Exercises: Domestic installations and utilities

Match each component of a dwelling's consumer unit with its function: (1) power limiter (ICP), (2) main circuit breaker (IGA), (3) residual current device (RCD) and (4) miniature circuit breakers (MCBs) for each circuit, with: (a) cuts off the supply if you exceed the contracted power; (b) allows the power to the entire dwelling to be cut off at once and protects the whole installation against overcurrents; (c) protects people against earth leaks or current leakages; (d) protects the wiring of each circuit against overloads and short circuits.

The residual current device (RCD) compares the current entering the dwelling with the current returning through the neutral: if a fraction is missing (it escapes to earth through an appliance or a person), it trips in milliseconds. Explain in your own words how this works and why it is said to protect people rather than wiring.

Explain the two hazards that a miniature circuit breaker (MCB) protects against in each circuit: overload (too many appliances running at once, which heats up the cables gradually) and a short circuit (direct contact between live and neutral, causing a sudden, massive current).

A dwelling's installation is divided into independent circuits (lighting, power sockets, cooker, washing machine, etc.). Explain two advantages of this division: locating faults without going completely into the dark and using cables with cross-sectional areas appropriate to each circuit's consumption.

Modern sockets have a third contact: the earth connection. Explain where this cable leads (to an earth rod buried next to the building) and what happens if an appliance with a fault in its metal casing is plugged in (it diverts the leakage current to earth so that the RCD trips instead of shocking whoever touches it).

Describe the path of the water supply in a dwelling: where it enters (main stopcock and water meter), what two internal distribution networks exist (cold water and hot water downstream of the water heater) and which device should be checked to prevent leaks.

Gas can be supplied via mains (mains natural gas) or bottled (butane or propane cylinder). Explain one advantage of each supply method and two essential safety measures in a gas kitchen (permanent ventilation and inspection of hoses and valves).

Describe the components of telecommunications in a dwelling: what comes in from the street (fibre optic or cable), what the router does (distributes the connection via Ethernet cable and Wi-Fi) and why the router's position has such a major impact on wireless coverage.

The cooker circuit powers an oven rated at $3.000\ \text{W}$ at $230\ \text{V}$. (a) Calculate the current flowing through the circuit, $I = \frac{P}{V}$. (b) Choose the appropriate miniature circuit breaker (MCB) from $10\ \text{A}$, $16\ \text{A}$ and $25\ \text{A}$, justifying your choice. (c) Reason why the MCB rating must be above the normal current of the appliance but as close to it as possible.

A power socket circuit feeds at most $2.500\ \text{W}$ simultaneously at $230\ \text{V}$. (a) Calculate the maximum current of the circuit. (b) Choose its MCB from $10\ \text{A}$ and $16\ \text{A}$. (c) Explain what would happen if a $2.000\ \text{W}$ heater and a $1.500\ \text{W}$ radiator were connected to that circuit at the same time.

A dwelling has a contracted power of $4{,}6\ \text{kW}$. In the kitchen, the oven ($3\ \text{kW}$) and the hob ($2\ \text{kW}$) are operating simultaneously. (a) Calculate the total power and reason what happens to the power limiter. (b) Propose two ways to solve the problem without changing electricity supplier: a management solution (not using both at the same time) and a technical solution (increasing the contracted power), commenting on what each implies for the electricity bill.

The figure shows the single-line diagram of a dwelling's consumer unit, with an element marked with a question mark, as shown in the figure.

(a) Identify the element marked with the question mark and explain its protective function. (b) Describe the order of the components in the consumer unit and reason why the RCD is placed before the MCBs of each circuit. (c) Explain what happens in each case: an MCB trips, the RCD trips, or the power limiter trips, and how to reset them.

The cooker circuit, with a $25\ \text{A}$ MCB, uses thicker cables than the lighting circuit, which has a $10\ \text{A}$ MCB. (a) Explain the relationship between the maximum current of a circuit and the cross-sectional area of its cables (more current requires more copper to prevent overheating), reasoning with $P_{perdida} = I^2 \cdot R$. (b) Reason what can happen if a thin cable is 'protected' by an excessively large MCB.

A dwelling's electric water heater is rated at $1.500\ \text{W}$ and operates for about $2\ \text{h}$ per day. (a) Calculate its monthly energy consumption ($30$ days), in $\text{kW} \cdot \text{h}$. (b) Calculate its monthly cost at $0{,}20\ \text{€}$ per $\text{kW} \cdot \text{h}$. (c) Check that the consumer unit's $16\ \text{A}$ circuit is sufficient for the water heater, by calculating its current at $230\ \text{V}$.

