I thought the simulation post that compared the small signal square wave response (to about 1V at 10kHz superimposed on a slow ramp from zero up to the rail voltage) of a switching UcD to the non switching equivalent (all schematic components the same, just lower gain) was very interesting because it highlighted only the non linear products produced by the switching action. I wonder if Bruno saw that one and if he might have any comments on the frequency droop effect (including mitigating this via higher order phase shift networks in the feedback loop).
Regards -- analogspiceman
PS: Is Michael Andersen aware of leapfrog? From his various papers on switching amplifier feedback schemes, I would guess he might find it somewhat an interesting read
(the three PDFs are attached to the first few posts from the leapfrog thread from a year and a half ago).
I guess you mean currend feedback with a DC component when you say leapfrog. What you have showed is a clever (4 bjt) version of mesuring the current with a DC component. Current feedback in generel and by means of a current sense resistor is nothing new. Michael Andersen and his fellow workers at Oersted DTU are using current feedback in one of their class d designs (tapped inductor, predicted current), but it dosn't have a DC component. So I guess the answer is no, he dosn't know about the leapfrog implementation. They are using current feedback in some of their switchmode power converters (resistor sense) and exploiting the current limiting properties. But no I haven't seen them use it in amplifiers.
You could try to emal Michael A. E. about it at: ma at oersted dot dtu dot dk
By the way does leapfrog provide current limiting when additional parralel THD shaping opamps are added to the control scheme (using voltage feedback)?
How exactly do you set the current threshold? Is opamp saturation/clipping the keyword?
You could try to emal Michael A. E. about it at: ma at oersted dot dtu dot dk
By the way does leapfrog provide current limiting when additional parralel THD shaping opamps are added to the control scheme (using voltage feedback)?
How exactly do you set the current threshold? Is opamp saturation/clipping the keyword?
Leapfrog is much more than simple dc current feedback. It is a method of choosing the optimum amounts of negative and positive voltage and current feedback from key points distributed throughout the output filter structure such that any number of output LC filter sections may be included within the control loop.
If you go back and read the pdfs originally posted at the start of the leapfrog thread, you will see how one can seamlessly limit (with absolutely no overshoot or rail sticking) current and or voltage at any point throughout the filter ladder by simple resistive scaling and Zener limiting of opamp voltage swing.
One of my goals for leapfrog is to run multiple phase staggered, self-oscillating stages in parallel in order to up the effective carrier frequency (as usual, for reduced ripple and increased bandwidth). Unfortunately, it seems that this would either require high side output current sensing or separate power supplies for each of the paralleled half bridge stages (if simple grounded sensing were to be used). This is what motivated the design of the 4 bjt current sense circuit.
Regards -- analogspiceman
Does anybody know a way to calculate the exact switching frequency versus modulation index for a phase-shift modulated self-oscillating amplifier(COM, UCD etc.).
It can be done for hysteretic modulators because of the simple linear carrier.
It can be done for hysteretic modulators because of the simple linear carrier.
Exact switching frequency? - I'd just like to know how you calculate the unmodulated mark/ space ratio of the carrier if you connect a UcD amp to an inductive load! -- Just try coupling a UcD 180 to the primary of an unloaded 100 volt line audio output transfomer - sends the carrier (and the power supply) bananas!
Ahh! -- the wonders of the academic world, versus the real one!!
Ahh! -- the wonders of the academic world, versus the real one!!
😱 Further to above post -- it should read that the Philips UCD demonstration amplifier doesn't like inductive loads. The Hypex UCD180 performs impeccably - as usual!
So apologies to Hypex for unfairly criticising the UCD180.
And apologies for hijacking the thread with a 'only partly relevant' rant 😱
So apologies to Hypex for unfairly criticising the UCD180.
