# Is there a differential equation that can represent a circuit with an arbitrary voltage source connected acrorss an antenna?

An RLC circuit with a voltage source can be characterized by the differential equation: $$LC\;\ddot{I}\left(t\right) + RC\; \dot{I}\left(t\right) + I\left(t\right)-C \;\dot{V}\left(t\right) = 0$$ ... but this will be true only as long as radiation is ignored.

My question is, what will this equation become in the general case, when the radiation cannot be ignored and also has to be accounted for? I ask this because the radiation resistance on any antenna depends on the frequency of the (sinusoidal) current passing through it -- so I suppose it cannot simply add it on to the $R$ term.

I expect $\dot{I}\left( t\right)$ to be involved -- as the far field $\vec{\mathrm{E}}$ depends on $\dot{\vec{\mathrm{J}}}$ which must derive from $\dot{I}$ somehow. But I don't seem to be able to go beyond that.

Looking for pointers -- are there any textbooks that address this part? So far my Googling hasn't turned up anything, so I guess I don't know what I should be searching for or maybe this is something that has fallen through the cracks between circuit theory textbooks and antenna theory textbooks (most electromagnetism texts seem to jump to $\cos{\omega t}$ or $e^{j\omega t}$ and stay there).

Another thing -- would the L & C values that can be computed from the antenna geometry be sufficient to address the reactance in the high frequency case, or is there something else to it?

I'm looking for an understanding within the bounds of classical electrodynamics (that is, leaving quantum mechanics out).

Thanks...

## 2 Answers

I'd like to build on Lionel Brits's answer and the electronic book reference you cite Sophocles J. Orfanidis, "Electromagnetic Waves and Antennas", Chapter 22.

The structure of the relationship is more readily understood if you think of the Laplace or Fourier transforms of all your impedances. A relationship between $I(t)$ and $V(t)$ that is given by a constant co-efficient linear differential equation always gives rise to an impedance $Z(i\omega)$ or $Z(s)$ ($\omega$ = Fourier transform frequency variable, $s$ = Laplace transform complex frequency) which is a rational function of $i\omega$ or $s$.

Now an antenna's impedance is in general made up of reactions from the electromagnetic field which contain pure delays. This is owing to the finite speed of light. Unless the antenna is perfectly matched everywhere, the impedance is made up of components arising from delayed reflexions from different parts of its structure. One cannot match the antenna everywhere at all frequencies. So there will always be terms of the form $\exp(-s\,\tau)$ in the numerator and denominator of the impedance $Z(s)$ (here $s$ is the complex frequency), where $\tau$ is a there-and-back delay for a reflexion. When expanded out, $\exp(-s\,\tau)$ corresponds to the infinite differential operator:

$$1 - \tau {\rm d}_t + \frac{\tau^2}{2!} {\rm d}_t^2 - \frac{\tau^3}{3!} {\rm d}_t^3 + \cdots$$

so you'll never exactly represent such a thing by a finite differential equation.

That's the theory. In practice, depending on the important timescales, you could likely get a good approximation using only a finite number of terms. How many is hard to answer: this would need to be answered by someone with experience in numerically modeling antennas. But, by dint of the $\exp(-s\,\tau)$ terms, there is no theoretical limit to the order of derivatives that will arise in your expressions.

• Thanks @WetSavannaAnimal this is great. I was looking for what happens to an arbitrary antenna anyway -- what happens for specific antenna designs are generally well known. This answers my question completely. Nov 16, 2013 at 11:44

Suppose you have a dipole radiator. Then you can look up or calculate it's radiation resistance $R_\mathrm{rad} = \frac{2 \pi}{3} Z_{0} \left( \frac{\ell}{\lambda}\right)^{2}$ , which might be off by a factor of 4.

I would say that you can use the superposition principle to just treat the problem in frequency space. There is also reactance to think about.

• I am aware of the expression for radiation resistance of the dipole -- but you'll notice that it contains $\lambda$, which pre-supposes a sinusoidal excitation. I was wondering if there was any way to way to deal with this entirely in the time domain Nov 15, 2013 at 18:16
• @Avijit, my antenna theory is rusty but, according to lionelbrits answer, if the radiation resistance is proportional to the square of the frequency, the antenna voltage goes as the square of the frequency of the current source. Which, thinking off of the top of my head indicates that: $v(t) = k \frac{d^2i(t)}{dt^2}$ Nov 16, 2013 at 0:06
• @AlfredCentauri ... unfortunately, that relation would only for short dipoles. For a small loop, we get $R_r \propto \lambda^{-4}$ ... and for a large dipole, it mutates to a large expression in $\lambda$, as shown here Nov 16, 2013 at 3:24