The "Leap-Frog" Method Of Switching Amplifier Control Loop Design - diyAudio
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Old 3rd September 2004, 07:11 AM   #1
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Default The "Leap-Frog" Method Of Switching Amplifier Control Loop Design

Hi all,

Yes, there is a way to include the entire output filter of a class d amplifier within the feedback loop. I first thought of this method (which I like to call the "Leap-Frog" method for reasons that should be obvious after reading the rest of this post) around 1996. (Unfortunately, this is a text only post so you won't be able to see the accompanying illustrations - see attached pdf.)

The leap-frog control loop design method extends active damping techniques to incorporate an unlimited number of output filter sections within a switching amplifier feedback loop. By working in steps from the power switching stage outward, the process of designing gain coefficients for each feedback filter component is simplified to a first order problem. At each stage the amplifier's impedance characteristic leap-frogs between that of current and voltage source (hence the name). The leap-frog method breaks the design problem into manageable steps, and turns what would otherwise be a practically intractable problem with four, six or more independent variables, into a series of straightforward choices for each feedback coefficient.

Switching amplifiers are attractive for high power audio applications because of their inherently low conduction/blocking losses. This results from maintaining the output power devices in either a fully saturated or cut-off state such that they never simultaneously support large currents and voltages as is typical of standard linear audio amplifiers. This switching characteristic can provide an important efficiency advantage over standard linear amplifiers if the losses from the switching transitions are also kept to a relatively low level. Toward this end is desirable to switch at as low of a frequency as is compatible with closed-loop system bandwidth and output impedance requirements. (A switching amplifier is actually a high level digital sampled data system with its ensuing Nyquist sampling effects which limit maximum bandwidth to no more than one half the switching frequency).

Another significant complication often arises because of the need to strictly limit the level of switching ripple components on the amplifier's output without restricting the amplifiers ability to deliver rail-to-rail audio signals at 20 kHz. This normally requires the use of an output recovery filter with multiple L/C sections and with pole locations just above the audio pass band. To optimize closed-loop system bandwidth and output impedance necessitates that the feedback system be able to track and compensate the rapid phase shift stemming from the output filter's high Q poles and zeros, the location of which will vary dynamically due to current and temperature dependent shifting of the component values. Note that, in high efficiency power applications, dissipative elements may not be readily used in the recovery filter to control L/C resonances.

This has not been an easy problem to solve using traditional techniques. Standard compensation methods with opamps, resistors and capacitors fail because it is not possible to match and track the frequency characteristics of the high Q L/C filter sections. Typically, the amplifier's feedback loop will include none or only the first of the output filter sections within its control loop. This approach degrades the accuracy of the amplified audio signal.

In some prior switching amplifiers, the control loop has been designed using active damping techniques to track filter component shifts, manage output filter Q and extend bandwidth. With this method, a sensed signal directly proportional to output filter capacitor current is an integral part of the feedback loop. This insures direct, accurate tracking and control of output filter resonances, and allows maximum loop gain-bandwidth in the face of a single L/C filter section.

The leap-frog design method described below extends the active damping technique to incorporate an unlimited number of output filter sections within the feedback loop, and describes how to choose the gain coefficients for each feedback filter component by working in steps from the power switching stage outward. As the gain coefficient for each component is chosen, and that component is incorporated within the amplifier's black box boundary, the impedance characteristic the amplifier presents at its output changes to a current source if the component is an inductor or to a voltage source if the incorporated component is a capacitor. As each component is swallowed up, the overall closed loop bandwidth must be reduced by a small factor (about 1.5).

Thus, the amplifier's output characteristic leap-frogs between that of a current and voltage source (hence the name). This simplifies the design process of each succeeding gain coefficient to a first order problem. The leap-frog method breaks the design problem into manageable steps, and turns what would otherwise be a practically intractable problem with four, six or more independent variables, into a series of straightforward choices for each feedback coefficient.

