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Wed Jan 5 08:30:41 EST 2011

## Simply dirty exponential converter

```As mentioned by Gert, it might be possible to do it "simply, dirty" by
driving the base of a BJT directly, and solve the rest with the
(digital) feedback controller.

My concern is then to at least make it safe enough so we don't blow up
the BJT due to controller bugs.  Let's analyze this minimalistic
circuit:

I_out    RL
o----/\/\/\---o Vcc
|
R              /
o---/\/\/\/----o----|/  Q  (i.e. 2n3904 NPN)
|    |\
|      V
PWM in         C =====    |
|      |
|      |
o--------------o------o  GND

* Safety

For safety there needs to be an R between the PWM drive and the
base of Q.  If PWM gets stuck high, R will limit the current once
C is charged.

* Filter

The 3dB point should probably be set around 200Hz.  At 10MHz PIC
clock resolution this gives 50000x oversampling, or a little over
15 bits of binary modulation resolution.

* PWM levels

The useful voltage range at the base of Q is determined by the
dynamic range of the current.  We need about 4 decades which
corresponds to roughly 10 V_T or about 250mV.

V_BE at max current is around 600-700 mV.  The useful control
range at the base will thus be about 300-700mV.  Allowing for some
modulation headroom to get sufficiently fast rise times, it seems
that PWM levels of 0-1V are going to be optimal.

* Impedance

RC product is fixed by bandwidth constraints, so what remains is
to pick the impedance or the R/C ratio.

The impedance of the filter circuit needs to be relatively high
because it needs to drive the transistor base current I_BE.  There
are two ways of looking at this: C needs to be large wrt I_BE such
that voltage drop due to I_BE should be much less than the
expected ripple introduced by the load/unload currents.
Alternatively one can say that the leak current due to I_BE should
be much less than the load/unload currents, leading to small R.

In any case: this load current is signal-dependent, so it needs to
either be completely eliminated through low R and C impedance, or
we need a compensation to deal with this nonlinearity on the fly.
This compensation needs to be fast relative to the speed at which
human hearing can detect de-tuning.  I'd say about 100-200ms for a
stable tone.

It seems C needs to be as high as possible as it needs to drive
the transistor base current.  This load is nonlinear so we
probably best keep its effects minimal in order to keep the
PWM->V_BE transfer average as linear as possible.

If C is large R needs to be relatively small.  R's low bound is
set by a safety limit determined by the max current setting.

Max I_BE is determined by the load of Q (say order 1mA) over Q's
current gain, which can be highly variable.

* R_L

Determined by max current output, probably around 1mA.  This is
for test only: in operation the load will be an emitter coupled
pair or an OTA input (requires PNP version of above schematic).

* Current filtering?

It might be necessary to filter the current ripple using a small
inductor at the collector of Q.  This to prevent "peaked" current
wave forms due to the exponential conversion.

However, when the voltage ripple at the base is small, the AC
transfer function is fairly linear and the output ripple's wave
shape should resemble the input, which will be +- triangular.

* Eliminate PWM reference?

As mentioned above, PWM needs only 0-1V, so driving from 0-5V is
mostly a waste of dynamic range.  However, driving from anything
else than Vcc requires switching circuitry that significantly
complicates the circuit.  I don't think the cheap buffer/inverter
chips operate from a 1V supply.

What about this hack: use 2 PIC pins attached to the buffer cap
through 2 different resistors: one used for charge and one for
discharge.  Picking a larger resistor for the "more distant" rail
should make it possible to keep the rise times equal in the range
of interest.  Say R_5V = 5 x R_0V.

Instead of charging / discharging through 2 different resistors,
we could also discharge through 2 resistors in parallel and charge
through one.  When more pins are available, this could also be
used to use some different rise/fall time configurations.

Problem: the transfer is then no longer simply an RC filter.  My
hunch is that it will be fine as long as you stay away far enough
from the rails.  At some DC point, the 2 resistors act as current
sources / sinks.

However, this analysis is signal-dependent so introduces some
nonlinearity in the transfer function that is measurable.

Overall this nonlinearity does seem predictable and probably can
be compensated once when the exact resistance values and cap value
are known.

* Feedback

This circuit is sensitive to the following disturbances, sorted
from low to high variation over time.

- Manufacturing spread due to V_BE spread and Q's current
gain.  In theory this could be calibrated once.

- Temperature drift: mostly 1st order during startup, after
that it might be signal dependent if there's not enough
thermal mass attached to the transistors.  All things equal,
output current varies 20% over 50 degrees variation in die
temperature.

- Non-linearities in the control loop from duty cycle to V_BE.
These are due to input bias current, the 2-resistor trick,
and possibly transistor nonlearities (probably small).  This
needs fast response in the order of 100ms, i.e. the time
necessary for the ear to figure out an oscillator is not
tuned correctly.

Bottom line: it's quite useless without compensation.

What measurements can or can't be used?

* Due to the high dynamic range of the output current, we can't
use current measuring resistors for feedback.  ADC resolution
is too low.

* We can use timing-based feedback derived from an oscillator.
This can go to arbitrary high resolution by averaging capture
times of oscillator transitions fed into the PIC over longer
time scales.

Note that we could use thermally coupled (and possibly V_BE
matched) transistors to use one for driving the circuit and
one for driving the feedback oscillator in case the circuit
doesn't implement an oscillator but i.e. a filter or an
envelope.

* We can measure V_BE at 12 bits resolution, which should be
enough for compensating the curved response, but of course
doesn't solve V_BE spread or thermal drift.

It seems best to split compensation into two parts:

* Between runs, or once after manufacturing, we can store the
compensation point in ROM.  This should capture the large but
relatively constant variation due to component spread and aging.

* The temperature spread can then be tracked on-line at a rate
of say 1 update per second.  Nonlinearities can also be
compensated this way, but this requires a tighter control
loop.

In general it seems better to aim for a tighter control loop that
can capture both temperature variations and control chain
non-linearities, than to design a laxer control loop and use
feedforward calibration.  The former might relax some more
high-frequency timing issues in the pulse modulator.

Caveat: using frequency-feedback necessarily introduces a large
delay in the feedback loop.  This might introduce instabiltities.

* Oscillator

A bare bones sawtooth oscillator can be constructed from this
circuit by

* Numbers

Starting point: I_out max 1mA.  Everything else scales with this.
Designing for PWM voltage of 1V.

Using 2N3904[1].  I'm not sure what to pick for nominal h_fe but
let's use 100.  Data sheet range is 40-300 over the useful range.
With 100, max I_BE is then 10mA.  Might need a different
transistor with higher current gain.  Scale design accordingly.

[1] http://www.fairchildsemi.com/ds/2N/2N3904.pdf

```
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