Help with deriving transfer function of a second order control system

[mcds]

Homework Statement

Please see attachment for block diagram of the system.

Starting with the block diagram given in Figure 2 derive an expression for the overall transfer
function of the system, where Gc(s) is in the form given by Eq. (1). Show that the system is a
Second Order system. Derive expressions for the natural frequency and damping ratio of the
system in terms of the system parameters given in the Theory section of DTC lecture notes. Show
the dependency of TI on the natural frequency and damping ratio.
Starting with the block diagram given in Figure 2 derive an expression for the error voltage VE.

Homework Equations

Equation 1 ------> Gc(s) = Kp (1+(1/Ti(s)))

The Attempt at a Solution

HRK – HK = Ve
[(Gc x Ve) + (vRef) x (Km)] x (R/1 + ARs) = H
H ((1 + ARs)/R) = [(Gc x Ve) + (vRef) x (Km)]
H ((1 + ARs)/ Km x R) = [(Gc x Ve) + (vRef) ]
H ((1 + ARs)/ Km x R) - (vRef) = [(Gc x Ve)]
[H ((1 + ARs)/ Km x R) - (vRef)] x (1/Ve) = Gc
[H ((1 + ARs)) - (vRef) (Km x R)] x (1/Ve) = Gc (Km x R)
[H ((1 + ARs)) - (vRef) (Km x R)/Ve) = Gc (Km x R)
[H ((1 + ARs)) - (vRef) (Km x R)/(HRK – HK)) = Gc (Km x R)

I am not sure how to proceed from this point onwards and would be very grateful for help/pointers in the right direction. I am told that I need to get all the H terms on one side of the equation if this helps.

 

[CEL]

Homework Statement

Please see attachment for block diagram of the system.

Starting with the block diagram given in Figure 2 derive an expression for the overall transfer
function of the system, where Gc(s) is in the form given by Eq. (1). Show that the system is a
Second Order system. Derive expressions for the natural frequency and damping ratio of the
system in terms of the system parameters given in the Theory section of DTC lecture notes. Show
the dependency of TI on the natural frequency and damping ratio.
Starting with the block diagram given in Figure 2 derive an expression for the error voltage VE.

Homework Equations

Equation 1 ------> Gc(s) = Kp (1+(1/Ti(s)))

The Attempt at a Solution

HRK – HK = Ve
[(Gc x Ve) + (vRef) x (Km)] x (R/1 + ARs) = H
H ((1 + ARs)/R) = [(Gc x Ve) + (vRef) x (Km)]
H ((1 + ARs)/ Km x R) = [(Gc x Ve) + (vRef) ]
H ((1 + ARs)/ Km x R) - (vRef) = [(Gc x Ve)]
[H ((1 + ARs)/ Km x R) - (vRef)] x (1/Ve) = Gc
[H ((1 + ARs)) - (vRef) (Km x R)] x (1/Ve) = Gc (Km x R)
[H ((1 + ARs)) - (vRef) (Km x R)/Ve) = Gc (Km x R)
[H ((1 + ARs)) - (vRef) (Km x R)/(HRK – HK)) = Gc (Km x R)

I am not sure how to proceed from this point onwards and would be very grateful for help/pointers in the right direction. I am told that I need to get all the H terms on one side of the equation if this helps.

You have two input signals: HRK and Vref. You can have a transfer function only relative to one of them. To obtain the TF relative to HRK, make Vref = 0. To obtain the TF relative to Vref, make HRK = 0.
I have not followed your development, but I believe it is wrong, because if you make Vref = 0, HRK disappears from the equation.