# Servo motor analysis

I'm studying a mathematical behaviour of a servo motor and I need some help to understand it.

The output signal is $\beta(t)$, representing the angle rotated by the axis at instant t, in relation to the equilibrium position. On the servomotor there are two torques:

• the first torque is produced by the electric current $i(t)$ that does through the motor. It is proportional to the electric current being given by $Ki(t)$.
• the other torque is given by friction and is given by $-2 \beta'(t)$ where $\beta'(t)$ is the derivative of $\beta(t)$, meaning it represents the angular velocity.

The feedback control system is given by:

Where Motor represents the differential equation $4\beta''(t)=100i(t) - 2 \beta'(t)$

And $\alpha(t)$ is the input angle that give the desired position (angle) for the servomotor.

$B$ is unknown but we will assume $A = -1$ so that if the input is constant we'll have the output equal to the input (unitary static gain).

The transfer function of the system is:

$$H(s)= \frac{100}{4s^2+(2-100B)s-100A}$$

By this expression we conclude that our system is a second order system with no zeros. By controlling the value of $B$ we will control where the poles are in our system and by that the type of system we get. More specifically:

• If $B<-0.38$ we have two real poles (and an overdamped system)
• If $B=-0.38$ we have o double pole (critically damped system)
• If $-0.38<B<0.02$ we have two complex conjugated poles (underdamped system).

So until this I don't have questions and everything makes sense to me. The next affirmation is what leaves me doubtful:

• By controlling the value of $B$ we are controlling the friction of the system.

No more explanation is given to this and I'm kinda worried if my interpretation of this fact is correct or not. Here it goes:

• We have a 2nd order system without zeros. By controlling $B$ we will control the type of poles our system will have. For low values (in module) of $B$ we have an underdamped system. For $B=-0.38$ the system is critically damped. For $B<-0.38$ (meaning, large absolute values of $B$) the system will be overdamped. The system damping is related to the existence of friction in the motor. For low absolute values we will have little friction and so the system will oscillate a lot until it reaches its final value. As we increase $B$, we will increase friction which will oppose to the natural oscillatory tendency of the system, making it monotonous until it reaches its final values. We have to be careful however because as we increase the friction the system will also take more time to reach the value (it has to "beat" the friction).

Is this interpretation correct? I'm not sure if this is the correct relationship between friction and the type of system I get? I'm also curious if there is any more implication or consequence of this relationship. Can anyone clarify me please?

Thanks!

• it turns out you cannot just copy from the Electrical Engineering SE because they like to have dollar signs in their text (because sometimes engineers have to talk about money). here and in the Math SE and the Physics SE, there is no backslash-dollar-sign. – robert bristow-johnson Dec 15 '17 at 17:56
• also, you haven't placed the URL link to your figure into the text. can you please do that? – robert bristow-johnson Dec 15 '17 at 17:57
• Hi! Yes I will edit the question! – Granger Obliviate Dec 16 '17 at 12:35

Yes your interpretation is correct. Your model of the servo motor creates a second order differential equation: $$I \beta'' + b \beta' + k \beta = \tau_{ext}$$
where $\beta$ is the position (in radian angle) , $\tau$ is the externally applied torque (given by the input current and the mechanical load if it exists and is being modeled), $I$ is the moment of inertia of the rotating part, $b$ represents the effect of frictive forces which depend on angular velocity such as your damping friction effect, and $k$ is the effect of position dependent torsional-spring force if it exists. In your system you only consider the angular velocity dependent frictive force and ignore the (non-existing indeed) spring force hence $k=0$. This yields the differential equation
$$I \beta'' + b \beta' = \tau_{ext}$$
The system can also be analysed by Laplace transforming the LCCDE, at this point I'm not sure how you reached your $H(s)$, but assuming it's correct then it can be seen that the coefficient $2-100B$ models the frictive force strength and as you have stated; by adjusting $B$ you will adjust the type of solutions from oscillatory underdamped to monotonic overdamped.