When deforming a resonant circuit into an antenna, why can we cut the inductor in half?

Question

The working principle and fields of an antenna can often be shown by considering a parallel LC circuit. The plates can be moved to opposite ends and the inductor can be stretched out, both of which decreases the capacitance and inductance of the circuit and increases the resonant frequency.

In the limiting case of a half-wave dipole, the antenna must be center-fed (if fed by a current source) and connected to the transmission line. To accomplish this, the inductor in the middle needs to be cut and there is no current flow directly through the inductor:

As can be seen in the picture above, the center is insulated, while this point in the LC circuit corresponds to the center of the inductor, which is conducting.

While in a regular LC circuit a current flows through the inductor, it now moves through the whole transmission line at each cycle. It seems to me that these circuits are not equivalent anymore, since that introduces all the impedance of the transmission line into the circuit.

It's not so clear to me what changes if the inductor is still connected (apart from presenting a short circuit to the transmission line), and does this "cutting" change the inductance of the antenna itself?

Answer 1

Consider the wire antenna itself, before you cut it to insert the feed line. It has a certain resonant frequency based on its length. When operated at this resonant frequency, there will be a standing wave on the antenna, with a certain voltage distribution (zero at the middle and high at the ends) and current distribution (zero at the ends and high in the middle).

You can think of the middle of the antenna as the "inductor" and the ends as the "capacitor", but these effects are really both now distributed along the entire length of the antenna.

The antenna also has a characteristic impedance that varies along its length that you can get by dividing the voltage by the current. Near the ends of the antenna, the impedance is high because the voltage swing is very high while the current is very tiny. At the center, the impedance is lowest because the voltage is much lower and the current much higher.¹

The point is, if you cut the antenna at any point and connect a feedline of the same impedance at that point, it does not fundamentally change the operation of the antenna at all. It just provides a means of coupling electrical energy directly into or out of the antenna. Both the "effective inductance" and the "effective capacitance" are not changed by doing this.

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¹ The impedance never goes all the way to zero or infinity because of the way that the fields of the antenna interact with the free space around it.