The figure shows the three branch circuits of a dwelling supplied from the consumer unit, with the current of one of them marked with a question mark, as shown in the figure.

(a) Calculate the current flowing through the cooker circuit, marked with the question mark. (b) Calculate the current of the other two circuits. (c) Assign each circuit its standard MCB rating ($10\ \text{A}$, $16\ \text{A}$ or $25\ \text{A}$), justifying your choice.

When plugging in the washing machine, the dwelling's RCD trips and cuts off all power; with the washing machine unplugged, it can be reset without any problem. (a) Explain what this symptom indicates (an earth fault or leakage current inside the washing machine). (b) Describe an ordered diagnostic procedure to confirm which appliance is at fault. (c) Reason why this must not be 'fixed' by bypassing the RCD.

In a home with a single-phase supply at $230\ \text{V}$ and a contracted power of $5{,}75\ \text{kW}$ (limited to $25\ \text{A}$), the main distribution board has four circuits: C1 lighting ($10\ \text{A}$, $1{,}5\ \text{mm}^2$ cable), C2 general-purpose sockets ($16\ \text{A}$, $2{,}5\ \text{mm}^2$ cable), C3 cooktop and oven ($25\ \text{A}$, $6\ \text{mm}^2$ cable), and C4 washing machine and water heater ($20\ \text{A}$, $4\ \text{mm}^2$ cable). A user observes that the sum of the ratings of the four miniature circuit breakers (MCBs) is $71\ \text{A}$ and claims: "The installation is oversized and poorly designed; furthermore, to prevent the power from tripping when I simultaneously turn on a $2.000\ \text{W}$ radiator in the living room, the $2.800\ \text{W}$ oven, and the $1.500\ \text{W}$ water heater, I will replace the $16\ \text{A}$ MCB with a $25\ \text{A}$ one."\n\n(a) Refute the first part of the statement by technically justifying why the sum of the rated currents of the MCBs can be three times that of the main switch (ICP), referring to the concepts of diversity factor (simultaneity) and protection selectivity.\n(b) Calculate the total current that the three appliances would demand operating at the same time and quantitatively determine which switch in the chain will cut off the supply and under what physical principle (thermal or magnetic).\n(c) Justify the real risks of fire and deterioration if the $16\ \text{A}$ MCB of C2 were replaced with a $25\ \text{A}$ one without changing the $2{,}5\ \text{mm}^2$ wiring (whose allowable current-carrying capacity $I_z$ in conduit embedded in a wall is about $18\ \text{A}$).

The figure shows the consumer unit (distribution board) of a home with the ratings of its protection devices, with the maximum power of one of the circuits marked with a question mark.

(a) In a home appliance connected to circuit C2, an insulation fault occurs whereby the live conductor touches the metal casing through a contact resistance $R_{\text{def}} = 4.600\ \Omega$. Knowing that the building's earth connection has $R_{\text{tierra}} = 20\ \Omega$, calculate the earth leakage current and determine with reasoned arguments whether the RCD in the diagram will trip or not.\n(b) If an induction hob and an oven demanding a combined $6.440\ \text{W}$ are simultaneously used in circuit C3, determine which device (the main limiter ICP or the MCB of C3) will trip first if in the rest of the house only $230\ \text{W}$ of lighting in C1 is turned on.\n(c) Explain why the residual current device (RCD) has a rated current of $40\ \text{A}$ marked on it when the upstream limiter is $25\ \text{A}$, and what thermal danger would exist if that RCD were rated at only $16\ \text{A}$ in the event of an overall overload.

To an outlet of a domestic general-purpose socket circuit ($16\ \text{A}$, copper conductors of $2{,}5\ \text{mm}^2$ protected by a C-curve MCB), a power strip is connected that already powers computer equipment and several peripherals with a total consumption of $2.500\ \text{W}$. The user plugs a $1.500\ \text{W}$ heater at $230\ \text{V}$ into the same power strip. Seeing that the circuit breaker does not trip instantly, they conclude that "the installation handles the power with no problem".\n\n(a) Calculate the total current flowing through the line and the percentage overload experienced by the MCB relative to its nominal rated current.\n(b) If the resistance of the conductors of that line is $0{,}014\ \Omega$ per metre (round trip), calculate by how much the power dissipated by the Joule effect increases per metre of cable compared to that dissipated when the circuit operates at the allowable limit of $16\ \text{A}$.\n(c) Based on the combined operation of the bimetallic strip and the electromagnet of an MCB, explain why it does not trip immediately and technically refute the user's conclusion, indicating what risk the cable insulation faces if the overload is maintained for several hours.\n(d) If the power strip's own cable is only $1\ \text{mm}^2$ (allowable current capacity around $10\ \text{A}$), predict which element will fail thermally first and why the consumer unit's MCB does not guarantee the protection of smaller cross-section cables connected downstream.

