How do Sinusoidal output comes out in the Wein-Bridge Oscillator

[darkxponent]

This question was asked to me in a VIVA.

What examiner asked.

How do Sinusoidal output comes out in the Wein-Bridge Oscillator.

...

I tried to solve the problem using the control system. That is, by deriving the transfer function of the Wein-Bridge Oscillator, but first i got confused on where to take the input as there is no input and oscillators work due to noise input(this is what written in the books). Then i assumed the input to be the negative Op-Amp terminal, this time the confusion was on what function the input Voltage should be?

I have studied two Sinusoidal Oscillators, another one is phase-shift oscillators, and I don't know the 'How' of Phase-Shift oscillators also. The book i follow is Sedra-Smith and doesn't say anything about the 'How' behind the generation of Sinosoidal output. I have one more book on Op-Amp Circuits and nothing is written about it this in that book also.

 

[NascentOxygen]

The input to the passive network is the output of the op-amp. As far as the transfer function goes, you need the combination to give 2.Pi radians of phase shift at a gain of just over unity, at some frequency, for oscillations to be self-sustaining.

I suggest that you include a scan of the particular circuit you have in mind, if you have further questions.

 

[darkxponent]

I suggest that you include a scan of the particular circuit you have in mind, if you have further questions.

This is the Circuit, the wein-bridge oscillator.

The input to the passive network is the output of the op-amp. As far as the transfer function goes, you need the combination to give 2.Pi radians of phase shift at a gain of just over unity, at some frequency, for oscillations to be self-sustaining.

I know the two conditions that you just told. This is the only two things that we have studied about oscillators. Which comes from A_v*β = 1

1. β is real.
2. ∠ A_v*β = 2n∏

I derived the Transfer function taking non-inverting input as the input, but the question is how does the Sine wave comes out. Without knowing input i can't predict the output.

 

[NascentOxygen]

The potential divider at the - input sets the gain of the amplifier stage (the OP-AMP), at a value that just makes up for the attenuation of the passive Wein bridge, restoring loop gain to 1. Input to the bridge comes from the output of the OP-AMP. Output of the bridge is seen at the - input of the OP-AMP.

So long as the loop gain is not much over 1 where phase shift is 2 Pi, and this condition is not met at another frequency, the system noise will develop into a self-sustaining sinusoidal oscillation within a short period of time. The more loop gain you allow, the wider the band over which feedback can give sustained oscillation, and the more frequency components you introduce into your signal, meaning it starts to depart from a pure sinusoid.

 

[darkxponent]

The potential divider at the - input sets the gain of the amplifier stage (the OP-AMP), at a value that just makes up for the attenuation of the passive Wein bridge, restoring loop gain to 1. Input to the bridge comes from the output of the OP-AMP. Output of the bridge is seen at the - input of the OP-AMP.

That means the principal of operation is same as that of a square-wave generator and monostable multivibrator. I understood this part.

So long as the loop gain is not much over 1 where phase shift is 2 Pi, and this condition is not met at another frequency, the system noise will develop into a self-sustaining sinusoidal oscillation within a short period of time.

Why sinusoid?. I mean when we study the square wave generator(whose working is similar to this), we derive how output comes out to be the square wave.

The more loop gain you allow, the wider the band over which feedback can give sustained oscillation, and the more frequency components you introduce into your signal, meaning it starts to depart from a pure sinusoid.

Again same question, why. Isn't there a General output Voltage as function of time and Circuit components as in the case of square wave generator?

 

[NascentOxygen]

It's a linear amplifier/network, with a loop gain great enough at only one frequency to stop oscillations dying out. A squarewave can't leap out of nowhere in a linear system. Squarewaves originate from abrupt non-linearities, there are none here.

 

[darkxponent]

It's a linear amplifier/network, with a loop gain great enough at only one frequency to stop oscillations dying out. A squarewave can't leap out of nowhere in a linear system. Squarewaves originate from abrupt non-linearities, there are none here.

So you mean there is no proof for the Sinosiodal output. Is is just an observed fact?

 

[Jony130]

Output signal is a sinewave because only for this one frequency wein-bridge and the amplifier meets the conditions necessary to start oscillations.

 

[darkxponent]

Output signal is a sinewave because only for this one frequency wein-bridge and the amplifier meets the conditions necessary to start oscillations.

