For a half wave dipole antenna, what's the value of the reactances that cancel out when the antenna is resonant?

Question

According to a few books and articles on antenna fundamentals I've read recently, including Wikipedia and the ARRL Handbook, when a half wave dipole is resonant, the inductive and capacitive reactances cancel out, and that's why there is no reactance in the impedance.

My understanding is that the amount of reactance seen at the feed point of a center fed half wave dipole is determined by the phase relationship in time between the applied AC electric potential at the feed point and the resultant AC current which arrives at the feed point after being reflected back from the ends of the antenna.

Some say that a half wave dipole can be thought of as a series RLC circuit. If so, what are the values of reactance that cancel out at resonance ?

Is an antenna resonant because the voltage and current are in phase, or because the reactances cancel out, or both ?

Also see my related question Exactly what is the cause for the zero reactance seen in the impedance at the center feed point of a resonant half wave dipole?

Answer 1

And why is an antenna resonant, because the voltage and current are in phase or because the reactances cancel out, or both ?

Reactance ($X$), resistance ($R$), and the phase difference between voltage and current ($\theta$) are related by:

$$ \tan \theta = { X \over R } $$

So if reactance is zero then $X/R$ is zero, so $\theta$ must be zero, meaning voltage and current are in phase.

In other words, "zero reactance" means "voltage and current are in phase". One does not cause the other: they are two ways of saying the same thing.

Likewise, resonance is defined as zero reactance. It's not "caused" by zero reactance or voltage and current being in phase: it simply is, by definition.

In other words, if you design an antenna with any one of these three objectives:

• current and voltage are in phase at the feedpoint,
• feedpoint reactance is zero, or
• antenna is resonant

you will find the other two are satisfied. There is no ordering or causality between any of them: they are each mathematically equivalent.

If you are concerned only about the feedpoint, asking what the inductive and capacitive reactances are which sum to zero reactance is like solving $x-x=0$ for $x$. It's not possible, and also it doesn't matter.

Besides, a dipole isn't an RLC circuit, so there is no inductance in henrys or capacitance in farads that can be found, although as Brian K1LI explains it is possible to approximate a dipole as an RLC circuit over a limited frequency range.

There is perhaps a slightly different question you could ask which does have an answer. A dipole, like any resonant system, has a Q factor. We know that a dipole must store some energy in its electromagnetic field, and some of this energy is lost to radiation and possibly other losses, like resistive losses in the conductor and ground. The Q factor relates to the ratio of these quantities.

One way to think of Q is as the ratio of reactive power to real power. The reactive power has current and voltage 90 degrees out of phase. The energy associated with this power doesn't go anywhere and doesn't do any real work: it just oscillates between the electric and magnetic fields, forever. The real power has voltage and current in phase, and does real work: mostly (ideally, entirely) radiation.

A good half-wave dipole will have a Q of about 10. Which means if the antenna is radiating 100 watts, there is about 1000 watts of reactive power. You could measure this with electric and magnetic field probes in the near field of the antenna. With the electromagnetic fields known you can calculate the Poynting vector. In the near field of the antenna you should expect to find the imaginary part of the Poynting vector is about 10 times that of the real part, meaning the electric and magnetic fields are almost but not quite 90 degrees out of phase. Of course the precise values will depend on just where you place the probes, and the construction and environment of the particular antenna.

While this is not exactly what you asked, it does seem closer to the understanding you're seeking. Unfortunately it does require thinking beyond just what's happening at the feedpoint, and instead considering what's happening in the electromagnetic fields in space around the antenna, and this requires some more complicated mathematics.

Answer 2

An antenna is not a series RLC circuit, but it can be modeled with lumped circuit elements over a certain frequency range. See, for example, Tang et al, "Equivalent Circuit of a Dipole Antenna Using Frequency-Independent Lumped Elements."

A thought experiment shows why: we observe that the feedpoint impedance of a half-wave dipole exhibits periodic behavior at harmonics of its design frequency, but a simple lumped circuit does not. For example, using the formulas of Tang et al, we can model the "input impedance" of the equivalent circuit of a 20-m half-wave dipole:

It "resonates" at about 14.3-MHz. The same antenna modeled by NEC-2 shows the very different behavior to which we are accustomed:

The input impedance behavior of the half-wave dipole is more reminiscent of a transmission line, which can also be modeled with lumped elements. But, since the object of the lumped-element model is computational simplicity, the number of elements required to produce an accurate transmission line model would probably defeat the purpose.

