For the SM2000 I need a 146MHz Band Pass Filter (BPF). This lead me to the Double Tuned Filter (DTF) or Double Tuned Circuit (DTC) – two air cored coils coupled to each other, and resonated with trimmer capacitors. To get a narrow pass band, the Q of the resonators must be kept high, which means an impedance of a few 1000 ohms at resonance. So I connect the low impedance 50 ohm input and output by tapping the coils at half a turn.
Here is the basic schematic (source Vasily Ivanenko on Twitter):
For 146MHz the inductors are about 150nH, and capacitors about 8pF. For 435MHz the inductors are about 50nH, and capacitor 3pF. The “hot end” of the inductors where the trimmer cap connects is high impedance at resonance, several 1000 ohms.
These filters are sometimes called Helical Filters – however this is a little confusing to me. My understanding is that Helical filters – although similar in physical construction – operate on transmission line principles and have the “hot” end of the inductors left open – no capacitor to ground.
Here is a photo of a 146MHz DTC filter, and it’s frequency response:
I understand how the band pass response is formed, but have been mystified by the notches in the response. In particular, what causes the low frequency notch just under the centre frequency? Here is a similar DTC I built for 435MHz and it’s frequency response, also with the mystery notches:
After a few days of messing about with Spice, Octave simulations, RF books, and bugging my RF brains trust (thanks Yung, Neil, Jeff), I accidentally stumbled across the reason. A small capacitor (around 1pF) between the hot end of the inductors creates the low frequency notch. Physically, this is parasitic capacitance coupling across the air gap between the coils.
Here is a LTspice simulation of the UHF version of the circuit. Note how the tapped inductors are modelled by a small L in series with the main inductance. The “K” directive models the coupling. Air cored transformers have a low coupling coefficient, I guessed at values of 0.02 to 0.1. You can see the notch just before resonance caused by the 1pF parasitic coupling between the two inductors. Without this capacitor in the model, the notch goes away.
The tapped inductor is used for an impedance match. An equivalent circuit is simply driving and loading the circuit with a high impedance, say 1500 ohms:
After some head scratching I found this useful model for transformers. It’s valid for low-k transformers where the primary and secondary inductance is the same (Ref):
Note it doesn’t model DC isolation but that’s OK for this circuit. While I don’t understand the derivation of this model, it does makes intuitive sense. A loosely coupled air core transformer can be modeled as a high (inductive) impedance between the primary and secondary. We still get reasonable power transfer (a few dB insertion loss) as the impedance of the primary and secondary is also high at resonance.
Using the model in Fig 55 with k = 0.1, the top L in the PI arrangement is about 10L1 or 500nH. I also removed the 3pF capacitors in an attempt to isolate just the components responsible for the notch. So we get:
Finally! Now I can understand how the notch is created. We have 1pF in parallel with 500nH, which forms a parallel resonant circuit at 225MHz. Parallel resonance circuits have very high impedance at resonance, which blocks the signal, causing the notch.
It took me a while to spot the parallel resonance. I had assumed a series resonance shorting the signal to ground, and wasted a lot of time looking for that. Parasitic inductance in the capacitors is often the reason for notches above resonance.
This suggests we can position the notch by adjusting the capacitance between the coils, either by spacing or adding a real capacitor. Positioning the notch could be useful, e.g. deeply attenuating energy at an image frequency before a mixer.