We assume that the output is Sine wave that is wave we put s=jω. Putting s = jω is only true for sinusoidal transfer functions. My question is why do we assume the output to be sinusoid?

 

[Jony130]

"Wein network" work as a highly selective bandpass filter. So even if you put square wave ta the input the output voltage will be sinusoidal. So the op amp input will see a sine wave at his non-inverting input. And next the amp will amplifier this sine wave 3x times. So the output also will be sinusoidal. Because just for one single frequency F = 1/(2 * pi RC) circuit meets conditions necessary to start oscillations.

On the other hand in multivibrator we have a all pass filter network so for all frequancy circuit meets the conditions necessary to start oscillations. And this is why we have a square wave output.

 

[sophiecentaur]

We assume that the output is Sine wave that is wave we put s=jω. Putting s = jω is only true for sinusoidal transfer functions. My question is why do we assume the output to be sinusoid?

For some reason you keep ignoring the explanation you are being given. The feedback is only 'right' at one frequency so no other frequencies will be sustained - that makes the output a single frequency with no harmonics (i.e. a sine wave).
PS In general, it's pretty fruitless to question why the (well established) analysis of a common system happens to be done in a particular way. Science and Maths is full of apparently arbitrary approaches to solving problems - which just happen to work. Very annoying until you know the particular crafty dodge. Why do you divide both sides of an equation by the coefficient of x (etc. etc.)? Because you know it will work. Why do you try a sinusoidal solution to this sort of problem? Because you know it works.

 

[NascentOxygen]

We assume that the output is Sine wave that is wave we put s=jω. Putting s = jω is only true for sinusoidal transfer functions. My question is why do we assume the output to be sinusoid?

It's a second order linear system, and the natural response of every second order system is a sinusoid, just as you see in a child's swing. Give it a jolt and it adopts SHM; give this filter/amplifier system a voltage jolt at switch-on, and it responds with SHM. The sinusoid doesn't decay away because of accurately setting the loop gain to just sustain the response.

 

[NascentOxygen]

That means the principal of operation is same as that of a square-wave generator and monostable multivibrator. I understood this part.

The principal of operation of a square-wave generator, and of a sinusoidal oscillator, are totally different. Granted, superficially their schematics may sometimes bear a resemblance, but one is highly non-linear, and the other very linear. The two are horses of a different colour!

 

[darkxponent]

It's a second order linear system, and the natural response of every second order system is a sinusoid, just as you see in a child's swing. Give it a jolt and it adopts SHM; give this filter/amplifier system a voltage jolt at switch-on, and it responds with SHM. The sinusoid doesn't decay away because of accurately setting the loop gain to just sustain the response.

A second order linear system. If you could tell me where did you get this from.

 

[darkxponent]

For some reason you keep ignoring the explanation you are being given. The feedback is only 'right' at one frequency so no other frequencies will be sustained - that makes the output a single frequency with no harmonics (i.e. a sine wave).

I am not ignoring that explanation. Okay the feedback is right at only one frequency-agree. Then you say feedback is right at only one function(sine wave). How do match these two things-frequency and function. I mean the output could have been a different function with the same frequency!

 

[sophiecentaur]

I am not ignoring that explanation. Okay the feedback is right at only one frequency-agree. Then you say feedback is right at only one function(sine wave). How do match these two things-frequency and function. I mean the output could have been a different function with the same frequency!

Yes, but the phase and amplitude characteristic of the feedback in this oscillator is such that only one component of your hypothetical waveform would satisfy the requirement for oscillation. There is only one function that has one frequency component and that is a sinewave. The amplifying device in this sinewave oscillator needs to be linear so you can use superposition and treat signals of all possible frequencies independently. When you have a non-linear amplifying device, other waveforms may result - but the same thing could happen if you followed the nicely sinusoidal Wien Bridge with a non linear amplifier.
In practice, of course, there will be a certain amount of harmonic content, even in the Wien Bridge output.

 

[NascentOxygen]

A second order linear system. If you could tell me where did you get this from.

It's from mathematics. Differential equations, etc., it's all a part of physics. The current through a capacitor is determined by the derivative of the voltage across it, and so on.

 

[darkxponent]

It's from mathematics. Differential equations, etc., it's all a part of physics. The current through a capacitor is determined by the derivative of the voltage across it, and so on.