In a home with a $230\ \text{V}$ supply protected by a $30\ \text{mA}$ residual current device ($t_{\text{disparo}} \le 40\ \text{ms}$), a washing machine with a metal casing suffers a direct insulation fault (the live wire comes into direct contact with the outer panel). Two different installations are analyzed:

• Home A: it has a protective conductor (earth connection) connected to the chassis, with an earth resistance of $R_A = 20\ \Omega$.

• Home B: an old installation where the socket has no earth connection (floating chassis).

In your calculations, consider that the only resistance in the path of the current is the one given in each case.

1. a) For Home A, calculate the fault current that is diverted to earth at the moment of the fault. Determine whether the residual current device will cut off the circuit without anyone needing to touch the appliance, and calculate the fault voltage of the casing with respect to earth before the cut-off ($V_d = I_d \cdot R_A$), assessing whether it exceeds the safety limit voltage ($50\ \text{V}$).

1. b) For Home B, describe what happens in the machine after the fault and calculate the current that will pass through a barefoot person with a body resistance of $1500\ \Omega$ who touches the panel (contact resistance with the floor: $500\ \Omega$). Determine whether the residual current device will trip and at what stage of the process.

1. c) Based on the physiological effects of electric current (perception threshold, muscle tetanization and ventricular fibrillation), justify why a $30\ \text{mA}$ residual current device on its own does not prevent the shock if there is no earth, refuting the common belief that "if I have a residual current device, the earth connection is unnecessary".

A family evaluates whether to replace their traditional butane gas cooker with an electric induction hob. To prepare their monthly meals, they need to transfer a net useful thermal energy of $100\ \text{kW}\cdot\text{h}$ to the cookware.\n\nTechnical and economic data:\n- Gas cooker: supplied by $12{,}5\ \text{kg}$ butane cylinders ($LCV = 12{,}8\ \text{kW}\cdot\text{h}/\text{kg}$). Due to thermal losses from convection and radiation to the environment, the efficiency of the gas burner is only $42\,\%$. The price of a cylinder is $17{,}50\ \text{€}$.\n- Induction hob: directly heats the ferromagnetic base of the vessel, with an overall energy efficiency of $84\,\%$. The cost of the electricity consumed is $0{,}19\ \text{€}$ per $\text{kW}\cdot\text{h}$.\n\n(a) Calculate the gross primary energy contained in a full cylinder and determine how many cylinders per month (or fraction of a cylinder) the family needs with the gas cooker.\n(b) Calculate the monthly electricity consumption in billed $\text{kW}\cdot\text{h}$ needed to provide exactly the same $100\ \text{kW}\cdot\text{h}$ of useful heat with the induction hob.\n(c) Determine the monthly cost of fuel/electricity for both alternatives and justify why comparing only the price per gross $\text{kW}\cdot\text{h}$ of gas versus electricity leads to an erroneous conclusion.\n(d) Analyse what impact the installation of the induction hob (typical maximum power of $7\ \text{kW}$) could have on the contracted electrical power of the home and the fixed term of the bill.

