Lab I · Industrial motor torque-speed characteristics

Objectives

As set by the Lab Works booklet:

1. Model and plot the steady-state torque-speed curve of a squirrel-cage induction motor (SCIM) as a function of slip.
2. Model and plot the torque-angle curve of an equivalent synchronous motor.
3. Model and plot the torque-speed characteristics of shunt-, series- and compound-wound DC motors.
4. Compute the equivalent RMS torque of an intermittent duty cycle and verify a continuous-duty motor selection against it.

Background

A motor's suitability for an application is captured by its torque-speed curve (Chapter 1). For the induction motor, the equivalent circuit gives the developed torque as a function of slip (eq. 1.1 of the Lab Works booklet):

$$\tau(s) = \frac{3 V_1^2\, (R_2'/s)}{\omega_s\left[(R_1 + R_2'/s)^2 + (X_1 + X_2')^2\right]}, \qquad \omega_s = \frac{2\pi n_s}{60}, \qquad n_s = \frac{120 f}{p}$$

Its maximum occurs at $s_\text{bd} = R_2' / \sqrt{R_1^2 + X^2}$, with $X = X_1 + X_2'$:

$$T_\text{bd} = \frac{3 V_1^2}{2\,\omega_s\left(R_1 + \sqrt{R_1^2 + X^2}\right)}$$

The synchronous motor follows $\tau_\text{em} = 3 V_t E_f \sin\delta / (\omega_s X_s)$. The DC motor follows $T = k\Phi I_a$ and $V_t = k\Phi\,\omega + I_a R$, with the field relation of each configuration: shunt, $\Phi$ constant; series, $\Phi \propto I_a$ below saturation; compound, a weighted combination.

Parameter sheet

Parameter	Symbol	Value
Supply frequency	$f$	50 Hz
Number of pole pairs	$p/2$	2 (4-pole)
Per-phase stator voltage (SCIM and synchronous)	$V_1$	230 V
SCIM stator resistance	$R_1$	0.5 Ω
SCIM referred rotor resistance	$R_2'$	0.4 Ω
SCIM stator + rotor leakage reactance	$X_1 + X_2'$	2.0 Ω
Synchronous reactance	$X_s$	5.0 Ω
DC motor rated armature voltage	$V_t$	240 V
DC motor armature resistance	$R_a$	0.8 Ω
Duty cycle: loaded torque / time	$T_1, t_1$	80 N·m, 10 s
Duty cycle: rest torque / time	$T_2, t_2$	0 N·m, 15 s

Procedure

The booklet's five steps, with the names the checker looks for shown in code font.

1. Set up the environment. Import NumPy and Matplotlib (already in the template).
2. Define parameters. Set V1, R1, R2, X, f, poles, then ns (rpm) and ws (rad/s).
3. Implement the models.
   • Induction motor: write scim_torque(s, R2=R2) from eq. (1.1). It must accept a single slip (the checker calls it with random values) and a NumPy array (for the plot); the second argument lets you try another rotor resistance. Compute s_bd, T_bd and the starting torque T_st ($s = 1$).
   • Synchronous motor: write sync_torque(delta), sweeping $\delta$ from 0 to $\pi/2$, and give the pull-out torque T_po and the angle delta_po (rad) where it occurs.
   • DC motors: write dc_motor(kind, Ia) returning (torque, speed_rpm) for kind equal to shunt, series or compound, using the assumptions above.
4. Generate the signals. Sweep each model over its range and compute the RMS torque Trms of the duty cycle:

$$T_\text{rms} = \sqrt{\frac{\sum T_i^2\, t_i}{\sum t_i}}$$

5. Plot and analyse. Plot all the curves on comparable axes. On the induction-motor curve, mark the starting, breakdown and full-load points. Report $T_\text{rms}$, then answer the questions under the workspace from your own results.

Report

The booklet requires, for each lab, one individual report and your code, in this order: Objective, Model, Parameters, Results, Discussion, Conclusion, Code. It is marked out of 20: model correctness 6, results and plots 5, critical discussion 6, report quality 3. The workspace builds that report for you from your code, figures, checks and answers.