And apologies for hijacking the thread with a 'only partly relevant' rant 😱
With a UcD type circuit, almost all of the circuitry around the feedback loop is linear. The exception is the nonlinear section from the input to the comparator to output of the MOSFET switches. This is where it is problematic to recognize the gain and phase of the signal at the amplifier's switching frequency. Here, these signals aren't sine waves, so how can gain have any meaning in the linear, small signal sense?
The "trick" is to first mentally pick out the fundamental Fourier components of the signals appearing at the comparator and at the output (which is always gain limited via saturation). From this Fourier analysis point of view, the comparator/output stage appears completely linear because there are only perfect, unsaturated sine waves components remaining. At a single frequency, gain and phase always make sense. (IMO, this process is made easiest if delay is pulled out as a separate effect.)
For example, consider the UcD at idle (i.e., no input). Then the signal at the comparator looks more-or-less like a small sine wave and the output signal is a square wave that is (except for a small delay) exactly in phase with the comparator input. But what is gain?
Since the UcD at quiescence is just a power oscillator, neither decaying nor increasing in amplitude, we know that total loop gain must be exactly unity. Now calculate the attenuation around the rest of the loop due to the output filter and other R/C networks before the comparator. The effective gain of the saturating comparator/output stage gain must make up exactly this amount of loss (the output square wave is fixed in size by the power supplies).
The other condition required to build a stable oscillation (now that's an oxymoron) besides unity gain around the loop is reinforcing phase (i.e., 0, 360, 720, etc. degrees). Since loop gain will always self adjust via output saturation at any operating frequency, it must be the loop phase characteristics that control the exact operating frequency. In fact, the UcD is a really a phase-shift power oscillator and its frequency of operation will be exactly at the point where phase shift through the output filter and other R/C networks back to the comparator is 180 degrees (the inverting hook up of the comparator provides the remaining required 180 degrees).
The reason the UcD suffers frequency droop at high level outputs is due to the changing phase of the fundamental Fourier component of the highly asymmetrical pulse waveform required to achieve a large "dc" output. As I recall, this waveform "geometry" phase shift approaches 90 degrees in the limit, so at very high output levels, the UcD will naturally operate at a lower frequency to where the additional phase shift through the output filter and feedback circuits is now only 90 degrees (in theory, anyway). With the UcD, oscillation frequency is all about phase.
Regards -- analogspiceman
I've always used the analogy that the fundamental of the output and the carrier are en phase, but I've never thought about taking the fourier analysis that step further.
So what you say is that I'll have to calculated the phase of the fundamental for a square wave as a function of modulation index and then use it to find the closed-loop oscillation frequency. It really sounds simple. Have you done any calculations on this?
When I look at the carrier for an UDC type modulator, it becomes less and less sinusoidal when the modulation index increases. Do you think that the harmonics might come into play or is everything still covered by the phase of the fundamental?
Regards Kaspar
So what you say is that I'll have to calculated the phase of the fundamental for a square wave as a function of modulation index and then use it to find the closed-loop oscillation frequency. It really sounds simple. Have you done any calculations on this?
When I look at the carrier for an UDC type modulator, it becomes less and less sinusoidal when the modulation index increases. Do you think that the harmonics might come into play or is everything still covered by the phase of the fundamental?
Regards Kaspar
Hmm according to the following page the phase doesn't change as a function of duty-cycle:
http://www.dspguide.com/ch13/4.htm
Ther's only a_n components in ther seris:
a_0=A*d
a_n=2*A/(n*pi)*sin(n*pi*d)
b_n=0
On the other hand this page has resolved ther series with both a_n and b_n components, but I do think that's because they didn't centered the wave before computing the series (in order to eleminate the odd terms).
http://www.engr.uidaho.edu/thompson/courses/ME330/lecture/FourierIntro.html
http://www.dspguide.com/ch13/4.htm
Ther's only a_n components in ther seris:
a_0=A*d
a_n=2*A/(n*pi)*sin(n*pi*d)
b_n=0
On the other hand this page has resolved ther series with both a_n and b_n components, but I do think that's because they didn't centered the wave before computing the series (in order to eleminate the odd terms).
http://www.engr.uidaho.edu/thompson/courses/ME330/lecture/FourierIntro.html
The notion that loop gain at the oscillation frequency is unity, leaves an error of around 7.6 dB (ie it's 7.6dB better than that).
