Analysis of a certain dipole animation

Modern people look for videos and animations for their learning, and these are often not from reputable sources and raise more questions than they answer.

An example is an animation of a half wave dipole on the Internet, and being discussed on QRZ.

Dipole_antenna_standing_waves_animation_461x217x150ms

Above, the animated graphic.

Without trying to understand the problem, lets just extract two cases for further discussion, an analysis in the limits if you like.

 

IMax

Above, the animation frame where I is maximum. Note that V is zero at all points.

Vmax

Above, the animation frame where V is maximum. Note that I is zero at all points.

The above two graphics would give the impression that:

  • the V distribution is sinusoidal;
  • the I distribution is sinusoidal; and
  • the phase of V/I is 90° everywhere.

What is I?

A reasonable assumption is that I is the current flowing in the antenna conductor. It is tangible, it can be measured in practice. The sinusoidal form reconciles fairly well with NEC models of a half wave dipole.

The arrows shown in the conductors might give the impression that at an elementary level, individual electrons flow from one end of the dipole, along the dipole, down the feed line, through the source, backup the feed line and along the dipole to the other end.

But is that correct?

Lets work a case for 1A flowing in 2mm diameter copper wire at 7MHz:

  • 1A is 1 Coulomb charge per second.
  • 1 Coulomb is equivalent to 6.242e18 electrons, the elementary charge q=1/6.242e18=1.602e-19C.
  • skin depth is 25µm, so if we assume that the effective CSA is 0.002*π*25e-6=157e-9m^2.
  • lets take the free electron density in copper to be 5e28e/m^3

Using the transport equation rearranged to make v the subject, lets find the electron drift velocity.

v=I/(nAq)=1/(5e28*157e-9*1.602e-19)=795e-6m/s

In a quarter cycle at 7MHz, the distance travelled by an electron is less than v/f/4=795e-6/7e6/4=28.4e-12m.

Individual electrons don’t travel very far during a cycle, a huge number just shuffle along a tiny bit and very slowly.

What is V?

Voltage is a general sense is the potential difference between two points.

The questions that arise with this graphic are:

  • is V voltage; and if so
  • what is the reference (ie when V is shown at some location along the dipole, what is it referenced to).

If V is the voltage wrt some reference, it is not practical to measure this in an antenna system as any wires to a volt meter will have voltage induced in them due to the antenna fields.

Many plots like this that are labelled V are actually plots of charge Q. Charge can be found by integrating current over time, it does not raise the question of a reference as does voltage.

An interesting similar topology that can be modelled simply

To some extent, as a standing wave antenna, a half wave dipole over ground can be split at the axis of symmetry (in the plane of the dipole), and one half modelled as a lossy single wire transmission line over ground, driven at one end at open at the other.

The voltage V (wrt ground) and current I can be calculated along such a transmission line.

The input impedance of the full dipole would be that the impedances of both sections added.

Lets consider a 2mm diameter copper wire, 3m above ground and a quarter wave long at 1MHz.

Zo can be approximated as 138*log(2h/r)=138*log(2×3/0.001)=521Ω.

The tool used to make these calculations cannot handle infinite impedance, so the far end of the transmission line is loaded with 1,000,000,000,000,000,000Ω to approximate the open circuit end.

Clip 198

Above is a plot of calculated V and I at displacements from the driven end (equivalent to the dipole centre), and calculated phase of V/I.

Without visiting the absolute phase of V and I, the phase difference is not constant. Though for much of the length the phase is -90°, near to the driven end it quickly approaches zero.

The lossy transmission line model suggests that the animation above oversimplifies the problem and probably give some false impressions.

ARRL perspective

Screenshot - 20_05_16 , 20_49_42Above is an explanatory diagram from the ARRL Handbook (Silver 2011) under the heading Current and voltage distribution. Though it is a half-baked graph (lacking proper axis scaling and labels), it preaches the same voltage distribution perspective: the current is minimum at the ends, regardless of the antenna’s length. … The opposite is true of the RF voltage. That is, there is a voltage maximum at the ends.

The voltage is the potential between what two points, can it be experimentally verified in a simple measurement, is the voltage plot actually meaningful?

Charge plot

If we were to accept that the current distribution is approximately sinusoidal, then the charge distribution is the time integral of current, and it also would be approximately sinusoidal with maxima at the dipole ends and minimum in the centre. The phase relationship between the charge distribution and current distribution is given by the integration operation, charge lags current by 90°.

If the original animation was changed so that V became Q, it has the approximate form of the I and Q distribution, and the correct phase relationship between them.

References

  • Silver, H Ward  ed. 2011. The ARRL handbook for radio communications. 2011 ed. Newington: ARRL.