The diagram above from the video shows the topology of the series stub match.
Let’s work an example using 50Ω coax where the load is 10-j40Ω at 146MHz.
Above is a naive Simsmith model of the scenario with line lengths adjusted for a ‘perfect’ match. The inset VSWR curve looks great!
Why did I say naive?
Well, when we look at the series stub more completely, along with the differential mode behaviour that is captured in the Simsmith model above, we have the outside surface of the outer conductor of the stub connected to the main line inner conductor on the source side of the stub and it is capable of carrying current.
This stub outer surface current may give rise to radiation and impresses a load impedance across the main line coax in shunt with the perfectly transformed 10-j40Ω load. The value of this unintended impedance depends on the stub physics.
In this case, the stub is just under a quarter of a wavelength, and modelled as a monopole over a ground plane would present a driving impedance of around 15-j80Ω.
Above is the Simsmith model with the addition of the estimated impedance of the stub outer as a radiator in shunt with the main line. The matching result is degraded by the stub exterior’s ‘antenna effect’ and that unintended radiation / pickup may be undesirable.
The VSWR curve is not so good now when the ‘antenna effect’ of the stub is brought to book. The match could be tweaked some, but it does not remove the radiation / pickup behavior of the stub.
Matchers using series or shunt stub tuners are nearly ideal in waveguide and can be quite good / practical in two wire lines; but while shunt tuners are nearly ideal in coax, series tuners in coax may not be so good and tend to be avoided.
]]>Above is the prototype 2631540002×2 wound with 3.5t of RG316.
Above is the plot of R and X components of Zcm from 1-30MHz. Self resonant frequency SRF is 5.4MHz.
|Zcm| is very high from 2-14.5MHz and high from 1-26MHz, and this should make an effective choke for most reasonable scenarios.
Having measured the SRF, we can calibrate the predictive model.
Above, the calibrated model is quite close in form to the measured, allowing for the rather wide tolerance of ferrite.
A follow up article will report thermal tests on the prototype balun.
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Above is the prototype 2843009902 binocular wound with 3.5t of RG316.
Above is the plot of R and X components of Zcm from 1-30MHz. Self resonant frequency SRF is 8.75MHz.
|Zcm| is very high from 3-22MHz and high from 1.8-30MHz, and this should make an effective choke for most reasonable scenarios.
Having measured the SRF, we can calibrate the predictive model.
Above, the calibrated model is quite close in form to the measured, allowing for the rather wide tolerance of ferrite.
A follow up article will report thermal tests on the prototype balun.
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This article presents the workup of a balun with similar design objectives using a low cost Fair-rite 2843009902 binocular core (BN43-7051).
Above, a pic of the core.
The design is a variation on (Duffy 2007) which used RG174 coax for the choke to give low Insertion VSWR.
For low Insertion VSWR, the choke uses 50Ω coax wound around a pair of ferrite tubes. The coax is a miniature PTFE insulated cable, RG316 with silver plated copper centre conductor (be careful, some RG316 uses silver plated steel and it is unsuitable for HF).
Matched line loss in the 350mm length of coax is 1.2% @ 30MHz, 0.4% @ 3.5MHz, and could be higher or lower with standing waves.
PTFE coax is used for high voltage withstand and tolerance of high operating temperature.
Above, an insulation test of the RG316. It withstood 7kV peak (5kV RMS) from inner to outer, and the jacket also withstood 7kV peak at a knife edge. Voltage breakdown is more likely to occur somewhere else in the balun.
For this design, the cores need to be large enough to accommodate 4 passes of RG-316 coax, but no larger.
Above, the cores will accommodate four round conductors of diameter 2.6mm, so they will comfortable accommodate the four passes of the RG-316 coax (2.45mm each). (For the mathematically minded, the minimum enclosing circle diameter for four equal circles is 1+√2 times the diameter of the smaller circles.)
Al (10kHz) is about 9µH.
The main contribution to loss and heating will be the ferrite core losses, and they are dependent on common mode current.
Above is a first estimate of common mode impedance of 3.5t (4 in one hole, 3 in the other – an approximation) assuming an equivalent shunt capacitance of 2pF. The latter is an experienced guess, and will be adjusted upon measurement of a prototype.
Implementation will be described in a follow up article.
