DC Motor Circuit: Predicting Vehicle Distance

[TMM]

So I've designed this simple circuit for an electric vehicle. The inductor and resistor that are boxed off are the motor. The way it works, as you can probably figure out from the diagram, is that the RC circuit on the left controls how long the right side of the circuit runs (via the transistor). The resistor in the RC circuit is a potentiometer, and the time that the whole circuit runs is easy: t = -R*C*ln(actV/V) where actV is the minimum base-emitter voltage drop on the transistor.

My question is about the second half of the circuit. Is there an easy way to predict how far the vehicle will travel if the only thing powering it is that motor? Does the transistor act as an amplifier as well as a switch?

My attempt to predict the outcome was to say that the power applied to moving the vehicle was proportional to that dissipated in the motor. I treated the motor as a resistor and found the Power = V^2/R = F*v. Is this realistic?

Here is the diagram:

The goal of this is to have an electric vehicle that will run on its own for a distance dependent only on the two variable resistors.

 

[TMM]

I improved on the RC side of the equation using the Ebers-Moll stuff:

V = Gain*ln((k*T*dq/dt)/e) + R*dq/dt + q/C

which I can't find a closed solution to, but I've gotten some approximations using SPICE.

For the DC motor side, I did some research and found a model for it:

V = LdI/dt + IR + Kv/r

cI = kdv/dtr + Fv/r

where c, k, and K are constants, v/r is the angular velocity, F is the frictional load, I is current, V is voltage, R is resistance, etc. So basically I've treated the dc motor as an inductor and resistor in series with a back emf proportional to the angular velocity, which is related to the current and frictional load.

Am I sort of maybe kind of moving in the right direction? Just a little feedback from someone with some experience with this would be awesome.

 

[berkeman]

I don't understand your schematic. What is I1? Is that the power source somehow? There would appear to be some connections, bias paths, and power sources missing from the schematic?

 

[TMM]

I've revised the schematic since then.

Thrown together in paint just now:

 

[berkeman]

It would be more traditional (and easier to understand) if you drew the schematic with the DC voltage source facing with + upwards, all by itself on the lefthand side of the schematic, and then the PNP transistor to the right, with the emitter connecting to the upper + power rail, and the collector facing down and the motor RL components below the PNP collector, and then your final R return to the negative power rail (show a ground symbol on that). The RC from the base of the PNP then goes down to that common ground rail.

So back to your original question, you have some initial condition (base capacitor shorted?), and then you unshort the cap and want to know the time constant of the output drive to the motor?