An electric water heater with an $80\ \text{L}$ capacity ($P = 1.800\ \text{W}$) supplies domestic hot water (DHW) to a home. Four people take showers consecutively in the morning. Each shower lasts $6$ minutes with a mixed water flow rate at the showerhead of $8\ \text{L/min}$ at a comfort temperature of $38\ ^\circ\text{C}$. Mains cold water enters at $14\ ^\circ\text{C}$, while the heater keeps the stored water at $62\ ^\circ\text{C}$ (recommended temperature for Legionella prevention). The standing thermal losses of the heater due to imperfect insulation are equivalent to $1{,}1\ \text{kW}\cdot\text{h}$ per day. ($c = 4.186\ \text{J}/(\text{kg}\cdot^\circ\text{C})$, water density $= 1\ \text{kg/L}$, $1\ \text{kW}\cdot\text{h} = 3{,}6\cdot 10^6\ \text{J}$).\n\n(a) Using a thermal mixing balance ($V_{\text{mezcla}} \cdot T_{\text{mezcla}} = V_{\text{termo}} \cdot T_{\text{termo}} + V_{\text{red}} \cdot T_{\text{red}}$), calculate how many litres of water at $62\ ^\circ\text{C}$ are drawn from the heater during each shower.\n(b) Deduce with reasoned arguments whether the heater will be able to supply hot water at the desired temperature to the third and fourth person if they shower consecutively without recovery time.\n(c) Calculate the total monthly electrical energy expenditure attributable to DHW (including the four daily showers and standing thermal losses) and its economic cost at $0{,}18\ \text{€}/\text{kW}\cdot\text{h}$.\n(d) A family member suggests: "If we turn down the water heater thermostat to $40\ ^\circ\text{C}$, we will save $50\,\%$ energy without noticing any difference". Quantitatively refute this statement by evaluating the volume of hot water that each shower would require at $40\ ^\circ\text{C}$ compared to $62\ ^\circ\text{C}$, and warn about the associated microbiological risk.

A detached single-family home of $150\ \text{m}^2$ located in the mountains (continental climate zone with cold winters) needs to replace its comprehensive heating and DHW system. The estimated annual thermal demand for the comfort of the home is $15.000\ \text{kW}\cdot\text{h}$ of heat. Three engineering alternatives are compared:\n- Alternative A (Bulk piped propane): Condensing boiler with a seasonal average efficiency of $91\,\%$. Propane cost: $0{,}095\ \text{€}/\text{kW}\cdot\text{h}$ of primary energy. Initial investment in boiler and tank: $3.200\ \text{€}$.\n- Alternative B (Biomass - Pellets): Pellet boiler with a seasonal efficiency of $84\,\%$. Calorific value of pellets: $4{,}8\ \text{kW}\cdot\text{h}/\text{kg}$; pellet price: $0{,}34\ \text{€}/\text{kg}$. Initial investment in boiler, hopper, and installation: $8.800\ \text{€}$.\n- Alternative C (Air-source heat pump with underfloor heating): Air-to-water heat pump with a seasonal average coefficient of performance ($\text{SCOP}$) of $2{,}75$ (affected by low outdoor temperatures in winter). Cost of electric $\text{kW}\cdot\text{h}$: $0{,}17\ \text{€}$. Initial investment (equipment and integration): $11.500\ \text{€}$.\n\n(a) Calculate the annual economic expenditure on fuel or electricity for each of the three alternatives to fully cover the demand of $15.000\ \text{kW}\cdot\text{h}$ of useful heat.\n(b) Calculate the simple payback period of the investment (amortisation) for the heat pump compared to the propane option based on the annual fuel savings.\n(c) Draft a technical assessment giving a reasoned choice for a family living in the house year-round, weighing not only economic cost and payback, but also operational constraints (space for a silo and ash maintenance in biomass vs reliability of COP during severe frosts in aerothermal systems).

In a two-story home with a reinforced concrete structure, a user has a $600\ \text{Mbps}$ symmetric fiber connection. The main router is located on the ground floor. In the upper-floor office, a room dedicated to critical remote work, the WiFi coverage in the $5\ \text{GHz}$ band drops to $-85\ \text{dBm}$ (with frequent disconnections), and in the $2{,}4\ \text{GHz}$ band it barely achieves $15\ \text{Mbps}$ with an average latency of $95\ \text{ms}$. The home electrical panel has independent circuits for both floors, each protected by a different residual current device (RCD).\n\n(a) The user purchases a "$1.000\ \text{Mbps}$" PLC (Powerline) kit and a standard plug-in WiFi repeater. Physically justify why both solutions will fail to deliver a high-speed, low-latency connection in this specific scenario (analyze the high-frequency filtering introduced by RCDs and coils in the electrical grid for the PLC, and the bandwidth loss due to half-duplex operation and floor slab attenuation in the WiFi repeater).\n(b) To install a network cable to the upper floor, running an Ethernet cable through the same corrugated conduits carrying the $230\ \text{V}$ phase, neutral, and ground wires is considered. Explain what electromagnetic induction issues (crosstalk/noise) and what regulatory impediments according to the REBT arise, and what type of shielded cable (FTP/STP) or physical alternative (such as plastic optical fiber, POF) would solve the problem safely.\n(c) Design the ideal network topology for this home, detailing the necessary components (router, switch, wired access point), the configuration of radio channels and bands to avoid overlap, and how to configure the identifier (SSID) to enable seamless roaming without connection drops.