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This article presents the workup of a balun with similar design objectives using a pair of low cost Fair-rite 2631540002 cores (FB-31-5621) which are similar in size to the LF1260 and have higher µi (1500 vs 1000).
Above, a pic of the cores from Amidon’s catalogue.
The design is a variation on (Duffy 2007) which used RG174 coax for the choke to give low Insertion VSWR.
For low Insertion VSWR, the choke uses 50Ω coax wound around a pair of ferrite tubes. The coax is a miniature PTFE insulated cable, RG316 with silver plated copper centre conductor (be careful, some RG316 uses silver plated steel and it is unsuitable for HF).
Matched line loss in the 350mm length of coax is 1.2% @ 30MHz, 0.4% @ 3.5MHz, and could be higher or lower with standing waves.
PTFE coax is used for high voltage withstand and tolerance of high operating temperature.
Above, an insulation test of the RG316. It withstood 7kV peak (5kV RMS) from inner to outer, and the jacket also withstood 7kV peak at a knife edge. Voltage breakdown is more likely to occur somewhere else in the balun.
For this design, the cores need to be large enough to accommodate 4 passes of RG-316 coax, but no larger.
Above, the cores will accommodate four round conductors of diameter 2.6mm, so they will comfortable accommodate the four passes of the RG-316 coax (2.45mm each). (For the mathematically minded, the minimum enclosing circle diameter for four equal circles is 1+√2 times the diameter of the smaller circles.)
The main contribution to loss and heating will be the ferrite core losses, and they are dependent on common mode current.
Above is a first estimate of common mode impedance of 3.5t (4 in one core, 3 in the other) assuming an equivalent shunt capacitance of 2pF. The latter is an experienced guess, and will be adjusted upon measurement of a prototype.
Implementation will be described in a follow up article.
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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) with Simsmith we can find a near perfect match.
The capacitors are 177.2 and 92.9pF for the match.
Also calculated is the impedance looking back from the load to the source shown here as L_revZ. The impedance looking back towards the 50Ω load is 17.28-j0.6216Ω, which is quite close to the value obtained by measurement, 18.0-j0.8Ω (which is dependent on the actual Q of the ATU elements).
Is there some smoke and mirrors in calculation of L_revZ? Lets turn the network around.
Now turning the network around by swapping the capacitors and changing the load to 50+j0Ω.
Above, the impedance looking back towards the 50Ω load is 17.28-j0.62Ω, which consistent with the L_revZ calculation and is quite close to the value obtained by measurement, 18.0-j0.8Ω (which is dependent on the actual Q of the ATU elements).
So, in answer to the question Should we have expected this outcome?
, the answer is yes, it is not surprising and quite similar to what we might expect from a network of this type.
Walt Maxwell’s Conjugate Mirror (Maxwell 2001 24.5) which imbues a magic system wide conjugate match with certain benefits, a utopia, which does not apply to systems that include any loss, it does not apply to real world systems. Maxwell does not state that limitation of his proposition.
Is a ham transmitter conjugate matched to its load? Watch for a follow up post.
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The MDF is located where the underground cable enters the building. From here it rises vertically and travels some 25m across the ceiling to the VDSL modem.
The choke can be seen in the pic, it is 7 turns of CAT5 data pair wound around a LF1260 ferrite sleeve.
Above is the measured common mode impedance R,X of the choke. It is designed to peak between 3.5 and 7MHz to afford some moderately high impedance at both frequencies.
Measurements were made with a nanoVNA-H and the graphic made in Python scikit-rf from a saved .s1p file.
On test, the choke is effective on 40m.
]]>An SWR shifting Tillustrates the pitfalls in naive design and implementation of transmission line matching systems. I say naive because the article does not address the matter of loss, yet QST publishes it as an example.
K2PO outlines an issue that looking into the end of the feed line at 3.58MHz he measures Z=135-j90Ω, and his solution using a single stub tuner of RG-8X right on the back of the power amplifier.
Let’s model that in Simsmith (though I must declare I have issues with Simsmith’s transmission line modeling).
The model above implements a near perfect match, and the source is set to supply 1500W. If we are to accept Simsmith’s loss model, the calculated power values as the signal flows right to left (huh!), 35W is dissipated in the open circuit stub, and 269W is dissipated in the 20m long series section. In all 20% of the 1500W transmitter output is converted to heat in the matching system.
