Motor Startup

List of principal symbols:

Time t
Location x
Velocity (Speed) v
Acceleration a
Force F
Mass m

$F=F_{pwm}-F_{bemf}=ma$

$F=C-Kv=ma \qquad with \qquad a=\frac{d v}{d t}$

C represents the Force originated by the PWM voltage, and Kv the BEMF (Back Electromotive Force).

We are going to suppose that Fpwm is constant and Fbemf is the only counteracting force and proportional to the speed.

At the beginning of the startup sequence, time=0, we have v(0)=0 and x(0)=0.

The maximum velocity of this motor will be Vmax=C/K.

$\frac{d v}{d t}=(C/m)-(K/m)v\qquad .$ (1.0)

$\frac{d^2 x}{d t^2}=(C/m)-(K/m)\frac{d x}{d t}$

$\frac{d x}{d t}=(C/m)t-(K/m)x+Const1,\qquad Const1=0$

$x=\frac{1}{2}(C/m)t^2-(K/m)xt+Const2,\qquad Const2=0$

$x=\frac{1}{2}(\frac{C}{m})t^2/(1+\frac{K}{m}t)\qquad .$ (1.1)

From (1.0):

$v=(C/m)t-(K/m)vt+Const3,\qquad Const3=0$

$v=(\frac{C}{m})t/(1+\frac{K}{m}t)\qquad .$ (1.2)

If we ignore Fbemf (if K=0) we get:

$x=\frac{1}{2}(\frac{C}{m})t^2\qquad and\qquad v=(\frac{C}{m})t\qquad .$ (1.3)

And we can simplify this way because Vmax-startup<Vmax/10 (typical). We need this model only for little startup speeds (t near zero).

At Motor CTRL Basics we said: "We control mainly 2 variables: the time for the next commutation and the voltage.". At startup we will begin with PWM around 30% or 40%, and the time for the next commutation is derived from (1.3):$t_n=k1. \sqrt{x_n}$

to be continued…

Links

Mountain Engineering- Sensorless Motor Control

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