The failure of lossless line analysis in the real world

(Moore 2013) gives a series of examples based on lossless transmission line and a lossless inductance to support his propositions about the behaviour of an Antenna Tuning Unit (ATU) in an antenna system.

His examples are based around:

• a load of 50-j500Ω;
• one wavelength of lossless transmission line;
• an ideal (ie lossless) inductor of 500Ω reactance;
• the transmitter requires a nominal load impedance of 50+j0Ω;
• the transmitter is well represented by a Thevenin equivalent circuit with Zs=50+j0Ω.

Moore suggests an equivalence between the loading coil and an ATU: [w]hat if we put the "loading coil" in a box at the source and call it an "antenna tuner"?

 Moore arrives at the conclusion:

It seems to the author that since the "antenna tuner" has the same effect on the load whether it is located at the load or at the source, an Old Wives' Tale has bit the dust.

The question is does the "antenna tuner" ha[ve] the same effect on the load whether it is located at the load or at the source in the real world?

Solution to a practical form of Moore's example

Let us work the same problem with practical system components. For the exercise, lets assume:

• frequency is 3.6MHz;
• transmission line is one wavelength of Belden 8267 (RG213 type);
• ATU as described by (Schmidt nd) A T-Network Tuner Simulator using the default settings; and
• the transmitter requires a nominal load impedance of 50+j0Ω.

Note that it is not possible to match the real world system with a simple inductor at the source end, so for consistency, the T match ATU simulation will be used for matching at the source and load end. In fact, only a lossless inductor can match at the source end with this example, so further reason why a more general ATU is used in this article.

Efficiency will be calculated as the power delivered to the feed point as a percentage of the power delivered by the transmitter.

Load end matching

ATU

Lets find the total system efficiency when the ATU is used to match the 50-j500Ω load and that is connected to the transmitter using one wavelength of Belden 8267.

"Matched" in this section means the ATU is adjusted to deliver an impedance of 50+j0Ω to the feed line.

So, first we must find the efficiency of the ATU in transforming the 50-j500Ω load to the nominal feed line Zo of 50+j0Ω.

Fig 1:

 Fig 1 shows the solution of the ATU problem using A T-Network Tuner Simulator. ATU efficiency is 100-23.0=77.0%.

Transmission line

Next step is to find the transmission line loss with the ATU's input impedance of 50+j0Ω as a load.

 Fig 2 shows the solution of the transmission line connecting the transmitter to the ATU input. Using RF Transmission Line Loss Calculator the line efficiency is 87.51%.

System efficiency

Combined efficiency of line and ATU is 77.0%*87.51%= 67.38%, ie 67% of the transmitter output power is delivered to the antenna feed point.

Note that the load delivered to the transmitter at 50.00-j0.06Ω is not perfect, a subtle effect of Zo of RG213 line theoretically not being 50+j0Ω and a consequent by product of adjusting the ATU for 50+j0Ω match at the load end of the line (as would usually be done in practice). The Mismatch Loss caused to a source with equivalent source impedance of 50+j0Ω is less than 0.000dB.

Fig 3:

Fig 3 shows distribution of power in the load end matched solution. 

Source end matching

Transmission line

Now lets find the total system efficiency when the ATU is used to match the source end of one wavelength of Belden 8267 with a load of 50-j500Ω.

"Matched" in this section means the ATU is adjusted to deliver an impedance of 50+j0Ω to the transmitter.

First step is to find the transmission line loss with the a load impedance of 50-j500Ω.

 Fig 4 shows the solution of the transmission line connecting the ATU output to to the load. Using RF Transmission Line Loss Calculator the line efficiency is 12.76% and the load presented to the ATU is 244.56-j314.27Ω.

ATU

Next step is to find the efficiency of the ATU in transforming the 244.56-j314.27Ω load to the nominal feed line Zo of 50+j0Ω.

Fig 5:

Fig 5 shows the solution of the ATU problem using A T-Network Tuner Simulator. ATU efficiency is 100-6.8=93.2%. The impedance at the input to the ATU is 50+j0Ω (it was adjusted for that goal) and since the assumed transmitter equivalent source impedance is 50+j0Ω, the is a conjugate match at this junction.

System efficiency

Combined efficiency of line and ATU is 12.76%*93.2%= 11.89%, ie 12% of the transmitter output power is delivered to the antenna feed point.

Fig 6:

Fig 6 shows distribution of power in the load end matched solution.

