Newbie question - sawtooth signal generation - capacitor discharge via bjt
Consider this circuit for a minute: -



Other than using a BJT to control the capacitor discharge, it self oscillates and produces a clean triangle waveform and a decent square wave. Now consider what happens when the 2k2 has a diode across it.

This is effectively what you are trying to achieve (saw-tooth) with a transistor - the diode massively reduces the charge time (or discharge time) dependent on the diode direction but think about how this discharge would occur - you need the transistor going to the negative rail and not the earth (mid-rail point) for this to work correctly. It would work in the above circuit (when the diode is added) because the output from the comparator has high negative and positive output voltages with respect to 0V (mid-rail).

Your circuit has the BJT discharging to 0V and this won't cut the mustard.

Here's what my circuit would look like for independent rise and fall ramps using a diode (or two): -



The rise slope is determined by the 100k resistor (R2) and the fall by the 5k6 (R1). You can use a transistor of course but what you end up with is the equivalent of a 555 timer with a constant current source (linear ramp): -



Inside the 555 is the discharge transistor you are trying to implement.