A) Construct the theoretical transfer function relating the voltage applied to m

Question

A) Construct the theoretical transfer function relating the voltage applied to motor, Va, to the velocity theta-dot (=omega) of the disk.

Hint: see equations 1 and 2.

B) Using the transfer function derived and the final theorm, find the symbolic expression for the steady-state disk velocity for a step input of Va=12V. Assume a zero-magnet configuration for hte damping coefficient B. How would the steady-state disk velocity change if you added magnets to the system?

C)The mechanical power of the motor is given by P_mech = T theta-dot. The electrical power of the motor is given by P_el = V_a i. What should the theoretical relationship by between K_t and K_e for the conservation of power? When considering the electrical equation of the motor (eq 2), assume no power losses in the motor windings (R_a = 0).

Construct the theoretical transfer function relating the voltage applied to motor, Va, to the velocity theta-dot (=omega) of the disk. Using the transfer function derived and the final theorem, find the symbolic expression for the steady-state disk velocity for a step input of Va=12V. Assume a zero-magnet configuration for hte damping coefficient B. How would the steady-state disk velocity change if you added magnets to the system? The mechanical power of the motor is given by P_mech = T theta-dot. The electrical power of the motor is given by P_el = V_a i. What should the theoretical relationship by between K_t and K_e for the conservation of power? When considering the electrical equation of the motor (eq 2), assume no power losses in the motor windings (R_a = 0). T = Kti = J + B Va = iRa + kc

Explanation / Answer

A)

From given circuit

theta-dot = w

theta-dot-dot = w-dot

Va = IRa + IRp + Ke*w

taking laplase transform ----

Va(s) = I(s)*(Ra+Rp) + Ke*W(s) -(1)

Kt*I = J*(w-dot) + Bw

taking laplase transform ----

Kt*I(s) = J*s*W(s) + BW(s)--------(2)

from equation (1)&(2)

Va(S) = ((Ra+Rp)/Kt)(Js+B)W(s) + KeW(s)

Transfer Function = W(s)/Va(s)

TF = 1/[((Ra+Rp)/Kt)(Js+B)+Ke]

= Kt/[(Ra+Rp)(Js+B)+KeKt]

B) Va=12V

Va(s) = 12/s

Final Value theroem

lim _{{t to infty }}f(t)= lim _{{sto 0}}{sF(s)}

W(s) = 12Kt/[s*((Ra+Rp)(Js+B)+KeKt)]

SW(s) = 12Kt/[(Ra+Rp)(Js+B)+KeKt]

steadt staet value at s=0

= 12Kt/((Ra+Rp)*B+KeKt)

If B=0

steadt stae value = 12

now if we add magnet then steadt state value will decrease.

= 12Kt/((Ra+Rp)*B+KeKt)

C)

P_mech = T*w

P_el =Va*I

pwer conservation

P_mech =P_el

T*w=Va*I ------(3)

for Ra=0

Va= Ke*w

now

(Kt*I)w = (Ke*w)*I

Kt=Ke

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