Re: PWM Amp Design


Chris Carlen wrote:

> [For] academic purposes I am still curious about what other
> approaches other than an equalizing network one might try to solve
> the reactance problem? It seems to me that one approach might be
> to try to sense the PWM output post-filter, and then compensate
> for the peaking within the PWM amplifier control loop.

One can incorporate any number of LC output filter sections within
the feedback loop of an amplifier. Doing this requires sensing a
more or less orthogonal set of states for the output filter and
delivering them in a properly weighted measure of feedback and
feedforward terms to the error amplifier. Here is a link to my
illustrated explanation of this technique: (You may have to become
our newest group member to access this file :) ...it's free to join.)

http://groups.yahoo.com/group/LTspice/files/adventures%20with%20analog/class%20d%20audio/Leapfrog_Design.pdf

Do you have access to the binaries group? I could also post the PDF
there as well. In any case, here is the text sans illustrations:

The ?Leapfrog? Method Of Switching
Amplifier Control Loop Design
analog@xxxxxxxx - 1996

The leapfrog 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 leapfrogs between that
of current and voltage source (hence the name). The leapfrog
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 it 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 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 tech-
niques. 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
ensures 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 leapfrog 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 or so). Thus, the amplifier?s output characteristic leapfrogs
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 leapfrog 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 leapfrog 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. 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 of any length within its feedback path.

The following schematics were simulated in LTspice in order to
demonstrate and confirm the principles of the leapfrog method of
switching amplifier design. As expected, the simulation output from
the three variations was absolutely identical, verifying the validity
of the topological manipulations.

Typical output from ac frequency response and 10kHz square wave
transient response is presented below, with each showing the effect of
stepping the load resistor from 1 to 8 ohms. Note that fs represents
the effective sampling frequency which may be quite different from the
nominal switching frequency. For example, in a free-running, self-
oscillating design, the effective sampling frequency would be very
close to the lowest switching frequency during dynamic excursions and
not the typically 3-to-4-times higher quiescent operating frequency.
Likewise td represents the effective worst-case delay rather than the
typical delay. Thus, the rather high fs and low td of the simulation
would be difficult to achieve in practice unless the design employed
multiple, parallel, staggered phase output stages feeding the recovery
filter. (This technique multiplies the sampling frequency by the
number of staggered phase output stages.)
.