Exercises: The dynamo and the DC motor

Explain in your own words what electromagnetic induction is: what the three requirements are for generating a voltage in a conductor and what happens if there is no relative motion between the conductor and the magnetic field.

A bicycle dynamo powers the lights by pressing against the wheel. Explain where the electrical energy it produces really comes from and why you notice more resistance when pedalling while the dynamo is connected.

Match each part of a DC motor to its function: (1) stator, (2) rotor, (3) split-ring commutator, and (4) brushes, with: (a) stationary part that creates the magnetic field; (b) rotating part containing the coils; (c) segmented cylinder that reverses the current at the right moment; (d) sliding contacts that supply current to the moving part.

Explain what problem the commutator-brush assembly solves in a DC motor: how current reaches a rotating part and why its direction must be reversed every half turn so that the torque does not stop.

A small direct current motor, powered by a battery, spins in the opposite direction to the one desired.

Challenge of the unit. Explain the simplest way to reverse its direction of rotation and justify why it works.

1. a) State what change you would make to the connection of the motor to the battery so that it spins in the opposite direction.

1. b) Justify your answer taking into account that a force acts on the motor's coils due to the magnetic field of the magnet, and its direction depends on the direction of the current and the direction of the magnetic field.

Hint. Think about what happens to the force on a conductor carrying current if you change the direction of the current but leave the magnet as it is.

You will be assessed on: whether you clearly state the action to carry out, whether you relate the direction of the current to the direction of the force on the coils, and whether you explain how this reverses the direction of rotation.

Match each appliance to the electrical machine inside it and the direction of conversion: (a) bicycle dynamo, (b) drill, (c) car window motor and (d) toy fan, stating in each case whether it converts mechanical energy into electrical energy or vice versa.

A direct current motor and a dynamo are very similar machines inside: both have a fixed magnet (or electromagnet) and a coil that rotates inside it.

1. a) Explain what differentiates them in terms of the direction of the energy conversion: what energy goes in and what energy comes out in each one.

1. b) Explain why a motor whose shaft is turned by hand can behave as a generator.

The brighter the bicycle light powered by a dynamo shines, the harder it is to pedal. Explain this relationship using the conservation of energy: the electrical energy coming out of the dynamo must enter from somewhere.

A direct current motor is powered at $12\ \text{V}$ and draws $3\ \text{A}$.

1. a) Calculate the electrical power it absorbs, $P = V \cdot I$.

1. b) Explain what this value means and how this power is divided between the two destinations it has in a motor.

A motor absorbs $100\ \text{W}$ from the mains and delivers $80\ \text{W}$ of mechanical power at its shaft.

1. a) Calculate its efficiency, $\eta = \frac{P_{util}}{P_{abs}} \cdot 100$.

1. b) Calculate the power that is lost and explain what it is transformed into.

The figure shows a schematic cross-section of a DC motor. The two identical parts marked with "?" are located next to the commutator.

1. a) Explain the function of the labeled parts of the motor: housing (stator), magnet, rotor, commutator and shaft.

1. b) Identify the two parts marked with "?" and explain their function in the motor.

1. c) Explain which forces make the rotor turn and why the commutator must reverse the current every half turn.

A cordless drill is powered by a $18\ \text{V}$ battery and consumes $5\ \text{A}$ at full load, with an efficiency of $75\ \%$. (a) Calculate the electrical power absorbed. (b) Calculate the useful mechanical power at the drill bit, $P_{util} = \eta \cdot P_{abs}$, expressing the efficiency as a decimal. (c) Calculate the power lost and reason why the casing heats up during use.

The figure shows the chain of energy transformations from pedalling to a bicycle light, with one stage covered by a question mark, as shown in the figure.

(a) Identify the machine and the form of energy in the stage marked with the question mark. (b) Explain why there is an output of heat to the surroundings and which law guarantees that all the energy from pedalling ends up in one of the two outputs. (c) Reason what would happen to the pedalling effort if the light bulb were disconnected.

