Varactor Linearization techniques

[djsb]

In the attached article author John Linsley hood states

Happily, a technique exists for
linearising the voltage/frequency
characteristics of a varicap tuned
circuit (Fig. 4b). The curvature in
the way in which the collector
current of a silicon transistor varies
as the base voltage is increased is
balanced against the varicap diode
voltage/frequency non-linearities,
and if the correct value of R is used
for the transistor and varicap diode
chosen the overall linearity can be
very good.

This circuit can be seen on page 2 of the attached article (Fig 4b). How well does this circuit work in practice, and can it be improved in any way? Are there any references anywhere to the theory behind how this works?Any thoughts? Thanks.


PS I have not yet tested the VCO circuit schematic in KiCad V8.00 as I have only JUST downloaded KiCad V8 today and ngspice simulation is something I am new to.

[CaptDon]

I have seen and currently use a 10 bit A to D / D to A chain with a lookup table in between. The A to D drives EEProm address lines directly and the lookup or truth table output drive the D to A chain which drives the varactor. Super Linear frequency sweep indeed. The drive for the A to D input is simply a 10 turn linear potentiometer. 8 bits would have been FAR easier but didn't give me the fine resolution I wanted. 10 bits resolves better than the markings on the ten turn dial! Using EEproms allowed me to make on the fly table adjustments until I got it really dialed in. The table can easily be adjusted for different varactors!

[djsb]

Thanks for your reply. It will take me a while to fully understand your setup, and I'll have to do some research into this. Might try building my prototype first and do some basic tests first.

[RoGeorge]

I have not yet tested the VCO circuit schematic

Didn't play much with it, but it works.  The amplitude varies a lot with frequency, but oscillators are tricky to simulate, and the results depend a lot with how good the models are.  Not all aspects are included in models, might work better in practice.

Didn't make any simulation tests regarding the frequency linearity.

If you are curious, install LTspice https://www.analog.com/en/resources/design-tools-and-calculators/ltspice-simulator.html (the Windows version works in Linux too, with WineHQ), unzip the attached VCO, then drag and drop the .asc file into LTspice (.asc is the schematic, the rest of the files are some included libraries, those transistors and diodes are now obsolete and not available in the default LTspice libraries).  Then press the 'running man' icon in the LTspice icon bar, to run the simulation.  After it finishes, click inside the schematic windowm and you can click any node or components to see the waveforms.  Except the graphical interface, the rest of SPICE background is all text, same as in ngspice, pspice, etc.

The complete circuit and transient simulation might not be very useful to study how it works.  The simplified schematic from the PDF would give more understanding of how the linearization of F with V varicap is done:



For example, set different DC points, then make an AC simulation (not transient), and look at the frequency response.  The oscillations will appear at the frequency where the peak in the AC response is.

Or even better, don't simulate at all.  Solder everything on a copper clad PCB, Manhattan style, and measure the real thing. 

[djsb]

Thanks. I'll have a play with this next weekend. It would also be a good excuse to try Manhattan style construction, I've got most of the parts I need. Bye for now.

[David Hess]

That will certainly work to linearize the VCO using the exponential transconductance of the bipolar transistor, but a considerable temperature coefficient is added, which does not matter in a PLL application.

[RoGeorge]

How well does this circuit work in practice, and can it be improved in any way? Are there any references anywhere to the theory behind how this works?Any thoughts?

Might be some ways to improve it, but I didn't built it in practice.  About the theory behind that:

- the goal is to get a VCO (Voltage Controlled Oscillator) with a very linear frequency versus control voltage characteristic.  F has to vary linearly with V because the VCO is used in a PLL (Phase Locked Loop) that will demodulate the FM signal.  The PLL loop will try to keep the VCO in sync, which will create a control voltage (for the vco) that copies the audio signal from the transmitter.  The more linear F with V in our VCO, the less deviations from teh original audio signal carried by FM (or else said, the less audio distortions), which is the final goal:  Hi-Fi (High Fidelity) audio.

- a varactor (or varicap diode) is a diode junction reverse polarized.  The more reverse voltage is applied, the more the gap inside the diode opens, the less capacitance it has.  This is from the BB105B datasheet (the diode used in the FM tuner):



Note how the X axis (reverse voltage on the varicap) is logarithmic (logarithm is the inverse of an exponential), the Y axis (capacitance) is linear, and the slope is almost linear.  That tells that capacitance decreases exponentially with the applied reverse voltage.

- a transistor is the BE diode (Base Emitter) with some additions, the BC junction (Base Collector), very roughly speaking, which means the BJT (Bipolar Junction Transistor) inherits the exponential behavior of a typical diode.



In this case, note how Ic (the collector current) grows exponentially with the Vbe (the base emitter voltage).  The Ic is turned by the Rc into the reverse voltage that will control the varactor's capacity.  The slide show picture is from https://inst.eecs.berkeley.edu/~ee105/sp08/lectures/ in lecture 4 slide 11.  The previous lectures might help, too, though those are only very brief slide-show presentation.  That class follows the textbook 'Fundamentals of Microelectronics' by Behzad Razavi, Wiley Press, January 2008.  https://inst.eecs.berkeley.edu/~ee105/sp08/


The frequency linearization idea from the initial VCO is to combine the two behaviors (exponential increase in the BJT current + exponential decrease of the varicap capacitance) in such a way thet the two exponential mostly compensate each other, and the C varicap will become linear with the Vbe applied on the transistor's BE.

There are more effects into play than only those 2 aspects (e.g. the transistor's internal capacitances), but I guess the exponential increase in Ic with Vbe + the exponential decrease of Cv with the VR (reversed voltage on the varactor) have the biggest contribution for F with V linearization.


However, the frequency of an LC tank does not vary linearly with C, so I guess the F range of interest is small enough (10.7MHz +/-75kHz), such that the quadratic variation of F with C doesn't matter.  Or maybe it was compensated already by some other mechanism I am missing. 

If not yet compensated, maybe inserting a FET (Field Effect Transistor) might help, because a FET has a quadratic behavior (similar with how a diode, or a BJT, has an exponential behavior).

[David Hess]

However, the frequency of an LC tank does not vary linearly with C, so I guess the F range of interest is small enough (10.7MHz +/-75kHz), such that the quadratic variation of F with C doesn't matter.  Or maybe it was compensated already by some other mechanism I am missing. 

The frequency change is small enough that the non-linearity is greatly reduced, but of course first order compensation from the transistor will still help.