The objective of this laboratory is to determine the

Question

Mathematical Modeling of a DC Motor

1 Objective:

The objective of this laboratory is to determine the mathematical model of a DC motor which will be used in the subsequent labs to control the velocity and position of a physical system.

2 Introduction:

Figure: DC motor model for position and velocity control.

The purpose of this laboratory is to develop a mathematical model of a servo system from Feedback Instruments Limited. The main component of the servo system is the DC motor. The electrical model of the motor is shown in Figure. The DC motor is basically a torque transducer that converts electrical energy into mechanical energy. The torque developed on the motor shaft is directly proportional to the field flux and the armature current. Assuming that the field flux is constant, the motor torque can be written as

Tm = K_i * i_a N.m

where i_a is the armature current and K_i is the motor torque constant. When the current carrying conductor moves in the magnetic field, a voltage proportional to the speed of the motor is developed across the terminals of the motor (back emf) that opposes the current.

Therefore, the back emf, e_b, can be written as

e_b = K_b * ω_m V

where ω_m is the speed of the motor in rad/s and Kb is the back emf constant. In the SI units, the back emf constant is equal to the motor torque constant. However, if the SI units are not used, a conversion factor to relate the torque constant and the back emf constant

K_i (N.m/A) = 14.8K_b (V/rad/sec)

Now, writing the electrical equation using Kirchoff Voltage Law (KVL), we get

where ea is the applied voltage to the armature, Ra is the armature resistance and La is the armature inductance. Since the motor is driven by the servo amplifier, the armature voltage is given by

e_a = K_sv * e_s

where e_s is the voltage applied to the servo amplifier and K_sv is the gain of the servo amplifier. If the flywheel is mounted directly to  the motor shaft, the mechanical output of the servo-mechanism shown in Figure is given by

T_m = J_eθ¨_m +B_mθ?_m

where θ_m is the angular position of the shaft in radians, B_m is the viscous friction of the motor and J_e = J_m + J + J_tg is the effective inertia that the motor sees. Note that J_m is the motor inertia, J is the flywheel inertia and J_tg is the inertia of the gearbox/tach attached to the motor. The output voltage of the tachometer, etach, is given by

e_tach = K_tach ω_m

where K_tach is the tachometer constant. The position of the output shaft is given by

θ_o = θ_m/N_g

where N_g is the gear ratio of the gear box. The output voltage of the potentiometer, e_pot is given by

e_pot = K_potθ_o V

where K_pot is the potentiometer constant. Note that the output shaft is the shaft of the gear box with a gear ratio of 30. The potentiometer is connected to the output shaft.

Pre-Lab Assignments:

1. Draw the block diagram of the servo system shown in Figure with es as the input; and tachometer and potentiometer voltages as the outputs.

2. Write the electrical time constant of the motor in terms of R_a and L_a, and the mechanical time constant in terms of J_m and B_m.

3. Find the following transfer functions: E_tach/E_s, and E_pot/E_s.

4. Assuming that the electrical time constant is much smaller than the mechanical time constant, simplify the transfer function E_tach/
E_s by neglecting the electrical time constant.

5. Assume that the term R_aB_m is much smaller than the term KiKb. Rewrite the transfer function found in part 4 and find the time constant of this transfer function.

6. etach and epot are the electrical quantities (in volts) which you will be able to measure in the lab. Given these quantities, please indicate the units (using SI) for the following system parameters: R_a, L_a, i_a, e_a, e_pot, e_tach, K_tach, K_b, K_i, K_pot, T_m, J_m, J, B_m, θ_m, ω_m and θ_o.

In-Lab Assignments:

1. Use an impedance meter to find the resistance, R_a, and inductance, L_a, of the armature winding.

2. Find the inertia of the flywheel.

3. Set up the circuit shown in Figure 4.1 and collect the data in Table I.

Table I. Voltage versus Speed data to find Kb of the motor.

Input Voltage	Armature Voltage	Armature Current	back emf	Tach Voltage	Angular Velocity Slow Shaft	Angular Velocity of the motor
es (volts)	ea(volts)	ia(amp)	eb (volts)	Vtach (volts)	rpm	ωm (rad/sec)
0 V
0.2 V
0.4 V
0.6 V
0.8 V
1.0 V
1.2 V
1.4 V
1.6 V
1.8 V
2.0 V
2.2 V
2.3 V
2.4 V
2.5 V

4. Plot eb versus the motor speed ωm and find the back emf constant Kb.

5. Plot etach  versus the motor speed ωm  and find the tach constant Ktach.

6. Apply 1 volt input to the servo amplifier and record the tachometer output on the O-scope. Determine the transfer function, E_tach(s)/E_s(s) and the mechanical time constant of the motor.

7. Using the results of parts 4 and 6 and part 5 of pre-lab, find the effective inertia Je seen by the motor.

8. If the inertia of the gearbox/tach is 4.10^-6 kg m², find the inertia Jm of the motor

Report

1. Substitute the values of the motor parameters in the block diagram of the system drawn in the pre-lab and find the transfer functions E_tach(s)/E_s(s) . How does this transfer function compare with the transfer function obtained in part 6 of in-lab? If they are different, can you explain why?

2. Based on the results obtained, draw the block diagram for the speed control of the motor.

Answer 1

The assignment is all about Mathematical Modeling of a DC Motor. The main objective of this laboratory is to find out the mathematical model of a DC motor that will be employed in the subsequent labs to control the velocity and position of the physical system. 1) Sketch the block diagram of the servo system illustrated by means of es as the input; and tachometer and potentiometer voltages as outputs. 2) Write down the electrical time constant of the motor in terms of Ra and La and the mechanical time constant in terms of Jm and Bm. 3) Determine the given transfer functions: Etach/Es, and Epot/Es. 4) Supposing that the electrical time constant is much smaller as compare to the mechanical time constant, solve the transfer function Etach/Es by avoiding the electrical time constant. 5) Suppose that the word RaBm is much smaller than the word KiKb. Rewrite the transfer function found in part 4 and determine the time constant of this transfer function.