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.

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 $Q = \sqrt{S^2 - P^2}$.

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.

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 $\cos \varphi = 0{,}95$.

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 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).

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.

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 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.