# Antenna system resonance and the nanoVNA

With the popularity of the nanoVNA, the matter of optimisation of antenna systems comes up and the hoary chestnuts of ham radio are trotted out yet again.

Having skimmed a presentation published on the net, an interesting example is presented of an 80m half wave centre dipole with feed line and various plots from the nanoVNA used to illustrate the author’s take on things.

The author is obsessed with resonance and obsessed with phase, guiding the audience to phase as ‘the’ optimisation target. Phase of what you might ask… all the plots the author used to illustrate his point are phase of s11.

## A model for discussion

I have constructed an NEC-4.2 model of a somewhat similar antenna to illustrate sound concepts. Since NEC-4.2 does not model lossy transmission lines (TL elements), we will import the feed point data into Simsmith to include transmission line loss in the model.

Above is the Simsmith model.

The dipole is fed with around 20m (66′) of RG58A/U 50Ω line with vf=0.67.

Above is the Smith chart looking into the feed line (click for a larger image).

There are three markers, from right to left:

1. 3.738MHz: phase of s11 is very low (0.006°, approximately zero), Z=129.9+j0.0153Ω, VSWR=2.60;
2. 3.636MHz: phase of s11 is very low (0.17°, approximately zero), Z=74.18+j0.0902Ω, VSWR=1.48; and
3. 3.600MHz: phase of s11 is nowhere near zero (30.8°), Z=63.22+j9.443Ω, VSWR=1.33

Lets look at the power delivered to the antenna.

Above, we see that the power is maximum at approximately the same frequency as where VSWR is minimum. That is no coincidence, standing waves result in higher line loss in this scenario.

Again the same three markers, and the two frequencies where phase of s11 is approximately zero have higher loss, less power reaching the antenna, than the one at minimum VSWR where phase of s11 is 30°.

## Optimisation

It turns out that with practical feed lines and antenna conductors with this type of antenna, the loss in the antenna conductors is small compared to the feed line:

• dipole radiation efficiency is not very sensitive to frequency, small departure from resonance does not change the radiation efficiency of the dipole itself much; and
• feed line loss will be lowest where VSWR is minimum.

So, this antenna system will have best radiation efficiency at 3.6MHz because feed line losses are least.

## Angle of complex reflection coefficient Γ or s11

Where the reference impedance for calculation of the complex reflection coefficient Γ or s11 is real, then when the angle of gamma is zero at some point, the ratio or V/I or impedance at that point is purely real (ie, zero reactance).

That says nothing for the value of the resistance, nor of the magnitude of Γ or s11 or of the VSWR.

Further, when the magnitude of s11 is very small (and VSWR is close to unity), the angle of Γ or s11 as measured is dominated by instrument noise and at the limit is essentially a random number and is meaningless.

So any optimisation based simply on angle of   Γ or s11 is naive in the extreme.

## Angle of complex feed point impedance

Optimisation for phase of load impedance at the source equal to zero means reactance X is equal to zero, but says nothing about the resistance component, and therefore about VSWR.

In the example above, at the frequency of minimum VSWR (least line loss, maximum power delivered to the antenna) the impedance looking into the line has a phase angle of -8.5°.

So any optimisation based simply on angle of impedance seen by the source is naive in the extreme.

## Relationship between angle of reflection coefficient and angle of impedance

It was stated above that the angle (or phase) of s11 or Γ is not the same as the angle (or phase) of Z.

Given Zo and Γ, we can find θ, the angle of Z.

$$Z=Z_0\frac{1+\Gamma}{1-\Gamma}$$

Zo and Γ are complex values, so we will separate them into the modulus and angle.

$$\left | Z \right | \angle \theta =\left | Z_0 \right | \angle \psi \frac{1+\left| \Gamma \right | \angle \phi}{1-\left| \Gamma \right | \angle \phi} \\ \theta =arg \left ( \left | Z_0 \right | \angle \psi \frac{1+\left| \Gamma \right | \angle \phi}{1-\left| \Gamma \right | \angle \phi} \right )$$

We can see that the θ, the angle of Z, is not simply equal to φ, the angle of Γ, but is a function of four variables: $$\left | Z_0 \right |, \psi , \left| \Gamma \right |, \& \: \phi$$ .

It is true that if ψ=0 and φ=0 that θ=0, but that does not imply a wider simple equality. This particular combination is sometimes convenient, particularly when ψ=0 as if often the case with a VNA.

## Take aways

So, pursuit of:

• feed point X equal to zero;
• feed point phase of Z equal to zero; and
• some notion of ‘resonance’ at the feed point

for this type of antenna system are all misguided.

Claims that phase alone is some magic quantity that drives optimisation is misguided.

This type of antenna system can be optimised with the nanoVNA, and the optimisation target is VSWR.

Read widely, don’t accept plausible looking explanations that appeal simply because they appeal, without understanding them, specious works are widespread and swallowed by the gullible.