The statement made by Prof.Cyril,"The final and maybe the most popular viewpoint, is to consider the overall feedback as composed of two partial feedbacks - a nonlinear negative feedback (the voltage divider Rb-Rf connected to the inverting op-amp input) and a frequency-dependent positive feedback (the Wien network connected to the non-inverting input). Thus the feedback voltage applied to the op-amp differential input is the difference between the two partial voltages" is perfectly correct and can be supported by our earlier. discussions on Virtual Ground and _ve ( or + ve Feed back in Op-amp)

The capacitor acts as bypass to R1, and thus the input is difference of both -ve and +ve feed back applied across Rb.

Professor's step by step development is appropriate.

Dr.P.S.

Hello Cyril,

I think, your contribution gives a good survey about the classical explanations how and why the WIEN oscillator operates as desired.

In particular the INIC approach shows that each oscillator in two-port topology (with feedback) can be seen also as a one-port negative-resistance oscillator.

(BTW, for my opinion the last two circuit interpretations seem to be equivalent.).

However, in this context it is interesting to investigate a modification of the circuit:

* All of the above explanations/interpretations also apply to a symmetrical modification of the circuit (as described in the following) - however, without the ability to oscillate.

* Interchange the complex impedances within the positive feedback path and interchange both resistors within the negative feedback path (resulting in a closed loop gain of +1.5). At the same time replace the PTC (lamp) by an NTC thermistor of the same nominal resistance.

* It is a simple task to prove that the oscillation condition (Barkhausen) is fulfilled - however, the circuit does not oscillate.

* This leads to a fundamental question that - up to now - seems to be not answered: Is there an oscillation condition, which is necessary as well as sufficient?

Dear Cyril and Lutz,

Thank you for your interesting discussion concerning the WB oscillator of Mr. Hewlett (1939).

We want to understand the mechanisms behind steady state oscillations in the time domain. We want to find a sufficient criterion for steady state oscillations.

So far our starting point for the oscillator is an unstable small signal model for the closed loop circuit (modified Barkhausen).

When we switch on the constant voltage power supply to the nonlinear closed loop circuit we may observe two situations: (1) steady state oscillations or (2) a transient from the zero bias point to a bias point different from zero. In situation (2) we may add an impulse and observe steady state oscillations or a transient to an other bias point (2 rails).

From our "linear" amplifier design practice we are used to a time invariant DC bias point. Oscillators have a time varying bias point and we should study how the power source currents behave. Some time we observe "sinusoidal" currents some times

we observe pulses similar to the escape mechanism in the pendulum clock oscillator.

If the closed loop circuit is of second order (two memory elements) limit cycle oscillations may occur. If the order is greater than two also chaotic steady state oscillations may occur because energy "do not know" where to go.

A circuit always try to be in a minimum energy state. When the signals reach their max/min values "control" disappear and the circuit try to go to a minimum energy state. We should study how energy is moving around.

Returning to the Hewlett-WBO of Cyril above:

Do you assume constant temperature for the lamp ?

Do you have a SPICE model for the lamp ?

The lamp may be replaced with a diode-resistor circuit which makes the analysis easier.

Hello Erik,

to continue the discussion: From your contribution I can derive that the bias point is defined as the dc output voltage of the opamp, right?

Therefore and in this context, I have two questions:

1.) Why do you think that this bias point does vary with time?

And if yes - why is this variation important to understand the oscillation principle?

2.) Why is it necessary to " study how the power source currents behave" ?

Is this an important aspect for a better understanding of the oscillation principle?

Regards

Lutz

Hello Lutz,

The bias point to me is all the voltages and currents at a certain instant. In case of no oscillations we speak about the DC bias point.

When we observe voltages and currents as functions of time they are a picture of the transport of flux and charge in the circuit.

Oscillations seem to be based on a balance of energy we receive from the power source with energy we loose in the circuit as heat and radiation.

Our assumptions are very important. We assume constant temperature in case of steady state behavior. We assume ideal constant voltage power source. We assume that a set of nonlinear differential equations could be used as a model for the circuit. We assume lumped elements so the equivalent scheme of the circuit is a mapping of the differential equations. We assume a quasi-stationary circuit i.e. the wavelength of the oscillations is much larger than the size of the circuit.

Best regards

ERIK

Pisupati, regarding your speculation "...'the feedback voltage applied to the op-amp differential input is the difference between the two partial voltages' is perfectly correct and can be supported by our earlier discussions on Virtual Ground..." I would like to note that here both the op-amp inputs change almost simultaneously. So, if you look for some resemblance with the virtual ground phenomenon, you should see here sooner a kind of "virtual zero voltage" between the op-amp inputs instead a steady virtual ground potential. In this connection, you should revise your assertion "...the capacitor acts as bypass to R1, and thus the input is difference of both -ve and +ve feed back applied across Rb..."

Regards, Cyril.

Erik, thank you for your reply.

I completely agree with the contents of your contribution - however, for my opinion all statements apply - for example - to simple amplifiers as well.

Therefore my question again: How and why can these points help to a better understanding of oscillators in particular?

Regards

Lutz

Lutz, Thank you for your question.

I hope that we by investigating these points in the start-up phase may observe behavior which might lead us to some kind of better understanding.

The pendulum clock is not self-starting but we know that we must start-up with a certain size of the angle in order to activate the escape mechanism.

Regards

ERIK

My explanation for the function principle of the WIEN oscillator is rather simple :

Positive feedback path: Z(series)=Rs+1/sCs and Zp=Rp||(1/Cp).

(A) Frequency domain: Oscillation possible for loop gain equal to (or slightly above) unity for one single frequency only (Barkhausen condition).

(B) Time domain (steady state): The above mentioned loop gain condition leads to a conjugate-complex pole pair in the RHP of the s-plane. Thus, the solution of the corresponding diff. equation (time domain) gives a rising sinusoidal solution.

(Remember the relationship between charact. equation, pole distribution and diff. equation of the system).

(C) Time domain, start-up at t=0:

*Assumption: V(out)>0 at switch-on (t=0)

*Thus: Also V(P)>0 and V(N)=V(out)/3>0 at t=0.

*However, always: V(P)

Hi Lutz

Thank you for a clear description. I agree completely but I will return to the DC bias point of the opamp ;-) Possibly with an email directly to you as a supplement to my comments here.

I think this will be useful

As I can see, the discussion about the circuit operation has become extremely interesting and creative. I would like to take gradually part in this undertaking by first commenting some of your thoughts:

Lutz: Regarding "...for my opinion the last two circuit interpretations seem to be equivalent..." - yes, they are just two ways of looking at the same bridge passive circuit: first, as a full bridge and second, as two separate half bridges. Or they are two viewpoints at the same op-amp input voltage: first, as a whole differential voltage and second, as two separate single-ended input voltages. In this connection, it is interesting to answer the question, "Can we connect a floating voltage source between the two inputs of a differential amplifier?" (I have already asked it).

I think it would be of use for our purposes to consider the bridge circuit (e.g., the humble Wheatstone bridge) as a converter with a single-ended input and a differential output. A few years ago, this viewpoint helped me to unmask the Deborah Chung's "apparent negative resistance":

http://en.wikibooks.org/wiki/Circuit_Idea/Deborah_Chung%27s_%22Apparent_Negative_Resistance%22#Presenting_the_carbon_fiber_network_as_a_bridge_circuit

See Fig. 1 in the link above. The output voltage of such a "bridge converter" is a floating voltage difference between two grounded voltages - VR and VRc. It is interesting that, depending on the proportion between R and Rc, this difference may be zero (R = Rc), positive (R > Rc) or negative (R < Rc). Since this converter is connected in the feedback loop of the op-amp, we may have respectively three kinds of feedback (FB): no FB, positive FB and negative FB. So, if the bridge is balanced, there is no feedback; if it is unbalanced, there are two possibilities - PFB and NFB... a unique property...

Now let's convey these observations to the more complex Wien bridge connected in the op-amp feedback loop. It is interesting here that the transfer ratios (and respectively, the output voltages) of both the half bridges vary through the operation... and we should guess what is the our case...

Now I try to realize your assertion "V(P)

Erik, I am looking at the Wien bridge oscillator diagram and wondering what is the "time varying bias point" here...The philosophy of biasing deserves a separate question(s) especially in the case of the differential (op-amp) amplifier. Well, generally speaking, "biasing" means to apply an appropriate constant input signal with the intention of adding another varying input signal. So, in an amplifier, the biasing circuit is simply a summer (series or parallel) of two quantites (voltages or currents). We can vary both the constant and the varying quantity; the amplifier amplifies its sum.

