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I am studying the textbook Fundamental of electric circuit by Charles and Sadiku. In section 10.9.2, the textbook discusses oscillator and uses Wien bridge oscillator as an example. I was not able to understand how the harmonic oscillations are generated and how they are alimented by the dc source. I searched google, wikipedia, and particularly, youtube, but cannot find a simple discussion on how it works.

I got the following "physical pictures" via the search:

(1) The small oscillations originates from some random fluctuations.

(2) The signal is amplified by the operational amplifier and then fed back to the source. This is the same physical content why the speakers howl when a microphone comes to close to the speaker boxes.

(3) Usually the signal passes through a frequency filter so that only the desired frequency is left. I kinda understand that in the case of Wien bridge oscillator it is done by the condition that the feed back oscillation has the same phase, and therefore it gets strengthened instead of getting cancelled out).

(4) On the other hand, the energy conservation must be satisfied. In this case it is the dc source that actually feeds the ac oscillation.

The circuit for Wien bridge oscillator

Now, my question is the following.

(1) If one only considers the steady state solution (the dc solution of the problem) I understand it is a simple circuit for an inverting amplifier. This is because in this case the capacitors simply correspond to open circuit.

(2) Now the solution of a non-homogeneous differential equation (or maybe a set of equations) is that a special solution of the non-homogeneous equation (with source) plus the general solution of the corresponding homogeneous equation (without source). Since the above steady state solution is clearly the special solution, we only need to find the general solution.

(3) I would guess the general solution shall be featured by a characteristic frequency determined by the second Barkhausen criteria (Eq.(10.12) in the textbook. However, since there is no source, my intuition implies that the general solution decays in time (maybe with the characteristic frequency decays the slowest). It is noted that the notion of impedance come from the special solution of the non-homogeneous differential equation with a source of given frequency in the first place.

(4) So what is the physics explanation for that small oscillation is amplified by the inverting amplifier circuit and is alimented by the dc source. Also, I simply do not see how the amplified signal is fed back to the source. In addition, what is wrong in the above discussion.

If possible, please also point me to some textbook with details, many thanks in advance!

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Edit: following ThreePhaseEel, Steward and Whit3rd's answers and comments, I came up with the following additional notes.

(0) I do not like the explanation in terms of feedback. Though it is intuitive, but such approach becomes even more obscure to me from a mathematical point of view. Since we do not solve the equation iteratively, it is essentially a problem of ordinary differential equation, and from math class we know how to solve such equations. So at least, some explanations alone this line of thought is desirable.

(1) The general solution of the homogeneous equation indeed dies out. For instance, if the corresponding differential equation is of second order, it possesses two characteristic frequencies. Since it dies out, this solution is not really relevant in practice.

(2) There is always harmonic (white) noises with different frequencies, so the dc source is in fact a ac source with different frequencies. The dc part of the source can be viewed as a part with null frequency.

(3) Now, the two sections with capacitors can be viewed as frequency filters, this page provides a very intuitive explanation on this point. As a results, the frequency which sees the smallest impedance is the one called resonance frequency.

(4) Since all these are related to the solution (temporal, not steady state) of the non-homogeneous equation, other frequencies do not actually die out in time (as the transient solution, the general solution of the corresponding homogeneous equation). Instead, they are much suppressed by the high-frequency and low-frequency filters, and therefore does not effectively play any role. It is noted that till this point the role of the inverse amplifier has not be discussed yet, which does play an essential role as in the analysis from the feedback circuit viewpoint.

(5) According to the first Barkhausen criteria, the ratio of the two resistors in the invert amplifier feed back must match the gain obtained from the second Barkhausen criteria. This is the ideal case for the oscillator to function properly. On the contrary, there may not be any solution of Kirchhoff equations for a given frequency, (because there is no current in the op-amp, it then implies a voltage difference.) However, a solution may still be obtained by considering a continuous frequency spectrum, this may corresponds to those square wave output as mentioned.

(6) It is indeed appreciable if any analytic study of the Wien bridge oscillator can be found in some textbook.

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  • \$\begingroup\$ See this question: Working of Wien Bridge Oscillator \$\endgroup\$ – KingDuken Aug 6 '17 at 22:44
  • \$\begingroup\$ Thx! I found this link electronics-tutorials.ws/oscillator/wien_bridge.html interesting, but I still am confused. \$\endgroup\$ – gamebm Aug 6 '17 at 23:52
  • \$\begingroup\$ so you got \$R_f+R_b \over rb\$ setting the gain and \$Z_1 + Z_2 \over Z_2\$ setting the frequency response curve, solve the latter with respect to frequency, then the correct operating point of the former will be obvious, \$\endgroup\$ – Jasen Aug 7 '17 at 0:21
  • \$\begingroup\$ Yes, I do follow the math which led to those expressions. But my question is about the physics. To start, if we do not have a ac source, it does not even make any sense to talk about impedance (for a given frequency) -- such concept is from the special solution of differential equation with ac source in the first place. :) \$\endgroup\$ – gamebm Aug 7 '17 at 0:34
  • \$\begingroup\$ The physics is: it's like a whistle. The random rush of air through the orifice powers it, and it's semistable only at the whistle frequency, and (because of the filament heating) always ends up at the amplitude where any-louder just turns down the volume control. \$\endgroup\$ – Whit3rd Aug 7 '17 at 1:37
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Startup can come from several things

In general, there is going to be a source out there for the initial excitation of an oscillator's feedback loop.