Above is a pic of the series section that dissipates 269W, fairly tightly coiled foam dielectric coax may get quite hot risking migration of the centre conductor.
Of course this is less a problem for low power or low duty cycle modes that high duty cycle modes like RTTY or even CW.
By contrast, a good ATU should provide the same impedance transformation at perhaps a quarter of the power lost.
A purpose specific L match does even better.
Above, a model of an L match with a good capacitor and mediocre inductor yields a loss of less than a tenth of the power of the single stub tuner solution described.
]]>Above is a demonstration circuit in Simsmith, a linear source with Thevenin equivalent impedance of 50-j5Ω. The equivalent voltage is specified by useZo, which like much of Simsmith is counter intuitive (as you are not actually directly specifying generator impedance):
Vthev and Zthev are chosen so that ‘useZo’ will deliver 1 watt to a circuit impedance that equals the G.Zo. Zthev will be Zo*.
So, in a lossless conjugate matched circuit we would expect the load power to be 0dBW.
In the circuit above, the load is not the conjugate of the source impedance so there will be some mismatch loss, to mean the ratio of power in a matched load to the power in the mismatched load.
In this scenario, the calculated power in the mismatched load is -0.307dBW, so mismatch loss is 0 – -0.307=0.307dB.
This mismatch load is often calculated as \( ML=10\cdot log_{10}(\frac1{1-|Γ|^2}) \) where Γ is calculated wrt 50+j0Ω, and the value is shown above at ML as 0.3856dB, quite different to the actual mismatch loss.
KML is mismatch loss calculated using Kurokawa’s power reflection coefficient, and at 0.3069dB (rounds to 0.307dB) it reconciles with the 0.307dB calculated from the displayed power levels.
The expressions used for the displayed G quantities are show above.
What if we could introduce a circuit element that hid the true nature of the source and made it look like an ideal 50+j0Ω source?
The scenario above inserts an ideal isolator between source and load (the N block as it is known in Simsmith).
Note that this is not your Hammy Sammy isolator. Isolator is one of the many terms with a well understood meaning in industry, appropriated by hams and give new / incompatible meaning just to confuse.
An isolator, this isolator, is a two port device that allows RF to flow in one direction and not the other for a specified Zo. So in this instance, Zo is specified as 50+j0Ω, and power can flow from source to load (with no loss), but no power can flow from load to source.
An isolator is characterised by \(S=\begin{bmatrix}0 & 0\\1 & 0\\\end{bmatrix}\) using conventionial notation, but Simsmith does not follow convention, you must transpose P1 and P2.
Above is Simsmith’s “backwards circuit” display option, the departure from convention in port labelling is more obvious, so to understand things in the Simsmith world, \(S_{simsmith}=\begin{bmatrix}0 & 0\\0 & 1\\\end{bmatrix}\) but to program it, you can undo the port transposition in the wiring and use the conventional S matrix, see below.
//Isolator P1 w2 gnd; P2 w1 gnd; sprm1 {w1 gnd w2 gnd} {{0,0},{1,0}} {50};
A consequence of the isolator is that the output of the isolator (P1… yes, more backwards confusion from Simsmith) appears to have a source impedance of 50+j0Ω. The displayed quantity L_sZ is the calculated impedance looking from load L back towards the source, it is 50+j0.0001247Ω… slightly off, probably due to rounding errors in a fairly complicated calculation.
Note that the effect of the isolator is that there is no reflected energy (wrt isolator Zo), so the source sees a load of 50+j0Ω.
Power into the isolator is -0.0108dBW (mismatch loss at this node is 0.0108dB) and power into the load is 0.396dBW, mismatch loss at the load input is 0.385dB. Both calculated (traditional) ML and KML reconcile with the calculated power levels.
Is an isolator the magic component that can deliver a transmitter with Thevenin source impedance equal to 50+j0Ω? Well, low loss isolators are practical at microwaves, and devices can be made to cover HF, the do it with considerable loss. For an ordinary HF SSB telephony transmitter there is little benefit and severe cost and efficiency issues.
… or where a little knowledge is dangerous.
A common newby online question is “my 50Ω VSWR is 3, surely that is really bad?”