Let's forget about UcD for a sec and move back to oscillator theory.
The oscillation criterion that says that phase shift equals 360º and gain equals 1 goes only for linear oscillators. If loop gain exceeds 1, the oscillation amplitude will increase. At some point in real life, the oscillation self-limits by clipping. Small signal gain has not decreased, but somehow the amplitude no longer increases. It is then that we introduce the notion of large signal gain, and we'll simply say that the large signal gain has become one to "recuperate" the oscillation criterion to describe an oscillator with not very well defined gain and yet stable output amplitude.
Now, see if you can define large signal gain outside the context of oscillators and that will still hold here. Voutrms/Vinrms? Voutpk/Vinpk??? Not, I'm afraid, because you need to know the shape of the signal being amplified. Large signal gain is simply retrofitted in an attempt to make the oscillation criterion apply in a place where it doesn't apply.
But it's a sine wave, isn't it? Well, the oscillator is clipping, do we really think only the fundamental gets fed back.
But then... the frequency? Indeed, in a clipped oscillator the frequency is no longer exactly equal to the 360º point because the harmonics now play a part in determining the oscillation frequency. How much it is off depends on the frequency determining network. In a crystal oscillator it's a few ppm. In an UcD it's off by as much as 10%.
So, throw out the oscillation criterion and its gain, they are defined and valid only in linear oscillators. Phase shift class D amps are patently non-linear.
OK so what is the low-frequency gain of the power-comparator once it's oscillating? The carrier that appears at the comparator replaces the triwave in an ordinary PWM system. One way of describing it is to say that gain equals the supply voltage over the probability density of the carrier. Another way is to say that at either edge, the time shift is dt/dVc (one over slope) of the carrier. Small signal gain becomes (dt/dVcr+dt/dVcf)*(Vdd-Vss)*fsw.
The carrier is not a triwave, so gain increases as the signal level increases. If you're smart with the modulation of fsw (which comes down) you can nearly compensate one with the other.
In the paper I calculate the gain based on an approximation of the carrier as a sine wave, and find that the loop attenuation is 6dB at the oscillation frequency. It's not a sinewave but thank G*d gain becomes even higher by some 1.6dB.
Let's forget about UcD for a sec and move back to oscillator theory.
The oscillation criterion that says that phase shift equals 360º and gain equals 1 goes only for linear oscillators. If loop gain exceeds 1, the oscillation amplitude will increase. At some point in real life, the oscillation self-limits by clipping. Small signal gain has not decreased, but somehow the amplitude no longer increases. It is then that we introduce the notion of large signal gain, and we'll simply say that the large signal gain has become one to "recuperate" the oscillation criterion to describe an oscillator with not very well defined gain and yet stable output amplitude.
Now, see if you can define large signal gain outside the context of oscillators and that will still hold here. Voutrms/Vinrms? Voutpk/Vinpk??? Not, I'm afraid, because you need to know the shape of the signal being amplified. Large signal gain is simply retrofitted in an attempt to make the oscillation criterion apply in a place where it doesn't apply.
But it's a sine wave, isn't it? Well, the oscillator is clipping, do we really think only the fundamental gets fed back.
But then... the frequency? Indeed, in a clipped oscillator the frequency is no longer exactly equal to the 360º point because the harmonics now play a part in determining the oscillation frequency. How much it is off depends on the frequency determining network. In a crystal oscillator it's a few ppm. In an UcD it's off by as much as 10%.
So, throw out the oscillation criterion and its gain, they are defined and valid only in linear oscillators. Phase shift class D amps are patently non-linear.
