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.
Common practice is to speak of a “source VSWR” to mean the VSWR calculated or measured looking towards the source, and very commonly this is taken wrt 50+j0Ω which may be neither the source or load impedance but an arbitrary reference.
If neither of the adjacent elements are real Zo=50+j0Ω transmission lines as is so often the case, then the value of VSWR is diminished. Often a calculated Mismatch Loss from that VSWR will be invalid.
The complex reflection coefficient at each Zo discontinuity is relevant and gives the correct value of reflected wave in a wave based analysis. Accuracy depends on use of the actual value of Zload and Zo for the calculation.
A better metric for some purposes in this type of scenario may be the (Kurokawa 1965) Power Reflection Coefficient |s|^2 of the actual source and load impedances. (Note that calculator input field Zref is not used in this calculation.)
Above calculation for this scenario (L2L1 junction) gives |s|^2 of -35dB.
Note that making changes that affect the mismatch at this point will probably affect the generator match.
Keeping in mind that C2 and L2 are an adjustable matching network, usually adjusted for minimum VSWR as seen at the source G. So, the questions are:
Does the system take maximum available power from the source G when the load impedance seen by source G is equal to the conjugate of its Thevenin equivalent source impedance (ie C2.Z=G.Zo in Simsmith speak)?
Does that ‘matched’ condition result in maximum power in the load L?
Above for reader’s convenience is the model conjugate matched at the GC2 interface. The calculated Po figure (lower right) is the power in the load L to high resolution.
I have already stated:
The answer to Q1 is easy, Jacobi’s Maximum Power Transfer Theorem tells us that for a valid Thevenin source (as in this simulation), then the maximum available power is obtained from the source when the Zl=Zs*. The mathematical proof of this is simple and in good text books.
Now to Q2.
Tweaking the matching components C2 and L2 by hand for best Po we get:
Po is very slightly greater and yet the input VSWR is higher.
This should be no surprise in this case as the matching inductor L2 is lossy, not grossly so but typical of an ATU, and adjusting it also changes the loss it brings to the system and the better Po solution above finds that although the entire network accepts less than available power from the source G, the reduction in L2 loss more than offsets it.
The difference in maximum power is very small in this scenario, but the maths of it proves the case that optimising VSWR seen by the source does not necessarily maximise power to the load.
Again the calculated values L2L1_lZ are the load impedance at the L2L1 junction (looking left as Simsmith is unconventional), and L2L1_sZ is the source impedance at the L2L1 junction (looking right).
The next installment will discuss a metric for the mismatch at this point… think about it.
]]>The replicated scenario with matching with an L network where the inductor has a Q of 100, no other loss elements is shown below. (Quality real capacitor losses are very small, and the behavior will not change much, the inductor loss dominates.)
Above is a model in Simsmith where I have adjusted the lossy L network for a near perfect match. I have used a facility in Simsmith to calculate the impedance looking back from L1, often known as the source impedance at a node but in Simsmith speak the calculated L1_revZ on the form, (ie back into the L network) from the equivalent load.
The calculated values L2L1_lZ are the load impedance at the L2L1 junction (looking left as Simsmith is unconventional), and L2L1_sZ is the source impedance at the L2L1 junction (looking right).
Complex conjugate, shortened to “conjugate”, has a well known meaning in mathematics, if x=a+jb then x*=a-jb. (Notwithstanding that fact, I see ham discussion redefining the meaning by talking about SWR* which in mathematical terms, because SWR is a scalar quantity (ie its imaginary part is zero), means SWR*=SWR.)
The previous discussion showed that even though the source G was conjugate matched to its load, along the network at the L2L1 interface L2L1_lZ is significantly different to L2L1_sZ* and so there was not a system wide conjugate match.
Keeping in mind that C2 and L2 are an adjustable matching network, usually adjusted for minimum VSWR as seen at the source G. So, the questions are:
The answer to Q1 is easy, Jacobi’s Maximum Power Transfer Theorem tells us that for a valid Thevenin source (as in this simulation), then the maximum available power is obtained from the source when the Zl=Zs*. The mathematical proof of this is simple and in good text books.
What about Q2, is it a no-brainer?
]]>It explains in great detail how the antenna tuner at the input terminals of the feed line provides a conjugate match at the antenna terminals, and tunes a non-resonant antenna to resonance while also providing an impedance match for the output of the transceiver.
Walt Maxwell made much of conjugate matching, and wrote often of it as though at some optimal adjustment of an ATU there was a system wide state of conjugate match conferred, that at each and every point in an antenna system the impedance looking towards the source was the conjugate of the impedance looking towards the load.