Answer 3

In my mind the difference is that:

• An RLC circuit "resonates" by bouncing energy back and forth within itself. And at a particular frequency this stored energy is retained for a particularly long time. (The Q of such a circuit represents the tradeoff between ringing for an extra long time at only one particular frequency, versus the energy bouncing around with still somewhat acceptable efficiency across a wider range of frequencies.)
• The "resonance" of an antenna usually has more to do with its radiation resistance at a particular frequency. Here's where I don't want to steer you wrong since maybe this is more conceptually related than I imagine, but the resistive energy loss in an RLC circuit is something of an inevitable "defect" whereas it's the entire point in a good antenna. You just want the energy to be lost as EMF radiation rather than thermally.

That said, a small loop antenna is decidedly both: it's an RLC circuit designed so that ideally all the R would be due to radiation rather than loss. This is again related to Q — and with resonance being at an exact "point" frequency means that any modulation sidebands need to be taken up by a lossy resistance somewhere but in practice with most antennas, and even many small loop antennas, the Q curve tends to be on the scale of e.g. the entire 80m "band" rather than anywhere near a concern for the bandwidth of a single CW/SSB transmission.

Answer 4

A resonant dipole antenna can be likened to a series RLC lumped constant circuit where the impedance seen across the two terminals contains zero reactance at resonance, and where operation at a frequency slightly above resonance results in an inductive reactance in the impedance, and a capacitive reactance in the impedance for frequencies slightly below resonance.

The conditions which determine resonance for a lumped constant RLC circuit which can be described using conventional circuit analysis are not the same as those seen for a system which is not very small compared to the wavelength of the frequency of operation and where wave theory mostly applies.

The formulas for inductive and capacitive reactance in a series RLC lumped constant circuit are shown below.

Xc = 1/2πfC

Xl = 2πfL

Where f = frequency, C = capacitance and L = inductance.

The formula for capacitance is C = e(A/d)

where e = dielectric permittivity, A = area of plates, d = distance between plates.

It's pretty obvious that a wire dipole antenna will have a very small amount of capacitance between the center feed points because the area of the plates, which is the surface area of the inner ends of the wire elements, is tiny. It is also obvious that this small amount of capacitance will not result in the same amount of reactance given by the self inductance of the wire elements, so these reactances will definitely not be equal and opposite at resonance.

In addition, see below the equation which describes the resonant frequency for a series RLC circuit.

fc = 1/2π√LC

A dipole antenna has multiple resonant frequencies, the above equation has only one solution for each combinations of L and C, and this equation cannot describe dipole resonance at harmonics of the fundamental resonant frequency.

See below equations using the Induced EMF method which are approximations which describe the impedance seen at the center feed points of a dipole.

where $a$ is the radius of the conductors, $k$ is the wave number 2πf/c, $η_0$ denotes the impedance of free space = 377Ω, and $γ_e$ is Euler's constant = 0.57721566

The terms in the equations cater for the length, diameter and self inductance of the dipole elements, as well as for the fact that each opposite point on the dipole elements is a different distance away from all other opposite points unlike the situation seen in a transmission line where the distance between opposite points is the same.

For a dipole antenna, the voltage of the standing wave on the antenna is at every point on the antenna about 90 deg out of phase with the current of the standing wave.

At resonance, the current of the standing wave is in phase with the voltage and current of the applied RF energy at the feed points.

At the feed points, the voltage of the applied RF adds vectorily to that of the voltage of the standing wave.

The reason a dipole has no reactance in the feed point impedance at resonance is because, at resonance, the electrical length of the dipole is such that the zero crossing point of the voltage of the standing wave (which is always out of phase with the applied RF), occurs exactly at the antenna center feed points and so contributes no reactance to the feed point impedance.

The electrical length is determined by the length and cross sectional area of the antenna elements, which includes self inductance and effects of the impedance of free space. This situation i believe is described in the equation above for feed point reactance.

Answer 5

If a half wave dipole is resonant, there is no reactance to cancel out. The definition of resonance is when there is no reactance, because there is no phase between voltage and current.

Answer 6

The value of reactance varies both in magnitude and sign as functions of the operating frequency, the radiating length of the antenna, and the physical location of the feedpoint terminals along that length.

At resonance, jX = 0.