Hertzian Dipole

The Hertzian dipole is the most fundamental radiating element in antenna theory. Although it is an idealized mathematical construct, every practical antenna can be analyzed as a superposition of Hertzian dipoles. Its radiation fields, radiation resistance, and pattern form the analytical starting point for all dipole antenna analysis in electromagnetics.

What is a Hertzian Dipole

A Hertzian dipole is an infinitesimally short current element of length dl (where dl much less than λ) carrying a uniform sinusoidal current I₀ e^(jωt) along its entire length. In reality, current in a short wire varies from maximum at the centre to zero at the tips, but the Hertzian dipole assumes uniform current throughout. This idealization makes its fields analytically tractable while capturing the essential physics of radiation.

The concept is important because any finite antenna such as the half-wave dipole can be decomposed into a series of Hertzian dipoles. The total field of the full antenna is then the superposition (integral) of fields from all infinitesimal elements along its length. This superposition principle underlies the array factor concept in antenna arrays and aperture integration in aperture antennas.

Radiation Fields

In the far field (r greater than 2D²/λ, where D = dl is negligibly small), the electric and magnetic fields of a Hertzian dipole oriented along the z-axis are purely transverse. The electric field has only a θ-component: E_θ = j(η₀ I₀ dl sin θ) / (2λr) × e^(−jkr), where η₀ = 120π ≈ 377 Ω is the intrinsic impedance of free space, k = 2π/λ is the wave number, and r is the radial distance. The magnetic field H_φ = E_θ / η₀. Both fields decay as 1/r in the far field, confirming outward power flow.

The factor sin θ in the field expression means that radiation is maximum in the equatorial plane (θ = 90°, perpendicular to the dipole axis) and zero along the dipole axis (θ = 0° and 180°). This gives the Hertzian dipole a doughnut-shaped (toroidal) three-dimensional radiation pattern — rotationally symmetric about the z-axis, with nulls along the dipole direction.

Mathematical Expression

The time-averaged power density (Poynting vector magnitude) in the far field is S_avg = |E_θ|² / (2η₀) = (η₀ / 8) × (I₀ dl / λ)² × sin²θ / r². Integrating this over a full sphere gives the total radiated power. The result is P_rad = (η₀ π / 3) × (I₀ dl / λ)². Since η₀ = 120π, this simplifies to P_rad = 40π² × (I₀ dl / λ)² × (1/2) I₀².

Using the definition P_rad = (1/2) I₀² R_rad, the radiation resistance of the Hertzian dipole is R_rad = 80π² (dl/λ)². For a Hertzian dipole with dl = λ/50, R_rad = 80π² × (1/50)² = 0.316 Ω. This extremely small value explains why electrically short dipoles are inefficient — their radiation resistance is much smaller than their ohmic loss resistance.

The directivity of the Hertzian dipole is D = 1.5 (1.76 dBi). This is obtained by evaluating 4π U_max / P_rad with U_max = (η₀/8)(I₀dl/λ)². The pattern function is F(θ) = sin²θ, and the HPBW is 90°.

Practical Understanding

The Hertzian dipole model is used in electromagnetic compatibility (EMC) analysis to estimate radiation from short PCB traces carrying high-frequency currents. A trace of length 5 mm at 1 GHz has dl/λ = 0.005/0.3 = 0.0167, giving R_rad = 80π² × (0.0167)² = 0.22 Ω. Since typical trace resistance might be a few ohms, most input power is dissipated rather than radiated — but at 10 GHz the same trace becomes much more efficient.

Given:
Hertzian dipole length dl = λ/10 (electrically short dipole), Frequency f = 300 MHz → λ = 1 m, Peak current I₀ = 1 A

Why this formula applies:
R_rad = 80π²(dl/λ)² gives the radiation resistance of a Hertzian dipole, and P_rad = ½ I₀² R_rad gives total radiated power.

Formula:
R_rad = 80π² × (dl/λ)²

Substitution:
dl/λ = (λ/10)/λ = 0.1
R_rad = 80 × π² × (0.1)²

Calculation:
R_rad = 80 × 9.87 × 0.01 = 7.896 Ω
P_rad = ½ × (1)² × 7.896 = 3.948 W

Final Answer with units:
R_rad ≈ 7.9 Ω
P_rad ≈ 3.95 W
For dl = λ/10 with 1 A peak current, the Hertzian dipole radiates approximately 3.95 W with radiation resistance 7.9 Ω.

Exam Tip: Remember R_rad = 80π²(dl/λ)² for Hertzian dipole — this is a very commonly tested formula. Also remember that directivity of Hertzian dipole = 1.5 (1.76 dBi), HPBW = 90°, and the pattern is proportional to sin²θ in power. Confusing sinθ (field pattern) with sin²θ (power pattern) is the most common GATE mistake in this topic.

• Hertzian dipole: infinitesimal current element dl much less than λ with uniform current I₀.
• Far field E_θ proportional to sinθ (field pattern); power pattern proportional to sin²θ.
• Radiation resistance R_rad = 80π²(dl/λ)² — very small for electrically short elements.
• Directivity D = 1.5 (1.76 dBi); HPBW = 90° in elevation; pattern is a torus shape.
• Nulls at θ = 0° and 180° (along dipole axis); maximum at θ = 90° (equatorial plane).

Quick Revision

• Hertzian dipole: dl much less than λ, uniform current assumption, analytical idealization.
• Far field: E_θ = j(η₀ I₀ dl sinθ)/(2λr) e^(-jkr); H_φ = E_θ/η₀.
• Power pattern: sin²θ. HPBW = 90°. Directivity = 1.5 = 1.76 dBi.
• Radiation resistance: R_rad = 80π²(dl/λ)². Radiated power: P_rad = ½ I₀² R_rad.
• Practical use: building block for all dipole and array antenna analysis by superposition.
• Trap: field pattern is sinθ, power pattern is sin²θ — do not mix these two in GATE MCQs.