OK so what is the low-frequency gain of the power-comparator once it's oscillating? The carrier that appears at the comparator replaces the triwave in an ordinary PWM system. One way of describing it is to say that gain equals the supply voltage over the probability density of the carrier. Another way is to say that at either edge, the time shift is dt/dVc (one over slope) of the carrier. Small signal gain becomes (dt/dVcr+dt/dVcf)*(Vdd-Vss)*fsw.
The carrier is not a triwave, so gain increases as the signal level increases. If you're smart with the modulation of fsw (which comes down) you can nearly compensate one with the other.
In the paper I calculate the gain based on an approximation of the carrier as a sine wave, and find that the loop attenuation is 6dB at the oscillation frequency. It's not a sinewave but thank G*d gain becomes even higher by some 1.6dB.
Bruno, thanks for taking a strong stand on this obscure, but interesting (at least to class d folks) subject. We might get to have some real fun on this forum (and maybe all learn a little in the process). 🙂
Hmmm... careful here. By including only "the oscillation frequency" you are eliminating all harmonics and other signals. That leaves only single frequency sine waves. Once you put on your Fourier goggles like that, gain through a circuit block can only be seen as one thing, the ratio of the magnitudes of input to output (and loop gain can only be unity, IMO).
I must admit to some trepidation here, because I haven't seen a lot in the literature on this subject, but I think the theory still holds, i.e., loop gain is indeed unity at the oscillation frequency (and also at its harmonics, in this "nonlinear" case).
Sorry, by the criteria you seem to be applying here (examining the small signal characteristics in various large signal operating zones), average gain is changing. In clipping it is zero and, in the rapidly narrowing transition zone, it is much, much greater than one. On average, it is decreasing.
With all due respect, that seems like hand waving to me. The science is not situational.
Now, here you are touching on one of the curious manifestations of this facinating nonlinear type of circuit: gain is not continuous with frequency. It "jumps" to different levels at the oscillation frequency (and its harmonics). Looked at from another way, the nonlinear action of saturation at the oscillation frequency acts to suppress gain at non harmonically related frequencies (the lower ones, for all practical class d purposes).
It is this gain suppression action that allows phase shift type crystal oscillators with poorly designed feedback frequency response to oscillate quasi-stably at one or more higher frequency modes. Once a particular mode has locked in, the circuit will run all day in that mode. Cycle power just right and it can lock "happily" into another mode.
Regards -- analogspiceman
Hmmm... careful here. By including only "the oscillation frequency" you are eliminating all harmonics and other signals. That leaves only single frequency sine waves. Once you put on your Fourier goggles like that, gain through a circuit block can only be seen as one thing, the ratio of the magnitudes of input to output (and loop gain can only be unity, IMO).
I must admit to some trepidation here, because I haven't seen a lot in the literature on this subject, but I think the theory still holds, i.e., loop gain is indeed unity at the oscillation frequency (and also at its harmonics, in this "nonlinear" case).
Sorry, by the criteria you seem to be applying here (examining the small signal characteristics in various large signal operating zones), average gain is changing. In clipping it is zero and, in the rapidly narrowing transition zone, it is much, much greater than one. On average, it is decreasing.
With all due respect, that seems like hand waving to me. The science is not situational.
Now, here you are touching on one of the curious manifestations of this facinating nonlinear type of circuit: gain is not continuous with frequency. It "jumps" to different levels at the oscillation frequency (and its harmonics). Looked at from another way, the nonlinear action of saturation at the oscillation frequency acts to suppress gain at non harmonically related frequencies (the lower ones, for all practical class d purposes).
It is this gain suppression action that allows phase shift type crystal oscillators with poorly designed feedback frequency response to oscillate quasi-stably at one or more higher frequency modes. Once a particular mode has locked in, the circuit will run all day in that mode. Cycle power just right and it can lock "happily" into another mode.
