Half Wave Dipole

The half-wave dipole is the most widely used and practically important antenna in all of electromagnetics. Its input impedance, gain, and radiation pattern are well characterized, making it the reference standard for antenna gain measurements (0 dBd) and a key topic in GATE and university examinations.

Structure and Current Distribution

The half-wave dipole consists of two straight conductors, each of length λ/4, aligned collinearly along the z-axis and fed at the centre with a transmission line or RF source. The total physical length is therefore λ/2, where λ is the operating wavelength. This specific length makes the antenna resonant — the input reactance is nearly zero, greatly simplifying impedance matching.

The current distribution along the half-wave dipole follows a sinusoidal (cosine) profile: I(z) = I₀ cos(πz/λ), where z ranges from −λ/4 to +λ/4 and I₀ is the peak feed current. The current is maximum at the feed point (z = 0) and falls to zero at the tips (z = ±λ/4). This realistic current distribution distinguishes the half-wave dipole from the idealized Hertzian dipole, which assumes uniform current.

Input Impedance

The input impedance of the half-wave dipole is obtained by integrating the contributions of all Hertzian dipole elements along its length with the cosine current weighting. The result, computed from the theory of thin wire antennas, gives a radiation resistance of approximately 73.1 Ω and an inductive reactance of approximately 42.5 Ω at the resonant length of exactly λ/2.

In practice, the dipole is slightly shortened to about 0.48λ to achieve resonance (zero reactance). At this shortened length, the input impedance is purely resistive at approximately 73 Ω, which is very close to the standard 75 Ω coaxial cable impedance. This near-perfect match makes the half-wave dipole one of the easiest antennas to feed efficiently in practice.

Radiation Pattern and Directivity

The far field electric field of the half-wave dipole is given by E_θ = j(60 I₀ / r) × [cos(π/2 × cosθ) / sinθ] × e^(−jkr). The pattern factor F(θ) = cos(π cosθ / 2) / sinθ is slightly sharper than the sinθ pattern of the Hertzian dipole. The pattern still has nulls along the dipole axis (θ = 0°, 180°) and maximum in the equatorial plane (θ = 90°). The HPBW for the half-wave dipole is 78°, compared to 90° for the Hertzian dipole, confirming slightly higher directivity.

The directivity of the half-wave dipole is D = 1.64 (2.15 dBi). This is the standard reference level for the dBd unit used in antenna specifications. A gain of 0 dBd means the antenna has the same gain as a lossless half-wave dipole. All base station antenna datasheets express gain in dBd or dBi, and the conversion is 0 dBd = 2.15 dBi.

Mathematical Expression

The total radiated power of the half-wave dipole is P_rad = (1/2) I₀² × 73.1 W (treating R_rad = 73.1 Ω). The directivity is D = 4π U_max / P_rad = 1.64. The effective length h_eff of the half-wave dipole is defined such that the far field equals that of a Hertzian dipole of length h_eff with the same peak current: h_eff = λ/π = 2/(π) × λ/2. This concept is important in computing the open-circuit voltage at a receive dipole antenna terminal.

Practical Understanding

The half-wave dipole is used as a standalone antenna for FM radio reception (87.5 to 108 MHz, λ/2 ≈ 1.4 to 1.7 m), VHF television, and as the driven element in Yagi-Uda arrays. Its balanced feed requirement means a balun (balanced-to-unbalanced transformer) is needed when connecting to an unbalanced coaxial cable. Without a balun, common-mode currents flow on the outer shield of the coax and distort the pattern.

In mobile communications, the half-wave dipole is used as a reference antenna for base station gain measurements. The folded dipole variant of the half-wave dipole has an input impedance four times that of a standard dipole (approximately 292 Ω) and is commonly used as the driven element in Yagi antennas to match the higher impedance of a folded element to standard feedlines via a simple 4:1 balun.

Given:
Half-wave dipole operating at f = 100 MHz, λ = c/f = 3×10⁸ / 10⁸ = 3 m, Peak feed current I₀ = 0.5 A, Radiation resistance R_rad = 73.1 Ω

Why this formula applies:
P_rad = ½ × I₀² × R_rad gives total radiated power from the antenna terminal quantities.

Formula:
P_rad = (1/2) × I₀² × R_rad

Substitution:
P_rad = 0.5 × (0.5)² × 73.1

Calculation:
P_rad = 0.5 × 0.25 × 73.1 = 9.14 W

Physical length:
L = λ/2 = 3/2 = 1.5 m

Final Answer with units:
P_rad ≈ 9.14 W
Physical antenna length = 1.5 m
The dipole radiates 9.14 W with 0.5 A peak current at 100 MHz. D = 1.64, HPBW = 78°.

Exam Tip: GATE frequently tests the half-wave dipole input impedance (73 Ω ≈ 75 Ω) and directivity (1.64 = 2.15 dBi). Remember: 0 dBd = 2.15 dBi. Also, the actual antenna is slightly shortened to 0.48λ to achieve zero reactance (true resonance). Confusing R_rad = 73 Ω with R_rad = 80π²(dl/λ)² (which is the Hertzian formula) is a very common GATE mistake.

• Total length = λ/2; each arm = λ/4; current distribution is cosine: I(z) = I₀ cos(πz/λ).
• Input impedance: R_rad ≈ 73.1 Ω, X_in ≈ 42.5 Ω (inductive) at exactly λ/2 length.
• Shortened to 0.48λ for resonance (X_in = 0); purely resistive at ~73 Ω — matches 75 Ω coax.
• Directivity D = 1.64 (2.15 dBi); HPBW = 78° — slightly better than Hertzian dipole (90°).
• Pattern factor: F(θ) = cos(π cosθ / 2) / sinθ — nulls at θ = 0°, maximum at θ = 90°.

Quick Revision

• Half-wave dipole: length = λ/2, fed at centre, current maximum at feed, zero at tips.
• Input impedance: Z_in = 73.1 + j42.5 Ω at λ/2; purely resistive 73 Ω at resonance (0.48λ).
• Directivity = 1.64 = 2.15 dBi. HPBW = 78°. Reference for 0 dBd scale.
• 0 dBd = 2.15 dBi — conversion between half-wave dipole and isotropic references.
• Radiated power: P_rad = ½ I₀² × 73.1 W.
• Effective length h_eff = λ/π ≈ 0.318λ (used in receive voltage calculation).
• Trap: do not use R_rad = 80π²(dl/λ)² for half-wave dipole — that formula applies only to the Hertzian dipole.