# Are there formulae to calculate capacitance and inductance of an antenna?

How to calculate internal impedance of a dipole antenna? Its resistance has two components ie wire resistance and radiation resistance, both these can be calculated using formula. Are there formula to calculate capacitance and inductance of an antenna? Or they are calculated empirically only?

• This previous answer might be what you're looking for. (There's never such a thing as capacitance and inductance together since they oppose, but you can compute one or the other from the impedance at a given frequency if you want it an answer in those units.) – Kevin Reid AG6YO Sep 10 '18 at 14:09
• It would help if you could explain what you envision or require as to the inputs to such an equation as this determines complexity of the answer. – Glenn W9IQ Sep 11 '18 at 23:17

A couple of minor comments on terminology. It isn't so much the "internal" impedance of the antenna but it is typically the "feed point" impedance that is of concern. This is the impedance seen where the transmission line connects to the antenna. Another common impedance of interest is the radiation resistance. This is the effective resistance attributable to converting the RF signal to a radiating electromagnetic wave. This impedance is often of interest when determining the efficiency of an antenna.

Yes, there are many such formulas just like the ones in the answer you saw linked in the comments. From an academic and research perspective, the formulas are under constant improvement and refinement. This usually translates into more complexity in the formulas in exchange for greater accuracy or detail.

Most professional antenna engineers and amateur radio antenna enthusiasts have transitioned to antenna modeling software. These software packages produce extensive data such as the input impedance but also go on to produce renderings of radiation patterns and other useful information. Within an hour or so, nearly any common antenna can be extensively analyzed using such software without any knowledge of calculus.

Several of these packages are available for free or on a trial basis. Here is one popular example: 4NEC2

• Another is EZNEC, which is what I use. – Mike Waters Sep 12 '18 at 14:00

In David Leeson's (W6NL) Nov/Dec 2018 QEX article, "The Story of the Broadband Dipole," he cites the work of Tang, et al, on the "Equivalent Circuit of a Dipole Antenna Using Frequency-indpendent Lumped Elements." (from IEEE Transactions on Antennas and Propagation, Vol 41, No. 1, Jan 1993, pp. 100-103).

Leeson recommends this schematic equivalent as an adequate representation for a center-fed dipole:

simulate this circuit – Schematic created using CircuitLab

with: $$h$$ as the dipole half-length and $$a$$ as the wire radius, both in meters, $$\omega_0$$ as the resonant frequency in Hz and $$R_{a0}$$ the radiation resistance at resonance in ohms:

$$C_{11} = \frac{27.82\mathsf x 10^{-12} h}{ln(2h/a)-1.693}$$ $$L_{11} = \frac{1}{\omega_0^2C_{11}}+C_{11}R_{a0}^2$$ $$R_{11} = \frac{L_{11}}{C_{11}R_{a0}}$$

• Interesting — is there a way to predict the radiation resistance (Ra0) term? That seems like a crucial missing piece to pure "back of envelope" estimation. – natevw - AF7TB Oct 18 '18 at 16:54
• This Dipole Antenna Calculator computes Ra0 in free space and gives the supporting equations. An example of the Feed impedance variation of a dipole against height above ground makes assumptions about the characteristics of the ground, which can strongly influence the results. – Brian K1LI Oct 20 '18 at 12:57

I assume you are referring to the feedpoint resistance and reactance. Stearns, K6OIK, Antenna Impedance Models, presented at ARRL Pacificon 2004 is an excellent treatment of the subject.

Keep in mind that a center-fed, infinitely thin, physical half-wave dipole in free space has an impedance of 73+j42.5 ohms; many of the calculations in the reference use this as the baseline for computation.

you can calculate the impedance of your actual antenna if you setup a test jig and take a number of readings. w2aew explains this very well in a youtube video https://www.youtube.com/watch?v=eYN7dhdt1Dw this method requires a signal generator and an oscilloscope, we don't all have the luxury to own these items but you can still learn the theory of what he's doing. you either need to be confident in complex number maths, have a nice scientific calculator or use online tools to solve the maths. the equation itself is just like solving current and resistance in simple voltage divider circuits... but each element has an amplitude and phase.

There is no formula to calculate reactance, it has to be empirically determined.

• If that were true, then you would have to wonder how antenna modeling software can do it with a great deal of accuracy. – Glenn W9IQ Sep 11 '18 at 5:00
• One could consider antenna modeling to be "empirical," inasmuch as it solves systems of equations rather than producing results from closed-form expressions. – Brian K1LI Sep 11 '18 at 5:58
• @BrianK1LI It would be quite a stretch to call integral equations like MFIE and EFIE "empirical" - iterative in this case but not empirical. The point was to gently guide this new contributor the the realization that the above answer is wrong. – Glenn W9IQ Sep 11 '18 at 7:12
• @GlennW9IQ I appreciate your point, Glenn, but the constantly falling cost of computing power blurs the distinction between physical experiments and simulated experiments. I have used antenna modeling programs to empirically determine the equivalent values of the RLC circuit which approximates an antenna's behavior near resonance, saving a lot of time and expense. – Brian K1LI Sep 11 '18 at 15:57
• em·pir·i·cal: based on, concerned with, or verifiable by observation or experience rather than theory or pure logic. Antenna modelling, based on the theory of classical electromagnetism, and the logic of mathematics and computer science, is not empirical. – Phil Frost - W8II Sep 11 '18 at 18:39