Exercises: Alternating current electric motors

The figure shows the schematic cross-section of a three-phase asynchronous motor, with one of its parts marked with the question mark (?). The parts housing, stator and shaft are labeled.

1. a) Identify the part marked with the question mark and describe its construction (what elements it is made of and how they are joined).

1. b) Explain the function of the stator, the shaft and the housing.

1. c) Explain how the marked part manages to rotate through the action of the stator's magnetic field and explain whether it needs an external electrical connection or brushes.

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 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.

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 ($3\,000$, $1\,500$, $1\,000\ \text{rpm}$).

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 figure shows the nameplate of a three-phase motor with a missing piece of data. a) State which data item is missing and what information it provides. b) Explain why the nameplate shows two voltages and two currents (delta or star connection depending on the supply). c) Explain what the data item $\cos \varphi$ represents in a motor.

Match each application to the most common type of motor and justify your answer: (a) the pump of an industrial treatment plant, (b) the ceiling fan in a home, (c) the conveyor belt in a factory, (d) the compressor of a refrigerator; with a three-phase asynchronous motor or a single-phase motor. Reason why the three-phase asynchronous motor dominates industry (robustness, price, absence of brushes, and direct-on-line starting).

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).

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).

A two-pole motor at $50\ \text{Hz}$ operates with a slip of $3\,\%$. a) Calculate its rotational speed under load. b) Calculate its angular velocity $\omega = \frac{2 \pi \cdot n}{60}$ in $\text{rad/s}$. c) If off-load it rotates at $2\,970\ \text{rpm}$, calculate its off-load slip and explain why it never reaches $3\,000\ \text{rpm}$ even with no load (friction and windage).

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 motor nameplate states "$230/400\ \text{V}$ — $\triangle/\text{Y}$". Explain what this dual rating means: to which mains voltage it should be connected in delta and to which in star, what voltage each winding experiences in both cases ($230\ \text{V}$), and why the power of the motor is the same in both connections.

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.

On starting, an asynchronous motor consumes between $5$ and $7$ times its rated current. Explain the reason using the slip (when starting $s = 1$: induced currents in the rotor are maximum) and describe two practical consequences (voltage drop in the installation, tripping of protective devices) and two methods to reduce it (star-delta starting, variable frequency drive starting).

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.

In an industrial plant supplied at $400\ \text{V}$ and $50\ \text{Hz}$, a three-phase motor drives a pump delivering $4\ \text{kW}$ of mechanical power with an efficiency of $88\,\%$ and operating at a power factor $\cos \varphi = 0{,}72$. The three-phase supply line has a resistance per conductor of $R_L = 0{,}25\ \Omega$. a) Calculate the line current drawn by the motor and the total power dissipated due to the Joule effect in the three conductors of the line. b) To reduce losses and avoid penalties from the electric utility, it is planned to connect a delta-connected capacitor bank in parallel with the motor to raise the overall power factor to $0{,}95$. Calculate the total reactive power that the bank must supply and the individual capacitance of each capacitor. c) Quantitatively evaluate the percentage reduction in the transport line's Joule losses after compensation and justify why it is not advisable to oversize the capacitor bank to achieve a $\cos \varphi = 1$ under variable load conditions.

In a production line, a $400\ \text{V}$, $5{,}5\ \text{kW}$ three-phase squirrel-cage motor drives a conveyor belt. Suddenly, the motor slows down sharply, emits an abnormal low humming sound, and the internal thermal sensor triggers a shutdown due to overheating. The maintenance team performs the following checks: 1) Line-to-line voltage at the terminals: $U_{L1-L2} = 400\ \text{V}$, $U_{L2-L3} = 400\ \text{V}$, $U_{L1-L3} = 400\ \text{V}$. 2) Line currents measured with a clamp meter: $I_1 = 0\ \text{A}$, $I_2 = 18{,}6\ \text{A}$, $I_3 = 18{,}5\ \text{A}$ (the rated current is $11\ \text{A}$). 3) With the motor uncoupled and de-energised, the rotor turns freely by hand without play or abnormal friction. a) Interpret the measured data and reasonably rule out the fault hypotheses of mechanical bearing seizure and a complete power supply outage. b) Deduce the exact cause of the fault ("single-phasing") and justify why the motor kept turning instead of stopping the moment the abnormality occurred. c) Using the theory of forward and backward revolving fields, explain the origin of the humming sound and reason why currents $I_2$ and $I_3$ greatly exceed the rated value, predicting what would happen to the winding if the protection relay failed to trip in time.