Let's now see our WB oscillator. The op-amp is steady biased by the internal current source and the biasing currents come from the ground, pass through Rb and R1, and enter the op-amp input (I assume n-p-n input transistors). Then, the sine oscillations superimpose on the bias signal. So, my question is, "Can we think of the oscillations as a kind of input signal that adds to the bias signal?"

Regards, Cyril.

To make the discussion even more intriguing, I pose another "provocative" speculation: I affirm that the lower part of the nonlinear network Rf-Rb has a capacitive nature like the lower part R1-C1 of the Wien network. Am I right?

Regards, Cyril.

Hi Cyril,

"capacitive nature"? Can you please explain how you arrived at this statement?

What about capacitive phase shift of the non-linear element?

Cyril, here comes my answer to your question:

"Now I try to realize your assertion "V(P)0 the voltage V(P) is rising, reaches a maximum, which still is smaller than V(N), and then decreases again until it is zero due to two effects:

*The charging current for Cp through Cs reduces continuously (until Cs is charged completely) and

* at the same time discharging of Cp takes place through Rp.

________________

This model description assumes a CONSTANT positive driving voltage for the Wien path (for the first microseconds after switch-on) in order to show why the opamp putput voltage goes from +Vcc to -Vcc (because of V(N) overrides V(P).

This "switching effect" - together with the mentioned charging/discharging effect of Cp results in a sinusoidal waveform for V(P) and, thus, for V(out).

I mean that the inertial lamp acts as a kind of integrating element - a "resistive capacitor" or a memristor (this is a good subject for discussion in the memristive section - "Is the classic Wien bridge oscillator a memristive circuit?") Of course, this resemblance is only superficial; such a virtual "capacitor" simulated by a resistance cannot store energy.

Do not take offence at my silence about the very circuit operation. I have a problem with understanding it; there is some missing detail that prevent my understanding...

Regards, Cyril

For clarification: What I have described above in detail - regarding V(P) - is nothing else than the classical step response of a passive RC band pass.

So, I begin realizing the problem - I assume zero output voltage in the beginning and then try to imagine how it gradually (sinusoidally) increases... And to increase, I thought, the proportion should be V(P)>V(N)...

But now another problem arises regarding your +Vcc speculation ("...Starting with a positive opamp output voltage (let`s assume: V(out)=+Vcc at t=0) the voltage at the inverting input is V(N)=Vcc/3 and V(P)=0 at t=0 (capacitor Cp empty)..."):

"Seeing" its huge differential input voltage, the op-amp will immediately decrease its output voltage up to zero (the noninertial negative feedback will "urge" the op-amp to do it-:) Am I right?

I just have performed an interesting simulation, which can help to better understand the function of the WIEN oscillator (see pdf-attachement).

I have simulated the step response of a passive circuit consisting of a series combination of 4 identical WIEN band pass circuits. The circuits are decoupled from each other using buffers (gain of 3, as in the oscillator).

This simulation arrangement emulates the first 4 "turn-around trips" of a signal starting at t=0 with a fixed dc voltage (power switch-on of a WIEN oscillator).

As can be seen, after 4 trips through the closed loop (that is the signal with the smallest amplitude) a sinusoidal-like signal can be already identified. The signal will improve if more stages (more "turn-around-trips") are used.

Final comment: Earlier I have explained why the sign of the driving opamp output voltage will change after power switch-on. Thus, driving this 4-element Wien bandpass with a suitable square wave voltage (between positive and negative) gives a rather clean sinusoidal signal at the output of the 4th stage.

Very sophisticated experiment, Lutz... "Series" probably means "consecutive" (cascaded)? It reminds me RC phase shift oscillators... but I cannot comment the experiment...

I have added above some initial material to prepare explaining the circuit operation. It was about the nice feature of the bridge circuits used as single-to-differential converters to change the magnitude and polarity of the bridge output voltage when varying with the balance. Here I add another basic statement about the function of the op-amp inputs:

"...In a non-inverting amplifier, the output voltage changes in the same direction as the input voltage...."

http://en.wikipedia.org/wiki/Operational_amplifier#Non-inverting_amplifier

So, in our case, the output voltage and the voltage V(P) across the bottom capacitor (Lutz) change in the same direction.

Also, the operation of the op-amp noninverting Schmitt trigger (during the transition) can be useful as well:

http://en.wikipedia.org/wiki/Schmitt_trigger#Non-inverting_Schmitt_trigger

Another question is, "Can we talk about a kind of an 'instant loop gain'?" I mean the product of the two instant gains - of the Wien network and the noninverting amplifier.

To explain the circuit operation, I would like to introduce more two concepts - "dynamical stable" and "static unstable" system. I mean that while V(P) and Vout change, the system is stable (it cannot reverse itself); when V(P) and Vout stop changing, the system becomes unstable (it can reverse itself).

Here are other two very important concepts - "noncontrollable" (self-reinforcing, avalanche like) positive feedback (e.g., in the case of the Schmitt trigger above) and "controllable" (slowed up, delayed), when we have connected an integrating element in the positive feedback loop and the loop gain is close to unity.

And another, final, concept - think of the Wien network as an imperfect integrating circuit driven via an imperfect differentiating circuit.

With these premises I have just prepared the explanation of the operation.

If you remove Cp or C1 in the figure above the WBO may be looked upon

as a modified common multivibrator !

An electronic circuit is normally a circuit with 2 input ports and 1 output port. One input port is the power source (battery) the other input port is the signal input port.

The output port is the signal output port. If the circuit is linear superposition gives that

the output signal is the sum of the input signals transferred and modified through the circuit.

An oscillator is a circuit with only two ports, the battery input port and the signal output port. If we observe steady state oscillations the circuit must be nonlinear. If we investigate the oscillator as a time varying linear circuit we may use all our linear methods with poles and zeros etc. When we switch on the battery at time minus zero all coils are open-circuits and all capacitors are short-circuits. At time plus zero we have the start-up of the DC bias point where all coils are short-circuits and all capacitors are charged based on time constants of their equivalent parallel resistors. If no steady state oscillation is observed we have a DC bias point with constant currents in the coils and constant voltage on the capacitors. If oscillations are observed we have a time varying DC bias point.

An interesting viewpoint...

Quote Cyril: "Another question is, "Can we talk about a kind of an 'instant loop gain'?" I mean the product of the two instant gains - of the Wien network and the noninverting amplifier."

Why the term "instant"? The loop gain AL is well-defined for linear circuits with feedback.

In this case, it is the product of the Wien bandpass (1/3 at fo) and the non-inv. gain - resulting in AL=1 (design value for f=fo)

Quote Erik:"If no steady state oscillation is observed we have a DC bias point with constant currents in the coils and constant voltage on the capacitors. If oscillations are observed we have a time varying DC bias point."

Of course, no objections - as far as the 1st sentence is concerned.

I disagree with the 2nd sentence (where is a proof?). Does this mean that the DC bias point would vary with a frequency of 1MHz in case the oscillation frequency is 1 MHz?

Rather I would say:

If oscillations are observed we have a superposition of two voltages:

A dc voltage (called bias voltage) and a sinusoidal voltage, which swings around the bias voltage. Can you agree?

Lutz, in the discussion about the Barkhausen criterion you have said, "...using the loop gain (your second contribution in this discussion) is a bit „problematic“ because the loop gain is a term defined in the frequency domain using the linear small-signal ac impedances of the WIEN path for f=fo. Thus, we cannot use it in the time domain..."

Cyril, I remember this statement - and I think it is true. Is there anything not clear?

Or did I misunderstand something?

Well, it's time to expose my explanation of this famous circuit. It is interesting that it is almost the same as this unhappy Wikipedia explanation two years ago:

http://en.wikipedia.org/wiki/Talk:Wien_bridge_oscillator#How_do_RC_oscillators_produce_sine_wave.3F

The sine wave consists of two half-sine waves; to explain the circuit operation, it is sufficient to explain one of them (I have chosen the upper). Six typical phases in forming the upper half-sine wave can be separated: increasing, delaying, stopping, reversing, accelerating and decreasing. They are closely related то the gain of the amplifier and the loop gain.