This was recently cited in a discussion about techniques to measure high impedances with a VNA:
WHEN the L and C’s of the tuner are set to produce a high performance return loss as measured by the vna, then in essence, if the tuner were terminated (where the vna was positioned) with 50 ohms and we were to look into the TUNER where the antenna was connected, we would see the ANTENNA Z CONJUGATE. Wow, that’s a mouth full. The best was to see this is to do an example problem and a simulator like LT Spice is a nice tool to learn. Or there are other SMITH GRAPHIC programs that are quite helpful to assist in this process. Standby and I will see what I can assemble.
The example subsequently described set about demonstrating the effect. The example characterised a certain antenna as having an equivalent circuit of 500Ω resistance in series with 4.19µH of inductance and 120pF of capacitance (@ 7.1MHz, Z=500-j0.119, not quite resonant, but very close). A lossless L network (where do you get them?) was then found that gave a near perfect match to 50+j0Ω. The proposition is that if you now look into the L network from the load end, that you see the complex conjugate of the antenna, Z=500+j0.119.
I asked where do you get a lossless L network? Only in the imagination, they are not a thing of the real world.
So lets replicate the scenario matched with an L network where the inductor has a Q of 100, no other loss elements. (Quality real capacitor losses are very small, and the behavior will not change much, the inductor loss dominates.)
Above is a model in Simsmith where I have adjusted the lossy L network for a near perfect match. I have used a facility in Simsmith to calculate the impedance looking back from L1, often known as the source impedance at a node but in Simsmith speak the calculated L1_revZ on the form, (ie back into the L network) from the equivalent load.
The impedance looking back is 471.3-j0.1274Ω. The resistance element is quite different to the proposed 500Ω. The L network in this scenario lacks reciprocity.
Although the L network is conjugate matched at its input, it is not conjugate matched at its output.
This example is yet another that disproves Walt Maxwell’s simultaneous system wide conjugate matching nonsense.
None of this should be taken to suggest that when an ATU or other matching network is adjusted for an input VSWR(50) that it is conjugate matched to the source, much less that a typical ham transmitter is well represented by a Thevenin equivalent circuit or that the Jacobi Maximum Power Transfer Theorem applies.
He asked the question
In the general case, if you are trying to match 50 Ohms, would you be better off feeding a normal backyard dipole with 75 Ohm coax if you are willing to prune it to a specific length after the fact?
Let’s look at the general case using a lossless line to simplify analysis and presentation.
If the objective is that Zin=50+j0Ω and Zo=75Ω, then the load impedance components can be plotted against the electrical line length.
Above is a plot of Rload and Xload vs the electrical length of lossless 75Ω to transform the load to 50+j0Ω.
Let’s say we have a dipole with feed point R of around 65Ω, drawing a horizontal line from R=65Ω across to the blue line, and then dropping that vertically we find that the electrical length needed is 40°, and Xload needs to be -27Ω. So the dipole needs to be shorter than a half wave to obtain Z at the feed point of 65-j27Ω.
Now in practical terms, R will change a little as the length is varied around a half wave, so it might take a few iterations to find the combination of R and X. The combination can be found more quickly by measuring the feed point impedance with an analyser that can display VSWR(75), just shorten the dipole until VSWR(75)=1.5. Use the measured Rload and Xload to find the expected electrical length of line from the chart above.
The electrical length of the line has to be converted to physical length, and it is LengthDegrees/360*300/FrequencyMHz*VelocityFactor metres.
The chart presented is for lossless line. Lossy line changes the curves slightly depending on line loss, but the figures a a good start for in-situ tuning.
Note that below 50MHz, the very popular RG6 line with CCS centre conductor may have significantly more loss than expected.
The above analysis assumes that the feed point impedance does not change when feed line is attached, but that may not be true if there is significant common mode current.
Understood, a length of 22m might be more convenient than 4m @7MHz as you may be able to use the 75Ω right to the radio.
Broadly it can be lengthened by integral electrical half waves, but the effect of loss becomes more significant on the tuning. Further, the longer the matching section, the narrower bandwidth.
There is also an intermediate solution where you make the dipole longer than a half wave to launch VSWR(75)=1.5.
That is true, it is an efficient matching solution for a single band, and suited to the highest power permitted to ham stations.
That is true, and if you ‘believe’ that is bad, there is little point presenting facts.
Jonathon Swift wrote:
Reasoning will never make a Man correct an ill Opinion, which by Reasoning he never acquired.
Certainly: An antenna for 7MHz local contacts but note that I used RG6 with a solid copper centre conductor.
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