Hi, first post to this site! I posted this question on SE earlier (https://electronics.stackexchange.c...vation-of-lc-impedance-transformation-formula) but got no responses. I thought I should take it to a more specialized audience, which is why I'm here.

Basically, I was looking at the original Vackar analysis document (https://web.archive.org/web/20120219062848/n1ekv.org/Oscillators/Vackar_wholepaper.pdf) and I came across this relation on page 4 (Equation 5): V0 = V2 * SQRT(Rd/R2'). Vackar describes this as "a well-known impedance transformation". But, I haven't seen this formula in any impedance transformation references that I have available.

Anyone recognise this equation?

 

Yeah. It is an expression for the voltage across the tuned circuit that looks like a transformer. Transformers transform voltages and currents as the square root of the impedance ratio. This is common in RF and audio circuits.

 

If you like oscillators.......check this out.

https://www.edn.com/design/analog/4368528/Series-LC-tank-VCO-breaks-tuning-range-records

Dick Capples posted this the other day on a different thread. I had never heard of it before.

 

Yeah. It is an expression for the voltage across the tuned circuit that looks like a transformer. Transformers transform voltages and currents as the square root of the impedance ratio. This is common in RF and audio circuits.

I was sort of coming to that conclusion when I wrote the original SE post. What threw me was that Vackar never mentioned that his 4 port oscillator section was tapped. Although, thinking about it later, it was implied by the fact that he distinguished the input impedance of the tank from the dynamic resistance at resonance.
A rather nice derivation of the tapped capacitor arrangement is given here: (http://www.ece.ucsb.edu/~long/ece145b/Resonators.pdf). In the end, Rd = R2'(C1 + C2/C1)^2 and if you consider V0 as the voltage across C1 + C2 and V2 as the voltage across C1 and rearrange, you get your answer.

Now to muddle my way through the rest of it! Thankyou!

 

If you like oscillators.......check this out.

https://www.edn.com/design/analog/4368528/Series-LC-tank-VCO-breaks-tuning-range-records

Dick Capples posted this the other day on a different thread. I had never heard of it before.

Very interesting! I'm not sure "like" is the right word! I'm actually trying to build a VFO signal source to test tuned filters. The VFO will have an upper frequency of 50Mhz. My first attempts (Hartleys mainly) were OK, but the amplitude stability was dreadful. You'd turn the variable capacitor and the output would fall off badly.

I've got about 3 or 4 different arrangements queued up to try and I'll add that one to that list. Hopefully, one of those will come good (in which case I may get to like them after all!)

 

Very interesting! I'm not sure "like" is the right word! I'm actually trying to build a VFO signal source to test tuned filters. The VFO will have an upper frequency of 50Mhz. My first attempts (Hartleys mainly) were OK, but the amplitude stability was dreadful. You'd turn the variable capacitor and the output would fall off badly.

I've got about 3 or 4 different arrangements queued up to try and I'll add that one to that list. Hopefully, one of those will come good (in which case I may get to like them after all!)

I was only trying to answer your question: "Anyone recognise [sic] this equation?", in a heuristic sense, by giving you some context of where I recognized similar equations. I am not claiming a deep understanding of the specific case, or it's particular derivation.