Regards -- analogspiceman
I'm trying to explain that the only correct analysis of gain follows from the carrier waveform that can be most practically derived through a time domain analysis, and that certainly the assumption that gain is unity at the switching frequency is incorrect, because the standard oscillation criterion does not apply for non-linear oscillators. The average gain that "could explain" the oscillation is not the same gain you get from the PWM modulation as a result. These are two different quantities and they are only vaguely related. Having done so myself, I leave to you to figure that out.
Bruno Putzeys said:
Bruno, not only are you, IMO, going against classical theory (perhaps you are just describing the same thing in a way that you are more comfortable with), but running away is not playing nice (and not nearly as much fun). 🙂
Regards -- analogspiceman
I'm not versed in the semantics of the way these things are taught academically, so it's quite common for misunderstandings to arise between me and someone who learned it the traditional way. That said, I do recognise when the misunderstanding is of a semantic nature, not of a fundamental one. This is largely the case here.
Apart from that I've been mixing up my figures in a previous post. In the oversimplified form (carrier=sine wave), loop gain at fsw is -6dB (ie 6dB worse than what one arrives at when the linearised gain of the power block is equated with "large signal gain of the power block" when viewed as an oscillator). In reality it's 1.7dB better, so the loop gain of the oscillating amplifier is around -4.3dB at the oscillation frequency.
I think, as long as we can agree that the loop gain is significantly less than 0dB at the switching frequency, we may leave the rest to semantics. When in doubt, run a transient simulation and insert an error source in the output.
Apart from that I've been mixing up my figures in a previous post. In the oversimplified form (carrier=sine wave), loop gain at fsw is -6dB (ie 6dB worse than what one arrives at when the linearised gain of the power block is equated with "large signal gain of the power block" when viewed as an oscillator). In reality it's 1.7dB better, so the loop gain of the oscillating amplifier is around -4.3dB at the oscillation frequency.
I think, as long as we can agree that the loop gain is significantly less than 0dB at the switching frequency, we may leave the rest to semantics. When in doubt, run a transient simulation and insert an error source in the output.
analogspiceman the gain is not unity. You can actually verify this by simulating and calculate the FFT ratios. Typically you have 6dB less gain that unity.
If you don't want to do the simulations, have a look at mine:
http://www.diyaudio.com/forums/showthread.php?s=&threadid=90404
shortcut:
http://www.student.dtu.dk/~s042302/diy/Small-signal_modelling_of_self-oscillating.pdf
I've included the calculations from the UCD paper and also a paper by Lars Risboe on how to find the equivalent gain of the comparator by looking at the slope of the carrier at zero crossing. He derives the formula that Bruno posted above. The calculations match the FFT analysis.
Putzeys have you thought of any way to calculate the drop in oscillation frequency as a function of modulation index M?
The carrier carrier in a UCD type modulator is not very sinusoidal at a high modulation index. In other words the harmonics comes into play and the whole thing becomes very nonlinear. It would be nice to be able to predict the small signal gain of the comparator when M>0, but so fair I've only been able to find it by simulation.
If you don't want to do the simulations, have a look at mine:
http://www.diyaudio.com/forums/showthread.php?s=&threadid=90404
shortcut:
http://www.student.dtu.dk/~s042302/diy/Small-signal_modelling_of_self-oscillating.pdf
I've included the calculations from the UCD paper and also a paper by Lars Risboe on how to find the equivalent gain of the comparator by looking at the slope of the carrier at zero crossing. He derives the formula that Bruno posted above. The calculations match the FFT analysis.
Putzeys have you thought of any way to calculate the drop in oscillation frequency as a function of modulation index M?
The carrier carrier in a UCD type modulator is not very sinusoidal at a high modulation index. In other words the harmonics comes into play and the whole thing becomes very nonlinear. It would be nice to be able to predict the small signal gain of the comparator when M>0, but so fair I've only been able to find it by simulation.
