# Why does an Oscillator produce sine waves?

By the definition of Oscillator: 'An Electronic circuit that produces a repetitive, oscillating electronic signal, often a sine wave or a square wave'
I'm familiar with RC Phase Shift Oscillator and Wien Bridge Oscillator. Both satisfy the Barkhausen criteria,so should produce an output, But why is the output Sine ?
If someone could elaborate using either of the two oscillators, or maybe some proof/result that give a better idea on the shape of the output wave.

I also know the principle of a square wave generator using op-amp, but that isn't an oscillator and works on the principle of saturation of op-amp. Am I correct in stating that oscillator and sq. wave generator are different ?

• Just because an op-amp saturates (often) to produce a square wave, it's still an oscillator. An oscillator can produce squarewaves but not all square waves are produced by an oscillator directly - ultimately something oscillates whether it is a clock or a DDS chip. – Andy aka May 14 '13 at 16:30
• The opposite question, combined with choice words, is far more common. – rackandboneman Jan 18 '19 at 17:06

## 3 Answers

If the gain of a system at every particular delay is constant, the system will produce oscillations with those periods which have a gain of one. At periods where the gain exceeds one, the strength of the oscillations will grow unless or until something causes the gain falls to below one. If there were one frequency where the gain stabilized at exactly one, and it were less than one at all other frequencies, the circuit would produce a sine wave at the frequency in question. The wave would be a sine wave because any other type of wave would have content at frequencies where the gain is less than one.

Note that in practice, many types of oscillating circuits have gains which vary widely during the course of each "cycle". Such variations make it very difficult to predict analytically the frequency content of their output. Because there is a very fine line between having oscillations die down to nothing, and having oscillations grow without bound, even circuits which are intended to produce sine waves generally end up having a gain which is sometimes greater than 1 and sometimes less than one, though ideally there's a gain control mechanism that will try to settle on the right value.

Incidentally, some circuits use incandescent light bulbs for that purpose, since their resistance varies with temperature. If the power fed to a light bulb is proportional to the strength of an oscillator's signal, and if an increase in resistance will cause a reduction in gain, then the light bulb's temperature will tend to reach an equilibrium where the gain is 1. If the frequency in question is fast enough, the light bulb will only heat up or cool down a little bit during each cycle, allowing reasonably-clean sine waves to be generated.

Addendum

Rather than using the term "constant gain", it may be more helpful to use the term "linear circuit". To borrow an analogy from a magazine I read some years back, comparing "linear circuits" to "non-linear circuits" is like comparing "kangaroo biology" to "non-kangaroo biology"; linear circuits are a particular subcategory of circuits, and non-linear circuits are everything else.

A one-input one-output linear black box is one which takes an input signal and produces an output signal, with the characteristic that if F(x) represents the signal produced by the box when it is fed signal x, and if A and B are two input signals, then F(A+B) will equal F(A)+F(B). There are many kinds of things a linear black box can do to a signal, but all must obey the above criterion. The output produced by a linear black box when given a combination of many different frequency signals will be the sum of the outputs that would be produced for each frequency in isolation.

The behavior of many practical circuits is close to that of a one-input one-output linear black box. Since any wave other than a sine wave is a combination of sine waves at multiple frequencies, for a circuit to oscillate with anything other than a sine wave, there must be multiple frequencies which, if fed in individually, would cause the output to precisely match the input in phase and amplitude. While it is certainly possible to construct such circuits, most practical circuits will only have one frequency where such behavior will occur.

• What do you mean by particular delay in the first line? Could you elaborate more on why the wave is sine ? What do mean by content at frequencies less than one ? – user23564 May 14 '13 at 16:37
• Suppose one has a black box that has an analog input and an analog output; suppose further the waveform it puts out depends in some fashion upon what is put in. If feeding a some waveform into the box will cause it to output an identical wave with precisely the same amplitude and phase, then if one were to connect the input of the box to the output it would keep on producing that same waveform. Many types of circuits that could play the role of the black box will have one possible input frequency where the output will match the input. – supercat May 14 '13 at 16:51
• If the input frequency were higher than the "perfect" frequency, the output would lag the input. If it were lower, the output would lead the input. If a circuit only has one frequency where the output neither lags nor leads the input, that will be the only frequency at which the device can oscillate. – supercat May 14 '13 at 16:56
• The Black box has one frequency where the input matches the output, but this doesn't necessarily the wave is sine, it could be a square of the same frequency ? I'm sorry if I'm missing something. – user23564 May 14 '13 at 16:58
• @user23564: A 1Khz square wave is the sum of many sine waves, with frequencies of 1Khz, 3Khz, 5Khz, 7Khz, etc. (all odd numbers). Most devices that would pass through the 1Khz part perfectly would not pass through the 3Khz, 5Khz, etc. parts perfectly. Note that some constant-gain devices could pass through multiple frequencies. For example, a device whose output would do whatever its input did precisely 1ms before could oscillate at any multiple of 1Khz and could thus output a square wave just fine. Most practical devices, however, don't work like that. – supercat May 14 '13 at 17:08

