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I am just into oscillators where I learned \$AB=1\$ for sustaining oscillations in positive feedback. Since \$A\$ and \$B\$ are both frequency-dependent, \$AB=1\$ is true only for a particular frequency.

  1. What happens to those frequencies for which \$AB>1\$ holds??

  2. Will these frequencies keep on getting amplified until the limiter circuit limits them?

  3. Then why don't we get those frequencies in our output??

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    \$\begingroup\$ I don't think "AB" is standard terminology. I'm guessing it means loop gain? \$\endgroup\$ – Hearth Dec 1 '18 at 13:46
  • \$\begingroup\$ Yes ! sorry about that. \$\endgroup\$ – Souhardya Mondal Dec 1 '18 at 13:54
  • \$\begingroup\$ @Hearth A is gain and B is the feedback fraction. When their product is one the denominator of the transfer function is zero. \$\endgroup\$ – user110971 Dec 1 '18 at 13:55
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    \$\begingroup\$ I think, we should be more specific: When the loop gain approaches unity (real, with zero phase shift) the CLOSED-LOOP function has a denominator that approaches zero. \$\endgroup\$ – LvW Dec 2 '18 at 10:13
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Why do we get only one frequency as output in oscillators?

Oscillators work at one frequency by ensuring two things: -

  • The signal fed back to sustain oscillations is exactly in phase with the signal it is trying to sustain. Think about lightly tapping a swinging pendulum at exactly the right place and, in the right direction.
  • The loop-gain is slightly more than unity. This ensures that a sinewave is produced without too much distortion and it is "sustained". If the loop-gain were less than 1, then it can't "sustain" an oscillation.

So, if we design a phase-shifting network that has a unique phase-shift for each frequency it handles, we will get an oscillator but, only if the signal fed back is sufficient in amplitude to sustain oscillation.

However, some phase shifting networks can produce a phase shift that is a multiple of the basic oscillation frequency. In other words if 1 MHz produces a phase shift of 360 degrees, maybe some higher-frequency might produce 720 degrees (2 x 360). This could potentially give rise to a sustained oscillation at two frequencies (usually deemed undesireable).

So, we design the phase-shifting network to ensure that the higher-frequency "in-phase" candidate is much lower in amplitude than the "basic" candidate and, given that we only allow the gain to be unity or slightly higher (to accommodate losses in the phase shift network) for the frequency we want, the higher-frequency candidate will not cause oscillation.

The above is also referred to as the Barkhausen criteria.

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  • \$\begingroup\$ So what happens to those frequencies which have AB>1?? \$\endgroup\$ – Souhardya Mondal Dec 1 '18 at 16:33
  • \$\begingroup\$ They won't sustain oscillation because they do not feedback a signal that is precisely in phase. Think of a pendulum; if you hit it (slightly) at exactly the point where it is starting to swing back you would not effect it's oscillating frequency AND you would sustain oscillations. \$\endgroup\$ – Andy aka Dec 1 '18 at 16:36
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    \$\begingroup\$ If the gain is greater than just more than unity at the oscillation frequency then the oscillator’s amplitude rises until it can’t rise any more because of supply voltage limitations or slew rate limitations. In other words the amplitude clips. \$\endgroup\$ – Andy aka Dec 1 '18 at 18:29
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So what happens to those frequencies which have AB>1??

Saturation.

Let's say there are several frequencies with loop gain \$AB \ge 1\$ and \$n2\pi\$ phase shift, but let's call the one with the highest loop gain \$f_x\$. For \$f_x\$, \$AB > 1\$ and you might expect it to produce an oscillation with amplitude increasing in time. But no real circuit can have its output increase in amplitude indefinitely. There is usually some saturation behavior that limits the output amplitude.

And when this happens, it tends to reduce the gain for all frequencies, not just the one that had the super-unity loop gain. So accounting for saturation, this frequency \$f_x\$ will end up with \$AB=1\$ and all the other frequencies that linear analysis told you had \$AB \ge 1\$ but less than at \$f_x\$, now have \$AB < 1\$, so they no longer oscillate indefinitely.

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  • \$\begingroup\$ The Photon...may I ask you: Have you ever seen a circuitry with a real loop gain >1 (zero phase !) at "several frequencies"? More than that, I have some severe doubts if the 1st sentence of the second paragraph in your answer is correct. The loop gain is frequency-dependent - and when it is reduced (due to some non-linearity) at one frequency, it will not automatically reduced for other frequencies with other amplitudes (because the non-linearity is by its nature amplitude-dependent). \$\endgroup\$ – LvW Dec 1 '18 at 21:25
  • \$\begingroup\$ I suppose you are referring to integrator-based oscillators, correct? But the magnitude condition is fulfilled at one single frequency only!! \$\endgroup\$ – LvW Dec 1 '18 at 21:33
  • \$\begingroup\$ I wonder if it’s viable to design an oscillator that naturally oscillates at two different sinewave frequencies? Reading your answer it shouldn’t be possible by mistakingly trying to design a regular oscillator but it has got me thinking. I think I remember messing around with a transmission line oscillator that produced two disparate sinewaves but I didn’t explore things. \$\endgroup\$ – Andy aka Dec 2 '18 at 12:36
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A short answer from my side:

You must not think in magnitude terms only. Don`t forget the phase. The product AB must be a REAL one . A frequency-selective circuitry has a magnitude as well as a phase which is a function of frequency. And - for a correct design - there will be only one single frequency which can fulfill both conditions at the same time (Barkhausens oscillation criterion with loop gain AB=1):

  • |A*B|=1 (for practical reasons somewhat larger than "1", for example "1.2") and

  • phaseshift exp(j*phi)=1 (phi=0).

