Exercises: Power in alternating current

Define the three powers in alternating current (active, reactive, and apparent), state the unit for each ($\text{W}$, $\text{VAr}$, $\text{VA}$), and explain which of them produces useful work and actually heats resistors.

Explain in your own words what the power factor, $\cos \varphi$, is, what values it can take between a purely resistive load and a highly inductive load, and why electricity companies are interested in it being close to $1$.

A workshop consumes active power $P = 3\ \text{kW}$ with apparent power $S = 4\ \text{kVA}$, as shown in the power triangle in the diagram. Calculate the power factor of the installation and, applying Pythagoras' theorem to the triangle, the reactive power $Q = \sqrt{S^2 - P^2}$.

Name two virtually resistive domestic loads (which only consume active power) and two loads with significant inductors (which also consume reactive power). What happens to the power factor of a home when mainly the former are running?

Explain how an inductor stores energy (magnetic field) and how a capacitor stores energy (electric field), and why this energy travels 'back and forth' to the grid twice per cycle without being consumed: this back-and-forth movement is reactive power.

Generator sets and transformers are specified in $\text{kVA}$ rather than $\text{kW}$. Explain the reason (their thermal limits depend on the total current, not on how much of it is useful) and calculate the maximum active power that a $20\ \text{kVA}$ set can supply with loads of $\cos \varphi = 0{,}8$.

An industrial bill shows active energy ($\text{kWh}$) and reactive energy ($\text{kVArh}$). Explain what each term records, why only active energy does 'work', and above what proportion of reactive energy penalties are usually applied.

The old electromechanical disc meter only recorded active energy. Explain why (the disc moves with the useful component of the current) and what difference exists compared to modern electronic meters, which also record reactive energy and time-of-use tariffs.

A single-phase motor operating at $230\ \text{V}$ draws $10\ \text{A}$ with a power factor of $0{,}8$. a) Calculate the apparent power, active power ($P = V \cdot I \cdot \cos \varphi$), and reactive power (with $\sen \varphi$ deduced from the triangle). b) Explain which part of the current measured by the ammeter produces no useful work.

A single-phase motor operating at $230\ \text{V}$ draws $10\ \text{A}$ with a power factor of $0{,}8$. Keeping the active power constant, calculate the current it would draw if its power factor were $1$, and reason what practical consequences the current reduction brings (cable cross-sectional area, heating, voltage drop).

Explain how capacitors compensate for the reactive energy of inductors: what energy exchange takes place between them in each half-cycle and why the capacitor bank is connected in parallel with the installation rather than in series.

An installation consumes $P = 12\ \text{kW}$ with $\cos \varphi = 0{,}6$. a) Calculate the apparent power $S$ and the reactive power $Q$ (use $\sen \varphi = 0{,}8$ for that angle). b) Calculate the current at $400\ \text{V}$ (equivalent single-phase, $I = \frac{S}{V}$) and compare it with the current that would flow with $\cos \varphi = 1$.

A workshop presents $S = 25\ \text{kVA}$ and $Q = 15\ \text{kVAr}$. a) Calculate the active power by applying Pythagoras' theorem to the power triangle. b) Calculate the power factor. c) Reason whether the installation would pay a penalty given a limit of $\cos \varphi = 0{,}95$.

Two loads of $2\ \text{kW}$ at $230\ \text{V}$ operate, one with $\cos \varphi = 1$ and the other with $\cos \varphi = 0{,}5$. a) Calculate the current for each one. b) Calculate the justified economic cross-section for each case, considering an allowable current density of $4\ \text{A/mm}^2$, and discuss why the distribution company passes on the cost of reactive power.

Old fluorescent tubes with electromagnetic ballasts had $\cos \varphi \approx 0{,}5$ if they did not include a power-factor correction capacitor. Explain why the ballast (an inductor) introduces reactive power, what the parallel capacitor does, and why LEDs with electronic drivers pose different problems (harmonics rather than traditional phase shift).

An industrial unit has $P = 40\ \text{kW}$ and $\cos \varphi = 0{,}8$, and wants to increase it to $0{,}98$. a) Calculate the current reactive power (with $\sen \varphi = 0{,}6$) and the permissible reactive power after the upgrade (with $\sen \varphi \approx 0{,}2$). b) Obtain by subtraction the required rating of the capacitor bank. c) Comment on why the bank is installed with automatic steps.

A factory has a power factor of $0{,}7$ and consumes $50\ \text{kW}$ of active power. The supplier penalises consumptions with $\cos \varphi < 0{,}95$. a) Calculate the current apparent power and reactive power. b) Calculate the reactive power remaining with $\cos \varphi = 0{,}95$ (you may use $\sen \varphi \approx 0{,}31$) and, by subtraction, the capacitive reactive power that the capacitor bank must supply. c) Comment on why correction does not reduce the active energy bill but does reduce other charges.

The monthly electricity bill of a metallurgical plant shows the following consumption records during operating hours: active energy $E_p = 40\,000\ \text{kWh}$ and inductive reactive energy $E_q = 28\,000\ \text{kVArh}$. Regulations establish that there is no penalty if $\cos \varphi \ge 0{,}95$ (which is equivalent to the reactive energy not exceeding $33\,\%$ of the active energy); if the reactive energy exceeds said limit but not $75\,\%$, the excess is billed at $0{,}0415\ \text{€/kVArh}$; and the bracket exceeding $75\,\%$ is penalized at $0{,}0623\ \text{€/kVArh}$. a) Determine the plant's average monthly power factor and interpret whether it incurs a penalty. b) Calculate the total amount of the surcharge for billable reactive energy in that month applying the official brackets. c) An installer quotes an automatic capacitor bank for $4\,200\ \text{€}$. Assuming that the operating regime is similar throughout the entire year, calculate the simple payback period of the investment (in months) and reasonably justify the purchasing decision.