  • The power-on ramp could get coupled into the output of the amplifier slightly, providing an edge that excites the feedback network
  • Some noise on the output lead could excite the feedback network at the right frequency to pass through to the input
  • Some noise could show up at the op-amp input and be amplified to be filtered down to the right frequency by the feedback network

Keep in mind that everything, even a humble resistor, has inherent noise sources when you're doing this, by the way!

The AC signal can be coupled back just fine for feedback

While DC can't pass through C2, AC can get through it just fine, and that's what matters for oscillator feedback.

As to that lamp? It's an AGC system

The incandescent lamp in the Wien-bridge oscillator is a gain control mechanism -- at startup, it's cold and offers a low resistance, causing the amp to have high gain and amplify tiny noises greatly. As it heats up from its own current draw, its resistance increases, causing the gain of the amp to drop significantly and stabilizing the oscillator at an even output level.

Likewise, if the output increases, the voltage across the lamp will increase due to voltage divider action, causing the lamp to heat up more and the gain to drop. If the output decreases, the voltage across the lamp will decrease, allowing the lamp to cool slightly and the gain to rise. It is a highly elegant system that is actually quite annoyingly hard to replicate using solid-state parts.

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  • \$\begingroup\$ Thx for the answer, so I understand the noise serves as a part of the dc source signal came together as a package. The gain at the beginning, before the system stabilizes, is huge so the noise is amplified. But I still would like follow my "solve the differential equation" line of thought, and added few comments to my original question. Please let me know it is correct to think that way. \$\endgroup\$ – gamebm Aug 6 '17 at 23:57
  • \$\begingroup\$ The analysis in terms of feedback, to me, is to solve a question using iteration, like a sum 1+1/2+1/4+1/8+...., it may or may not apply rigorously. \$\endgroup\$ – gamebm Aug 7 '17 at 0:30
  • \$\begingroup\$ if Vout/Vin is 1 and (Vout) feeds back to the source (Vin), then intuitively, we have 1+1+1+1+... in this case, which is obviously divergent. In this context, it is very difficult for me to understand the reasoning. \$\endgroup\$ – gamebm Aug 7 '17 at 0:50
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The Wien Bridge Oscillator is my least favorite and it works poorly as a stable sine oscillator , but it works.

How does it work?

At steady state (SS) both +/- inputs are matched signals equal to the R Ratio divider relative to the output and ground. For this to oscillate the inverting R ratio gain must be > 1 Then then impedance ratio of the positive feedback at 0 deg phase shift must match this ratio to get the same amplitude and same phase.

Rb can be a PTC or tiny bulb regulates the AC gain by heating up with the Rf Current . Since the current is low the R ratio must be only slightly greater than the positive feedback ratio so it doesn't have to heat up much. The output level will modulate over many thermal time constants until stabilizes. It starts with excess gain like a square wave then heats up the PTC with ~100mW or so to elevate the R and reduce the gain.

Here is a 250 Ohm PTC in a Wien Bridge Osc ( see image at bottom). The phase lead HPF is shunted by the phase lag LPF and when the impedance ratio matches the Rf/Rb -ve side , at 0 deg phase shift, that is the frequency it should oscillate at.

The original bridge applied a signal across Uwe and Uwy had zero across it, meaning matched signals. \$\omega^2=\dfrac{1}{R_xC_xR_2C_2}\$ ... if one chooses \$R_2 = R_x \text { and }C_2 = C_x\$; the result is \$R_4 = 2*R_3\$.

In your case, the output is at the bottom, top is 0V or ground and the differential Amp inputs (-+) are across Uwy .

enter image description here By Zureks - Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=6790678

The initial conditions for oscillation start with a DC input offset and minor thermal noise.

Note like in all bridges the impedance matching is essential for linear operation.

enter image description here

Due to the instabilities and low Q ( wide band noise for phase, frequency and amplitude I might call it the Wein Not Osc.

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  • \$\begingroup\$ Thx for the answer and sorry for not able to thumb up the posts of you guys. But would you please comments/correct directly what I was writing in the question :) The thing is, probably, I simply do not know some of the assumptions/facts you think to be trivial. \$\endgroup\$ – gamebm Aug 7 '17 at 0:28
  • \$\begingroup\$ hmm. a) inv gain must be >=-1 input DC offset is the initial step input which contains all spectrum for this low Q BPF. b) linear impedance equations apply. c) second Barkhausen criteria applies with slightly 1 gain at startup then stabilizes after many thermal oscillations to exactly unity gain if close, otherwise no osc. if too much gain, its a square wave. \$\endgroup\$ – Tony Stewart Sunnyskyguy EE75 Aug 7 '17 at 0:37
  • \$\begingroup\$ It is clear now I misunderstand something. Are there any source I can study which discusses your points a) and c) in detail? \$\endgroup\$ – gamebm Aug 7 '17 at 0:43
  • \$\begingroup\$ did you try my simulation link? \$\endgroup\$ – Tony Stewart Sunnyskyguy EE75 Aug 7 '17 at 0:49
  • \$\begingroup\$ It looks fun :) It seems other frequencies (except the resonance one) die out in time, but I don't understand why?! :( \$\endgroup\$ – gamebm Aug 7 '17 at 0:53

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