For this purpose, most known oscillators use lowpass, highpass or bandpass filters as feedback elements. But there are also other (more advanced) topologies.

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  • \$\begingroup\$ @ LvW Can you provide one or two examples (links) to more advanced topologies? please. \$\endgroup\$ – analogsystemsrf Dec 2 '18 at 2:47
  • \$\begingroup\$ Examples: (a) Two integrators in series (inv./non-inv), (b) notch-filter, (c) double-T-topology, (d) Allpass, (e) Active negative-resistance structure, (f) GIC resonator (FDNR resonance). \$\endgroup\$ – LvW Dec 2 '18 at 8:51
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  • Assuming you mean a classic crystal oscillators (XOs) with a square wave output (either series or parallel mode).

When saturation occurs, the loop gain (GH or AB) drops to zero, except during the linear transition of the output. The crystal acts as a bandpass filter to produce a sine wave at the input which may also contain harmonics, but the slew rate of the square wave output is generally much faster than the sine wave input, so the harmonic energy has insufficient outline linear time to amplify when it is not saturated and the gain is zero, thus suppressed.

More information

  • However in linear oscillators the harmonic content may contribute to phase noise, so those with the lowest phase noise have the highest Q at the fundamental, such as SC-cut crystals e.g. 10 MHz oven-controlled crystal oscillators (OCXOs) vs. standard AT cuts commonly used everywhere. That's all I'll say about this for now.

However, for smaller crystal structures >= 33 MHz resonance the gain of the harmonics tends to be higher than the fundamental. Thus you will find these classified as "overtone crystals".

For CMOS feedback oscillators, often a series R (3 kΩ ~ 10 kΩ) from the output is used to limited uW power dissipation in microslice crystals AND in high frequency >> 10 MHz also create additional attenuation of harmonics from the RC effects with the first load capacitor. The most common is third harmonic or "overtone", but higher overtones are used >> 150 MHz.

But when selective harmonics are desired for oscillation (3, 5, 7, etc.) then either how the crystal is processed or additional passive LC tuning helps to boost the harmonic of choice.

The most common warning for XO designs "Never use a buffered inverter" (three linear gain stages vs. one) to avoid amplification of spurious harmonics. When they saturate the inverter and the gain drops to zero, they suppress the fundamental frequency except for a short transition interval. They can behave like an injection locked loop (ILL) where it may randomly oscillate at the fundamental or harmonic depending relative gains and startup conditions. But with a buffered inverter there is more chance during the output transition time to cause spurious harmonic glitches on the transitions and lock onto harmonics.

However, those who successfully used a buffered inverter (myself included) for an XO can now understand that the type of crystal and relative lower gain of the harmonic protected the XO from locking onto the desired fundamental frequency. In some cases, this can be an advantage, but that's a different question.

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Although all answers are correct, I believe these are all missing the spirit of your question.

The term "oscillator" generally applies to a circuit specifically designed to produce an AC waveform at a specific frequency. This entails some design choices intended to minimize unwanted effects. This is particularly true for linear oscillators (which is the loop-gain case stated in your question).

You specifically design the gain to be slightly larger than 1 at a specific frequency and you design/rely on non-linearities in the system to keep the oscillation stable. If you allow the gain to be much larger than 1 then you stop having a linear oscillator.

However, this useful engineering simplification comes from having the loop gain only be slightly larger than one which allows you to treat it as a linear oscillator, when in reality it is not. What you actually have is the simplified border case of a non-linear dynamical system with a stable periodic orbit that approaches a sinusoid.

If you further develop that dynamical system (for example by making AB>>1) you can reach another extreme, a very non-linear but stable relaxation oscillator or in intermediate cases you will find a period doubling sequence that creates a chaotic oscillator such as Chua's circuit or a Van Der Pol oscillator.

This image is from an implementation of Chua's circuit you can see that it behaves somewhat as a combination relaxation oscillator/linear oscillator. But the "relaxation component" is non-periodic and long-term unpredictable.

Chua's circuit waveforms

There are uses for all of those alternatives, but linear oscillator theory specifically stays away from those conditions.