Why does an oscillator produce sine waves?

Often, sine waves are the desired output, for example in a radio system where we want to only transmit on a specified frequency band. We actually have to work hard to design an oscillator that produces a pure sine wave output.

Generally, by fourier analysis, any repetitive signal can be seen as a sum of harmonically related sine waves. If what we want is a sine wave, then all the harmonic content is considered harmonic distortion, and you need to use careful design and/or output filters to remove it.

I also know the principle of a square wave generator using op-amp, but that isn't an oscillator and works on the principle of saturation of op-amp. Am I correct in stating that oscillator and sq. wave generator are different ?

If it oscillates, I would still consider it an oscillator. Specifically, the op-amp circuit you're probably thinking of is a form of relaxation oscillator. Wikipedia even uses a diagram of an op-amp oscillator as the first illustration on their page on Electronic Oscillators.

Edit In reply to your comment, harmonic content is the content at frequencies that are harmonics (multiples) of the fundamental operating frequency. For example, from a 1 kHz oscillator you will get output at 1 kHz, 2 kHz, 3 kHz, etc.

The oscillator doesn't filter it out, but you can add a filter at the output of your oscillator to try to filter it out.

If you want a sine wave output, you design the oscillator carefully to generate as little harmonic content as possible. But in general, no matter how careful you are, there will be some harmonics in the output. They don't die out over time, they are part of the output as long as the oscillator runs. Supercat's answer explains some of the reasons why harmonic distortion in the output is unavoidable.

Harmonic content is often undesirable, so you try to design your oscillator to not produce it, but there are no perfect components, so you are always stuck with some harmonic distortion in the output.

• What is the Harmonic content ? And how does the oscillator filter it ? The frequencies with gain less than one die out ? And only remaining frequency is the one with which the oscillator oscillates ? – user23564 May 14 '13 at 16:40
• The idea that an oscillator's phase shift should be multiple of 360 degrees only really applies in cases where the circuit's gain and phase shift are at least somewhat stable. A typical relaxation oscillator will often have sections where during the course of each wave a section's gain and phase delay will vary from nearly nothing to nearly infinite. As such, the normal oscillation criteria can't really say much about the circuit beyond the fact that there's a wide range of frequencies where it may or may not oscillate. – supercat May 14 '13 at 17:24
• @supercat, I'm not sure I'm getting your point. The phase shift analysis doesn't apply to a relaxation oscillator because it's not (even close to) a linear system. And yet it still oscillates, so we can still call it an oscillator. – The Photon May 14 '13 at 17:58
• @supercat, did you mean to comment on Andy's answer? – The Photon May 14 '13 at 17:59
• @ThePhoton: I was responding to your comment "if it oscillates, I consider it an oscillator". I think the gain and phase-shift analysis applies if one figures that the range of gain and phase-shift values taken on during the course of each cycle must include values conducive to oscillation. The analysis is generally only helpful if variations are slight (e.g. indicating a circuit where phase shift at some frequency would during the course of a cycle vary from 354 to 359 degrees can't oscillate at that frequency; a circuit where it varies from 358 to 362 might or might not be able to). – supercat May 14 '13 at 18:12

Theoretically, RC and Wien bridge oscillators generate a single output frequency at a determinable precise point in the spectrum. This precise point in the spectrum will yield a feedback signal that is either:

• a 180º phase shift (plus an amplifer inversion = 360º) or,
• a 360º phase shift with a non-inverting amplifier

At other frequencies the phase shift will be different and will not sustain oscillation. A sinewave is a single point in the spectrum therefore a sinewave is produced by these types of oscillator.