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## Introduction

We have Memorial Day off from work and that can only mean one thing – time to work on some circuits for home projects. I have another inductive sensor project for which I want to generate a frequency that depends on the inductance value. The Colpitts oscillator is a good circuit for this type of application.

This is a quick note to document how I analyzed a circuit that I found on a tutorial and verified a statement made on the tutorial about the minimum gain required to startup the oscillation. I also derived an expression for the oscillation frequency.

My Mathcad, Mathcad PDF, and LTSpice source are here.

## Background

There are excellent Colpitts oscillator references available:

## Analysis

### Node Labeling

Figure 2 shows how I labeled the circuit nodes for analysis. It also shows where I opened the feedback loop (i.e. at the output).

### Circuit Analysis

Figure 3 shows my circuit analysis.

### Derivation

In Figure 4, I derive expressions for (1) the oscillation frequency, and (2) the critical amplifier gain needed to ensure the circuit oscillates. The analysis process was straightforward:

- set V
_{IN}= V_{3} - set s = j·ω because I am only interested in the steady-state sinusoidal solution.
- set the imaginary part of the loop equation to zero and solve for the frequency at which the phase shift is zero – a required condition for oscillation. This will give me my oscillation frequency.
- set the real part of the loop equation equal to one and solve for
*R*– this effectively sets the gain condition for oscillation. Since the gain is given by_{2}*R*, this will allow me to solve for the required gain in terms of_{2}/R_{1}*R*. This gain will be a minimum, and we need to exceed that gain slightly to ensure the startup of the oscillator._{1}

I computed a minimum gain of ~2.7 and an oscillation frequency of 11. 7 kHz. My simulation (below) confirmed these results.

When people discuss Colpitts oscillators, they usually state that the oscillation frequency is given by the equation $latex {{f}_{{Standard}}}\left( {{{C}_{1}},{{C}_{2}},L} \right)=\frac{1}{{2\cdot \pi }}\sqrt{{\frac{{{{C}_{1}}+{{C}_{2}}}}{{{{C}_{1}}\cdot {{C}_{2}}\cdot L}}}}&s=-2$. This is not the equation I derive in Figure 4 $latex \displaystyle \left( {{{f}_{{Mine}}}\left( {{{C}_{1}},{{C}_{2}},L} \right)=\frac{1}{{2\cdot \pi }}\sqrt{{\frac{{L+{{R}_{1}}\cdot {{R}_{3}}\cdot \left( {{{C}_{1}}+{{C}_{2}}} \right)}}{{{{R}_{1}}\cdot {{R}_{3}}\cdot {{C}_{1}}\cdot {{C}_{2}}\cdot L}}}}} \right)&s=-3$. This is because most Colpitts discussions are about current source-based, transistor designs versus the voltage source-based, opamp design considered here. This difference in topology makes the circuit equations slightly different. I show in Figure 5 that when $latex {{R}_{1}}\cdot {{R}_{3}}\cdot \left( {{{C}_{1}}+{{C}_{2}}} \right)>>L&s=-2$ (usually the case), the standard Colpitts equation and my equation are approximately equal.

### Simulation

The nodes in my simulation are labeled differently than in my mathematical work above. LTSpice wants to name the nodes in a manner that it finds convenient.

#### LTSpice Circuit

Figure 6 shows my implementation of the circuit using LTSpice. To force the oscillator to start, I set node 2 = 0.3 V at startup with the Initial Condition command (IC) – in real circuits, noise and the power supply ramp on startup drives the oscillator into action.

#### Results

Figure 7 shows my simulation results. The oscillation frequency is slightly lower (11.5 kHz) than the mathematical analysis showed because the analysis ignored the opamp's frequency response – the phase shift of the amplifier contributes to the where the circuit oscillates.

While not shown here, I tried various gains, and I confirmed that I needed an amplifier gain of ~3 to get the oscillator to startup.

I should note that all oscillators have some level of distortion caused by their amplitude-limiting circuitry, which is intrinsically a nonlinear function. For example, Bill Hewlett used a lamp with a nonlinear resistance characteristic to stabilize the old HP200's amplitude level – as elegant a solution as I can imagine.

In this case, amplitude stability is maintained because of nonlinearities in the amplifier, which is NOT the way to achieve a quality sinusoid. However, I am not building a lab grade project here – purely a home project.

## Conclusion

I will be building this circuit this weekend. I am hopeful that the analysis and simulation matches reality. My first step is ALWAYS to get a valid mathematical analysis and working simulation before I build anything.

Pingback: A Lesson Learned from Lab Testing a Colpitts Oscillator | Math Encounters Blog

Pingback: A Lesson Learned from Lab Testing a Colpitts Oscillator | Math Encounters Blog

Could you do this in common collector amplifier ?

Thanks

You can build this oscillator using Common-Base, Common-Collector, and Common-Emitter. The figure below shows the basic configurations.

There is a very good discussion of these configurations on this web page, where I got the figure above.

mark

Why have you considered a 200 ohms resistor between Vin and V1?