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On 9/7/20 11:34 AM, Torbj?rn Toreson wrote:

For the sake of making an instruction for an antenna tuner I placed a dipole of length about 2 times 8 meters in my garden about 1 meter up from the ground. The idea was to get some not ideal values that could be handled by an antenna tuner. I used my NanoVNA to get a Smith-diagram between 3 and 8 MHz according to the enclosed picture. Point 1 is 3 MHz and point 8 is 8 MHz.



It is usually stated that resonance occurs when +/- jX is zero in Z=R+jX. I began to think about where I had resonance for my dipole. There are four transitions of the curve over the pure resistance line so is it correct to say that all those are points of resonance? Point 7 is obviously close to 50 ohm, SWR is actually 1.2, but nevertheless I have pure resistance at three other points, all of which by the way could be handled by the antenna tuner to get 50 ohm for the transmitter.

yes, there are resonances at approximately any multiple of a half wavelength. The even multiples have fairly high resistance, the odd multiples tend to be lower resistance.? When you get to a very large number of multiples, the impedance converges to a value similar to twice that of a balanced transmission line, where the "other half" of the transmission line is the image in the earth under the antenna.
This is like any resonant system which is based on the propagation of something (sound or EM wave) - Organ pipes, trumpets, wind instruments in general all can resonate at multiples (that squeak on a clarinet or recorder when you overblow).
In acoustics or string vibration, the wave equation that controls things is slightly different than the wave equation for EM resonance, but ultimately, they're all solutions of a 2nd order differential equation.
Just like with stringed instruments (where particular harmonics/overtones can be suppressed by striking the string at a particular point), you can do the same with an antenna by driving it off center.
However, the value of an adjustable matching network for most applications is in being able to adjust out the reactive component.? And then on top of that, transform the R to something more suited for your source.
You can actually model this as two separate networks. One is purely a L or C (either shunt or series) to "cancel" the reactive part. And the other is a LC network to transform the resistive impedance.
Then, you can collapse the two networks into one.
if the two port impedances are R1 and R2
T network Z1 Z2 are series, Z3 is shunt
Z1 = -j * (R1 *cos(beta) - sqrt( R1 * R2))/ sin (beta)
Z2 = -j * (R2 *cos(beta) - sqrt(R1 * R2))/sin(beta)
Z3 = -j * sqrt(R1*R2) / sin(beta)
Pi network
Za and Zb are shunt, Zc is series
Za = j *R1*R2 * sin(beta) / (R2 * cos(beta) - sqrt(R1 * R2))
Zb = j * R1*R2 * sin(beta) / (R1 * cos(beta) - sqrt(R1 * R2))
Zc = j * sqrt( R1 * R2) * sin (beta)
You can transform impedance AND get a phase shift (beta) except beta can't be zero or pi.? Typically, what's done is some sort of iterative approach to get the Zs "doable", because in the general matching problem, you don't care about phase shift.
If you're building a phase array, you do.
In many practical applications, the load or generator impedances may be reactive (i.e. Z (port 1) and Z (port 2) are some general R+jX). This can be accomodated by absorbing the external reactive impedance into the network, reducing or increasing the series or shunt impedance as requred. For instance, if a T network is required to connect between two impedances: 50+j0 and 100-j20 with 45 degrees of phase shift:
First, calculate the Z's assuming resistive impedances: R1=50, R2= 100
Z1 = -j * (50 * .707 - sqrt(50*100))/.707 = +j 50 ohms
Z2 = -j * (100 * .707 - sqrt(50*100))/.707 = 0 ohms
Z3 = -j * sqrt( 50 * 100) / .707 = -j 100 ohms
(the example is somewhat contrived, and it winds up creating an L network for the resistive case).
Now, a reactive component is added to Z2 to exactly cancel the external reactive component. This changes Z2 from 0 ohms to +j20 ohms. The final network is then:
Z1 = +j50, Z2= +j20, Z3 = -j100 ohms
If you are working with a pi network, you would want to transform the external impedances into their corresponding shunt forms first, so that the reactive component is a shunt value, which would be absorbed (or combined) with the corresponding shunt component of the pi network.
This produces a "point solution"? - in a lot of systems, you might want to evaluate the design over a band of frequencies. And, if you've got non-ideal components with loss, some solutions will have more or less loss.

I am used to say that an antenna only has one resonance frequency apart from multiples of the basic resonance. My Smith-diagram clearly shows that is not correct. Point 3 is 4.6 MHz, Point 6 is 6.5 MHz and point 7 is 7.2 MHz. (I should have noted the frequency for the transition between points 6 and 7, but it is a bit less than 7 MHz.)

Can someone please shed some light on the definition of resonance in conjunction with an antenna.

73/Torbjorn

PS. I manually added the SWR 2 and SWR 3 circles afterwards, but it could be handy if the FW had the option to show them.