KL7AJ on the Conjugate Match Theorem asked the question
Should we have expected this outcome?
Let us solve a very similar problem analytically where measurement errors do not contribute to the outcome.
Taking the load impedance to be the same 10.1+j0.2Ω, and calculating for a T match similar to the MFJ-949E (assuming L=26µH, QL=200, and ideal capacitors) we can find a near perfect match.
The capacitors are 177.2 and 92.93pF for the match.
Now turning the network around by swapping the capacitors and changing the load to 50+j0Ω. Continue reading KL7AJ on the Conjugate Match Theorem – analytical solution
KL7AJ proposed a little test for his readers on QRZ:
One of the most useful (and sometimes astonishing) principles in radio is the Conjugate Match theorem. In the simplest terms, what this says is that the maximum power will be transferred between a source (like a transmitter) and a load (like an antenna), when the source impedance is the COMPLEX CONJUGATE of the load impedance (or vice versa).
Here’s a neat little experiment to prove the conjugate match theorem. You need four basic ingredients: an antenna analyzer like the MFJ259 (or an actual impedance bridge, if you know how to use one). A good low loss antenna tuner. A good 50 ohm resistor. And a good 200 ohm resistor. And some appropriate connecting hardware, namely some short bits of coax.
Step 1) connect the 50 ohm resistor to the OUTPUT of the antenna tuner. Connect the antenna analyzer to the INPUT of the antenna tuner.
Step 2) Adjust the antenna tuner to get precisely 50 ohms, zero reactance on the antenna analyzer. This step simply confirms everything is working.
Step 3) Replace the 50 ohm resistor with the 200 ohm resistor. Readjust the antenna tuner to get 50 ohms, zero reactance on the antenna analyzer. Do not disturb the antenna tuner adjustments after this point.
Step 4) Remove the 200 ohm resistor and insert the antenna analyzer in its place (at the OUTPUT of the antenna tuner).
Step 5) Insert the 50 ohm resistor at the INPUT of the antenna tuner.
Step 6) Take a careful reading of the antenna analyzer. (What do you think it will say?)
10 points for anyone who will correctly explain why this works.
Jacobi maximum power transfer theorem
Jacobi published his maximum power transfer theorem in 1840. It states that maximum power is transferred from a (Thevenin) source to a load when the load resistance is equal to the (Thevenin equivalent) source resistance.
It was later adapted to apply to AC circuits with sinusoidal excitation, maximum power is transferred from a (Thevenin) source to a load when the load impedance is the complex conjugate of the (Thevenin equivalent) source impedance.
Walt Maxwell’s Conjugate Mirror
(Maxwell 2001 24.5) states
To expand on this definition, conjugate match means that if in one direction from a junction the impedance has the dimensions R + jX, then in the opposite direction the impedance will have the dimensions R − jX. Further paraphrasing of the theorem, when a conjugate match is accomplished at any of the junctions in the system, any reactance appearing at any junction is canceled by an equal and opposite reactance, which also includes any reactance appearing in the load, such as a non-resonant antenna. This reactance cancellation results in a net system reactance of zero, establishing resonance in the entire system. In this resonant condition the source delivers its maximum available power to the load. …(1)
Note that it states that if a conjugate match is established an any junction, then a conjugate match occurs in any (all) other junctions, simultaneously a conjugate match exists everywhere. Continue reading KL7AJ on the Conjugate Match Theorem
Measurements of Insertion VSWR of UHF series connectors consistently show increasing Insertion VSWR with frequency, an issue that often impacts measurement accuracy.
My own article Exploiting your antenna analyser #12 is but one of many.
Measurements consistently hint that the defect is that the characteristic impedance is typically somewhere between 30 and 40Ω.
Above is a dimensioned drawing from Amphenol (https://www.amphenolrf.com/connectors/uhf.html). Continue reading UHF series coaxial connector characteristic impedance
The Amidon AB_200_10 2-30MHz, 1KW balun and knock-offs have been around for a very long time, I recall Dick Smith selling them in the early 1970s in Australia.
They were regarded as the epitome of the art… but it was not a very well understood art.
Lets analyse the common implementation as a Ruthroff 4:1 voltage balun in a 50:200Ω scenario.
Ruthroff 4:1 voltage balun
In this implementation, Amidon’s instructions show 16 bifilar turns on a T200-2 core.
A very simple model is to consider the device as an ideal transformer with a shunt magnetising impedance equal to the impedance of the 16t winding that appears across the 50Ω terminals. This has its greatest effect at low frequencies and although it is specified from 2-30MHz, lets analyse it at 3.5MHz.
The powdered iron core has very low loss at 3.5MHz, sufficiently so that we can ignore the imaginary component of µr for this analysis and take µr to be 10+j0.
Above is a calculation of the magnetising impedance and admittance under those assumptions. The magnetising admittance (0.00-j0.0134S) appears in shunt with the transformed load admittance (0.02S) so we can simply add them to find the admittance seen by the transmitter (0.02-j0.0134S). Continue reading Review of the Amidon AB_200_10 balun
This article expands on the detail behind A low Insertion VSWR high Zcm Guanella 1:1 balun for HF.