1. INCREASING. I suppose that, because of some initial noise, the output voltage begins increasing. 1/3 of it is applied to the noninverting input. It is amplified by a gain that is close to but a little more than three (G > 3). So, the loop gain is close to but a little more than one; the output voltage is a little more than needed for the equilibrium magnitude; so it continuously increases; also, V(P)>V(N). But this self-reinforcing process is not uncontrollable as in the case of the noninverting Schmitt trigger above where the op-amp switches in an avalanche like manner. Here it is controllable (held) by C1 in the top figure. This is the key point of this phase of "moving" - the output voltage slightly exceeds the needed voltage so that the voltage across C1 (and, after an amplification, the very output voltage) gradually but quickly increases and this process is held by the integrating lower part of the Wien network. Figuratively speaking, the op-amp output "pulls up", via the upper part of the Wien network, the voltage at the lower part of the network. This state of continuous increasing is stable; the voltage "movement" cannot change its direction. So, until moving, the system is stable; it is "dynamically stable".

2. DELAYING. At some moment, the nonlinear negative feedback begins decreasing the amp gain (it is still above 3 but it is very closely to 3). The output voltage produced is less than in the previous phase (but still above the needed magnitude). The voltage "movement" slows up its "speed" and the curve begins rounding. This state continues to be "dynamically stable".

3. STOPPING. At the top of the half-sine wave, the nonlinear negative feedback decreases the gain up to (exactly) 3; accordingly, the loop gain becomes exactly 1. There is no self-regeneration and the voltage "movement" stops. This state is stable only if the voltage continues to increase; it is not stable if the voltage begins decreasing... and exactly this is our case. The voltage at the noninverting input begins decreasing: first, since the upper capacitor is almost charged and the current through it decreases; second, the bottom capacitor C1 discharges through the resistor R1 (as Lutz has said as well).

4. REVERSING. So, at the top, the output voltage reverses the direction of its "movement" and begins decreasing. The upper charged capacitor C2 conveys the output voltage change (Lutz). The nonlinear negative feedback slowly increases the gain above 3 (as before) and the loop gain becomes again more than 1. The same self-reinforcing process starts again but now downwards.

5. ACCELERATING. The voltage "movement" pick up speed and forms the round right part of the half-sine wave. The amp gain is very close to 3 and the loop gain is very close to 1. The state is "dynamically stable".

6. DECREASING. The nonlinear negative feedback increases more the gain and the rate of the voltage change increases more thus forming the steep region of the sine half-wave. The state is "dynamically stable"; there is no chance for reversing...

Regards, Cyril.

The open loop circuit is a circuit closely related to the oscillator circuit.

A study of the open loop circuit may give us a lot of information, but it is

the closed loop circuit which we should investigate.

The oscillator is the closed loop circuit for which Barkhausen observed that

gain around the loop must be one and phase shift around the loop must be

zero or a multiple of 2pi. Barkhausens basic assumption was a linear circuit.

Steady state oscillators are nonlinear circuits. Linear oscillators are damped

circuits. Superposition is only valid for linear circuits.

Cyril, I must confess that I have some problems to follow the first part of your explanation.

Quote: " ...the output voltage begins increasing. 1/3 of it is applied to the noninverting input. It is amplified by a gain that is close to but a little more than three (G > 3)."

1.) For my opinion, the opamp output will not start from zero and "begins increasing".

Instead - after switch-on the opamp does not have the desired dc bias point, but will have (at t=0+) an output voltage equal to one of the power rails. That was the reason that my explanation as given in an earlier post did start at +Vcc.

2.) I doubt if immediately after switch-on 1/3 of the output voltage is applied to the noninv. input. This factor 1/3 is derived from the ac analysis of the WIEN network under steady-state conditions. However, for investigating the start-up behaviour we must not assume steady-state conditions. Instead, my explanation did start with an empty grounded capacitor (zero voltage for t=0+).

3.) I think, it is necessary to add some comments to my simulation results (given as a pdf attachement in my former post):

* I have simulated the step response for a cascade of 4 identical passive WIEN bandpass circuits isolated from each other (buffers, gain of 3). This scheme emulates the first 4 trips of the originated signal (starting with +Vcc) through the closed-loop of the real oscillator.

The largest signal is the output of the first stage - and the smallest signal is the output of the last stage (4th round trip). As can be expected from the last stage output - the more stages we have, the more will the signal approach a sinusoidal form.

Regards

LvW

Question by Cyril Mechkov, Technical University of Sofia,

What is the basic idea of Wien bridge oscillator? How does it operate?

Cyril: I have stated several times that we can find the answer of

this question in the time domain by following the sine "movement"

of the output voltage between the supply rails.

Erik: I agree but we should also investigate how energy is supplied

from the power source and how energy is dissipated in the circuit.

We may treat the nonlinear circuit as a time varying linear circuit

an make use of our linear circuit theory in time intervals where

we assume picewise linear nonlinearities. In the limit we should

linearize the circuit at a certain time instant corresponding to

the integration step in our circuit analysis program.

Hello Erik, please excuse me because I cannot resist to repeat some arguments, which you know already:

I see no necessity to

"treat the nonlinear circuit as a time varying linear circuit

an make use of our linear circuit theory in time intervals where

we assume picewise linear nonlinearities"

At least during start-up (and I think, that is the main point of our discussion) the circuit behaves linear in my opinion. Which non-linearity are you referring to?

I am sure, it is possible to explain the main property of the oscillator (the ability to oscillate) based on linear considerations only.

The non-linearity comes into play only for calculating the resulting oscillator amplitude.

Why do you think, we have to count with non-linearities from the beginning?

What do we gain using this approach?

What do you think about my attempt (former post) to describe the start-up phase ?

Kind regards

Lutz

Hello Lutz,

As usually I agree with you and I think it is time to study a specific case. In my talk in Palanga in 2012 I presented a "linear" Wien Bridge Oscillator (WBO) and demonstrated

that it was possible to reduce the harmonics by means of adding a simple diode circuit in the negative feed-back path. Please see:

In order to make the circuit as linear as possible I chose a frequency well below the dominating pole frequency of the opamp AD712.

First the values of the linearized circuit are controlled by means of ANP3. The amplifier is assumed to be an ideal Opamp. Resistor RCL is infinite and RC is 2*RD for a complex pole pair on the imaginary axis and 2*Rd + 10ohm for a complex pole pair in RHP. Please see:

.It is difficult to upload files

Some of the output from ANP3 is a netlist for SPICE programs. The ideal opamp is modeled as a perfect amplifier with gain 1e+12. Please see:

From which we now create the following input file for PSpice:

We now have a tool for investigating the WBO.

My first experiment shows that the oscillations are damped for the linear WBO with poles on the imaginary axis. It is of course necessary to start with charge on at least one of the capacitors CA and CB.

And

Next experiment is to replace the perfect amplifier with gain 1e+12 with an AD712 in the case with a complex pole pair in RHP. Initial conditions: zero voltage on CA and CB.

And

It is seen how the output voltage V(3) of the opamp rises exponentially from zero as expected. The currents in the power sources I(VN) and I(VP) are also shown. It is seen that an impulse of about 6mA is delivered in each period from VN. The impulse disappear when the diode circuit is activated.

This is very clumsy. Apparently only one file at a time could be uploaded.

The input port of the oscillator is the power source. The output port is e,g. the output node V(3) of the opamp i.e. we observe a step response.

We should see what happens when the power source is a sine wave function ;-)

Best regards to all of you

ERIK

PS

Now I see it is possible to edit an entry !!!!

with new text but not with upload of more files !!

E

Erik, I like your diode circuit but I would like to show what Wikipedians thought about the non-linear diode network that I (Circuit dreamer, shortly CD) considered two years ago:

http://en.wikipedia.org/wiki/Talk:Wien_bridge_oscillator#Please_clarify_.22voltage_peak.22.2C_.22voltage_changing.22

"...When CD uses sources, they are poor: consider your http://www.ecircuitcenter.com/circuits/opwien/opwien.htm "reference" (which is now doubled in the external links sections). Why do you believe that is a reliable source? It's some random SPICE blog that doesn't cite to any sources that it uses. Nothing on the webpage is traceable to a reliable source..."

"...The bridges are balanced with slowly varying elements - the lamps. The time constant is significant. If you look at Williams application note, his WBO uses a balanced bridge. Sometimes the lamp is replaced with a FET, but the reaction time of the FET is slowed down. (BTW, Williams' app note is all about "Bridge Circuits".) Williams is not using fast acting (e.g., diode) limiters. (Oliver is not using explicit fast acting limiters, but he points out the need for a very subtle fast acting limiter for the lamp balancing.)