I am pretty okay at maths and physics. The EE curriculum at my university is quite mathematical. For me understanding circuits with mathematical equation is the easiest method. I mean this is way we are taught circuits at our University. So if you could tell me those Differential equations describing wein-bridge oscillator, i would be very keen to solve those myself.

 

[sophiecentaur]

I googled "wein bridge oscillator analysis". Did you try that? You get loads of hits with treatments at various levels of difficulty. I suggest you look at them. It's a far more efficient way of learning than just asking questions and getting off - the - cuff answers, many of which will come straight off the net as people refresh their memories of stuff they knew quite well in the past.

 

[darkxponent]

Yes, but the phase and amplitude characteristic of the feedback in this oscillator is such that only one component of your hypothetical waveform would satisfy the requirement for oscillation.

Why are the phase and amplitude characteristics satisfying only sine function? If you could go into some mathematical proof, that would be easier for me to understand.

There is only one function that has one frequency component and that is a sinewave. The amplifying device in this sinewave oscillator needs to be linear so you can use superposition and treat signals of all possible frequencies independently.

And finally we assume it to be linear. I don't understand why this are assumptions is made. Not many Op-Amp circuits are linear, including the square generator, comparator, monostable multivibrator. I will definitely agree that this circuit is linear just because you and others saying this. I just want to know how do we came to conclusion that this is linear.

 

[darkxponent]

I googled "wein bridge oscillator analysis". Did you try that? You get loads of hits with treatments at various levels of difficulty. I suggest you look at them. It's a far more efficient way of learning than just asking questions and getting off - the - cuff answers, many of which will come straight off the net as people refresh their memories of stuff they knew quite well in the past.

I did googled "wein bridge oscillator" before posting this thread. Didn't find much. Didn't googled "wein bridge oscillator analysis". I am going to do that now!

 

[NascentOxygen]

And finally we assume it to be linear. I don't understand why this are assumptions is made.

We don't assume anything. We design the circuit to be linear throughout. Resistors are linear, capacitors are linear, and the OP-AMP-based amplifier is given a linear feedback network so it presents as a linear amplifier. The only non-linearity possible occurs if the OP-AMP output hits a supply rail, and in a practical circuit steps are taken at the design stage to ensure this doesn't happen (not shown in the simplified circuit you attached).

 

[sophiecentaur]

Why are the phase and amplitude characteristics satisfying only sine function? If you could go into some mathematical proof, that would be easier for me to understand.




And finally we assume it to be linear. I don't understand why this are assumptions is made. Not many Op-Amp circuits are linear, including the square generator, comparator, monostable multivibrator. I will definitely agree that this circuit is linear just because you and others saying this. I just want to know how do we came to conclusion that this is linear.

If you read the articles you will find, you will notice they deal with the problem of linearity and specify that the amplifier must be operated well below its maximum output power. You will find this sort of analysis throughout electronics. It's a general Engineering approach to build in as many ideal components as possible to the model (and the realisation of it). It's the only way, really.

 

[f95toli]

Why are the phase and amplitude characteristics satisfying only sine function? If you could go into some mathematical proof, that would be easier for me to understand.

Maybe you already know this, and I missunderstood this comment; but the point is that if you have a single frequency you have -per definition- a sinewave.
Every other waveform will -again by definition- contain multiple frequencies. This is no such thing as a "single frequency square wave".
You can of course have many other periodic functions, but if you look at their frequency distribution you will find that they always contain other frequency components in addition to the fundamental.
.
This is just a consequence of the math (Fourier analysis).

 

[darkxponent]

If you read the articles you will find, you will notice they deal with the problem of linearity and specify that the amplifier must be operated well below its maximum output power. You will find this sort of analysis throughout electronics. It's a general Engineering approach to build in as many ideal components as possible to the model (and the realisation of it). It's the only way, really.

We don't assume anything. We design the circuit to be linear throughout. Resistors are linear, capacitors are linear, and the OP-AMP-based amplifier is given a linear feedback network so it presents as a linear amplifier. The only non-linearity possible occurs if the OP-AMP output hits a supply rail, and in a practical circuit steps are taken at the design stage to ensure this doesn't happen (not shown in the simplified circuit you attached).