A bicycle dynamo generates alternating current and the motor of a cordless drill works with direct current.

1. a) Explain the difference between direct current and alternating current.

1. b) Explain why a battery can only supply direct current.

Challenge of the unit. In a simple direct current motor, supplying it with a higher voltage makes it spin faster.

1. a) Explain this relationship by reasoning through the chain of causes: more voltage, more current through the coils, greater forces on them and, as a consequence, higher rotation speed.

1. b) Describe how you would vary the speed of a toy fan driven by a direct current motor using an adjustable voltage source (what setup you would build, which quantity you would change and what you would observe).

You will be assessed on: whether you correctly link each step of the chain (voltage, current, force, speed), whether you mention the relationship between voltage and current in the coil (Ohm's law: $I = V/R$), and whether the procedure in part b) is clear and well organized.

A car window motor is rated at $60\ \text{W}$ and operates at $12\ \text{V}$. (a) Calculate the current it draws under normal operation, $I = \frac{P}{V}$. (b) Choose the appropriate fuse from $5\ \text{A}$, $7{,}5\ \text{A}$ and $10\ \text{A}$, justifying your choice. (c) Reason why the fuse rating should be slightly above normal current but not far above it.

A direct current motor connected to a $12\ \text{V}$ line vertically lifts a $20\ \text{kg}$ load at a constant speed of $0{,}2\ \text{m/s}$ while drawing $4\ \text{A}$ ($g = 9{,}8\ \text{m/s}^2$). The internal resistance of the armature is known to be $R_i = 0{,}4\ \Omega$.

(a) Determine the overall electromechanical efficiency of the system and break down the power losses: calculate how much power is dissipated by Joule effect in the armature winding and how much corresponds to the remaining losses (mechanical friction and magnetic losses).

(b) An operator claims that if the load mass is tripled ($60\ \text{kg}$), the motor will consume exactly triple the current ($12\ \text{A}$) and lift it at the same speed. Reasonably refute this claim by analyzing how the back electromotive force ($E'$) and the rotational speed vary with the increase in load torque in a DC motor.

(c) Calculate the current that the motor would draw if the load were to jam, stalling the rotor ($n = 0$), and justify what immediate consequences this would have for the integrity of the winding if a circuit breaker does not trip in time.

The diagram represents the rated power balance of a DC motor supplied at a constant voltage of $24\ \text{V}$. During a test on the test bench at full load, an internal armature resistance of $R_i = 0{,}8\ \Omega$ is measured.

(a) Quantify the total flow of losses ($?$) and break it down quantitatively by calculating the power dissipated by the Joule effect in the winding versus the remaining stray losses (mechanical and iron losses).

(b) After an extended test without forced ventilation, the armature copper heats up, rising from $20\ ^\circ\text{C}$ to $70\ ^\circ\text{C}$ (temperature coefficient $\alpha = 0{,}00393\ ^\circ\text{C}^{-1}$). Predict the new value of the power lost through the Joule effect if the motor continues to draw the same $100\ \text{W}$ of electrical input power, and calculate how the overall efficiency changes.

(c) A student claims that if the mechanical load is reduced so that the motor delivers only $40\ \text{W}$ of useful power, the efficiency will remain $80\ \%$, since "losses are cut in half". Refute this statement by distinguishing between fixed and variable losses in a DC electrical machine.

A team of students builds an educational prototype of an elementary DC motor consisting of a single rectangular loop and a cylindrical commutator with only two segments connected to two fixed brushes. When starting it up, they observe two anomalous behaviours:

1. At certain initial angular positions, when energising the motor with DC voltage, it does not start autonomously and remains stationary even though current flows.

2. When they manage to start it by giving it a manual push, strong sparks occur between the brushes and the segments during each half-period of rotation.