The fast diode limiter of ecircuitcenter.com should never be mistaken for a slow balancing of the bridge. That circuit should never be mistaken for a delicately balanced bridge. That bridge is out of balance by 10 percent. HP operated their bridges at 0.33 percent..."

Regards, Cyril.

Hello to all,

at this moment, I must confess that I am not very happy with the way we discuss the "secrets" of a "linear" oscillator. I regret that it is not really a sequence of questions and answers. Instead, we have a mixture of different opinions and different statements regarding different points like:

oscillation condition, ability to oscillate , start-up phase, varying bias poinnt (?), non-linear diff. equations, reversal of voltage directions, amplitude stabilization, currents through the power lines,...

Of course, all these aspects are not independent on each other, but as mentioned:

For my opinion, it is not a straight-forward discussion of some new insights or new approaches to explain/understand the behaviour of such an interesting circuit (a circuit that must contain some non-linearity in order to operate it its quasi-linear range)

Regards

Lutz vW.

Lutz, let's freely exchange ideas and finally systematize them...

To make the background of my contribution above more clear:

I think, all of us know

* the working principle of the WIEN oscillator

* the basic equations for designing the oscillator

* the two different views (time vs. frequency domain) to describe the oscillator function

* the necessity of an amplitude stabilization network

* the different methods for amplitude stabilization.

So what are we discussing here?

For my opinion, there are just two basic questions that are NOT yet answered in the textbooks satisfactorily up to now and which are worth to be discussed here:

1.) Why do some circuit do not oscillate - in spite of the fact that the oscillation condition (Barkhausen) is fulfilled? This question (which I have asked already earlier here in RG), of course, touches the problem of a still missing sufficient oscillation criterion.

2.) How to describe the start-up phase - that means: What effect causes the circuit to start self-excitement resulting in a sinusoidal signal form.

In this context, some textbooks simply state: The noise spectrum and/or the power switch-on transient contain a line that is identical to the oscillation frequency (loop gain slightly larger than unity) - and, thus, will be amplified again and again...

My opinion: I doubt if this description is correct. It assumes that the system is from the beginning (t=0+) under steady-state conditions. But that is not true!

Did I forget an important point worth to be discussed?

Regards

Lutz vW

"Lutz, let's freely exchange ideas and finally systematize them.."

OK - hopefully it works

(I went roughly through the wiki-discussion linked by you - and I missed such a systematic summary).

L.

Lutz, you are right - actually, I have considered the circuit under steady-state conditions. It is extremely interesting to consider the start-up as you have done. Let's begin gradually to do it.

I am not familiar with the op-amp behavior after switching on the power supply. I suppose it depends on the supply behavior and the initial output voltage (immediately after the switch-on) can be various (near to the rails and why not to ground). But, I have explained it, as the voltage across the bottom capacitor will be zero, the output will momentarily become zero (the arrangement is a non-inverting amplifier with zero input voltage). The nonlinear feedback has set such ratio that the loop gain is more than one. As a result, the circuit is unstable and the voltage begins increasing (if the initial "jump" of the output voltage was to the positive rail).

IMO the VBO will ever run since, as you have also said, at the switch-on of the power supply, the output voltage "jumps" up to the supply rails; then, as I have said, it returns almost to zero because of the negative feedback. But this jump injects a portion of charge into the bottom capacitor and its voltage slightly changes... and since the system is unstable (loop gain > 1), the output voltage begins "moving" from zero to the respective rail...

Regards, Cyril.

Lutz, this was my unhappiest wiki discussion that put an end to my Wikipedia contributions. My idea was to reveal the secret of the Wien bridge oscillator together with Wikipedians... but I didn't manage... After that, I abandoned Wikipedia and completely moved to Wikibooks...

Now my hope is on ResearchGate-:)

Lutz, IMO there are mainly high-level explanations in textbooks; there is a lack of low-level intuitive explanations saying what exactly happens there. I agree with your second conclusion about the start-up phase:

"...some textbooks simply state: The noise spectrum and/or the power switch-on transient contain a line that is identical to the oscillation frequency (loop gain slightly larger than unity) - and, thus, will be amplified again and again...

My opinion: I doubt if this description is correct. It assumes that the system is from the beginning (t=0+) under steady-state conditions. But that is not true!"

You are absolutely right about the pre-existence of a "a line that is identical to the oscillation frequency" (f0). This is a vicious circle - to have a sine wave with f0, we have to have some sine wave with f0. But if there is no such a wave?

IMO the oscillations begin from zero frequency and gradually increase its frequency until reaching the equilibrium. In terms of time, the voltage across the bottom capacitor begins changing slowly and then gradually accelerates until the swing reaches its maximum rate of change (frequency). After that, there is some frequency stabilization phenomenon performed by the Wien network. It is extremely interesting to explain this process of a frequency stabilization...

Regards, Cyril.

Hello Cyril,

I think we are - more or less - on the same line again.

Some comments to

(Quote Cyril):"I am not familiar with the op-amp behavior after switching on the power supply. I suppose it depends on the supply behavior and the initial output voltage (immediately after the switch-on) can be various (near to the rails and why not to ground). But, I have explained it, as the voltage across the bottom capacitor will be zero, the output will momentarily become zero".

With reference to the simulation results as given in one of my previous posts (pdf attachement) I like to add that I have simulated the step response for parts values R=10k and C=4.7nF. These values belong to a oscillation period of approx. 300µsec.

I don`t believe that both supply rails for the opamp can be switched-on within this small time period. Thus, immediately after switch-on the negative feedback will not cause an opamp output voltage, which is around zero.

Thus, as you have mentioned, it is very probable that the dc output at t=0+ (input step for the WIEN path) will be anywhere between Vcc and ground.

Therefore, I am pretty sure that my simulation (with four identical WIEN networks for emulating the first four non-sinusoidal events) can give a pretty good visualization of the start-up process.

After this first period with self-excitement is completed, I think we can treat the whole circuit simply as an amplifier who is able to produce its own input signal. And the non-linearity (lamp, NTC, Diodes, FET-AGC, amplitude clipping by the supply rails) comes into play for limitation of the amplitude only.

I like to repeat: There is no part in the electronic world - and in particular no amplifier - that is really linear. Nevertheless, we are operating all these devices within a range, which we can consider as being "quasi-linear". Why shouldn`t we do the same with an oscillator - if it is well designed with sufficient low THD values?

Thus, I am really convinced that neither the description nor the design of these oscillators require non-linear differential equations.

With regards

Lutz vW

Lutz, I have considered your simulation of the cascaded Wien networks. I would like to know if the buffers connected between them are the real nonlinear amplifiers or they are simply amplifiers with a gain of three. The experiment is very interesting but, for now, I have no idea how to use it for explaining the process of sine wave creating. Cyril.

Hello Cyril and Erik,

in the enclosed pdf attachement you can find - supported by simulations - my attempt to explain the start-up behaviour of the WIEN oscillator in the time domain.

As you can see - there is no fundamental difference between LC and RC oscillators because in both cases the response of the passive and frequency-determining circuit on a short exciting impulse is "something like a decreasing wave".

Comments are welcome.

Lutz vW

Hi, Lutz! Very profound scientific investigation... I begin studying it...

It was interesting to me to peep into the first 200 μsec and to see how the op-amp-output "jumps" up and down, and finally calms down by the because of the negative feedback. Well, I agree that there is such an impulse after the switching the power supply. But IMO its role is only to help the oscillator to run; it cannot explain how the sine oscillations are produced during the steady state (particularily, the reversal)... Am I right?

Regards, Cyril.

Dear colleagues,

I will try to answer Cyril question what is the basic idea of the wien bridge oscillator?

The oscillator is an electronic circuit producing periodic wave forms.These wave forms may be sinusoidal, or rectangular or triangular or any arbitrary waveform.To produce such waveforms the oscillator circuit must contain a timing circuit. The timing circuit may be LC , or RC, or a crystal.

According to the oscillation theory an oscillator is either a positive feed back amplifier with the condition of Av* B=1, where Av the amplifier gain and B the feed back factor, or a negative resistance connected across the timing network.

Normally the feed back circuits makes out the timing network.

The wien bridge oscillator is a sine wave oscillator based on a positive feed back amplifier with an RC feed back network as frequency determining element.

