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,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.

This example is yet another that disproves Walt Maxwell’s simultaneous system wide conjugate matching nonsense.

- Duffy, O. Mar 2013. The failure of lossless line analysis in the real world. VK1OD.net (offline).
- Everitt, W L. 1937 Communications Engineering, 2nd ed. New York: McGraw-Hill Book Co.
- Everitt, W L, and Anner, G E. 1956 Communications Engineering, 3rd ed. New York: McGraw-Hill Book Co.
- Maxwell, Walter M. 2001. Reflections II. Sacramento: Worldradio books.

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.

]]>Jacobi’s law (also known as Jacobi’s Maximum Power Transfer Theorem) of nearly 200 years ago stated

Maximum power is transferred when the internal resistance of the source equals the resistance of the load.

Implied is that the internal resistance of the source is held constant, it does not work otherwise. The source must be one that can validly be represented by a Thevenin equivalent circuit. This was in the very early days of harnessing electric current, direct current initially.

Later adaptation dealt with alternating current and it became

Maximum power is transferred when the load impedance is equal to the complex conjugate of the internal impedance of the source.

Again a necessary condition is that the source must be one that can validly be represented by a Thevenin equivalent circuit.

It is one of the principles of basic circuit theory / analysis and the mathematical proof is something a high school student should be able to perform.

So, if you have a transmitter with a known Thevenin equivalent source impedance, you can seek to provide a load that ensures maximum power transfer.

The practical problem is that it is quite difficult to control the equivalent source impedance of a high power transmitter so that it is a known constant over the entire range of transmitter output power, frequencies etc. So for many purposes, source impedance is not controlled, and the transmitter cannot be validly represented by a Thevenin equivalent circuit, and as a result, the Jacobi Maximum Power Transfer Theorem does not apply.

Do not be confused about transmitter specifications, a requirement that the transmitter load impedance be some value (eg 50Ω) is not specification of a Thevenin equivalent source impedance.

Very few ham transmitters have a controlled equivalent source impedance. Though some claim to have proven by one or few measurements that it is 50Ω, many other valid experiments have shown otherwise and it takes only one valid experiment to disprove the claim.

For the most part, calculations and other claims of mismatch loss in transmitter / antenna systems are misguided, popular but misguided.

The objective with most high powered transmitters is to provide them with a load impedance at their output terminals that complies with the design requirements as embodied in published specifications so as to safely obtain stability, rated power and rated distortion performance.

The obsession with conjugate matching is mostly with the many hams who are devotees of Walt Maxwell’s teachings.

A little knowledge is a dangerous thing; read widely and think.

]]>Walt Maxwell (W2DU) made much of conjugate matching in antenna systems, he wrote of his volume in the preface to (Maxwell 2001 24.5):

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 is popularly held to be some nirvana, a heavenly state where transmitters are “happy” and all is good. Happiness of transmitters is often given in online discussion by hams as the raison d’être for ATUs, anthropomorphism over science.

(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)

Let us look at a very simple example in SimSmith.

The scenario is:

- a Thevenin source at 1MHz with a source impedance of 50+j0Ω;
- a nominal half wave of RG59 transmission line; and
- an adjustable load impedance.

This should not be taken to imply that ham transmitters are commonly well represented as a Thevenin source.

The load impedance has been adjusted for a nearly perfect match at the source.

Above is the SimSmith model. The load R and X were adjusted for extremely low |Γ| at the source. |Γ| at the source is extremely low (0.0000173), Return Loss is 95dB, this is a match better than instruments could ever measure. We have achieved an almost perfect conjugate match at the interface between source and T1.

So let us now examine the impedance looking both ways at the load to T1 interface.

SimSmith has an internal feature to calculate the impedance looking backwards into an element, and it is used to calculate the impedance looking back from the load into element L. It is shown under the generator element as L_revZ.

So at the load to T1 interface:

- Z looking into the load is 38.75+j1.813Ω; and
- Z looking into T1 is 58.43-j1.534Ω.

They are not conjugates of each other, not nearly, in fact the mismatch is characterised by Return Loss (in terms of the load Z) is just 14dB (or VSWR=1.5).

In this very simple configuration, a near perfect match at the source does not result in a similar quality match at the other node in the system.

(Maxwell 2001 24.5) relies on a quotation:

If a group of four-terminal networks containing only pure reactances (or lossless lines) are arranged in tandem to connect a source to its load, then if at any junction there is a conjugate match of impedances, there will be a conjugate match of impedances at every other junction in the system. (Everitt 1937 243) and (Everitt and Anner 1956 407)

The problem is that Maxwell silently dropped from his statement (1) above the requirement that networks and lines must be lossless, and the example calculated here shows that Maxwell’s proposition does not apply to real world networks that have loss.