(a) Geometrically justify, with respect to the field lines of the stator, the exact position where the starting "dead spot" occurs, and analytically prove why the instantaneous motor torque is zero in that position.

(b) Propose and justify a constructive solution in the armature (regarding the number of coils and the design of the segment commutator) that eliminates the starting dead spots and reduces the fluctuation of the motor torque delivered at the shaft.

(c) Explain the electromagnetic origin of sparking during commutation using the phenomenon of self-induction in the coils, and state what consequences it will have on the commutator in the medium term if it is not resolved.

To control the opening and closing of an automatic gate driven by a $12\ \text{V}$, $3\ \text{A}$ DC motor, an H-bridge circuit is designed with four power switches ($S_1, S_2$ in the upper and lower left branch; $S_3, S_4$ in the upper and lower right branch; with the motor connected between the intermediate nodes).

(a) Draw up the truth table for the switches determining the switching combinations for the four operating states: clockwise rotation, counterclockwise rotation, coasting to a stop (freewheeling), and fast dynamic braking. Physically justify what happens in the armature in the latter case to make the motor stop almost instantly.

(b) Reasonably identify which combinations of switches lead to a catastrophic short-circuit failure (shoot-through). If, due to a control logic fault, $S_1$ and $S_2$ are closed simultaneously, and the power supply has an internal resistance of $0{,}08\ \Omega$, quantitatively predict the short-circuit current and the resulting damage.

(c) When abruptly switching from running to stopped, destructive voltage spikes are detected at the switch terminals. Explain the electromagnetic origin of these spikes and design the necessary protection by integrating flyback diodes, sketching or detailing their forward-bias orientation in the bridge.

A DC dynamo whose electromotive force follows the equation $E = 0{,}012 \cdot n$ (with $E$ in volts and the rotational speed $n$ in $\text{rpm}$) and whose internal resistance is $R_d = 0{,}4\ \Omega$, is used to charge a battery with a constant EMF $E_b = 12\ \text{V}$ and an internal resistance $R_b = 0{,}2\ \Omega$. The two-wire connection line between both machines has a total resistance of $R_c = 0{,}4\ \Omega$.

(a) Reasonably determine above what rotational speed ($n_{\min}$) of the dynamo the battery will begin to charge. Predict what would happen electrically and mechanically to the dynamo if the speed drops below that threshold and the circuit lacks a reverse-current blocking diode.

(b) When the dynamo is mechanically driven at $1400\ \text{rpm}$, calculate the charging current, the total electrical power induced in the dynamo, the net chemical power stored in the battery, and the total power dissipated by the Joule effect in the entire installation, checking the power balance.

(c) Quantitatively justify what percentage of the power generated by the dynamo is lost as heat and propose two specific technical measures in the design of the line or the machine to optimise the charging efficiency.

For the winch of a light industrial hoist, a motor-gearbox unit must be selected capable of vertically lifting a $40\ \text{kg}$ load at a constant linear speed of $0{,}3\ \text{m/s}$ ($g = 9{,}8\ \text{m/s}^2$). The cable is wound on a $10\ \text{cm}$ diameter drum coupled to the output shaft of a gear reducer with a mechanical efficiency of $80\ \%$. Two commercial DC motors are available:

• Motor A: Rated voltage $12\ \text{V}$, rated current $10\ \text{A}$, rotational speed $2400\ \text{rpm}$, electromechanical efficiency of $65\ \%$.

• Motor B: Rated voltage $24\ \text{V}$, rated current $9\ \text{A}$, rotational speed $1800\ \text{rpm}$, electromechanical efficiency of $75\ \%$.

(a) Determine the useful mechanical power that must be delivered by the motor shaft to lift the load at the required speed, taking into account the losses in the reducer. Justify which of the two motors is viable for this application.

(b) For the selected motor, calculate the gear ratio $i = \omega_{\text{motor}} / \omega_{\text{drum}}$ required in the reducer and the rated useful torque (in $\text{N}\cdot\text{m}$) that this motor delivers at its own shaft.

