Normalized power gain of dipole antennas

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Key focus: Briefly look at linear antennas and various dipole antennas and plot the normalized power gain pattern in polar plot and three dimensional plot.

Linear antennas

Linear antennas are electrically thin antennas whose conductor diameter is very small compared to the wavelength of the radiation λ.

Viewed in a spherical coordinate system (Figure 1), for linear antenna, the antenna is oriented along the z-axis such that the radiation vector has only components along directions of the radial distance Fr and the polar angle Fθ. The radiation vector is determined by the current density J which is characterized by the current distribution I(z) [1].

Electrical and magnetic fields from a current source
Figure 1: Electrical and magnetic fields from a current source
Linear antenna element
Figure 2: Linear antenna element

Hertzian dipole (infinitesimally small dipole)

Hertzian dipole is the simplest configuration of a linear antenna used for study purposes. It is an infinitesimally small (typically L < \lambda/50 [2]) antenna element that has the following current density distribution

\[I(z) = I \; L\; \delta(z)\]

The radiation vector Fz (θ) is given by [1]

\[F_z(\theta) = \int_{-L/2}^{L/2} I(z^{‘}) e^{jk_z z^{‘}} dz{‘} = \int_{-L/2}^{L/2} I L \delta(z{‘}) e^{jkz{‘}cos \theta} dz{‘} = IL\]

The normalized power gain of the Hertzian dipole is [2]

\[g(\theta) = C_0 sin^{2} \theta \]

where, C0 is a constant chosen to make maximum of g(θ) equal to unity and θ is the polar angle in the spherical coordinate system.

Center-fed dipole (standing wave antenna)

For the center-fed small dipole antenna, the current distribution is assumed to be a standing wave. Defining k = 2π/λ as the wave number and h = L/2 as the half-length of the antenna, the current distribution and the normalized power gain g(θ) are given by

\[I(z) = I \; sin \left[ k \left(L/2 – |z| \right) \right]\]
\[g(\theta) =C_n \left|\frac{cos (k h \; cos \theta) – cos( k h) }{sin \theta} \right| ^2\]

where, Cn is a constant chosen to make maximum of g(θ) equal to unity and θ is the polar angle in the spherical coordinate system.

Center-fed small dipole
Figure 3: Center-fed small dipole

For half-wave dipole, set L = λ/2 or kl = π. Therefore, the current distribution for half-wave dipole shrinks to

\[I(z) = I \; cos(kz)\]

The normalized power gain is

\[g(\theta) = C_2 \frac{cos^2 (\frac{\pi}{2} cos \theta)}{sin^2 \theta} \approx C_2 sin^3 \theta\]

Plotting the normalized power gain

Let’s plot the normalized power gain pattern of Hertzian & Half-wave dipole antennas in polar plot and three dimensional plot.

Check out my Google colab for the python code to plot the normalized power gain in polar plot as well as three dimensional plot. The results are given below.

Hertzian dipole power gain polar plot
Figure 4: Hertzian dipole – power gain pattern (polar plot)
Hertzian dipole power gain 3d plot (cartesian coordinates)
Figure 5: Hertzian dipole – power gain pattern (3D plot)
Half-wave dipole - power gain pattern (polar plot)
Figure 6: Half-wave dipole – power gain pattern (polar plot)
Half-wave dipole power gain 3d plot (cartesian coordinates)
Figure 7: Half-wave dipole power gain 3d plot (cartesian coordinates)
Normalized power gain pattern for dipole of length L
Figure 8: Normalized power gain pattern for dipole of length L=2.5 \lambda
Normalized power gain pattern for dipole of length (3D projection)
Figure 9: Normalized power gain pattern for dipole of length (3D projection)

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References

[1] Orfanidis, S.J. (2013) Electromagnetic Waves and Antennas, Rutgers University. https://www.ece.rutgers.edu/~orfanidi/ewa/

[2] Constantine A. Balanis, Antenna Theory: Analysis and Design, ISBN: 978-1118642061, Wiley; 4th edition (February 1, 2016)

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