A technical report proposes replacing the drive for a continuous $1{,}2\ \text{MW}$ raw mill in a cement plant: "It is recommended to replace the existing synchronous motor with a standard squirrel-cage asynchronous motor, since the synchronous motor consumes too much inductive reactive energy penalising the bill and, furthermore, its speed fluctuates excessively with mineral shock loads". a) Refute the two conceptual errors contained in the statement regarding speed stability and the behaviour of the synchronous motor concerning reactive energy. b) Explain using Mordey's "V" curves how the rotor excitation current of a synchronous machine can be adjusted so that it operates as a synchronous condenser that improves the plant's power factor. c) Design and justify the choice of the most suitable machine for this high-power drive, explaining the starting difficulty presented by the synchronous motor directly connected to the grid and what technical solution is usually adopted to start it.

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.

You must specify the electric motor to drive a conveyor belt that operates outdoors in an aggregate quarry (environment with high suspended dust and subject to periodic high-pressure water jets for cleaning). The belt requires a net mechanical power of $2{,}2\ \text{kW}$ at a speed between $1\,400$ and $1\,450\ \text{rpm}$. You have a three-phase $400\ \text{V}$, $50\ \text{Hz}$ supply. In the manufacturer's catalogue you find these three $2{,}2\ \text{kW}$ models: Motor A (2 poles, IP54, IE1 efficiency, rating plate $230/400\ \text{V}$); Motor B (4 poles, IP65, IE3 efficiency, rating plate $400/690\ \text{V}$); Motor C (4 poles, IP44, IE3 efficiency, rating plate $230/400\ \text{V}$). a) Reasonably select the ideal option, ruling out the other two one by one based on the required rated speed, suitability for the environmental conditions (interpreting the IP code), and current industrial energy efficiency regulations. b) For the chosen motor, determine the connection you must make on its terminal board (star or delta) to connect it to the $400\ \text{V}$ mains and draw the arrangement of the copper links across terminals $U_1, V_1, W_1, U_2, V_2, W_2$. c) If the selected motor has $\cos \varphi = 0{,}81$ and an efficiency of $86{,}5\,\%$, calculate its rated line current and determine the adjustment range of the motor protection circuit breaker's thermal release to ensure its protection.

Several values on the metal nameplate of a three-phase asynchronous motor have been erased by corrosion. Only the following parameters are legible: $P_{\text{useful}} = 2{,}2\ \text{kW} \quad | \quad 50\ \text{Hz} \quad | \quad \cos \varphi = 0{,}84 \quad | \quad \eta = 86{,}0\,\% \quad | \quad \text{Connection } \text{Y: } 400\ \text{V}$. When tested on the test bench at full load, a digital tachometer measures a shaft speed of $1\,430\ \text{rpm}$. a) Deduce in a justified manner the number of stator poles, proving the impossibility of it having 2 or 6 poles, and calculate with precision its nominal percentage slip and the useful torque transmitted to the shaft. b) Determine the total active power that will be displayed by a three-phase wattmeter connected to the motor input and the rms current value measured by a clamp meter on any of the three line conductors. c) An operator proposes connecting the motor in delta ($\Delta$) to the same $400\ \text{V}$ grid, claiming that this will allow the motor to run cooler and deliver more power without suffering damage. Refute their claim by calculating the voltage that each phase winding would withstand in delta compared to its design nominal value in star, and predict the thermal and inductive consequences on the motor insulation.

Topic Challenge. Design a mini electromagnetic audit in your home by analyzing two everyday electrical machines.

1. Data collection: Find a household appliance with an AC motor (washing machine, fan, or blender) and a transformer or power adapter. Record their nameplate ratings: voltages ($V_1$, $V_2$), power ($P$), and frequency ($f = 50\text{ Hz}$).

2. Technical calculations: In LibreOffice Calc, find the transformation ratio ($m = V_1 / V_2$) and estimate the nominal secondary current. For the asynchronous motor, calculate the line current ($I$) and apparent power ($S$) assuming $\cos \varphi = 0,85$, as well as its synchronous speed ($n_s = 60 \cdot f / p$) for $p = 1$ and $p = 2$ pole pairs.

3. Proposal and context: Check the peninsular daily demand curve on the Red Eléctrica de España (REE) website and write a brief report (one page) justifying why compensating for the reactive power of motors avoids overloading distribution transformers.

Evaluation criteria:

• Rigor and accuracy in calculating transformer and motor parameters ($m$, $I$, $S$, $n_s$).

• Correct handling of data and units in LibreOffice Calc.

• Quality of the report and technical consistency when linking the power factor to efficiency in the REE grid.