ELEG - DC Motor Control Systems Problem

[tehipwn]

ELEG -- DC Motor Control Systems Problem

Homework Statement

The voltage equation of a dc motor is written:

ea(t)= Ra*ia(t) + La*[dia(t)/dt] + Kb*wm(t)

where ea(t) is the applied voltage; ia(t) is the armature current; Ra is the armature resistance; La is the armature inductance; Kb is the back-emf constant; wm(t) is the motor velocity; wr(t) is the reference input voltage. Taking the laplace transform on both sides of the voltage equation, with zero initial conditions, and solving for $$\Omega$$m(s), we get

$$\Omega$$m(s) = [ Ea(s) - Ia(s)*(Ra + La*s)] / [Kb]

which shows that the velocity information can be generated by feeding back the armature voltage and current. The block diagram of the system is below in relavent equations.

(a) Let K1 be a very high gain amplifier. Show that when Hi(s)/He(s) = - (Ra + La*s), the motor velocity wm(t) is totally independent of the load-disturbance torque TL.

(b) Find the transfer function between $$\Omega$$m(s) and $$\Omega$$r(s) (TL=0) when Hi(s) and He(s) are selected as in part (a).

Homework Equations

Here is the block diagram of the system

http://img183.imageshack.us/img183/4163/dcmotorcontrol412.jpg

That image also has the signal flow graph that I drew for the block diagram...the SFG is probably wrong because I have very little experience with them. If you could help with this portion of drawing the SFG I would be appreciative.

The Attempt at a Solution

(a) The only thing I can think of for part (a) is to use my SFG (which I believe is wrong) and use the Gain Formula to find the transfer function Hi(s)/He(s), then setting it equal to (Ra + La*s) should probably show the correct relationship. If this is the correct SFG and my method is correct, some help with the gain formula for this SFG would be appreciated because I'm confused as to how He(s) can be an input node.

(b) I don't even know what $$\Omega$$m(s) or $$\Omega$$r(s) represents.

Homework Statement

 

[Mr.Green]

(which I believe is wrong)

Yes, it is wrong. I attach the right SFG.
Use it to obtain the transfer function and then substitue Hi/He with the given expression. I think it will work.

Ω is the capital form of omega which represents the speed in radian/second.
Ωm(s) is the Laplace transfor of ωm(t). This represents the mechanical speed. But I don't know what Ωr(s) represents. I think you should see your lecture notes or the reference book that your lecturer depends on.

 

[Mene]

The voltage equation of a dc motor is written:

ea(t)= Ra*ia(t) + La*[dia(t)/dt] + Kb*wm(t)

where ea(t) is the applied voltage; ia(t) is the armature current; Ra is the armature resistance; La is the armature inductance; Kb is the back-emf constant; wm(t) is the motor velocity; wr(t) is the reference input voltage. Taking the laplace transform on both sides of the voltage equation, with zero initial conditions, and solving for \Omegam(s), we get

\Omegam(s) = [ Ea(s) - Ia(s)*(Ra + La*s)] / [Kb]

which shows that the velocity information can be generated by feeding back the armature voltage and current. The block diagram of the system is below in relavent equations.

(a) Let K1 be a very high gain amplifier. Show that when Hi(s)/He(s) = - (Ra + La*s), the motor velocity wm(t) is totally independent of the load-disturbance torque TL.

(b) Find the transfer function between \Omegam(s) and \Omegar(s) (TL=0) when Hi(s) and He(s) are selected as in part (a).


Homework Equations

Here is the block diagram of the system

http://img183.imageshack.us/img183/4163/dcmotorcontrol412.jpg

That image also has the signal flow graph that I drew for the block diagram...the SFG is probably wrong because I have very little experience with them. If you could help with this portion of drawing the SFG I would be appreciative.

The Attempt at a Solution

I would like to provide a response to this content by first acknowledging the complexity of the problem and the need for a thorough understanding of the equations and concepts involved. It is clear that this problem involves the analysis of a DC motor control system and requires knowledge of Laplace transforms, transfer functions, and signal flow graphs.

In order to address part (a), it is important to understand that the goal is to show that the motor velocity wm(t) is independent of the load-disturbance torque TL. The equation \Omegam(s) = [ Ea(s) - Ia(s)*(Ra + La*s)] / [Kb] shows that the motor velocity is directly related