# NPN transistor model for use with differential equations-LC oscillator

#### Sravoff

Hello,

I am trying to get a better understanding of how oscillator circuits (clapp, hartley, colpitts etc) work, so I have been trying to solve the differential equation for a very simple one--the one most the way down the page here: http://www.electronics-tutorials.ws/oscillator/oscillators.html.

I can do the diffy-q for the LC and RLC circuits, but as soon as I add in a bigger circuit my equation breaks down, which I assume is due to my model for the npn transistor:

-Base-Emitter voltage drop is 0.6 v
-Collector Current is Beta times Base Current
-Work around Collector-Emitter voltage since I don't have a relationship for that

I feel like those assumptions work for steady state dc circuits, but not for this time-domain analysis. Is there a more complicated model that allows for the differential equations?

I ideally I would like to solve the circuit with with a small five-ohm speaker attached to V-out so I can guess the frequency of the sound and compare to a guitar tuner... But I'm a little stuck at the transistor.

Thanks for looking!

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#### jim hardy

Gold Member
2018 Award
I dont know what you mean by
I can do the diffy-q for the LC and RLC circuits, but as soon as I add in a bigger circuit my equation breaks down, which I assume is due to my model for the npn transistor:
I'm guessing this is the circuit you're modeling ?

I dont think that one would oscillate successfully. It's too linear , would have to have very precise and stable gain. No bias provision for the transistor, either.

I was taught in vacuum tube days that oscillators don't operate as a linear circuit. They develop enough feedback to turn off during part of every cycle. They spend a substantial fraction of their time in "Cutoff", ie zero current through the tube or transistor..
That lets them find a stable operating point , because the approach to cuttoff is nonlinear and clever circuit designers of the 1920's used that property of tubes to achieve the variable gain necessary for stable oscillation.

So i'm not surprised that linear DE's don't work on an oscillator model. The oscillator isn't linear.

http://www.electronics-tutorials.ws/oscillator/hartley.html
It mentions that trick from the 1920's:
If the amplitude of the oscillations decreases the bias decreases and the gain of the amplifier increases, thus increasing the feedback. In this way the amplitude of the oscillations are kept constant using a process known as Automatic Base Bias.

One big advantage of automatic base bias in a Voltage Controlled Oscillator, is that the oscillator can be made more efficient by providing a Class-B bias or even a Class-C bias condition of the transistor.
Harrumph, more efficient indeed - it makes it self stabilizing, too.
As you well know class B or C means cutoff for part each of cycle.

In my day (1962) it was a grid not a base,
but same principles apply.
Observe the extra capacitor in this Hartley from that link, at node R1 R2 Base:

That capacitor lets the transistor develop enough DC at its base to hold itself cut off for part of every cycle.
In vacuum tube days that's how we'd tell if the oscillator is running - with our VTVM look for lots of negative on the grid.
We called it "Self bias" and when oscillation ceases you lose it. 10 or 20 volts negative was typical at a grid, i think 2 or 3 is more typical for a transistor base.

I'm no expert - just i noticed you said that your differential equations weren't working with the transistor model, so i wondered whether your circuit model expects a garden variety transistor oscillator circuit to stay in its linear range. It shouldn't.

Even the mathematically exquisite Wein bridge oscillator needs a variable gain in its feedback.

hope above is of help and not way off the mark...

old jim

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#### Sravoff

Thanks for the analysis Jim. I think that non-linearity is my problem. So how does the frequency get determined for the LC-tank circuits? Most of the frequencies are in a form that looks like f = 1 / (2pi*sqrt(LC)), where LC are often functions of the capacitors and inductors in the circuit--like the Hartley. I want to be able to understand how those got designed, and how to figure out their frequencies. So far every book I have come across just tosses the circuits out and the formulas, then leaves it at that.

LC, LRC: I was speaking of the idealization of a charged capacitor connected in series to an inductor in a circuit without resistance, so the energy transfers between the electric field in the capacitor and the magnetic field in the inductor indefinitely. Adding a resistor simulates the energy loss in the wires, so the oscillations decay to zero.

Now, what's wanted to do was take that basic theory and figure out how to compensate for that loss with a transistor and a voltage source, which is more or less what these circuits do. Well, also the loss of whatever load is applied to v-out.

Thanks for tossing up those circuit diagrams, I'm on an iPad for now and it doesn't let me do much more than text, let alone numerical analysis on a nonlinear circuit.

Just to make sure I'm clear on that nonlinearity, it is when the base current isn't just zero, but the base voltage is less than the emitter--that's the 2 or 3 volts you're talking about? That would make sense, since the linear model I was using (I think) assumes that voltage is always 0.6 or 0.7, depending on the book.

I guess I should have been born earlier so I could play with these things, instead of all the code-monkeying. We do in school these days.

dlgoff:

Thanks for the link, I'll take a look and see if I can get any of those to work in a circuit.

#### jim hardy

Gold Member
2018 Award
Thanks for tossing up those circuit diagrams,
well they were from your links - thanks go to you for finding such clear ones.

Just to make sure I'm clear on that nonlinearity, it is when the base current isn't just zero, but the base voltage is less than the emitter--that's the 2 or 3 volts you're talking about? That would make sense, since the linear model I was using (I think) assumes that voltage is always 0.6 or 0.7, depending on the book.
Yes. It develops more than enough "self bias" to cut itself off.

The LC determines frequency. Its damping is low so it'll ring after receiving a pulse. That's the frequency of oscillation.

Look at the bottom Hartley:
When feedback voltage drives base positive the transistor conducts
and that unnamed capacitor in series with base charges + on its left side and - on right .
Note right side can't go more positive than 0.6 volt, the E-B junction is an effective clamp.
So you could get several volts across the capacitor.
When the feedback voltage starts back toward negative a bit later on in the AC cycle, the right side of capacitor gets driven negative however far the AC wave takes it.
So collector current ceases to flow until next cycle.
And the base is driven below 0 volts. The average will be negative when it's oscillating because negative peaks are bigger than positive..
That means the tank circuit gets not sinewave excitation but just a pulse of current every cycle.
That pulse is not even a half cycle wide. But due to the tank circuit's low damping it's plenty to sustain the oscillation. The sinewave will have some distortion from the pulsed excitation current, but it works fine for radio circuits.
When you have your oscillator stable it'll find the conduction angle that gives steady oscillation.
Too much loop gain and it'll go in and out of oscillation, we call that "squegging". Wiki has a piece on that.

There's a practical limit on circuit design - most transistors have a reverse bias limit on EB junction of about 5 volts. So we don't see the -20 volts of self bias you get in tube radios.

Anyhow the Wein bridge oscillator is linear and follows your equations beautifully. So it is a favorite among authors. Search on "TI sine wave techniques" and you'll get lots of application notes.
http://www.ti.com/lit/an/snoa665c/snoa665c.pdf
http://www.ti.com/lit/ml/sloa087/sloa087.pdf [Broken]
http://www.ti.com/sc/docs/apps/msp/journal/aug2000/aug_07.pdf

Sinewaves can even be approximated pretty closely by summing square waves, try a search on "Magic sinewaves tinaja" . We had a recent thread on that.

Good luck with your project - i envy you guys whose math is so fresh and so powerful.

old jim

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