(c) If an installer mistakenly mounts the discarded motor while maintaining the same gear ratio calculated in section (b), predict whether the system will be able to lift the load at rated conditions and analyze the thermal and operational consequences for that motor.

In the technical service shop, an angle grinder with a universal DC motor is inspected, showing a marked loss of torque under load, abnormal heating in the head, and a continuous shower of sparks inside. Upon dismantling the equipment, it is found that the graphite brushes are worn below their minimum limit and the commutator exhibits pronounced blackening and irregularly grooved segments.

(a) Justify from a tribological and electrical point of view why brushes are traditionally made of graphite or electrographite blocks rather than highly conductive metals such as solid copper. Explain how the force of the brush-holder spring relates to contact resistance and the micro-sparking phenomenon.

(b) An inexperienced user suggests filing the copper segments smooth and replacing the brushes with two solid brass rods "so that they last much longer and don't wear out." Technically refute this proposal by predicting what failure would occur in the commutator during commutation between adjacent segments.

(c) Many current cordless tools replace this system with brushless (BLDC) motors. Compare both technologies by critically analyzing overall energy efficiency, initial cost, maintenance needs, and the complexity of the associated electronic control circuit.

A cycle tourist plans to power a navigation system during a trip using a hub dynamo installed in the front wheel. The dynamo delivers an average constant power of $3\ \text{W}$ at their cruising speed of $18\ \text{km/h}$. The energy is transferred through an electronic charge controller (efficiency of $80\ \%$) to recharge an external lithium battery with a rated voltage of $3{,}7\ \text{V}$ and a capacity of $5000\ \text{mAh}$.

(a) Calculate the useful energy stored in the battery when fully charged, expressing it in watt-hours ($\text{W}\cdot\text{h}$) and in joules ($\text{J}$). Determine the uninterrupted pedaling time and the route kilometers required to complete the recharge from $0\ \%$ to $100\ %$.

(b) If the hub dynamo has an electromechanical conversion efficiency of $60\ \%$, calculate the extra mechanical power that the cyclist's legs must continuously apply to the pedals exclusively to keep the recharge active, and evaluate the additional mechanical work (in $\text{kJ}$) performed during a $4\ \text{hour}$ ride.

(c) When climbing a mountain pass at $6\ \text{km/h}$, the voltage at the dynamo terminals drops below the controller's minimum activation threshold ($4{,}5\ \text{V}$) and charging is cyclically interrupted. Conceptually design an electronic solution in the power conditioning stage (or in the storage scheme) to solve this problem without forcing the cyclist to pedal harder on the climbs.

Topic Challenge. Imagine you work as a technical advisor for sustainable mobility and need to analyze the feasibility of regenerative braking on an electric scooter, where the DC motor operates reversibly as a dynamo when going down slopes.

1. Real data collection: Consult the hourly average electricity price ($€/\text{kWh}$) on Red Eléctrica de España's (REE) ESIOS platform and look up the technical specifications of a commercial scooter (note down nominal voltage $U$, power, and estimate $R_i \approx 0,8\text{ }\Omega$).

2. Modeling and calculation: In LibreOffice Calc or Tinkercad Circuits, model the equivalent circuit in traction ($U = E' + I \cdot R_i$) and in regenerative braking as a generator ($E = U + I \cdot R_i$). Calculate the induced EMF and the energy recovered on a section with a $4,5\,\%$ slope at $18,5\text{ km/h}$, applying an electromechanical efficiency of $68,5\,\%$.

3. Reasoned proposal: Write a brief technical report with a power balance table, the annual cost savings calculation, and a physical justification of how the braking torque varies with speed.

Assessment criteria:

• Rigor in using the electrical equations of the dynamo and the DC motor.

• Precision in incorporating real data from REE and the manufacturer.

• Consistency in the technical justification of the machine's reversibility.