Why is the 3λ/2 dipole's directivity lower than the λ dipole's?

Phase reversal in the current splits the beam into tilted lobes, reducing broadside gain

Its current is zero at the ends

It has a higher radiation resistance

Why is a centre-fed λ dipole hard to feed?

Current is a minimum at the centre, so input impedance is very high

Its radiation resistance is 73 Ω

2070 Magh · 6 marks

Comparing λ/2, λ and 3λ/2 dipoles

Short answer

λ/2 dipole: two broadside lobes, HPBW 78°, R_rad 73 Ω, Z_in 73+j42.5 Ω, D=2.15 dBi. λ dipole: two narrower broadside lobes, HPBW 47.8°, R_rad 199 Ω, very high Z_in, D=3.82 dBi. 3λ/2 dipole: six lobes (four major at 42.6°), HPBW 33°, R_rad 105 Ω, Z_in 105 Ω, D=3.48 dBi.

Differentiate between λ/2, λ and 3λ/2 length dipoles in terms of their individual radiation pattern, self impedance and directivity; where λ is the wavelength of operation frequency.

For a centre-fed dipole of length L with sinusoidal current I(z) = I0 sin[k(L/2 − |z|)], the far-field pattern is

E(θ) ∝ [cos((kL/2)cosθ) − cos(kL/2)] / sinθ

The values below are computed from this formula (R_r referred to the current maximum).

Property	λ/2 dipole	λ dipole	3λ/2 dipole
Current	One half-loop, max at feed	Two in-phase loops, zero at feed	Three loops, centre loop reversed
Pattern formula	cos(π/2·cosθ)/sinθ	[cos(π cosθ)+1]/sinθ	cos(3π/2·cosθ)/sinθ
Lobes (E-plane)	2 (broadside)	2 (broadside, narrower)	6 (4 major at 42.6°, 2 minor at 90°)
Max direction	θ = 90°	θ = 90°	θ ≈ 42.6°, 137.4°
HPBW	≈ 78°	≈ 47.8°	≈ 33° (major lobe, half-power points at ≈ 24.4° and 57.2°)
Radiation resistance	≈ 73 Ω	≈ 199 Ω	≈ 105 Ω
Input impedance at centre	≈ 73 + j42.5 Ω (resonant when slightly shortened)	Very high (kΩ), current minimum	≈ 105 Ω
Directivity	1.64 (2.15 dBi)	2.41 (3.82 dBi)	2.23 (3.48 dBi)

   λ/2              λ               3λ/2
 .-. | .-.      .--.|.--.       \  |  /
(   )|(   )    (    |    )     ( ) | ( )
 '-' | '-'      '--'|'--'     ( )--+--( )
                               ( ) | ( )
                              /    |    \

Key points:

The λ/2 dipole is resonant, easy to match to 75 Ω coax, and is the reference antenna.
The λ dipole has higher directivity and a narrower beam, but its very high input impedance makes it hard to feed at the centre (it is often end-fed instead).
The 3λ/2 dipole has a phase reversal in its current; the main beam splits into tilted cones, so broadside gain falls and directivity is lower than for the λ dipole.

✓ Answer checked by Claude Opus 5.5 (high effort)

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