As shown in the circuit of the oscillator given by Cyril. this RC circuit is composed of RC in series followed by RC in parallel.The feedback factor B= Z2/(Z2+Z1) while the gain of the amplifier Av= 1+Rf/Rb.

The feed back factor B frequency response resembles that of the LC circuit except that the quality factor is low and the feed back factor is =1/3 at peak frequency determined by w=1/RC assuming equal resistors and equal capacitors in the feed back network. The conditions of oscillations will be satisfied selectively at the peak frequency w provided that the gain of the amplifier is independent of frequency and its magnitude is =>3 as the colleagues stated before.

This oscillator must produce a sine wave. This will be discussed in a separate comment. It is observed experimentally, Without gain control this circuit tend to produce square waves.

Thank you

Quote Cyril: "... it cannot explain how the sine oscillations are produced during the steady state".

Hi Cyril, yes -that`s true. My attempt to describe the oscillator in the time domain refers primarily to the start-up behavior.

This is because - I think - during steady state conditions the situation is rather clear. is it not?

(a) Frequency domain: Pole location in the RHP and - as a consequence - the solution of the differential equation is a rising sinusoidal signal.

(b) Time domain: I think, the oscillator can simply seen as an amplifier that produces its own input signal (with the correct damping and the correct gain, resulting in an overall gain of unity.

Quote Prof. A. Zekry: "Without gain control this circuit tend to produce square waves."

For my opinion, it is more correct to say: The circuit tends to produce a clipped sinusoidal wave. The amount of clipping depends solely on the excess gain (as an example, a loop gain of 1.05 will cause a small clipping effect only.)

Referring to Prof. A.Zekry's statement, " The conditions of oscillations will be satisfied selectively at the peak frequency w provided that the gain of the amplifier is independent of frequency and its magnitude is =>3 as the colleagues stated before.This oscillator must produce a sine wave " it is true that for the circuit can act as an oscillator the Voltage gain of the amplifier must be at least 3. Refer Link :

http://www.electronics-tutorials.ws/oscillator/wien_bridge.html

Wien Bridge Oscillator Summary

Then for oscillations to occur in a Wien Bridge Oscillator circuit the following conditions must apply.

1. With no input signal the Wien Bridge Oscillator produces output oscillations.

2. The Wien Bridge Oscillator can produce a large range of frequencies.

3. The Voltage gain of the amplifier must be at least 3.

4.. The network can be used with a Non-inverting amplifier.

5. The input resistance of the amplifier must be high compared to R so that RC network is not overloaded and alter the required conditions.

6. The output resistance of the amplifier must be low so that the effect of external loading is minimised.

7. Some method of stabilizing the amplitude of the oscillations must be provided because if the voltage gain of the amplifier is too small the desired oscillation will decay and stop and if it is too large the output amplitude rises to the value of the supply rails, which saturates the op-amp and causes the output waveform to become distorted.

8. With amplitude stabilisation in the form of feedback diodes, oscillations from the oscillator can go on indefinitely.

What Prof Lutz has stated that, " Quote Prof. A. Zekry: "Without gain control this circuit tend to produce square waves."

For my opinion, it is more correct to say: The circuit tends to produce a clipped sinusoidal wave. The amount of clipping depends solely on the excess gain (as an example, a loop gain of 1.05 will cause a small clipping effect only.) " has to be studied further.

Dr.P.S.

I may also add that the effect mentioned by Prof. A. Zekry that Wein Bridge Oscillator produces Square Wave, without Gain Control , may be due to "Amplitude Bouncing" as stated in Wikipedia on the subject.

http://en.wikipedia.org/wiki/Wien_bridge_oscillator

Amplitude stabilization

The key to the Wien bridge oscillator's low distortion oscillation is an amplitude stabilization method that does not use clipping. The idea of using a lamp in a bridge configuration for amplitude stabilization was published by Meacham in 1938 The amplitude of electronic oscillators tends to increase until clipping or other gain limitation is reached. This leads to high harmonic distortion, which is often undesirable.

Light bulbs have their disadvantages when used as gain control elements in Wien bridge oscillators, most notably a very high sensitivity to vibration due to the bulb's microphonic nature amplitude modulating the oscillator output, a limitation in high frequency response due to the inductive nature of the coiled filament, and current requirements that exceed the capability of many op amps. Modern Wien bridge oscillators have used other nonlinear elements, such as diodes, thermistors, field effect transistors, or photocells for amplitude stabilization in place of light bulbs. Distortion as low as 0.0003% (3 ppm) can be achieved with modern components unavailable to Hewlett.

Wien bridge oscillators that use thermistors also exhibit "amplitude bounce" when the oscillator frequency is changed. This is due to the low damping factor and long time constant of the crude control loop, and disturbances cause the output amplitude to exhibit a decaying sinusoidal response. This can be used as a rough figure of merit, as the greater the amplitude bounce after a disturbance, the lower the output distortion under steady state conditions.

Dr.P.S.

Thank you professor Subramanyam. You always complete my answers.

Quote: Prof. Subramanyam:

""The amount of clipping depends solely on the excess gain (as an example, a loop gain of 1.05 will cause a small clipping effect only.) " has to be studied further."

Hello Prof. Subramanyam;

I suppose, this subject (investigation of the clipping effect) has been studied in the past already in detail. Please refer to the keywords "1st harmonic analysis" or "harmonic balance" or "describing function". There are numerous publications dealing with the subject of oscillator amplitude clipping caused by hard-limiting (power rails). Which new results do you expect?

Regards

Lutz vW

Important remark: Oh - for a moment I forgot that the method of "harmonic balance" cannot be applied to the WIEN oscillator with good results because of its bad harmonics damping. This method requires that all harmonics of the fundamental frequency are sufficiently damped in the frequency-determining network - and this requirement cannot be met by the Wien network (however, for many other passive LC or RC networks it works very good).

Amplitude stabilization

Dear collegues, inspired by Prof. Subramanyam´s last contribution I like to add some general - and partly new - aspects/comments to the question of amplitude stabilization for harmonic oscillators.

1.) Quote Prof. Subramanyam: “Modern Wien bridge oscillators have ...distortion as low as 0.0003% (3 ppm)."

For my opinion, this statement supports my previous claims that it is (a) not necessary and (b) not meaningful to use non-linear differential equations (as proposed elsewhere sometimes) for investigating the ability of a circuit to oscillate and/or for describing the principle of oscillation. I think, non-linear effects are to be considered for calculation of the resulting amplitudes only.

2.) Several articles, Internet contributions and textbooks contain statements like: „During continuous oscillation the system poles are not located on the imaginary axis, but instead, swing between the RHP and the LHP. “

For my opinion, this applies only for stabilization methods containing a memory element (time constant) like thermistors (NTC, PTC) and FET-control loops (all these methods provide „gain control“). It does NOT apply for amplitude limiting mechanisms that react to each single half-wave like diode stabilization or amplitude clipping due to the power rail („amplitude control“).

Explanation:

In contrast to the above mentioned methods of „gain control“ the methods for amplitude limitation do NOT alter the loop gain of the feedback loop. It is a non-linear effect that limits each single amplitude separately.

During this „non-linear phase“ of the half wave the definitions for loop gain and system pole distribution do not apply anymore (and cannot be used to observe or assume any pole movement).

For example, consider an oscillator with (a) a frequency-dependent network with good selectivity (harmonics damping) and (b) with amplitude clipping due to the power rails. I think, in this case we can state that the opamp input is a „good“ sinusoidal and that the output (due to a designed loop gain>1) is a clipped sinusoidal. However, as far as the first harmonic (fundamental frequency) is concerned the real loop gain based on the describing function DF will be exactly unity. Therefore, if we consider this fundamental wave as the only relevant frequency within the system, the resulting pole location is exactly on the imaginary axis (Erik Lindberg`s formulation: „balance on rasors edge“ ??).

Your critics and comments are welcome.

Regards

Lutz vW

Lutz, I highly appreciate your way of creative thinking... we have to be bold to demolish the tradition...

It is very interesting phenomenon that we continue thinking in the same direction. Although I have not still posed emphatically my final conclusion about the truth behind the Wien bridge oscillator, I have been continuously thinking about it and waiting for the final great insight... I am just waiting the ideas to become ripe...

The main question to be answered is, "What does actually shape, form, mould the single half sine wave - the nonlinear network, the Wien network or the both?" And the next logical question is, "Is there some memory effect in this process (i.e., does it depend on the previous half waves)?" Thinking about these questions, I have decided to ask a separate question about the basic idea behind the very Wien network... and I am preparing this question...