I am very much familierr with the notion of taking resistors, capacitors to be linear, we do that in analysis of almost all the circuits. and i know these assumptions are necessary in circuit analysis. I asked for the non=linearity because we don't take op-amps as linear element in all circuits, specially comparators and wave-generators are mostly non-linear ones(among those i have studied). However most of the op-amp circuits i studied are linear.

Maybe you already know this, and I missunderstood this comment; but the point is that if you have a single frequency you have -per definition- a sinewave.
Every other waveform will -again by definition- contain multiple frequencies. This is no such thing as a "single frequency square wave".
You can of course have many other periodic functions, but if you look at their frequency distribution you will find that they always contain other frequency components in addition to the fundamental.
.
This is just a consequence of the math (Fourier analysis).

Thanks. Did know this but couldn't connect Fourier series to this circuit till now. Things are becoming clear now for me.

 

[skeptic2]

Isn't this the same circuit that was Hewlett and Packard's first product, a sine wave oscillator? The trick is to have barely enough feedback to oscillate. They accomplished that by using a variable resistance, a tungsten bulb, for R3. If the gain is too high and produces a squarewave, the bulb shines brighter and increases its resistance which in turn reduces the gain.

 

[davenn]

Isn't this the same circuit that was Hewlett and Packard's first product, a sine wave oscillator? The trick is to have barely enough feedback to oscillate. They accomplished that by using a variable resistance, a tungsten bulb, for R3. If the gain is too high and produces a squarewave, the bulb shines brighter and increases its resistance which in turn reduces the gain.

ahhh indeed :)

I went through this a couple of years ago when building a Wein bridge oscillator. Watching the output on a scope was very informative, the smallest of adjustments had it going from a nice sine wave to a square wave

Dave

 

[skeptic2]

Note that the reactance of the capacitor is equal to the resistance or 10K. The combination of R1 and C1 in series has an impedance of ~14K ohms. The parallel combination of R2 and C2 has an impedance of ~7K causing 1/2 the output voltage to be fedback to the non-inverting input. To compensate for the 0.5 gain, R3 and R4 have to produce a gain of barely over 2 in order for the circuit to oscillate. The more the overall gain goes over 1, the more of a squarewave will be produced.

 

[jim hardy]

Now you're getting there.

Mother Nature loves sines - they're the only function I know of whose shape does not change when you differentiate or integrate them. So pendulums and other harmonic systems produce them.

So I think of it this way - a sinewave appears for the same reason it does in a spring/mass system - the feedback system is linear(until amplitude grows to point it hits limits as pointed out above) and contains a restoring force that's proportional to rate-of-change of perturbing force.
a quick search took me to this page which shows the principle:(I'll not attempt the derivation right now)
http://www.calpoly.edu/~rbrown/Oscillations.pdf

Any help ?

 

[sophiecentaur]

Isn't this the same circuit that was Hewlett and Packard's first product, a sine wave oscillator? The trick is to have barely enough feedback to oscillate. They accomplished that by using a variable resistance, a tungsten bulb, for R3. If the gain is too high and produces a squarewave, the bulb shines brighter and increases its resistance which in turn reduces the gain.

I seem to remember circuits that incorporated a thermistor, to achieve stability in a similar way. Something of the sort is essential is you want to make the oscillator frequency sweepable over a wide range for lab work.

 

[sophiecentaur]

Now you're getting there.

Mother Nature loves sines - they're the only function I know of whose shape does not change when you differentiate or integrate them. So pendulums and other harmonic systems produce them.

So I think of it this way - a sinewave appears for the same reason it does in a spring/mass system - the feedback system is linear(until amplitude grows to point it hits limits as pointed out above) and contains a restoring force that's proportional to rate-of-change of perturbing force.
a quick search took me to this page which shows the principle:(I'll not attempt the derivation right now)
http://www.calpoly.edu/~rbrown/Oscillations.pdf

Any help ?

Asking what's so 'special' about a sine wave is a bit like asking what's so special about ∏ and e. They just creep out of the analysis. If one is not careful, one could start asking why the mathematical world seems so mimic the physical world so well. You'd soon be into the dimensions of the Great Pyramid and the lost gold of the Incas etc. etc. Move over, Dan Brown.

 

[darkxponent]

I seem to remember circuits that incorporated a thermistor, to achieve stability in a similar way. Something of the sort is essential is you want to make the oscillator frequency sweepable over a wide range for lab work.

