Pi Day Special: A Short Dipole with Radiation Resistance of 3.14 Ohms

Welcome to The Antenna Lab, where we explore antenna concepts and simulations in a practical and intuitive way. Today is March 14th—Pi Day! What better way to celebrate than by demonstrating how a short dipole can have a radiation resistance of 3.14 Ohms?

The Radiation Resistance of a Hertzian Dipole

The radiation resistance of a Hertzian dipole—an idealized, infinitesimally thin short dipole with a uniform current distribution—is given by:

$\displaystyle R_r = 80\,\pi^2\, \left(\frac{L}{\lambda}\right)^2 \quad \text{Ohms} \qquad (1)$

where:

• $L$ is the length of the dipole,
• $\lambda$ is the wavelength.

This formula applies in the limit where $L/\lambda$ tends to zero. Therefore, for equation (1) to be valid, $L/\lambda$ must be small enough, meaning the dipole antenna must be electrically short. The smaller the value of $L/\lambda$, the closer the radiation resistance will approach the value given by equation (1).

Adjusting for a Realistic Short Dipole

A practical short dipole exhibits a triangular current distribution when its ends are free, rather than a uniform one, because the current must drop to zero at the ends. As a result, the effective length $L_e$ used in radiation resistance calculations is half of the total dipole length:

$\displaystyle L_e = \frac{L}{2} \qquad (2)$

Thus, the modified radiation resistance formula becomes:

$\displaystyle R_r = 80 \, \pi^2 \, \left(\frac{L_e}{\lambda}\right)^2 \qquad (3)$

Solving for $L_e/\lambda$ to achieve a radiation resistance of $\pi \approx 3.14\,\text{Ohms}$:

$\displaystyle \frac{L_e}{\lambda} = \sqrt{\frac{R_r}{80\pi^2}} = \sqrt{\frac{1}{80\pi}} \approx 0.0631 \qquad (4)$

Since $L_e = L/2$, we find that the required total dipole length is:

$\displaystyle L = 2 \times 0.0631 \, \lambda = 0.1262 \, \lambda \qquad (5)$

Validating with AN-SOF Simulation

To verify this result, we set up a short dipole simulation in AN-SOF with the following parameters:

• Frequency: $299.791\,\text{MHz}$ (where $\lambda = 1\,\text{m}$ to 6 significant figures)
• Dipole Length: $L = 0.1262\,\text{m}$
• Dipole Radius: $a = 2 \times 10^{-7} \, \lambda$, approaching near-zero thickness

Results:

• Radiation Resistance: $3.14\,\text{Ohms}$
• Current Distribution: Triangular, as expected for a center-fed short dipole
• Radiation Pattern: Omnidirectional, characteristic of a short dipole

The results, shown in the image below, confirm the theoretical prediction. The left plot displays the current distribution along the dipole, while the right plot shows the 3D radiation pattern.

Conclusion

This experiment demonstrates how theory and simulation align to produce an elegant result: a short dipole with a radiation resistance of 3.14 Ohms on Pi Day!

Stay tuned for more insights in The Antenna Lab! Happy Pi Day!

See Also:

• Linear Antenna Theory: Historical Approximations and Numerical Validation

• Complete Workflow: Modeling, Feeding, and Tuning a 20m Band Dipole Antenna