Regards, Cyril.

Hello Lutz,

Thank you so much for your start-up-Wien.pdf above. It is in agreement with my

findings. Please provide your SPICE netlist !!

Best regards to all of you

ERIK

Hi Cyril,

(Is there some memory effect in this process (i.e., does it depend on the previous half waves)?")

Don`t you think that the impuls response of the WIEN network - in conjunction with positive feedback - could give a sufficient answer?

By the way - do you know the following paper (see Fig. 4)?

http://www.google.de/url?sa=t&rct=j&q=an%20analytical%20solution%20to%20a%20class%20of%20&source=web&cd=4&ved=0CE0QFjAD&url=http%3A%2F%2Fwww.oocities.org%2Fyeshpa%2Ftcas.pdf&ei=hl87UYn3FI3NswbIgoHwDA&usg=AFQjCNFwj69pxVPkKX22oMArthU23J2r2g&bvm=bv.43287494,d.Yms

I agree with What Prof. Lutz stated,"During continuous oscillation the system poles are not located on the imaginary axis, but instead, swing between the RHP and the LHP. “

For my opinion, this applies only for stabilization methods containing a memory element (time constant) like thermistors (NTC, PTC) and FET-control loops (all these methods provide „gain control“). It does NOT apply for amplitude limiting mechanisms that react to each single half-wave like diode stabilization or amplitude clipping due to the power rail („amplitude control“)."

For the response to be Sinusoidal we should have a simple pole on imaginary axis balancing on razor edge as prof .Erik has pointed out..

PSpice analysis of open loop and closed loop Analysis of Weinbridge Osillator is given in the following Link also

http://www.ecircuitcenter.com/Circuits/opwien/opwien.htm

Regarding Gain control it states, " NON-INVERTING AMPLIFIER

The RC network falls short of the oscillation conditions in that the gain is only 1/3 V/V. How is the gain of 1V/V around the loop to be achieved? As you might have guessed, the non-inverting amplifier provides the needed gain. How much? A gain of 3 V/V makes the total gain 1/3 x 3 = 1 V/V. Setting the correct op amp gain is critical. Not enough - oscillations will cease. Too much – oscillation amplitude will grow until the output saturates.

What’s needed is a mechanism to guarantee oscillations will start (GAIN > 3), yet, limit the gain (GAIN=3) at steady state. Enter our heros - D1, D2 and R12. The circuit adjusts its gain depending on the signal level. For small signals, the diodes do not conduct and the gain is set by

Gain=(1+((R11+R12) / R10 )) >3

For larger signals, the voltage across R12 is big enough to make D1 and D2 conduct. The shunt resistance of the conducting diodes effectively reduces the R12 resistance, consequently, reducing the overall gain to GAIN=3.

CIRCUIT INSIGHT What happens if there’s not enough gain around the loop? Reduce R12 to 1k making the total loop gain less than 1. Run a simulation. The circuit rings briefly, but there’s not enough gain to sustain oscillations.

Dr.P.S.

Referring to Prof. Cyril'"What does actually shape, form, mould the single half sine wave - the nonlinear network, the Wien network or the both?" And the next logical question is, "Is there some memory effect in this process (i.e., does it depend on the previous half waves)?" s question, " I may say that High Pass and the Low Pass Filters which are part of the Wein Bridge Oscillator produce the Half sine wave.

The Filter Circuits and the Response are given in the Link :

http://www.electronics-tutorials.ws/oscillator/wien_bridge.html

The Wien Bridge Oscillator uses a feedback circuit consisting of a series RC circuit connected with a parallel RC of the same component values producing a phase delay or phase advance circuit depending upon the frequency. At the resonant frequency ƒr the phase shift is 0o. Consider the circuit below.

The above RC network consists of a series RC circuit connected to a parallel RC forming basically a High Pass Filter connected to a Low Pass Filter producing a very selective second-order frequency dependant Band Pass Filter with a high Q factor at the selected frequency, ƒr.

The Amplitude depends on the amplitude of the Fundamental only at the resonant frequency.

Dr.P.S.

Lutz, this article is too difficult for me; it stays at too high level for my low-level thinking-:) Actually, I am a layman in this area; I have only some happy guessworks... nothing more...

Lutz, I would like to ask you, "What and where is memorized from the previous cycle, when the sine wave crosses the zero level?" I do not mean the bulb "memory". Regards, Cyril.

I like to place two comments (correction) to the former post from Dr. P.S.:

(The above RC network consists of a series RC circuit connected to a parallel RC forming basically a High Pass Filter connected to a Low Pass Filter producing a very selective second-order frequency dependant Band Pass Filter with a high Q factor at the selected frequency, ƒr.)

1.) I think, the WIEN network does not consist of a "high pass filter connected to a low pass filter". It is just an RC bandpass, which cannot be split into two separate filters.

2.) More important: This bandpass is not "very selective" with a "high Q factor".

Just the opposite is true. It has a very low Q factor (Q=1/3). As I have mentioned earlier, this bad selectivity causes a bad damping of harmonics and is the reason that the method of "harmonic balance" (based on the describing function) cannot be applied.

We may figuratively say that, in the Wien network, a high-pass and a low-pass filters are assembled (integrated, interlaced...) into a bandpass filter. I suggest to move the discussion about the Wien network to the new question about it:

https://www.researchgate.net/post/What_is_the_basic_idea_behind_the_Wien_network_How_does_it_operate_in_the_frequency_and_time_domains#share

BTW this resource (electronics-tutorials) is not so reliable.

Regards, Cyril.

Quote Prof. Cyril Mechkov: "BTW this resource (electronics-tutorials) is not so reliable"

Yes - I strongly support this statement. For this forum (RG), I suggest to rely on classic scientific sources only (textbooks, int. journals,...).

Question: "What and where is memorized from the previous cycle, when the sine wave crosses the zero level?" I do not mean the bulb "memory".

My answer:

I can imagine nothing else than the discharging effect of the grounded capacitor and feedback of the corresponding voltage (via amplifier) to the input of the whole network.

Only it seems, when the sine wave crosses the zero level, the output voltage and, respectively, the voltage across the grounded capacitor, should be zero as well... so there is no charge memorized somewhere...

Quote Cyril: "We may figuratively say that, in the Wien network, a high-pass and a low-pass filters are assembled (integrated, interlaced...) into a bandpass filter."

My answer: Yes - that`s also my opinion.

With other words: It is a FUNCTIONAL superposition of a low and a high pass function (but NOT a cascade of both blocks).

Thus, one can say that the WIEN network belongs to the class of parallel filters - similar to the bridged-T or the double-T networks, which also are frequently used in RC oscillator circuits.

I agree with Prof. Lutz in that it is also possible some times that the info found in the Internet may not be correct.

Some action is taking place and some times different persons try to explain it in different ways. Only thing is that it should be convincing and also not go against Fundamentals.

On the same topic, if I remember correctly, I remember to have seen another explanation in terms of Lag and Lead Networks.

I chose this two Filters Theory to stress the point that it is a Narrow Pass Band action around the Resonance Frequency.

The statement of Prof.Lutz, " My answer: Yes - that`s also my opinion.

With other words: It is a FUNCTIONAL superposition of a low and a high pass function (but NOT a cascade of both blocks).

Thus, one can say that the WIEN network belongs to the class of parallel filters - similar to the bridged-T or the double-T networks, which also are frequently used in RC oscillator circuits." lends support to this way of thinking.

The same reference goes on to explain further in the following way.

http://www.electronics-tutorials.ws/oscillator/wien_bridge.html

Wien Bridge Oscillator

The output of the operational amplifier is fed back to both the inputs of the amplifier. One part of the feedback signal is connected to the inverting input terminal (negative feedback) via the resistor divider network of R1 and R2 which allows the amplifiers voltage gain to be adjusted within narrow limits. The other part is fed back to the non-inverting input terminal (positive feedback) via the RC Wien Bridge network.

The RC network is connected in the positive feedback path of the amplifier and has zero phase shift a just one frequency. Then at the selected resonant frequency, ( ƒr ) the voltages applied to the inverting and non-inverting inputs will be equal and "in-phase" so the positive feedback will cancel out the negative feedback signal causing the circuit to oscillate.

