S4-2.2 Antenna Analysis: Dipoles & Loops

What this is and why it exists

Nobody manufactures an infinitesimal dipole, yet every antenna analysis begins with it — it is the hydrogen atom of the subject, the one radiator with clean closed-form fields. Real antennas are analysed by treating them as stacks of these elements and summing. The unit's destination is the half-wave dipole and the small loop, and one number worth carrying for life: 73 ohms.

The vocabulary

• Infinitesimal dipole — a current element much shorter than a wavelength, with uniform current; the building block of antenna analysis.
• Half-wave dipole — a wire half a wavelength long, fed at the centre, with a sinusoidal current distribution; the most-built antenna on earth.
• Radiation resistance — the fictitious resistance that would dissipate the radiated power at the feed current; how radiation appears to the circuit.
• Small loop antenna — a current loop much smaller than a wavelength; the magnetic twin of the electric dipole.
• Directivity — for these canonical antennas, computed exactly from their patterns: 1.5 for the tiny dipole, 1.64 for the half-wave dipole.

The mental model

Radiation resistance is the bridge between field theory and circuit theory. Integrate the radiated power over a far-field sphere, set it equal to the square of the feed current times a resistance, and that resistance is what the transmitter actually experiences — the antenna, to the circuit, is a resistor whose dissipation is escape into space. For the infinitesimal dipole this resistance is tiny and proportional to the square of length-over-wavelength: an electrically short wire radiates almost nothing, while its ohmic losses stay real, so short antennas are inefficient. That single result explains why antennas want to be a good fraction of a wavelength.

The half-wave dipole is the short element grown to resonance. The current distribution becomes a half sinusoid — maximum at the feed, zero at the tips — and summing the element contributions gives the classic result: radiation resistance near 73 ohms and directivity 1.64, with the doughnut pattern slightly flattened compared with the short dipole's. That 73 ohms is why coaxial cable and RF instruments standardised near 50 and 75 ohms — the infrastructure of radio is shaped around matching this one antenna.

The small loop completes the pair by symmetry: where the dipole is an electric current radiating with the electric field dominant, the loop is a magnetic moment with the roles of the fields exchanged — same doughnut pattern, rotated. Its radiation resistance grows with the fourth power of size-over-wavelength, so small loops are even weaker radiators than short dipoles, which is why loop antennas in practice use many turns or ferrite cores, and why your AM radio's internal antenna receives adequately but could never transmit.

What you should now be able to explain or do

Sketch the derivation route: element fields, far-field terms, power integral, radiation resistance. State radiation resistance and directivity for the short dipole, half-wave dipole and small loop, and rank them as radiators. Explain why electrically short antennas are inefficient. Say why 73 ohms matters beyond the exam.

Check yourself