lambda}\right)^2 \;\approx\; 790\,\left(\frac{L}{\lambda}\right)^2 \;\Omega

For a half-wave dipole ( $L = \lambda/2$ ), the corresponding integral gives $R_\text{rad} \approx 73\,\Omega$ . For $L = \lambda/10$ , $R_\text{rad} \approx 7.9\,\Omega$ . As the antenna shrinks below $\lambda/2$ , $R_\text{rad}$ collapses as the square of the size.

Why small antennas are inefficient

The real ohmic resistance of a thin wire is fixed by the conductor’s geometry and skin depth, typically a fraction of an ohm. For a $\lambda/2$ dipole with $R_\text{rad} \approx 73\,\Omega$ and $R_\text{ohm} \approx 0.5\,\Omega$ , the radiation efficiency $R_\text{rad}/(R_\text{rad} + R_\text{ohm})$ is over 99%. For a $\lambda/20$ dipole with $R_\text{rad} \approx 2\,\Omega$ , the same $R_\text{ohm} \approx 0.5\,\Omega$ now drops efficiency to 80%. At $\lambda/100$ , efficiency is below 10%.

Mutual coupling

Two antennas near each other each radiate into the other’s near field. The radiation from antenna B exerts a force on the charges in antenna A, and vice versa. This appears as a mutual impedance $Z_{12}$ in addition to each antenna’s self-impedance. In an array, the input impedance of every element depends on the currents in every other element. Treating array elements as independent is wrong; Chapter 6 makes this precise.

6. Phase encodes distance

The connection between the current at the antenna and the field at distance $r$ is the retarded-time relation: the field at $(r, t)$ depends on what the source was doing at the earlier time $t - r/c$ .

E(r, t) \;\propto\; \frac{I(t - r/c)}{r}

For sinusoidal current $I(t) = I_0 \cos(\omega t)$ , this becomes

E(r, t) \;\propto\; \frac{I_0}{r}\cos\!\bigl(\omega t - k r\bigr), \qquad k = \omega/c = 2\pi/\lambda

The phase $\phi(r) = -kr$ is a linear function of distance. Move the source (or the target) by $\Delta r$ and the phase shifts by $\Delta\phi = -k\,\Delta r = -2\pi\,\Delta r/\lambda$ . This is the same relationship Chapter 2 used to extract micron-scale chest motion from a returned radar phase, with the factor of two for the round trip.

Range and phase resolution are two different views of this same retarded-time relationship. Range counts wavelengths in chunks of size $c/(2B)$ , set by the waveform bandwidth. Phase measures a fraction of one wavelength, set by SNR. They live on completely different scales and are limited by completely different things.

What the wavelength is the ruler of

Wavelength $\lambda$ shows up in three independent roles:

Phase per unit distance. The wavenumber $k = 2\pi/\lambda$ sets how fast phase accumulates with distance. Small $\lambda$ (high $f$ ) means a tiny motion produces a measurable phase shift, the basis for phase-based motion sensing.
Resonant antenna size. A half-wave dipole has $L = \lambda/2$ ; a patch is $\sim\lambda/2 \times \lambda/2$ ; an aperture is sized in units of $\lambda$ . Chapter 5 unpacks why.
Diffraction limit on beamwidth. A beam of width $\theta$ requires an aperture of width at least $\lambda/\theta$ . Chapter 6 unpacks why.

The three roles are physically independent but numerically connected through $\lambda$ . Choosing an operating frequency simultaneously sets the phase sensitivity, the resonant antenna size, and the achievable beamwidth.

next: ch5: why antennas are sized in λ →