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
(Franklin 1924) described a technique to cophase sections of a long antenna by “concentrating alternating half wave length portions of the wire within a small space, by winding such portions as inductance coils or by doubling such portions back on themselves so that there is practically no radiation from these portions”.
Let’s explore his second option, as unlike the first, it does work reliably.
Above is an NEC-4.2 model with current shown (magnitude and phase). The stubs conductors are all defined from top to bottom. Continue reading Franklin antenna – how does it work?
A reader of End Fed Half Wave matching transformer – 80-20m asked if a good transformer could be made with with a FT114-43 core.
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. Continue reading End Fed Half Wave matching transformer – 80-20m – LO1238 variant
A reader of End Fed Half Wave matching transformer – 80-20m asked if a better transformer could be made with a stack of 2 x FT240-43 cores and using half the turns.
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. Continue reading End Fed Half Wave matching transformer – 80-20m – 2xFT240-43 variant
Recently I have had difficult reaching the local DMR repeater on 70cm, and needed to check that the antenna system had not deteriorated.
I took a baseline measurement with an AA-600 after some refurbishment work in Jan 2018, and was able to compare a current sweep to that baseline.
Above, a wide Return Loss sweep of the Diamond X-50N with feed line compared to the baseline (the thin blue line). Continue reading Diagnosing a possible antenna problem by comparison with a baseline
A chap seeking details for a matching inductor for his 5/8λ vertical on 20m reported “my AA54 RigExpert analyser gives the following reading (SWR 8,2). (R 81,5). (X -158) ” measured looking into a “length of rg58 about 15-20 cm” and asked “is the inductor coil going to be enough or will I need an L match to bring the real resistance to 50 ohms”. Continue reading Matching a 5/8λ ground plane – a single stub tuner example
Some time ago I wrote some articles on so-called Coax Traps, and an example design of an Inverted V dipole for 80 and 40m.
A coax trap (before cross connection).
The whole subject of trapped antennas elicits a lot of online discussion that is often more about semantics than understanding. Continue reading Trapped dipole
Ferrite cored inductors and transformers saturate at relatively low magnetising force.
#61 material example
Lets work through an example of a FT50-61 core with 10t primary at 3.5MHz.
Magnetic saturation is one limit on power handling capacity of such a transformer, and likely the most significant one for very low loss cores (#61 material losses are very low at 3.5MHz).
Let’s calculate the expected magnetising impedance @ 3.5MHz.
Zm=0.966+j144Ω, |Zm|=144Ω. Continue reading RF transformer design with ferrite cores – saturation calcs
A review of transformer design
In a process of designing a transformer, we often start with an approximate low frequency equivalent circuit. “Low frequency” is a relative term, it means at frequencies where each winding current phase is uniform, and the effects of distributed capacitance are insignificant.
Above is a commonly used low frequency equivalent of a transformer. Z1 and Z2 represent leakage impedances (ie the effect of magnetic flux leakage) and winding conductor loss. Zm is the magnetising impedance and represents the self inductance of the primary winding and core losses (hysteresis and eddy current losses). Continue reading RF transformer design with ferrite cores – initial steps