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    \$\begingroup\$ Relaxation Effects are due to nonlinear negative resistance also such as tiny hysteresis which is positive AC feedback together with DC negative feedback. This effect is common in cascaded Buck PWM then Boost-PFM converter control system noise, an example causing chaos theory noise. \$\endgroup\$ – Tony Stewart Sunnyskyguy EE75 Dec 1 '18 at 18:50
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    \$\begingroup\$ @TonyEErocketscientist those are all "conceptual buckets" that we use to make the effects easier to understand, analyze, and design around. But in reality these are all particular cases of more generalized non-linear dynamical systems. Note that you can tune Chua's circuit to present all of those behaviors just by tweaking the non-linear element characteristics. \$\endgroup\$ – Edgar Brown Dec 1 '18 at 18:55
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    \$\begingroup\$ Sorry, which examples are "those". Not familiar with Chua's publications except for the name, as I discovered how to make stable-linear low THD sine oscillators with nonlinear ccts long before Chua in the 70's. My 90's cascade Buck>Boost inductor sounded like bubbling water in the lab from the piezo acoustics, until I fixed it. \$\endgroup\$ – Tony Stewart Sunnyskyguy EE75 Dec 1 '18 at 19:11
  • \$\begingroup\$ @TonyEErocketscientist all behaviors from linear oscillator, through chaotic oscillator, to relaxation oscillator. Although Chua's circuit is the simplest possible physical dynamical system to produce chaos, in essence it is nothing more than a third-order transfer function attached to a non-linear negative resistor. \$\endgroup\$ – Edgar Brown Dec 1 '18 at 19:17
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    \$\begingroup\$ Yes of course. I see. Because all behaviours with harmonics are due to nonlinearities, even the structural properties of Xtals, buildings. So the "conceptual buckets" refers to specific linear approximations. I have learnt how to use linear piecewise approximations for non-linear theory for good uses like when the bulk Rs ( or ESR as I call it) exceeds the nonlinear incremental resistance with rising current in LED's or the soft limiting of sine wave Osc. to attenuate harmonics and raise Q as gain converges on unity. \$\endgroup\$ – Tony Stewart Sunnyskyguy EE75 Dec 1 '18 at 19:41
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The Barkhausen stability criterion says: \$|A\ \beta| = 1\$ and \$\angle A\ \beta = 0\$.

Where \$A\$ = amplifier gain,

and \$\beta\$ = feedback attenuation.

The Barkhausen stability criterion

So if \$|A\ \beta| = 1\$, then the oscillator will be stable. The feedback loop feeds a portion of the output \$v_o\$ back to the input, \$v_f\$. The amplifier amplifies the input \$v_f\$ to make a larger output, \$v_o\$.

If \$|A\ \beta| > 1\$, then the oscillator will drive itself into saturation and clip the output waveform. The amplifier is an operational amplifier with \$\pm\$ power rails. The amplifier cannot drive the output past the power rails.

Unstable Oscillator

The gain and attenuation are not stable and the amplifier output increases to the power rails of the amplifier. If it is a sine wave oscillator, the output increases until the amplifier saturates, and it is no longer a sine wave. Tops get clipped.

If \$|A\ \beta| < 1\$ the oscillations will fade out. This is called damping.

Damping

Given that: an oscillator circuit is designed to oscillate at a fixed frequency if \$|A\ \beta| = 1\$ and \$\angle A\ \beta = 0\$ (phase angle is 0\$^\circ\$).

So the crux of your question is: Why don't oscillators oscillate at other frequencies? This is governed by the components used (resistors, capacitors, inductors, and amplifiers).

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    \$\begingroup\$ Yes - I agree to the (short) explanation. Just one further comment: Because it is impossible to achieve per design a loop gain that is exactly "one", we always realize a loop gain somewhat larger than "1" (at one single frequency) and make use of an automatic amplitude regulation mechanism (non-linearity, diodes , NTC, FET as resistor,...), which can bring the loop gain back to "1" before clipping occurs. \$\endgroup\$ – LvW Dec 2 '18 at 10:18
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    \$\begingroup\$ @LvW I'd go for simple. When I answer a question, I look at 4 criteria. Can I answer the question? What is the level of OP? Do answers meet that level? What misconceptions must be corrected? We have an OP who doesn’t know his Greek \$\beta\$ from B. So I create a simple answer with graphics to not overload the cognitive load of the OP. Then I compete with better quality answers to see if I can connect with the OP. I'm happy if the OP understands \$\beta\$. \$\endgroup\$ – StainlessSteelRat Dec 2 '18 at 22:49
  • \$\begingroup\$ Again - I agree to your approach for answering questions like this. May I add another comment (correction)? The criterion named after Heinrich Barkhausen is not a "stability criterion" (such a criterion was formulated by Strecker and in parallel by Nyquist). Barkhausens criterion is a so called "oscillation condition" - to be more specific: It is just a "necessary" condition for a circuit to oscillate - not a sufficient one (Wikipedia is not always correct). \$\endgroup\$ – LvW Dec 3 '18 at 8:07
  • \$\begingroup\$ @LvW Thanks. That was an edit by another. I'll tweak it. I'm not a fan of wiki links. \$\endgroup\$ – StainlessSteelRat Dec 3 '18 at 8:14

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