[Amps] Resonance in PiNetworks

[RFlabnotes at aol.com]

The following facts about Pi Networks, especially those in typical HF power 
amplifiers, can be readily deduced by analyzing the circuit using 
straightforward circuit theory.

1. The network transfer function contains three poles, as you would expect 
from three non-degenerate reactances (C1,C2, and L). These three poles occur at 
frequencies very close to the resonances you would observe with simple LC 
circuits of (C1,L), (C2,L) and (C1+C2,L).

2. The peak of the power transfer for this network occurs at the 3rd of 
these. Namely, at omega=1/sqrt(Cseries x L)  where Cseries=C1+C2. Since this is the 
peak in the response of the network, it is the dominant resonance.

3. The other two resonances are also present but of little significance.

Conclusion: the tuned PINET is a highly resonant network with resonant 
frequency as I stated above. If it wasn't, you would never see the type of peaking 
that happens as the optimum plate resistance is reached.

As a reality check of this, I analyzed a number of typical PINETs using 
values from both my own designs and from the ubiquitous ARRL Handbook charts. As 
suspected, they all showed a dominant resonance at a frequency within about 2% 
of the simple omega formula (C1+C2,L).

I would like to offer one opinion relevant to all discussions of resonance. 
Namely, there is no universal definition of resonance that holds for all 
systems. The "resonant frequency" of a simple LC or mechanical circuit is 
universally accepted (it is the peak or null of some network parameter, such as the 
impedance) but more complex systems may contain numerous resonances. The only 
generally accepted definition of resonance is an extreme value (max or min) of 
some system quantity. 

In the case of an amplifier with a PINET tank, the most important of such 
quantities is the transmission coefficient of the network, which occurs at the 
dominant resonance frequency.

Eric von Valtier K8LV