Question

A permanent magnet brushed DC motor (the type of motor that spins when connected to a constant voltage) can be described by the simplified model tö+) = Kv, where o is the angle of the motor shaft, v is the voltage applied to the motor, 7 is a positive constant known as the motor time constant, and K is a positive constant known as the motor speed constant. Dots indicate time derivatives, so è = do and dt di? a) The initial angle and angular velocity of the motor are zero. A unit step in voltage is applied to the motor. (That is, the voltage to the motor is v(t)= u(t).) Determine the Laplace domain position of the motor ©(s)= L{0(t)}. (Hint: transform the differential equation for the motor to the Laplace domain and solve for ©(s).) b) Use (s) from part (a) to determine o(t) for 1>0. c) What is the steady state speed (or velocity) of the motor? (Note that the motor velocity is ė) Hint: you should be able to determine this from part (b) by inspection. d) For the solution from part (b), at steady state the motor will spin at a constant speed. How long does it take the motor to reach 63.2% of its final speed? (Note that 1-é' < 0.6321) e) The initial angle and angular velocity of the motor are zero. A step in voltage with magnitude v, is applied to the motor. What is the steady state speed of the motor and how long does it take the motor to reach 63.2% of its final speed?

Laplace Analysis of the Permanent Magnet Brushed DC Motor

and the right final answer should be like this

Answer 1

dao Sols Given, Y do de = K V d a) aina VIU) On both sides, Laplan transform By taking ps out)+ SOL) S k k 013) 8² (1+5%) s 1st qus?) k 16) ( 013) T s(1+57) "P sº (1457) ** (- ī 1457 k = ols) s? (۱۹۲۹)؟ k q2 K + 015) s IST on both sides, inverse Laplace transform BY taking T ult) + e (+) k tult)- krult) -IT ] t -T + те (+) Kulu) dort) ( c ) velocity K ult) [!] dt kult) [o] -tln ult) + Km (-) e -tla do - Velocity Kult) K e dt

WON to Find steady state speed put dolt) Velocity Steady state d! + 00 uit) kulal 1 speed = kult). Steady stale -LITY de ^ ] [ice 11 kulu (8) dh v = 0.632 K 44). at tati El ] Kult) 0.63) kull) [i-e -LITY I-e'' 11 -> 0.032 speed will . sº Final value oc Its Reach VO) - Vault) (e) Ginen VE) -410 Kva e do K Va 414) du dol) speed = K Va 41). = steady state dl 2 sec to reach Moto this Case 9110, even value. to 53.2 Final ot speed IUS