AC electric motors

The figure shows the stator of a three-phase motor with its three coils R, S and T arranged at $120^\circ$ and supplied by a three-phase mains. In the center there is an arrow, and the curved arrow indicates the direction of rotation.

1. a) Explain what type of current reaches each coil and how the three currents are phase-shifted with respect to each other.

1. b) Explain what the central arrow represents and why it rotates.

1. c) State the speed at which this field rotates, in rpm, if the mains is $50\ \text{Hz}$ and the motor has two poles (one pole pair).

Explain what a single-phase motor is and where it is used (washing machine, fan, drill, refrigerator): why the field of a single coil supplied in single phase is pulsating rather than rotating, why the motor does not start on its own, and what element (starting capacitor or shaded pole / short-circuited turn) solves the problem.

The synchronous speed of the rotating field is $n_s = \frac{60 \cdot f}{p}$, where $f$ is the mains frequency and $p$ is the number of pole pairs. Calculate the synchronous speed at $50\ \text{Hz}$ for $2$-, $4$-, and $6$-pole motors, and explain why manufacturers offer these "round" speeds ($3\,000$, $1\,500$, $1\,000\ \text{rpm}$).

Calculate the synchronous speed of $2$-, $4$-, and $6$-pole motors connected to a $50\ \text{Hz}$ grid, and that of a $4$-pole motor connected to an American $60\ \text{Hz}$ grid. Comment on the implication for a machine designed for $50\ \text{Hz}$ working on a $60\ \text{Hz}$ grid (speed $20\,\%$ higher).

Explain step by step why the rotor of an asynchronous motor rotates: what field passes through the squirrel cage, what currents are induced in it, and what forces appear on its bars. Reason why it is called "asynchronous" (the rotor never reaches the speed of the field) and what would happen to the torque if it did reach it.

Define the slip of an asynchronous motor, $s = \frac{n_s - n}{n_s}$, explaining what each speed is. State the value of the slip at the instant of starting ($n = 0$) and explain why the induced currents in the rotor are maximum at that instant.

A four-pole asynchronous motor connected to $50\ \text{Hz}$ rotates under load at $1\,440\ \text{rpm}$. a) Calculate its synchronous speed. b) Calculate its slip as a fractional value and as a percentage. c) Explain what happens to the slip if the mechanical load on the shaft is increased and how the motor responds (more induced current, more torque).

A four-pole motor at $50\ \text{Hz}$ delivers a useful power of $1{,}5\ \text{kW}$ with a slip of $4\,\%$. a) Calculate its rotational speed. b) Calculate its angular velocity. c) Calculate the torque delivered to the shaft, $M = \frac{P}{\omega}$, and express it in $\text{N}\cdot\text{m}$.

A motor absorbs $2{,}2\ \text{kW}$ from the mains and delivers $1{,}9\ \text{kW}$ to its shaft. a) Calculate its efficiency. b) Calculate the lost power and divide it qualitatively into copper losses in the stator and rotor, iron losses, and mechanical losses (friction and windage). c) Reason why a motor heats up more the heavier its load.

A four-pole, three-phase asynchronous motor ($50\ \text{Hz}$) drives an industrial fan whose resistive torque follows the quadratic law $M_r(n) = k \cdot n^2$. At the rated operating point, the motor delivers its rated useful power of $3\ \text{kW}$ rotating at $1\,425\ \text{rpm}$. a) Calculate the rated useful torque and the value of the load constant $k$ (expressed in $\text{N}\cdot\text{m}/\text{rpm}^2$). b) Due to an unexpected drop in the grid, the supply voltage decreases by $10\,\%$ ($U' = 0{,}90 \cdot U$). Knowing that the motor torque in the stable linear operating region is proportional to the slip and to the square of the voltage ($M \propto U^2 \cdot s$), predict and justify whether the motor will reach a new stable equilibrium point or stall. c) Determine the approximate new equilibrium rotational speed of the motor-load set after the voltage drop and calculate the new slip.

A centrifugal pumping system driven by a four-pole three-phase asynchronous motor ($400\ \text{V}$, $50\ \text{Hz}$) operates at rated conditions requiring $15\ \text{kW}$. Due to process requirements, the flow rate needs to be reduced to $70\,\%$. Management compares two options: 1) Keep the motor at $50\ \text{Hz}$ and partially close a throttling valve. 2) Install a variable frequency drive (VFD) and reduce the rotational speed while keeping the valve $100\,\%$ open. a) Relying on pump affinity laws (where flow rate is proportional to speed and power varies with the cube of speed), estimate the mechanical power required by the pump in option 2 and predict the relative energy savings compared to option 1. b) If the drive sets the output frequency to $35\ \text{Hz}$, determine the line voltage that must supply the motor under a constant $V/f$ scalar control law. Justify what would happen to the magnetic flux in the core and to the no-load current if the $400\ \text{V}$ were kept constant at $35\ \text{Hz}$. c) Predict what technical limitations regarding useful torque and cooling arise if the motor is operated above its rated frequency of $50\ \text{Hz}$ (field weakening) and below $15\ \text{Hz}$ with shaft-mounted self-cooling.

A single-phase induction motor of a concrete mixer has a faulty (open-circuit) starting capacitor. When connected to the $230\ \text{V}$ mains, it remains locked, emitting a humming sound. However, if the operator spins the pulley hard by hand in a clockwise direction, the motor starts and accelerates almost to its rated speed in that direction; if pushed to the left, it rotates counterclockwise. a) Demonstrate analytically and conceptually, using Ferraris' double revolving field theory, why the pure starting torque is identically zero ($s = 1$) and why a net motor torque arises as soon as the rotor experiences an initial speed in either direction. b) The damaged capacitor is replaced with another whose capacitance is one-fourth of the design value. Reasonably predict how the phase shift between the main and auxiliary winding currents, the resulting spatial magnetic field, and the starting capability under load will be affected. c) Draw or describe with precision the functional electrical wiring diagram that must include a two-position switch to allow reversing the direction of rotation of this single-phase motor at start-up in a fully autonomous way.

Explain how the direction of rotation of a three-phase asynchronous motor is reversed (by swapping any two of its three phases) and justify your answer using the rotating magnetic field: what changes in the sequence in which the coils are supplied. Would the same trick work on a single-phase motor? Justify your answer.