The figure below will be used to illustrate the leap-frog design process for a four element ladder filter network. Working from the power switch to the output (left to right), the voltage command to the power stage/modulator is the sum of the positive feedback signal of the voltage appearing on the output side of L1 and the negative feedback signal of the inductor current. Note that the modulator and totem pole output stage is approximated as a voltage controlled voltage source with delay due to sampled data nature of the pulse width modulation process. The unity gain positive feedback term of the load side voltage from the inductor serves to keep the voltage across the inductor (and hence its current) constant in the face of load side voltage perturbations. The negative feedback signal of inductor current roles off with a single pole due to the rising impedance of inductor L1. Gain K1 is set so that loop gain falls to zero somewhat before half the switching frequency (where the switching delay adds 180 degrees phase shift). As the inductor is merged into the black box of the amplifier on the left hand side, the resulting equivalent voltage controlled current source is shown below feeding the next filter element C2.

Next, capacitor C2 is incorporated into the equivalent circuit in exactly a dual nature. Looking at the following figure, the unity gain positive feedback term of load side current out of the capacitor functions to null net current through the capacitor in spite of sudden changes in load current, minimizing the resulting voltage fluctuations. The negative feedback term of capacitor voltage roles off with a single pole due to the falling impedance that capacitor C2 presents to the controlled current source. Gain K2 for this feedback path is set so that loop gain falls to zero at about two thirds of the current source's bandwidth. The resulting equivalent voltage controlled voltage source is shown below feeding the next filter element L3.

Now the leapfrog method has come full circle to the starting conditions of a controlled voltage source feeding an inductor element in an LC filter ladder. Just as before, this element is incorporated into the system by applying the appropriate amounts of positive and negative feedback. Gain K3 for this feedback path is set so that closed loop gain is about two thirds of what it was before. The resulting equivalent voltage controlled current source is shown below feeding the next filter element C4.

The process continues until all the filter elements are incorporated into the amplifier, yielding a well controlled, component insensitive, switching amplifier with the maximum possible bandwidth. These advantages come at a cost of an extensive feedback network distributed throughout the switching amplifier's recovery filter ladder.

In practice, both the sensing and feedback amplifier circuitry can be greatly simplified by combining adjacent signal paths. In particular, combining stages removes the need to reproduce dc signals in the sensing circuitry. Recognizing that the difference of inductor currents must flow through the capacitor on the common node between adjacent stages justifies using a simple current transformer to sense this difference current (or a fraction of this current by using a capacitive current divider). Likewise, recognizing that the difference of capacitor voltages must appear across the interposing inductor justifies using a simple floating winding to sense the difference voltage.

All of the distributed gain terms are easily consolidated into a single summing amplifier by simply accounting for the cumulative gain terms in the path for each signal as shown above. Following these constructs results in a switching amplifier system that is both practical and simple, yet easily accommodates a recovery ladder filter network any length within its feedback path.

analog(spiceman)
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Old 3rd September 2004, 12:01 PM   #2
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Clearly an elegant algorithm for turning high-order loop design into a repetitive, manageable exercise. I myself tend to use direct mathematical analysis, but your method is a much simpler way of obtaining the same result. Nice work!
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Old 3rd September 2004, 02:11 PM   #3
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Default Leapfrog method (pdf pages 3-4)

Pages 3-4 of the pdf should be attached. And, yes, I also have LTspice files. They are absolutely fascinating to play with. I have both averaged models and switching models. One can plot such things as output impedance, frequency response as well as time domain response.

For small signals, the averaged and switched models respond identically as would be expected, but with large signals the switched model saturates, effectively lowering the switching frequency (one way of looking at it) such that the increased delays cause the system to go unstable and "explode". This can be prevented by preclipping the input to prevent hard saturation and by limiting its slew rate to a bit greater than that required by a rail-to-rail 20kHz sine wave.

I am a switched mode power electronics specialist who has worked in a variety of fields, including consumer and pro audio (although I'm currently working in a non-audio field). I have designed a couple of multi-kilowatt class d sub amps as well as mixer circuitry, some preamp stuff and switching power supplies compatable with audio.

I always thought it would be fun to do a 10kW, self-oscillating (non clocked), multi-phase, full bandwith class d amp with an input power factor corrected power supply and enough energy storage to provide several seconds of 10kW+ audio without tripping a 15 amp circuit breaker. What a great demo such an amp would make integrated into the base (foot) of a pole stand for a three foot or so diameter sphere of a dozen or so 8 or 10 inch speakers driven in parallel. The sound would be both amazing and deafening.