When i did this experiment in lab i got all sorts of waveforms until tuning the resistance finally gives a sinusoid.

Now you're getting there.

Mother Nature loves sines - they're the only function I know of whose shape does not change when you differentiate or integrate them. So pendulums and other harmonic systems produce them.

So I think of it this way - a sinewave appears for the same reason it does in a spring/mass system - the feedback system is linear(until amplitude grows to point it hits limits as pointed out above) and contains a restoring force that's proportional to rate-of-change of perturbing force.
a quick search took me to this page which shows the principle:(I'll not attempt the derivation right now)
http://www.calpoly.edu/~rbrown/Oscillations.pdf

Any help ?

I am not going to ask for any derivation for mechanical osscillations as i am very much familier with almost all types of mechanical oscillations and i know how to derive those also. The question i would ask is how does these mechanical oscillations relate with Wein-Bridge oscillator?(exept for the fact that the output is sine)

 

[NascentOxygen]

how does these mechanical oscillations relate with Wein-Bridge oscillator?(exept for the fact that the output is sine)

They are characterised by identical second-order differential equations. With each, when given a step or pulse input, their reaction is seen to be a sinusoidal response (superimposed on a decaying characteristic transient).

 

[NascentOxygen]

Isn't this the same circuit that was Hewlett and Packard's first product, a sine wave oscillator? The trick is to have barely enough feedback to oscillate. They accomplished that by using a variable resistance, a tungsten bulb, for R3. If the gain is too high and produces a squarewave, the bulb shines brighter and increases its resistance which in turn reduces the gain.

The Wein Bridge has a reputation for being capable of delivering a high purity sinusoid. But my experiments with a pea bulb revealed, from memory, that its best sensitivity occurred below where it had begun to glow.

 

[sophiecentaur]

I am not going to ask for any derivation for mechanical osscillations as i am very much familier with almost all types of mechanical oscillations and i know how to derive those also. The question i would ask is how does these mechanical oscillations relate with Wein-Bridge oscillator?(exept for the fact that the output is sine)

If you are familiar (at a sufficiently high level) with the maths of mechanical oscillators then you only need to replace Masses with Ls and Spring constants with Cs (in principle) etc. etc. to translate your knowledge into electrical oscillators. Oscillators and feedback systems of all types share exactly the same analyses. Could there ever be any surprise that the sinewave appears in both contexts?

 

[davenn]

I seem to remember circuits that incorporated a thermistor, to achieve stability in a similar way. Something of the sort is essential is you want to make the oscillator frequency sweepable over a wide range for lab work.

yeah that is an alternative to a bulb.
I used a thermistor in my construction

Dave

 

[Averagesupernova]

I don't think a wein bridge oscillators bulbs are EVER supposed to glow.

 

[skeptic2]

But my experiments with a pea bulb revealed, from memory, that its best sensitivity occurred below where it had begun to glow.

It seems to me that the operating point could be altered by adjusting the value of R4.

http://hypertextbook.com/facts/2004/DeannaStewart.shtml

 

[darkxponent]

If you are familiar (at a sufficiently high level) with the maths of mechanical oscillators then you only need to replace Masses with Ls and Spring constants with Cs (in principle) etc. etc. to translate your knowledge into electrical oscillators. Oscillators and feedback systems of all types share exactly the same analyses. Could there ever be any surprise that the sinewave appears in both contexts?

Glad you put forward this point. I have studied Control Systems, where we replace mechanical systems by a circuit in Force-Voltage or Current-Voltage analogy. Now let's see this problem(the Wein-Bridge Oscillator) in a different way. The question is to draw the equivalent mechanical system for the wein-Bridge oscillator. Can this be done.

I haven't studied the equivalent mechanical thing for an op-amp. Moreover the op-amp is made up of transistors. Do transistors have a mechanical equivalent?

 

[technician]

Now you're getting there.

Mother Nature loves sines - they're the only function I know of whose shape does not change when you differentiate or integrate them. So pendulums and other harmonic systems produce them.

So I think of it this way - a sinewave appears for the same reason it does in a spring/mass system - the feedback system is linear(until amplitude grows to point it hits limits as pointed out above) and contains a restoring force that's proportional to rate-of-change of perturbing force.
a quick search took me to this page which shows the principle:(I'll not attempt the derivation right now)
http://www.calpoly.edu/~rbrown/Oscillations.pdf

Any help ?