Also the voltage gain of the amplifier circuit MUST be equal to three "Gain = 3" for oscillations to start. This value is set by the feedback resistor network, R1 and R2 for an inverting amplifier and is given as the ratio -R1/R2. Also, due to the open-loop gain limitations of operational amplifiers, frequencies above 1MHz are unachievable without the use of special high frequency op-amps.

Dr.P.S.

I would like only to note that if "the voltages applied to the inverting and non-inverting inputs are exactly equal and 'in-phase'", there is no any feedback and the differential input voltage is zero. What does it mean? Maybe, this is an open-loop op-amp configuration...

Also, "...the voltage gain of the amplifier circuit MUST be equal to three 'Gain = 3' " seems to be not sufficient "for oscillations to start..." if we assume that the transfer ratio of the Wien network cannot be more than 1/3.

No separate external input need be given to an Oscillator . The Omnipresent Random Noise being Non Sinusoidal contains Sinusoids of different Frequencies. The Frequency to which the Oscillator is tuned selects the appropriate Frequency from the Random Noise as the input.. Earlier we used to call it as Parametric input or so.

Also the Thermal Noise generated by the Resistors like Rb etc.given by Prof Cyril in his figure adds to the Random Noise.

For the second part of the Question, I think, the following Link on the subject using Pspice Simulation gives the answer

http://www.ecircuitcenter.com/Circuits/opwien/opwien.htm

What’s needed is a mechanism to guarantee oscillations will start (GAIN > 3), yet, limit the gain (GAIN=3) at steady state. Enter our heros - D1, D2 and R12. The circuit adjusts its gain depending on the signal level. For small signals, the diodes do not conduct and the gain is set by

Gain=(1+((R11+R12) / R10 )) >3

For larger signals, the voltage across R12 is big enough to make D1 and D2 conduct. The shunt resistance of the conducting diodes effectively reduces the R12 resistance, consequently, reducing the overall gain to GAIN=3.

I think the Discussions are becoming too lengthy making it difficult even to locate our own answers and links given earlier by us.

So,I suggest that Further Discussions may be taken as a new topic with connection to this.

Regards

Dr.P.S.

.

Quote Dr. P.S.: "I think the Discussions are becoming too lengthy making it difficult even to our own answers and links given earlier by us."

My answer: Yes - most probably, you are right. A new topic should concentrate upon the most important question that needs to be answered. But I don`t know what this can be.

For my own purposes, I have enough approaches for explaining the principle of harmonic oscillators.

Quote Dr. P.S.: "The Omnipresent Random Noise being Non Sinusoidal contains Sinusoids of different Frequencies. The Frequency to which the Oscillator is tuned selects the appropriate Frequency from the Random Noise as the input."

My answer: I don`t think that this description tells the full truth (I know that several textbooks contain similar explanations).

About 5 days ago I have expressed my opinion within this topic: The start-up process must not be explained based on the assumption that the noise spectrum contains a suitable frequency component that is amplified again and again....

This explanation is based on the (false) assumption that the circuit works already with the proper loop gain (slightly larger than unity) - that means that it would work already under steady-state conditions. And this is obviously not the case.

Supported by some simulation results I have tried to demonstrate that the power switch-on transient causes a relatively large impuls. Then, the classical impulse response function of the network serves as start of self-sustained oscillations.

For me there is no more beautiful thing in the world from the new idea – to come up with your own new idea or to explain an existing else’s great idea (in my opinion, the latter is no small achievement than the first).

Below, I have tried to reveal finally the great idea behind the legendary Wien bridge oscillator. I have done it in an intuitive manner in the time domain at the lowest level of thinking.

GENERAL IDEA. RC oscillators are based on a continuous charging and discharging of a capacitive storage element. While in relaxation oscillators it is made by switching a constant voltage, in harmonic oscillators (such as the Wien bridge oscillator) it is made by a smoothly changing voltage that leads the voltage across the storage element. This voltage is created by means of a "self-reinforcing positive feedback" (K > 1).

STRUCTURE. To understand a circuit, we have first to realize its structure. For this purpose, we have "to see the forest for the trees", i.e. to group the particular elements in well-known functional blocks. Thus, in the Wien bridge oscillator, we may discern the Wien bridge circuit (Rf, Rb, R1, C1, R2, C2) and to consider the whole circuit as a combination of an op-amp and a Wien bridge connected in the positive feedback loop between the op-amp output and its differential input. The loop gain is a product of the very high op-amp gain and the very low bridge ratio. At the oscillating frequency, the bridge is slightly unbalanced and has a very small transfer ratio; so, the loop gain is about unity.

If you prefer, you may break down the Wien bridge into two half-bridges, and consider the overall feedback as composed of two partial feedbacks - a nonlinear negative feedback (the voltage divider Rb-Rf connected to the inverting op-amp input) and a frequency-dependent positive feedback (the Wien network connected to the non-inverting input). Thus the feedback voltage applied to the op-amp differential input is the difference between the two partial voltages.

OPERATION. There are two requirements to an oscillating circuit: first, it has to run (during the start-up); then, it has to oscillate continuosly (during the steady state).

START-UP. I share the Lutz von Wangenheim’s speculation that the initial impuls caused by the power switch on excites the oscillations. Let’s, for concreteness, imagine that in the first moment after switching on the op-amp output voltage “jumps” to the positive rail (+Vcc); of course, with the same success, it could “jump” to the negative rail (this depends on which of the two power supplies will appear first). The voltage Vn at the inverting input immediately “jumps” to less than 1/3Vcc (since the bulb is still cold and its resistance is small) and then, under the action of the negative feedback, becomes equal to the small voltage across C1 (C1 manages to charge to this voltage by the short current impulse through the Wien network while the output voltage is near +Vcc). Then the self-reinfosing positive feedback (the gain along the loop between the output and the non-inverting input has to be > 1) begins to act through the Wien network and makes the voltage Vp at the non-inverting input continuously increasing.

STEADY STATE. We can separate ten typical phases in forming one period of the sine wave: increasing -> delaying -> stopping -> reversing -> accelerating -> decreasing -> delaying -> stopping -> reversing -> accelerating... and so on and so forth... But maybe, it is useful first to consider how the op-amp output voltage Vout depends on the input voltages Vp (at the non-inverting input) and Vn (at the inverting input).

If Vp > Vn (e.g., Vp = 5.1 V and Vn = 5 V, or Vp = -4.9 V and Vn = -5 V), Vout begins vigorously increasing thus “moving” towards the positive supply rail and finally settles somewhere in the positive area.

If Vp < Vn (e.g., Vp = 5 V and Vn = 5.1 V, or Vp = -5.1 V and Vn = -5 V), Vout begins decreasing thus “moving” towards the negative supply rail and finally settles somewhere in the negative area.

If Vp = Vn (e.g., Vp = 5 V and Vn = 5 V, or Vp = -5 V and Vn = -5 V), Vout begins “moving” towards the ground and finally becomes zero.

1. INCREASING (Vp >> Vn). I mean the middle region between the bottom and top peaks. Here the bulb is still cold and its resistance is small, so the transfer ratio Kp of the Wien network is bigger than the transfer ratio Kn of the non-linear voltage divider (Kp > Kn). The loop gain (between the single-ended op-amp output and the differential input) is close to but a little more than one and, as a result, there is a “self-reinforcing positive feedback”. This means that the voltage of some point along the loop (e.g., the non-inverting op-amp input) is amplified and applied again to this point; so, the output voltage continuously increases. But this self-reinforcing process is not uncontrollable as in the case of the noninverting Schmitt trigger where the op-amp switches in an avalanche like manner. Here it is controllable (held) by C1. This is the key point of this phase of "moving" - the output voltage slightly exceeds the needed voltage so that the voltage across C1 (and, after an amplification, the very output voltage) gradually but quickly increases and this process is held by the integrating lower part of the Wien network. Figuratively speaking, the op-amp output "pulls up", via the upper part of the Wien network, the voltage at the lower part of the network. This state of continuous increasing is stable; the voltage "movement" cannot change its direction. So, until moving, the system is stable; it is "dynamically stable"; it is “moving” confidently towards the positive rail...

https://www.researchgate.net/file.PostFileLoader.html?id=521b1bc3cf57d7ed6188ba79&key=e0b49521b1bc22822a

2. SLOWING (Vp > Vn). At some moment, the bulb begins increasing its resistance and the nonlinear negative feedback begins decreasing the loop gain (it is still above 1 but it is very close to 1). The output voltage produced is less than in the previous phase (but still above the needed magnitude). The voltage "movement" slows up its "speed" and the curve begins rounding. This state continues to be "dynamically stable".

