Jacobi’s Maximum Power Transfer Theorem
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. Continue reading Transmitter / antenna systems and the maximum power transfer theorem
I have written on Walt Maxell’s proposition about simultaneous system wide conjugate matching in antenna systems. I will repeat a little to set the context…
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. Continue reading Walter Maxwell’s teachings on system wide conjugate matching – a SimSmith example
Average power of SSB telephony used 80 year old research by (Holbrook and Dixon 1939) to come up with a ratio of peak voltage to RMS voltage of a voice waveform, and from that derive the ratio PEP/Pav..
(Holbrook and Dixon 1939) explored the subject measuring the voice characteristics of many talkers (as there is variation amongst talkers) to come up with an average characteristic.
Whilst in its day, obtaining instantaneous samples of voice was a challenge, it is trivial today and if you can’t believe the numbers given, try your own experiment (but realise it is for your own voice rather than the general population).
Many modern PC sound applications are capable of the measurement, I will demonstrate it with the feed Windows application Audacity with the stats.ny addin.
Above is a screenshot of a 6s recording of my voice made without stopping for breath. The statistics window shows a peak of -8.9dBFS and RMS of -27.4dBFS, giving a peak voltage to RMS voltage ratio of 18.5dB. Continue reading Average power of SSB telephony – experimental verification
Some components used for SSB telephony need not be capable of handling the Peak Envelope Power (PEP) continuously, many components for instance respond to the average power (Pav) which is quite less. Essentially, components that are subject to voltage breakdown (usually as good as instantaneous) must withstand the PEP, those that heat relatively slowly must withstand Pav.
In estimating the power dissipated in components due to an SSB telephony waveform, a good estimate of the ratio of Average Power (Pav) to Peak Envelope Power (PEP) is very useful.
Long before hams had used SSB, the figure has been of interest to designers of FDM or carrier telephone systems to size amplifiers that must handle n channels of FDM multiplex without overload which would degrade S/N in other channels of the multiplex. The methods are applicable to SSB telephony, it uses the same modulation type and the overload challenges are the same.
(Holbrook and Dixon 1939) gave the graph above which characterises the ratio of instantaneous peak to RMS voltage of voice telephony for different numbers of channels in a multiplex and different expectation of overload or clipping. They recommend a very low probability of clipping at 0.1% to avoid significant intermodulation noise in adjacent channels. Continue reading Average power of SSB telephony
Quality of termination used for calibration
Some of us use a resistor as a load for testing a transmitter or other RF source. In this application they are often rated for quite high power and commonly called a dummy load. In that role, they usually do not need to be of highly accurate impedance, and commercial dummy loads will often be specified to have maximum VSWR in the range 1.1 to 1.5 (Return Loss (RL) from 26 to 14dB) over a specified frequency range.
We also use a known value resistor for measurement purposes, and often relatively low power rating but higher impedance accuracy. They are commonly caused terminations, and will often be specified to have maximum VSWR in the range 1.01 to 1.1 (RL from 46 to 26dB) over a specified frequency range.
It is more logical to discuss this subject in terms of Return Loss rather than VSWR.
Return Loss is defined as the ratio of incident to reflected power at a reference plane of a network. It is expressed in dB as 20*log(Vfwd/Vref). Continue reading Exploiting your antenna analyser #30
Elecraft produces a directional coupler that may interest QRP aficionados. It comes with instructions for 20dB and 30dB coupling factors rated at 25 and 250W respectively from 1 to 30MHz.
This article reviews the magnetics design of the -20dB / 25W coupler.
The coupler uses a type of Sontheimer coupler (Sontheimer 1966) and these are commonly poorly designed. The first question is whether the magnetising impedance of T2 which appears in shunt with the load is sufficiently high to not give rise to poor insertion VSWR. Continue reading Elecraft CP-1 directional coupler – magnetics review
Should you trust your VSWR meter? asked an interesting question, and Should you trust your VSWR meter – detector linearity discussed a problem apparent in may VSWR meters.
This article illustrates one method of linearisation of the detector response of a practical VSWR meter.
Radio-kits SWR meter
This article contains an analysis of the analogue circuitry of the Radio-kits SWR meter.
The directional coupler at top left contains half wave peak detectors for forward and reflected waves. They are wired to the two compensated op amps at lower right (the connections are not shown on the circuit as the coupler may be remote, follow the terminal designations). Continue reading Should you trust your VSWR meter – linearisation
Should you trust your VSWR meter? asked an interesting question, and based on experience, including a relevant example, concluded:
The answer is no, like any measurement instrument, prove that it is trustworthy in the intended application.
It went on to ask:
If the VSWR meter is designed to fail, why does it fail?
This article contains an analysis of the analogue circuitry of the IC-7300 directional coupler to explain the likely cause of its poor behaviour.
IC-7300 directional coupler schematic
Above is an extract of the IC-7300 circuit in the area of the directional power coupler used for VSWR measurement. The circuit is a quite conventional Bruene coupler, and its response is similar to several types of directional couplers that produce a DC output voltage from a half wave detector. Continue reading Should you trust your VSWR meter – detector linearity
One often sees newbies ask about their VSWR meter readings, and a common observation is that the measured VSWR is better at low power and as power is increased, VSWR increases.
With the evolution of the ‘shack in a box’, and knowledge and experience to match, the problem is often reported observed with the transceiver’s internal VSWR meter.
Some of these ‘shack in a box’ have some pretty nifty features, for example the very popular Icom IC-7300 not only has an internal VSWR meter for the HF bands, but it can perform an assisted sweep and display the results graphically.
Isn’t that a great idea, so convenient, all good!
Or is it? Continue reading Should you trust your VSWR meter?
From time to time one sees discussion online about consistency of ‘measured’ VSWR at different power levels (on the same instrument).
A question often asked is:
I tune up at 10W and achieve VSWR=1.5, and when I increase power to 100W, the VSWR increases. Which should I believe?
The first thing to note is that good antenna systems SHOULD be linear, VSWR should be independent of power, it is if the system IS linear.
For the most part they are linear, even though many antenna systems contain elements such as ferrite cored inductors that may exhibit some small level of non-linearity in ‘normal’ operation.
Non-linearity caused by for instance saturation of magnetic materials, loss of permeability where the temperature of ferrite cores reaches Curie point, arcing of capacitors or other insulating materials is NOT normal linear operation of a GOOD antenna system. If high indicated VSWR at high power is caused by any of these effects, it is flagging a problem that requires attention.
That said, a significant non-linear element may be the VSWR meter itself.
A common, if not most common way to make these meters is to use a half wave detector to convert the direction coupler RF outputs into DC to drive an ordinary moving coil meter. These meters commonly assume that the detector DC output voltage is exactly proportional to the RF input voltage.
Lets look at the accuracy of that process.
Above is a plot of the detector output vs RF input voltage for a commercial 200W VSWR meter. The measurements cover input power from 10 to 100W.
Continue reading VSWR meter trap for the unwary