Jim...you see where others see dark ! you are spot on...mother nature is simple

 

[technician]

I don't think a wein bridge oscillators bulbs are EVER supposed to glow.

whether they glow or not is irrelevant...they are non-linear

 

[Averagesupernova]

But there are parts of that curve that are more linear than others.

 

[technician]

But there are parts of that curve that are more linear than others.

The word is non- linear

 

[carlgrace]

Glad you put forward this point. I have studied Control Systems, where we replace mechanical systems by a circuit in Force-Voltage or Current-Voltage analogy. Now let's see this problem(the Wein-Bridge Oscillator) in a different way. The question is to draw the equivalent mechanical system for the wein-Bridge oscillator. Can this be done.

I haven't studied the equivalent mechanical thing for an op-amp. Moreover the op-amp is made up of transistors. Do transistors have a mechanical equivalent?

There is no mechanical equivalent for an op amp directly. You need to think in terms of macromodels. So think of it as a device with infinite gain and you go a long, long way to understanding its behavior.

You know, these types of circuits are called analog circuits because they are electrical analogs to mechanical systems, since both mechanical and electrical systems can typically can be described by a set of differential equations.

So yes, the mechanical equiv or the Wein-Bridge oscillator can be drawn. The mathematics are identical.

 

[sophiecentaur]

There are plenty of examples of 'mechanical amplifiers' , of course. They can be incorporated into mechanical oscillators to maintain oscillations - such as the escapement system in a clock (essentially a non-linear amplifier), which meters a small amount of energy into the pendulum / hairspring at the right phase, to maintain the oscillation. the Q of the pendulum is very high and acts as a filter to 'smooth' the impulses of energy into the sinusoidal motion of the oscillator. There are fluid and also magnetic amplifiers which are fairly linear and which can give a Wien equivalent oscillator function.

 

[darkxponent]

There is no mechanical equivalent for an op amp directly. You need to think in terms of macromodels. So think of it as a device with infinite gain and you go a long, long way to understanding its behavior.

You know, these types of circuits are called analog circuits because they are electrical analogs to mechanical systems, since both mechanical and electrical systems can typically can be described by a set of differential equations.

So yes, the mechanical equiv or the Wein-Bridge oscillator can be drawn. The mathematics are identical.

suppose i want to draw the mechanical equivalent of an op-amp in Force-Voltage or Torque-Voltage analogy. How do i start?

There are plenty of examples of 'mechanical amplifiers' , of course. They can be incorporated into mechanical oscillators to maintain oscillations - such as the escapement system in a clock (essentially a non-linear amplifier), which meters a small amount of energy into the pendulum / hairspring at the right phase, to maintain the oscillation. the Q of the pendulum is very high and acts as a filter to 'smooth' the impulses of energy into the sinusoidal motion of the oscillator. There are fluid and also magnetic amplifiers which are fairly linear and which can give a Wien equivalent oscillator function.

I want to draw the mechanical equivalent of Wein-bridge oscillator with springs, masses etc. Can this be done?

 

[carlgrace]

suppose i want to draw the mechanical equivalent of an op-amp in Force-Voltage or Torque-Voltage analogy. How do i start?

The opamp is there to make the circuit more ideal, that is to approximate a mechanical system more closely. You wouldn't need the mechanical equivalent of an opamp in your drawing.

I think this is what you're looking for:

http://lpsa.swarthmore.edu/Analogs/ElectricalMechanicalAnalogs.html

Have fun! :)

 

[darkxponent]

The opamp is there to make the circuit more ideal, that is to approximate a mechanical system more closely. You wouldn't need the mechanical equivalent of an opamp in your drawing.

I think this is what you're looking for:

http://lpsa.swarthmore.edu/Analogs/ElectricalMechanicalAnalogs.html

Have fun! :)

I tried this in a Force-current analogy. The mechanical system i got was a first order differential equation for velocity. For velocity stands for voltage, it should have been a second order equation in v for the solution to be sine. Where am i doing wrong?

 

[sophiecentaur]

Voltage corresponds to displacement and force (aka acceleration) is the second derivative. That ends up as a second order equation to solve.