https://www.researchgate.net/file.PostFileLoader.html?id=521b1c17cf57d74d628721d4&key=e0b49521b1c17177fa

3. STOPPING (Vp = Vn). At the top of the half-sine wave, the nonlinear negative feedback decreases the gain and the loop gain becomes exactly 1. There is no self-regeneration and the voltage "movement" stops. This state is stable only if the voltage continues to increase; it is not stable if the voltage begins decreasing... and exactly this is our case. The voltage at the noninverting input begins decreasing since: first, the upper capacitor is almost charged and the current through it decreases; second, the bottom capacitor C1 discharges through the resistor R1.

https://www.researchgate.net/file.PostFileLoader.html?id=521b1c67cf57d711721dd02d&key=e0b49521b1c669c64d

4. REVERSING (Vp < Vn). So, at the top, the output voltage reverses the direction of its "movement" and begins decreasing “moving” towards the negative rail. The upper charged capacitor C2 conveys the output voltage change. The nonlinear negative feedback slowly increases the gain and the loop gain becomes again more than 1. The same self-reinforcing process starts again but now downwards.

https://www.researchgate.net/file.PostFileLoader.html?id=521b1d31d4c118ef6e27f70e&key=60b7d521b1d30db1f4

5. ACCELERATING (Vp < Vn). The voltage "movement" pick up speed and forms the round right part of the half-sine wave. The loop gain is very close to 1; the state is "dynamically stable".

https://www.researchgate.net/file.PostFileLoader.html?id=521b1d6ad4c1180f6f0ed34c&key=60b7d521b1d69dbe01

6. DECREASING (Vp

7. SLOWING (Vp < Vn). At some moment, the bulb begins increasing its resistance and the nonlinear negative feedback begins decreasing the amp gain (it is still above 1 but it is very close to 1). The output voltage produced is less than in the previous phase (but still above the needed magnitude). The voltage "movement" slows up its "speed" and the curve begins rounding. This state continues to be "dynamically stable".

https://www.researchgate.net/file.PostFileLoader.html?id=521b1dd3d11b8bfb20039968&key=3deec521b1dd307692

8. STOPPING (Vp = Vn). At the bottom of the half-sine wave, the nonlinear negative feedback decreases the gain and the loop gain becomes exactly 1. There is no self-regeneration and the voltage "movement" stops. This state is stable only if the voltage continues to decrease; it is not stable if the voltage begins increasing... and exactly this is our case. The voltage at the noninverting input begins increasing.

https://www.researchgate.net/file.PostFileLoader.html?id=521b1e4bd4c1189d72d41668&key=60b7d521b1e4aa2caf

9. REVERSING (Vp > Vn). So, at the bottom, the output voltage reverses the direction of its "movement" and begins increasing “moving” towards the positive rail. The upper charged capacitor C2 conveys the output voltage change. The nonlinear negative feedback slowly increases the gain and the loop gain becomes again more than 1. The same self-reinforcing process starts but again upwards.

https://www.researchgate.net/file.PostFileLoader.html?id=521b1e8bd4c118106fb97ed7&key=60b7d521b1e8a9a240

10. ACCELERATING (Vp > Vn). ). The voltage "movement" pick up speed and forms the round right part of the half-sine wave. The loop gain is very close to 1; the state is "dynamically stable". The output voltage surges upward the positive rail... and so on and so forth...

https://www.researchgate.net/file.PostFileLoader.html?id=521b1ed4d4c118ed6e0a429a&key=60b7d521b1ed390295

Lutz, I am very happy because I was finally able to truly understand how it is made the "movement" in the right direction, the "reverse" at the peaks and the "movement" in the opposite direction in all these oscillating circuits... Cyril

Hi Cyril - congratulations!

A very nice and detailed analysis of the system in the time domain.

It would be very interesting if you could do a similar analysis for a symmetrical circuit modification that has been proofed NOT able to oscillate continuously - in spite of the fact that the Barkhausen criterion is fulfilled.

In this context, I make reference to version 6 in Table 1 of my article on a "rigorous oscillation criterion" (crosswise exchange of the bridge elements and - at the same time - exchange of both opamp inputs).

As far as I know, up to now only the frequency domain has been analyzed in the literature.

Regards

Lutz

Hi Lutz! Thank you for the reaction although I am not sure if you have really seen something new in this "essay" or you wrote it by courtesy:-) I say this because, for completeness, I have repeated much of the old explanations above and it is not well seen what exactly is the new...

You know very well that I would say, "I have understood this circuit", if I REALLY have understood it at the lowest intuitive level (where there is no a lower level beneath it, there is no simpler explanation). Here I had some problems with understanding how the op-amp output voltage goes down after the reversal... and what will drive Vp to change when the output voltage crosses the "zero line".... To overcome them, as strange as it may seem, I had to imagine how the op-amp output voltage depends on the two particular input voltages Vp and Vn when they change in various combinations. For this purpose, I began simultaneously "moving" them (common mode) and slightly wiggle them in each position (differential mode) of the entire "voltage space" between the negative and positive rail and observing how the output voltage was changing ("moving")...

Finally (it was today), I "saw the light in the tunnel" when I realized that at the moment when Vp and Vn reached the top and Vp began going down below Vn, they both were positive but the op-amp output changed downwards the negative rail... this was its place... its final goal... And v.v., at the moment when Vp and Vn reached the bottom and Vp began going up above Vn, they both were negative but the op-amp output changed downwards the positive rail... its final goal... This was the mechanism ensuring the passing of the zero line... and this continuous "pulling" of Vp by Vout giving this endless "voltage movement"... what is the other name of the oscillation (to produce an oscillation is nothing else than to ensure an endless movement...)

Maybe, all this "mediation" looks quite strange for "normal" (non-thinking but "all knowing") people... but I needed it... just because I deeply understand circuits and explain them to students while the usual practice is just to know something about circuits and to say it to students... and because for me teaching is an interpretation but not a simple reproduction of knowledge... That was the reason to try more than 70 years after the Hewlett's invention to look for a real explanation of its oscillator.

Lutz, your questions and the article are clever... but I had first to clarify the basic truths about oscillating circuits... Now its time maybe first to answer your main question about the existence of a both necessary and sufficient condition... Can you extract something useful for this purpose from my intuitive explanations?

Best regards, Cyril

Quote: "That was the reason to try more than 70 years after the Wien invention to look for a real explanation of the oscillator..."

May I add "...during start-up phase - strictly in the time domain" ?

I think, this was the main advantage of your contribution.

The WIEN oscillator (like all analog oscillators) is relatively simple to explain in the frequency domain (Loop gain real and unity at the desired frequency).

However, this applies to steady-state conditions only.

The start-up process needs a separate consideration - and , mostly, the given explanation is based on thermal noise. But this is not correct for my opinion, because this approach assumes already a loop gain slightly larger than unity which, however, does NOT exist from the beginning (t=0).

Thus, a time domain analysis is necessary - similar to the approach you have used in your analysis.

Hi Lutz,

I have begun writing an incredible story (like a fairy circuit tale:) about the creation of RC sinusoidal oscillations. At the same time, I began to think about your fundamental questions and your "reverse Wien oscillator" ("a symmetrical circuit modification that has been proofed NOT able to oscillate continuously - in spite of the fact that the Barkhausen criterion is fulfilled"). I have already some explanation of the latter based on the comparison between the two systems - standard and reversed, at the moments of the peaks.

In the case of the standard system, when the output voltage reaches the peak and stops changing, the output voltage of the Wien network begins decreasing (due to the discharge through the shunt "pull-down" resistor) and the positive feedback reinforces this process. So the system is unstable towards the ground and it reverses the direction of the movement..

Contrary, in the case of the inverse system (Wien network), at the peaks the output voltage of the Wien network continues to increase (again due to the discharge through the "pull-up" shunt resistor) and the positive feedback reinforces this process but now towards the supply rail. So the system does not show a tendency to reverse and, I suppose, it stays at the rail (like a latch).

It seems, the RC network connected in the positive feedback loop of a sine oscillator has to ensure (at the peaks) an instability towards the opposite direction of the movement. Maybe this is a possible sufficient condition - the system has to be "dynamically stable" until moving towards the peaks and to be "static unstable" (only in the opposite direction) at the peaks? I continue thinking about your questions...

Regards, Cyril