Question

Question 3: DC motors can be simplistically modeled as shown in Figure 5 MoTor back emf Vi: +1 Resistor Ra Ia Figure 5: Simplified model of DC Motor (a) Write the three characteristic equations that determine the behavior of the DC motor. Denote torque constant and speed constant as Ka, K, respectively. (b) Motor parameters are the quantities that define the behavior of the motor. List the motor parameters from the described model. What are their units? (c) For given value of driving voltage V, and motor parameters, find torque T as a function of angular speed w (d) Find the no-load speed wNL and the stall torque T, from part (c) (e) Experimental data from a permanent magnet DC Maxon motor operated at 18 V is shown in Table 1. Determine the torque constant Ka, speed constant Kb, armature resistance Ra Stall torque (mNm)24 No load speed (rpm 3000 Table 1: Output torque at different speeds observed at Va 18V 36 32 28t-- 24 E 3 20 12 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Angular speed (rpm) Figure 6: Torque-angular speed plot (f) Write the relationship between torque T and w for the same motor operating at armature voltage of Va-24V. Draw this on the plot in Figure 6. For this operating voltage, what torque will the motor generate at 1500 rpm?

Answer 1

a)

If speed constant if $K_b$ back emf constant is

$V_b = K_e \dot \theta$

$I_a R = V_a - V_b \Rightarrow I_a R= V_a - K_e\dot \theta$

b)

Motor Parameters:

R = Resistance measured in ohms (Ω)

Ka = Torque Constant measured in Ņ-mA

Ke-back - em f Constant measured in V - s/rad

Kb-speed Constant measured in rad,s-1/V

c)

$I_a R= V_a - K_e\dot \theta$

Ka

Torque is given by:

$T=\frac{K_a(V_a - K_ew)}{ R }$

d)

$T=\frac{K_a(V_a - K_ew)}{ R }$

No load speed: When T = 0

$w_{nL}=\frac{V_a}{ K_e }$

Stall Torque: when angular velocity = 0

KaVa Tstall

e)

000rpm 3000 *2T 60 314.16rad/s

wnL =-= 314.16

Ke0.0573Vs/rad

in a DC motor, usually:

Back emf constant = Torque constant:

KaVa Tstall

KaVa0.0573 18 Tstall = 0.024

R-42.97 5Ω

f)

$T=\frac{K_a(V_a - K_ew)}{ R }$

0.0573(24 0.0573 *w) 42.975

0.032 416.727

$416.727 rad/s= 416.727*60/2\pi = 3979.5 rpm$

No load speed = 3979.5 rpm

Stall torque = 32mNm

36 Stall Orgue-ㅢ + + + 28 24 3 20 1tI 12 No Load ' speed 500 1000 1500 2000 25003000 3500 4000 45005000 Angular speed (rpm)

Torque at 1500 rpm is: 20mNm