Recourse to simple linear circuit analysis will reveal that lossy networks do not have the property Everitt ascribed to lossless networks.

Walt Maxwell’s conjugate mirror

does not apply in the real world, and the concept is of limited use in understanding real world antenna systems.

When you see people sprouting the Walt’s conjugate mirror

you can expect that they have not read widely or thought about the subject much.

- Duffy, O. Mar 2013. The failure of lossless line analysis in the real world. VK1OD.net (offline).
- Everitt, W L. 1937 Communications Engineering, 2nd ed. New York: McGraw-Hill Book Co.
- Everitt, W L, and Anner, G E. 1956 Communications Engineering, 3rd ed. New York: McGraw-Hill Book Co.
- Maxwell, Walter M. 2001. Reflections II. Sacramento: Worldradio books.

]]>

The original transformer above comprised a 32t of 0.65mm enamelled copper winding on a FT240-43 ferrite core, tapped at 4t to be used as an autotransformer to step down a load impedance of around 3300Ω to around 50Ω.

The FT114 core has a quite low ΣA/l value (0.000505), essentially a poor magnetic geometry.

A better choice for his enclosure is the locally available LO1238 core from Jaycar (2 for $5) with ΣA/l=0.0009756/m which is comparable with the FT240 form (though smaller in size) and nearly double that of the FT114. The LO1238 is a toroid of size 35x21x13 mm, and medium µ (L15 material).

A more detailed analysis of a 3t primary winding of the effects of magnetising impedance on InsertionVSWR and system loss when it is in shunt with a 50Ω load was performed.

Above is the expected core loss.

Above is the expected InsertionVSWR.

These both look encouraging, and the next step would be to build and measure some prototypes.

Above, VK4MQ’s prototype in development. (I do not recommend the pink tape.)

]]>The original transformer above comprised a 32t of 0.65mm enamelled copper winding on a FT240-43 ferrite core, tapped at 4t to be used as an autotransformer to step down a load impedance of around 3300Ω to around 50Ω.

A very rough approximation would be that with two stacked cores, the number of turns would be around the inverse of square root of two, so 70% of the original.

A more detailed analysis of the effects of magnetising impedance on InsertionVSWR and system loss when it is in shunt with a 50Ω load was performed.

Above is the expected core loss.

Above is the expected InsertionVSWR.

These both look encouraging, and the next step would be to build and measure some prototypes.

To the original question, would half the turns be enough? No. Notwithstanding that, you are likely to find such being used, being sold.

]]>Note that the measurements are of a particular implementation and should not be taken to imply generally to 5/8λ verticals, but the solution method can be applied more generally. Lets assume that the measurement is not affected by common mode current.

The answer to the last question first is that a series inductor will not bring the VSWR much below 3. It is a common belief that a 5/8λ vertical can be matched simply with a series inductor.

There are many ways to match the measured antenna, and there are articles on this site describing some of them, but a simple and effective method in this case is the single stub tuner.

Above is a graphical solution using Simsmith. The section of line nearest the measurement load is -ve length, it is to back out the effect of the line section into which measurements were made (antenna feed point is at the cursor, 139-j191Ω). The next line section is the series section, followed by the S/C stub. In this case the series section and stub use RG213 to reduce loss. Total matching system loss is a little under 0.3dB, and the stub can easily be weatherproofed with hot glue and heat shrink tube.

One could use RG58, an exercise for the reader is to assess the loss of that option.

Obviously the length of the measurement section plays into the solution, and using its length to the mm in the model gives a more accurate result.

]]>As pointed out in the articles, the solutions cannot be simply extended to real antenna scenarios. Nevertheless, it might provoke thinking about the performance of some types of so-called balanced ATUs, indeed the naive nonsense of an “inherently balanced ATU”.

(Witt 2003) goes to some length to calculate his IMB figure of merit based on a similar load of two not necessarily equal series resistors with the mid point grounded to the ATU chassis. Witt’s IMB is equivalent to the factor |2Ic/Id|that was calculated in earlier articles in this series, and equally useless in inferring behavior in a real antenna system.

(Duffy 2010) gives an explanation of the behavior of baluns in an antenna system, and it becomes apparent that simple linear circuit solutions of a couple of resistors does not give insight into the behavior in real antenna systems.

The bottom line though is that while NEC models might be informing, there is no substitute for direct measurement of common mode current (Duffy 2011)… and it is so easy.

- Duffy, O. Dec 2010. Baluns in antenna systems. https://owenduffy.net/balun/concept/cm/index.htm (accessed 21/02/12).
- ———. May 2011. Measuring common mode current. https://owenduffy.net/module/icm/index.htm (accessed 21/02/12).
- ———. Feb 2012. Balanced ATUs and common mode current. https://owenduffy.net/balun/concept/BalancedAtu.htm (accessed 18/03/2019).
- Witt, Frank. Apr 1995. How to evaluate your antenna tuner In QST May 1995. Newington: ARRL.
- ———. May 1995b. How to evaluate your antenna tuner In QST May 1995. Newington: ARRL.
- ———. Sep 2003. Evaluation of Antenna Tuners and Baluns–An Update In QEX Sep 2003. Newington: ARRL.