Conjugate matching

Fig 7:

Fig 7 shows the T match looking back towards the transmitter (right hand side of Fig 6). The impedance looking into the T match towards the transmitter is 258.95+j281.85Ω. Note that the at this junction looking towards the load has already been calculated to be 244.56-j314.27Ω. The impedances looking each way are not conjugates, not nearly, there is NOT a conjugate match at this junction.

 Fig 8 shows the solution of the transmission line looking back from the load to the ATU. The impedance seen looking into the line is 236.07+j143.42Ω, and the stated load impedance is 50-j500Ω. The impedances looking each way are not conjugates, not nearly, there is NOT a conjugate match at this junction.

(Moore 2013a), makes the statement:

If we disconnect the antenna and measure the impedance looking back toward the shack, in a low-loss system, we will measure pretty close to the conjugate of the antenna feedpoint impedance at the transmit frequency.

This statement depends on the meaning of low-loss system and it is true if low-loss system means zero loss, ie lossless. The above work through of the example that (Moore 2013) uses, but applied to a practical low loss transmission line and practical ATU model shows the folly of applying lossless assumptions to the real world without having quantified or otherwise adequately addressed the error.

Summary

Firstly, the above analysis is of Moore's example load but with real world transmission line and ATU losses for a typical 3.6MHz implementation brought to book. The outcome applies to the scenario modeled and may not be directly applicable to other scenarios.

Moore's conclusion [i]t seems to the author that since the "antenna tuner" has the same effect on the load whether it is located at the load or at the source might be supported by his choice of an example, but the working of an example based on the same load impedance and line length but using practical transmission line loss and ATU loss reveals a quite different picture where load end matching results in more than five times the radiated power.

Moore makes the statement:

For the sake of simplicity, transmission line losses, which make the calculations much more complex, have not been taken into account. Failure to include losses does not negate the concepts presented in this article. Note that the conjugate matching theorem applies only to lossless networks and just comes close for low-loss networks.

Well no, [f]ailure to include losses has led Moore to the false conclusion that the "antenna tuner" has the same effect on the load whether it is located at the load or at the source.

He states in the following discussion [s]ince the conjugate matching theorem assumes lossless transmission lines which cannot exist in reality, I would be an absolute fool to assert that a conjugate match is even possible in the real world.

The last statement is somewhat confused, the Conjugate Matching Theorem is probably that popularised by (Maxwell 2001) and his disciples, but originally from (Everitt 1937) who proposed:

If a group of four-terminal networks containing only pure reactances are arranged in tandem to connect a generator to a 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

To the extent that a lossless transmission line can be represented by a four-terminal networks containing only pure reactances, lossless transmission lines could form part of such a system.

Moore's statement is confused because Everitt's theorem does not to prevent a conjugate match existing at one or more points in a practical (ie lossy) system, just that it does not guarantee that a conjugate match exists at all points in the system simultaneously. 

Moore has used a carefully contrived example to support (peddle) his 'concepts', an example he acknowledges cannot exist the real world and without accurately quantifying and adequately addressing the error that results from his approximation, a somewhat confused approach.

It is yet another example where inferring solutions to real world problems from lazy application of lossless analysis may give a false result.

The analysis here did not depend on the Maximum Power Transfer Theorem or conjugate match concepts, much less Everitt's simultaneous system wide conjugate match though much of (Moore 2013) tries to explain these concepts at length as if they underpin the correct solution. Before the Maximum Power Transfer Theorem can be applied to such a problem, it is necessary to demonstrate that a plot of source V,I is a sufficiently straight line that a Thevenin equivalent circuit of the source is a valid model, and that was not done.

Links / References

• Duffy, O. July 2001. RF Transmission Line Loss Calculator. https://owenduffy.net/calc/tl/tllc.php  (accessed 30/03/13).
• Everitt, WL. 1937. Communications engineering. New York and London: McGraw Hill book company, inc.
• ———. September 1938. Coupling networks In Communications September 1938 p12.
• Maxwell, Walt. 2001. Reflections II. Worldradio Books.
• Moore, C. March 2013. A Tuner's Effect At The Antenna. http://www.eham.net/articles/29821 (accessed 30/03/13).
• ———. April 2013a. Comments to Resonant listening antennas -- why?. http://www.eham.net/articles/29848 (accessed 09/04/13).
• Schmidt, K. nd. A T-Network Tuner Simulator. http://fermi.la.asu.edu/w9cf/tuner/tuner.html (accessed 30/03/13).
  

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