Dipole Antennas — Interactive Tutorial

Theory of Dipole Antennas

Dipole antennas are cheap, easy to manufacture and install. They are used as TV antennas, for example. Let us consider a dipole antenna of length 2L at the origin of the spherical coordinate system shown in the figure below.

If we assume that the current flowing in the dipole is along the z-axis and is of the form:

\[ I(z) = I_m \sin(\beta(L - |z|)) \]

where:

\(I_m\) is the maximum current amplitude
\(\beta = 2\pi/\lambda\) is the wave number
\(\lambda\) is the wavelength
\(L\) is the half-length of the dipole (total length = \(2L\))
\(z\) is the position along the dipole (-L ≤ z ≤ L)

then the amplitudes of the electric and magnetic components of the field radiated by the dipole in the far field zone may be approximated by the formulas (in free space the intrinsic impedance may be approximated by \(120\pi \approx 377\Omega\)):

\[ E_\theta = \frac{60 I_m}{r} \left[ \frac{\cos(\beta L \cos\theta) - \cos(\beta L)}{\sin\theta} \right] \] \[ H_\phi = \frac{I_m}{2\pi r} \left[ \frac{\cos(\beta L \cos\theta) - \cos(\beta L)}{\sin\theta} \right] \]

where:

\(r\) is the distance from the antenna (far-field condition: \(r \gg \lambda\))
\(\theta\) is the polar angle measured from the antenna axis (z-axis)
\(\phi\) is the azimuthal angle

The average radiated power density (Poynting vector magnitude) is given by:

\[ P_{av} = \frac{15 I_m^2}{\pi r^2} \left[ \frac{\cos(\beta L \cos\theta) - \cos(\beta L)}{\sin\theta} \right]^2 \]

The radiation pattern depends on the dipole length relative to the wavelength. Below are common configurations:

Short Dipole (\(L \ll \lambda/2\))

Current is approximately uniform. The radiation pattern is a simple figure-8 (doughnut shape) with maximum radiation perpendicular to the dipole axis.

Half-Wave Dipole (\(L = \lambda/4\), total length = \(\lambda/2\))

The most common configuration. Maximum radiation is perpendicular to the antenna axis, with nulls along the axis. Directivity = 2.15 dBi.

Full-Wave Dipole (\(L = \lambda/2\), total length = \(\lambda\))

Pattern splits into multiple lobes. Still has maximum radiation broadside but with additional side lobes. Directivity = 3.82 dBi.

Interactive Simulation

Use the interactive simulation below to examine the distribution of the power density in terms of angle theta (\(\theta\)) as the length 2L of the antenna dipole is changed. Move the slider to change the length 2L. For each value of the length, you obtain an intensity pattern that displays the length of the dipole in terms of the wavelength. The half power beam width (the angle between the two red lines) which indicates the directivity of the antenna is also displayed. The beam width changes with the length 2L of the antenna.

Radiation Pattern (Polar Plot)

Dipole Half-Length L (in λ) 0.25 λ

Peak Current Iₘ (A) 1.0 A

Radiation Pattern

Half-Power Points (-3dB)

Pattern Characteristics:

Total Dipole Length: 0.5 λ

Half-Power Beamwidth: 78°

Directivity: 2.15 dBi

Maximum Power Density: 4.77 W/m² at r = 1m

Field Components Visualization

Observation Angle θ (degrees) 90°

Distance r (meters) 10 m

Field Values at Observation Point:

Electric Field E₀: 0.60 V/m

Magnetic Field Hᵩ: 1.59 mA/m

Power Density Pₐᵥ: 0.48 W/m²

Wave Impedance: 377 Ω

Current Distribution Along Dipole

Current Distribution I(z)

The current distribution along the dipole follows a sinusoidal pattern:

\[ I(z) = I_m \sin(\beta(L - |z|)), \quad -L \leq z \leq L \]

This distribution is visualized below for the current dipole length:

Maximum Current (at center): 1.0 A

Current at ends: 0.0 A

Wavelength: 3.0 m (at 100 MHz)

Power Density vs. Angle

The normalized power density pattern as a function of θ:

Key Observations:

The pattern is symmetric about θ = 90°
Nulls occur where the pattern function goes to zero
Side lobes appear for dipoles longer than λ/2
Maximum radiation is at θ = 90° for L ≤ λ/2