Unfortunately the marketing guys were never interested.
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Old 3rd September 2004, 02:46 PM   #4
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Default Leapfrog design (pdf page 5)

All practical power amplifiers are limited as to the voltage and current they can provide and the rates at which they can provide them. For a given power switch V/I capability and a given output filter configuration, the leapfrog design method should lead to better performance than with any other design approach.

When it comes to gracefully dealing with large signal effects, most designers fail to realize the importance the impedance of the output stage (as measured both by the square root of the ratio of output filter L over C and by the V/I limits of the output switch). Some loudspeaker loads with a nominal impedance of 6 or 8 ohms may dip down into the 1 ohm range when driven at certain frequencies. An amp that occasionally current limits into such a load will sound much worse than one that only occasionally voltage limits at other points, even if the clipping action in both cases is flat and clean.

Output filter root(L/C) and output stage V/I capability should both be many times lower than the nominal load impedance, IMO. This will lead to a very low damping factor and ensure that current limiting only occurs during a true fault. Usually designers overlook these considerations or are misguided by fear of high currents circulating through the output switch and filter.

By the way, many of my LTspice simulation files (including the 5 page design pdf in single file) are available in the files section of either of the following two Yahoo Groups:

http://groups.yahoo.com/group/audioexperiment/ (Johan Sörensen's Yahoo Group dedicated to design of Audio Power amplifiers in general, and designs using class D in particular.)

http://groups.yahoo.com/group/LTspice/ (Yahoo Group dedicated to exchanging information about SwitcherCADIII/LTspice, a fully functional schematic capture and circuit simulation program that is freeware available from Linear Technology at http://www.linear-tech.com/software/ The group includes relevant information about LTspice and simulation in general, and it also contains links to models, hints and tips beyond that which comes with the free program.)

I have yet to draft a full, complete component level simulation of a leapfrog based design because, the last time I worked on this problem, I was still unsatisfied with my scheme for auto phase staggering paralleled multiple free running output stages. Because it runs so slowly, there is little point to running such a simulation until all the pieces have passed muster. However, using an idealized output stage, I have run many large signal simulations with full limiting effects, both with averaged and switched models and the results were very encouraging. -- analog(spiceman)
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Old 3rd September 2004, 03:13 PM   #5
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1.) Neat
2.) Another indicator showing that post-filter feedback takeoff is possible

Regards

Charles
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Old 3rd September 2004, 06:21 PM   #6
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It looks to me that if it were a linear amplifier, this technique
would be called nested feedback.
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Old 3rd September 2004, 06:32 PM   #7
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Quote:
Originally posted by Nelson Pass
It looks to me that if it were a linear amplifier, this technique
would be called nested feedback.
It's called nested feedback in class D as well. A given circuit can be arrived at through many means. The extremes are to throw the whole thing into a big set of equations on the one hand and the algorithm proposed by analogspiceman. For a certain spec, the outcome of both methods will be identical, but a.s.m. offers a more elegant way of arriving at it.
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Old 3rd September 2004, 06:37 PM   #8
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Quote:
Originally posted by phase_accurate
1.) Neat
2.) Another indicator showing that post-filter feedback takeoff is possible

Regards

Charles

I think you'll find that post-filter derived feedback has been 'out there' for many years....

There is a JAES paper circa 1986 on the subject...by...some Englishman....
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Old 3rd September 2004, 06:41 PM   #9
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Default Alternate descriptions of leapfrog design

Quote:
Originally posted by Nelson Pass
It looks to me that if it were a linear amplifier, this technique
would be called nested feedback.
Yes, that name could well be used to partially describe this technique. Active damping, state variable feedback or matrix feedback might also be equally reasonable descriptions.
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Old 3rd September 2004, 06:41 PM   #10
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Default Re: Leapfrog design (pdf page 5)

Quote:
Originally posted by analogspiceman
I have yet to draft a full, complete component level simulation of a leapfrog based design because, the last time I worked on this problem, I was still unsatisfied with my scheme for auto phase staggering paralleled multiple free running output stages.
I've addressed the problem of getting multiple phases to cooperate peacefully using a high-order loop, followed by a quantiser (n+1 level flash ADC) and an FPGA circuit that takes care of equalising pulse counts on all n power stages. I invented this scheme for precisely the same reason as you did yours: to make a 10kW amp. Never taken it that far though. An open loop version of this idea was implemented to take DSD data directly (aes preprint 5631). Efficiency: 97%. The stuff of 10kW amps
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