This article reports the same asymmetric load using the MFJ-949E internal voltage balun.

The test circuit is an MFJ-949E T match ATU jumpered to use the internal balun and resistors of 50Ω and 100Ω connected from those terminals to provide a slightly asymmetric load.

The voltage between ground and each of the output terminals was measured with a scope, and currents calculated.

Above are the measured output voltage waveforms at 14MHz.

Lets work out the current amplitudes. Above, V1 (yellow) is 5.9divpp, V2 (cyan) is 7.2divpp. I1=V1/50=5.9*0.2/50=23.6mApp. I2=V2/100=7.2*0.2/100=14.4mApp.

Expanding the timebase allows better measurement of the phase difference.

V2 lags by a half cycle less 8.25µs, so V2 phase is -180+8.25e-9*14e6*360=-180+42=-138°.

Lets calculate the common mode and differential component of current in each load resistor. We will use Python as it handles complex numbers.

>>> i1=0.0236

>>> i2=0.0144*(math.cos(-138/180*math.pi)+1j*math.sin(-138/180*math.pi))

>>> ic=(i1+i2)/2

>>> abs(2*ic)

0.016100289594275147

>>> id=(i1-i2)/2

>>> abs(id)

0.01781446515461856

>>> abs(2*ic)/abs(id)

0.903776198417105

>>> 20*math.log(abs(2*ic)/abs(id))/math.log(10)

-0.8787820061070818

So, the differential component of current is 17.8mApp, and the total common mode current is 16.1mApp, the total common mode current is 90% of the differential current or 0.9dB less than differential current.

By any standard, this is appalling balance, and demonstrates why voltage baluns are unsuited to the application.

The fact that the “inherently balanced” topology is only 1.8dB better that this voltage balun experiment speaks volumes for the failure of the “inherently balanced” topology.

The measurements reported here are for a specific scenario (components, frequency and load), and should not be simply extrapolated to other scenarios.

The calculated imbalance if you like applies to the specific test circuit, and cannot really be extended to use of this balun in an antenna system scenario.

]]>This article reports the same equipment reversed so that the common mode choke is connected to the output of the MFJ-949E.

The test circuit is an MFJ-949E T match ATU followed by A low Insertion VSWR high Zcm Guanella 1:1 balun for HF. A banana jack adapter is connected to the balun output jack, and resistors of 50Ω and 100Ω connected from those terminals to provide a slightly asymmetric load.

The voltage between ground and each of the output terminals was measured with a scope, and currents calculated.

Above are the measured output voltage waveforms at 14MHz.

Lets work out the current amplitudes. Above, V1 (yellow) is 4.0divpp, V2 (cyan) is 8.0divpp. I1=V1/50=4.0*0.2/50=16.0mApp. I2=V2/100=8.0*0.2/100=16.0mApp.

Expanding the timebase allows better measurement of the phase difference.

V2 lags by a half cycle and 1.0µs, so V2 phase is -180-1.0e-9*14e6*360=-180-5=-185°.

Lets calculate the common mode and differential component of current in each load resistor. We will use Python as it handles complex numbers.

>>> i1=0.016

>>> i2=0.016*(math.cos(-185/180*math.pi)+1j*math.sin(-185/180*math.pi))

>>> ic=(i1+i2)/2

>>> abs(2*ic)

0.0013958203956907485

>>> id=(i1-i2)/2

>>> abs(id)

0.015984771545309726

>>> abs(2*ic)/abs(id)

0.0873218858170239

>>> 20*math.log(abs(2*ic)/abs(id))/math.log(10)

-21.177537875409207

So, the differential component of current is 16.0mApp, and the total common mode current is 1.40mApp, the total common mode current is 9% of the differential current or 21.2dB less than differential current.

Calculation of the common mode component of current involves the addition of two almost equal and almost opposite phase currents and is very sensitive to uncertainty in each of the measurements using this measurement method. This balun should achieve |2Ic/Id|>35dB in this scenario, but it would take a higher accuracy measurement system to measure it.

The fact that the “inherently balanced” topology measures 18dB worse that this experiment speaks volumes for the failure of the “inherently balanced” topology.

The measurements reported here are for the specific scenario (components, frequency and load), and should not be simply extrapolated to other scenarios.

The calculated imbalance if you like applies to the specific test circuit, and cannot really be extended to use of this balun in an antenna system scenario.

Continued at Inherently balanced ATUs – part 3 .

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