Two workshops consume the same active power of $20\ \text{kW}$ at $400\ \text{V}$ (equivalent single-phase), but one operates with $\cos \varphi = 1$ and the other with $\cos \varphi = 0{,}6$. a) Calculate the current drawn by each and the ratio between them. b) Calculate the Joule losses in a $1\ \Omega$ line for each case ($P_p = R \cdot I^2$) and express them as a percentage of the useful power. c) Draw a conclusion regarding transmission efficiency at a low power factor.

A bill penalises reactive energy that exceeds $33\,\%$ of the active energy consumed (equivalent to $\cos \varphi \approx 0{,}95$). A factory consumes $30\,000\ \text{kWh}$ and $16\,000\ \text{kVArh}$ per month. a) Check whether it exceeds the limit. b) Calculate the billable excess of $\text{kVArh}$. c) With a surcharge of $0{,}04\ \text{€/kVArh}$, calculate the annual penalty and compare it with the cost of a $50\ \text{kVAr}$ capacitor bank ($3\,000\ \text{€}$): what is the payback period?

A factory that only operates from Monday to Friday installs a fixed $60\ \text{kVAr}$ capacitor bank permanently connected to the main switchboard. Over the weekend, with the machinery turned off, the industrial building only registers a residual active consumption of $4\ \text{kW}$ due to security systems and servers. a) Qualitatively draw the power triangle over the weekend, calculate the resulting power factor, and classify it (inductive or capacitive). b) On Monday morning, technicians discover that several electronic devices have broken down due to overvoltage, and the distribution system operator notifies a financial penalty for capacitive injection into the grid during off-peak hours. Physically justify the cause of the terminal voltage rise (local Ferranti effect) and why distribution companies penalize this injection. c) Design a modification to the compensation system (type of bank and control logic) that solves both problems while ensuring proper compensation during weekdays.

An industrial building is fed from a main switchboard. $120\ \text{m}$ away, an internal copper line supplies a workshop with four large water pumps of $18{,}4\ \text{kW}$ each, with $\cos \varphi = 0{,}75$ at a three-phase voltage of $400\ \text{V}$ (take $\sqrt{3} \approx 1{,}732$). The cross-section of the internal conductors operates at its allowable thermal limit. Two options are proposed: installing an automatic bank at the main supply switchboard or installing individual capacitors at the terminals of each pump. a) Calculate the current flowing through the $120\ \text{m}$ internal line when all four pumps operate with and without individual compensation to $\cos \varphi = 0{,}98$, and determine by what percentage the Joule losses ($R \cdot I^2$) in that line are reduced with each of the two alternatives. b) Predict what would happen if a single fixed capacitor sized for the four pumps were installed next to the workshop, but due to low demand only one of them were running. c) Justify which is the optimal technical solution considering cost, flexibility, and the thermal relief of the lines.

In a bottling plant, conventional motors are replaced by variable-frequency drive motors (non-linear loads that inject harmonics, notably the $5\text{th}$ harmonic at $f_5 = 250\ \text{Hz}$). After connecting a standard compensation capacitor bank of $C = 120\ \mu\text{F}$ per phase associated with a transformer whose equivalent network inductance is $L = 3{,}38\ \text{mH}$ per phase, the capacitor bank fuses blow due to continuous overheating and the voltage distortion shoots up. a) Calculate the parallel resonant frequency of the system, $f_r = \dfrac{1}{2\pi \sqrt{L \cdot C}}$, and interpret the physical cause of the failure when interacting with the currents generated by the drives. b) A worker suggests replacing the capacitors with higher capacitance ones ($C' = 250\ \mu\text{F}$), claiming that "this way they will withstand more current." Technically refute his proposal by calculating the new resonant frequency and deducing which other typical harmonic ($3\text{rd}$ at $150\ \text{Hz}$) it would resonate with. c) Justify the correct technical solution: analytically demonstrate what condition the equivalent reactance of a series branch formed by a capacitor and a detuning reactor ($X_{\text{series}} = 2\pi f L_d - \dfrac{1}{2\pi f C}$) must satisfy at $50\ \text{Hz}$ and at $250\ \text{Hz}$ to prevent this phenomenon.

A power quality analyzer records the following operating regimes in a three-phase industrial carpentry workshop ($400\ \text{V}$, $50\ \text{Hz}$) throughout a workday: regime A (intensive production, $8\ \text{h}$): $P = 40\ \text{kW}$, $\cos \varphi = 0{,}70$ inductive ($\tan \varphi \approx 1{,}02$); regime B (auxiliary cutting and assembly, $6\ \text{h}$): $P = 20\ \text{kW}$, $\cos \varphi = 0{,}60$ inductive ($\tan \varphi \approx 1{,}33$); regime C (stand-by and idle, $10\ \text{h}$): $P = 2\ \text{kW}$, $\cos \varphi = 0{,}50$ inductive ($\tan \varphi \approx 1{,}73$). The objective is to maintain $\cos \varphi \ge 0{,}95$ at all times without ever overcompensating (avoiding capacitive $\cos \varphi$). a) Calculate the compensation reactive power $Q_c$ strictly required to achieve $\cos \varphi = 0{,}98$ ($\tan \varphi \approx 0{,}20$) in regime A and in regime B. b) Design the composition of a stepped automatic bank (number of steps and power of each step in $\text{kVAr}$) that adequately covers the analyzed load regimes. Justify why overly powerful steps should be avoided. c) Justify why oversizing the bank by taking the starting peak of the motors or installing a fixed bank calculated for regime A would cause severe technical and billing problems in regime C.