Choice of core
Online experts all have a preferred core material, but there is a dearth of measurement data to show the difference in actual use. If someone recommends a particular core material and cannot provide measured Zcm data to support the recommendation, regard it as a weak recommendation.
Beware the magic of unobtainium… just because something is hard to get is not an indication that it is desirable.
Above is the complex permeability characteristic of the #43 material used. Inductance calculators that do not take that frequency dependent complex characterisitic into consideration produce invalid results. (Duffy 2015) gives a suitable approximation, and there are links to calculators that do work properly at the bottom of this article. Continue reading A low Insertion VSWR high Zcm Guanella 1:1 balun for HF – more detail
This article describes a Guanella 1:1 current balun which has high common mode impedance (Zcm) and low Insertion VSWR. It is for application on antennas that have low VSWR(50) on at least some bands, especially if they would be used without an ATU on some bands.
The purpose of the balun is to minimise common mode feed line current which may contribute to EMC problems when transmitting, and contribute to increased ambient noise when receiving. Reduction of feed line common mode current also helps in achievement of expected load impedance characteristic, radiation pattern and gain. This article gives measured Zcm, but the definitive test of the effectiveness of such a balun is direct measurement of common mode current Icm… and it is so easy.
Example applications are half wave centre fed dipoles, fan dipoles, trapped dipoles, G5RV with hybrid feed, ZS6BKW, trapped verticals, monopoles, ground planes.
To obtain low Insertion VSWR, the choke will be wound with 50Ω coax, to demonstrate the practicality of the design budget (but good quality) regular (ie solid PE dielectric) RG58C/U will be used. Foam dielectric is NOT recommended. Solid PTFE coax could be used, but avoid coax with steel cored inner conductor, it may be lossier than you think at low frequencies with the silver cladding is relatively thin.
The candidate core is the readily available FT240-43 (Fair-rite 2643803802, 5943003801), it is a low cost NiZn ferrite with medium µr, and its µr and loss characteristic contributes to a broad high impedance choke well suited to this application.
Above is a model of the expected Zcm with 11 turns of RG58C/U coax and an equivalent shunt capacitance of 4.6pF. Continue reading A low Insertion VSWR high Zcm Guanella 1:1 balun for HF
This article shows just how easy it is to make an inexpensive low VSWR load for antenna analyser validation / measurements.
Above is an AA-600 sweep of the prototype from 10kHz to 100MHz. VSWR reads 1.02 in ‘All’ mode at 100MHz… better than the inherent accuracy of the instrument.
It is made from two 100Ω 1% 1206 SM resistors purchased on eBay for about $2/100, so about $0.04 for the resistors, and 40mm of bare copper wire (0.5mm phone / data wire in this case).
In use, it is held in contact with the coax socket (in this case an N type) with a pair of disposable plastic first aid tweezers (yep, you can buy them on ebay for about $0.20/pair).
While you are at it, make a good short circuit termination by scrunching up a bit of (clean) kitchen aluminium foil and press that against the coax socket conductors.
Try both of these on your antenna analyser and see how it stacks up.
A recent online posting asserted that an antenna is optimal when itself resonant, and fed with a resonant feed line length so delivering a purely resistive load to a source, and further that implementors needed to be careful that a shorter dipole could be offset to some extent by a longer feed line but it would be inferior because:
no short antenna is more efficient than a resonant-length antenna
… but does that stand scrutiny?
An NEC experiment
Lets walk though an experiment using NEC-4.2 models of a dipole of 2mm copper wire at 10m height at 7.1MHz over average ground (σ=0.005, εr=13).
- source has a Thevenin equivalent source impedance of 50+j0Ω;
- feed line is lossless.
The results are sensitive to the model assumptions.
We will calculate the ratio of radiated power to the power delivered by the transmitter to a matched load, let us call it TransmitEfficiency for the purposes of this article. Continue reading “No short antenna is more efficient than a resonant-length antenna”
A ham recently posted a graph on QRZ to educate his fellow hams on the behaviour of transmission lines under mismatch.
Above is one of his graphs (the red arrow is my annotation). It plots Impedance variation along a mismatched 75Ω transmission line. The curves look graceful, but are they science or just pretty artwork? Continue reading Failure to treat impedance as a complex quantity leads to…
An online expert recently reported:
I tried to make an antenna loop for longwave with cat 5 and after it did no good I realized the twisted wires canceled each other out.
Or did they really cancel?
I constructed a loop of one Cat 5 pair and measured its inductance when both wires are bonded at the ends.
The conductors are 0.5mm diameter and spaced 0.9mm. To estimate the inductance we use the geometric mean radius (GMR) as the equivalent radius of the pair. GMR=(0.5*0.9)^0.5=0.67, diameter=1.34mm. So let’s calculate the inductance of a single turn circular loop of 0.8m perimeter and round conductor of 1.34mm diameter.
The estimate above is 850nH.
Above is the measurement, the screen is not readable, but it is 852nH, very close to the estimated 850nH. Continue reading Inductance of a loop of CAT5 pair