The expert answers tend to go along the line “Don’t worry, reflected power is 25%, which means your transmitted power is 75%, that is just 1.3dB lower, a quarter of an S-point and no one will notice it.” Some may provide a link to a handy dandy table of these magic values.
The experts have assumed the transmitter is well represented by a Thevenin equivalent circuit, and that Zs is 50+j0Ω, an unwarranted assumption. Not only is the transmitter unlikely to be sufficiently linear to apply linear circuit theory in that way, practical transmitters often include protection systems that may reduce power output to limit ‘reflected power’ to 10% of rated power or less.
The fact that in many transmitters, protection circuits would have kicked in is a salutary warning. Operation well outside of specified load range might well result in degraded distortion products.
Calculation of mismatch loss requires an understanding of the characteristics of the source.
It is unsafe to assume that a transmitter that is designed to work into a 50Ω load is well represented as a Thevenin source with Zs=50Ω.
Traditional mismatch formula fails under some load and source impedance combinations.
Kurokawa’s power reflection coefficient may be a usable metric for linear systems, provided that actual source and load impedance are known.
The answer is on the face of it quite simple.
Jacobi’s Maximum Power Transfer Theorem extended to alternating current tells us that in a linear circuit, maximum power is transferred from a source to its load when the load impedance is the complex conjugate of the Thevenin equivalent source impedance.
This would seem to have obvious direct application in getting maximum power radiated from a transmitter.
But in most practical cases there are two important points that invalidate this thinking:
In the days before common use of VSWR meters, it was common practice to use an antenna current meter or a remote field strength meter to indicate maximum radiation power than transmitter adjustments were made to maximise that (within limits such asdesign or specification limits on PA device current).
Of course maximising power radiated regardless of any and everything else is a fairly inadequate technique, but with its roots in A1 Morse code transmitters and a user base who were not progressive, it survived until the ‘magic’ of VSWR meters penetrated the user base.
Modern SSB telephony transmitters will usually be solid state and designed to work into some fixed nominal load impedance. Some incorporate a wider range matching network (often known as an Antenna Tuning Unit) for greater flexibility.
For the purpose of this discussion, the term ‘transmitter’ is taken to mean the power amplifier and any necessary filters, but does not include an internal Antenna Tuning Unit.
Transmitter design includes a very wide range of parameters, and whilst it is possible to control source impedance at the output, it has almost no advantage and a lot of disadvantage to other aspects of the implementation… so it is not usually done for this type of transmitter.
Transmitters are usually designed to suit a given range of load impedance, often specified as a nominal value with some notional VSWR tolerance, eg a nominal 50Ω antenna with maximum VSWR of 1.5. It should be safe to operate the transmitter into a load that satisfies that criteria, and the buyer might hope that it should deliver its rated performance including distortion performance. That said, it is not common practice to test at these limits, and in practice specified power, distortion etc might only occur at very close to the nominal load impedance. None of this is to imply that maximum output power occurs in the nominal load (eg 50+j0Ω).
So, to obtain the rated performance, the objective is to deliver a load that is within it rated range, and preferably as close to the nominal value as practicable.
Given the power delivered to that transmitter load, the matter of how much power is radiated falls to the other system elements.
That being the case, the first optimisation objective should be to deliver the transmitter its rated load. Next, address performance of the rest of the system in terms of maximising radiated power.
Is there room for mathematics and theory here?
The greatest problem in applying conventional linear circuit theory to the problem is that in most cases, ordinary ham SSB telephony transmitters are not well represented by a Thevenin equivalent circuit and analysis based on that is simply invalid.
There is a whole vocabulary that flags woolly thinking, conjugate matching, flywheel effect, mismatch loss, re-reflection, an alternate expression for Γ (the complex reflection coefficient), the new SWR* (conjugate SWR, yes complex conjugate of a scalar) to name a few.
There is discussion in pseudo maths, or junk maths pretending to be science. All grist for the mill for a hobby that is less and less science based with the passage of time.
The proceeding discussion did not need to talk about “happy” transmitters, antennas or other system components. Antropomorphism is a technique often used to ‘communicate’ with hams who do not have a clue… and often by those who also do not understand the problem and often want to preserve their status by having others satisfy their thirst for knowledge with their own inadequate understanding.