I just want to review the equations for motor current in terms of spec motor constants when the voltage is other than the spec 12 volts for the Tetrix motor.
tq_nm = output torque
tq_stall = spec stall torque at 12 volts
w_free = spec free speed at 12 volts
i_free = spec current at w_free speed at 12 volts
i_stall = spec stall current at 12 volts
V = motor voltage
kt = motor torque constant
ke = motor speed constant
R = motor resistance
Lets start with the pure resistive (no inductance assumed) motor current formulation :
i = (V- ke*w)/R
Substituting known values at the spec 12 volts , and using R = 12/i_stall
ke =(12- i_free*R )/w_free = 12*(1 – i_free/i_stall)/w_free
plugging this back into equation for i gives
i = (V- 12(1-i_free/i_stall)*w/w_free)*i_stall/12
= i_stall*V/12 – (i_stall-i_free)*w/w_free or
1) i = i_stall*V/12( 1 -(1-i_free/i_stall) * w/ (w_free*V/12))
Now with a little manipulation we can write this in terms of parameters that might have been spec’d at some other voltage V.
2) i = i_stall@V*( 1 – (1-i_free@V/i_stall@V)* w/w_free@V)
where i_stall@V = i_stall*V/12
i_free@V = i_free*V/12
w_free@V = w_free*V/12
In other words if we just multiply all the spec values by a ratio V/12 and plug them into the current normalized formula 2) we will get the same results as using 1). This is artificial for i_free and w_free . i_free doesn’t change with voltage… ie it is based upon the drag torque and since kt is independent of supply voltage then i_free must also be.
Also, w_free@V is not just w_free*V/12 it is slightly different. It is ok if i_free is << i_stall. The exact formula can be derived from equating i at both free speeds.
So i_free@12 = i_free@V
12-ke*w_free = V – ke*w_free@V
so w_free@V = (V-12 + ke*w_free)/ke and plugging in for ke gives
w_free@V = w_free*(V/12 – *i_free/i_stall)/(1-i_free/i_stall)
One can see that as i_free/i_stall -> 0 then w_free@V -> w_free*V/12
For the minibot, i_free/i_stall =.5/7.5 = 1/15 so it is not a bad approximation.
At V = 14 volts, w_free@14 = w_free*(V/12)*1.01 so only a 1% error.