I'm pretty confident the result is non-agebraic, so I prefer to use my time inventing new and better ways of doing things 🙂sovadk said:
btw Putzeys is my last name. I prefer being called by my first name 😉
Even if the carrier remained sinusoidal the modulation would not become linear. After all, for a modulator to be linear a necessary condition is that the slope of the comparator input voltage at the zero crossings is inherently constant.sovadk said:
There doesn't seem to be an easy fix for this, and any working fix is most likely to end up being more complicated than a new method designed from scratch. From that point of view, interest in a closed-form solution to the operating parameters of a phase shift oscillating amplifier is mostly academic.
sovadk said:
Yes and no. Yes, the gain is not unity for frequencies right up to the switching frequency, but no, it still must be unity for the switching frequency itself. This is the surprising gain jump I mentioned earlier.
That's how I see it, anyway.
Regards -- analogspiceman
PS: If not this (the gain jump effect at the oscillation frequency), what is the mechanism that explains how certain crystal oscillator circuits may lock onto one of several different possible operating frequencies?
See, I told you this was going to be fun. 🙂
There is no gain jump. There are simply two gains at play. One is the large signal (or average) gain which is unity by virtue of the existence of a sustained oscillation. The other is the linearised gain of the power comparator which is defined by the slopes of the carrier.
A nonlinear oscillator can oscillate at any 360º phase transition provided dphi/df is positive and small signal loop gain >=1. In this condition, the oscillation will stabilise when large signal loop gain has become 1. Since the attenuation of the feedback network is not necessarily the same at all 360º transitions, the large signal gain of the gain block, according to its definition, will have changed.
Under normal loading conditions a UcD amp has only one 360º transition and therefore only one oscillating mode. There is no gain jump.
So what's happening is that there are simply two different gains at play at the same time. One explains the operation of the oscillator, the other governs the operation of the amplifier.
I understand it may be difficult to conceive of the same circuit as having two different gains at once, so I hope you will consider my non-academic solution which is the realisation that "large signal gain" for a nonlinear oscillator is a broken analysis tool for a situation which is much more complicated than that.
If you want to gain further insight into the matter, try constructing a wien bridge oscillator around a UcD amplifier. It's possible. It will be oscillating at two frequencies at once. One is the carrier, the other the sine wave. This should help understanding the difference between the two "gains".
On a philosophical note I'm always complaining that too many people use their language cortex for engineering and science, whereas it's the visual cortex that literally provides insights. The confusing dual-use of the word "gain" is a typical example of a problem resulting from this. It had never occurred to me that anyone might ever be baffled by a nonexistent problem. Yet, you are by no means the first person to have asked about this.
A nonlinear oscillator can oscillate at any 360º phase transition provided dphi/df is positive and small signal loop gain >=1. In this condition, the oscillation will stabilise when large signal loop gain has become 1. Since the attenuation of the feedback network is not necessarily the same at all 360º transitions, the large signal gain of the gain block, according to its definition, will have changed.
Under normal loading conditions a UcD amp has only one 360º transition and therefore only one oscillating mode. There is no gain jump.
So what's happening is that there are simply two different gains at play at the same time. One explains the operation of the oscillator, the other governs the operation of the amplifier.
I understand it may be difficult to conceive of the same circuit as having two different gains at once, so I hope you will consider my non-academic solution which is the realisation that "large signal gain" for a nonlinear oscillator is a broken analysis tool for a situation which is much more complicated than that.
If you want to gain further insight into the matter, try constructing a wien bridge oscillator around a UcD amplifier. It's possible. It will be oscillating at two frequencies at once. One is the carrier, the other the sine wave. This should help understanding the difference between the two "gains".
On a philosophical note I'm always complaining that too many people use their language cortex for engineering and science, whereas it's the visual cortex that literally provides insights. The confusing dual-use of the word "gain" is a typical example of a problem resulting from this. It had never occurred to me that anyone might ever be baffled by a nonexistent problem. Yet, you are by no means the first person to have asked about this.
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