The transformer

Explain why a transformer only works with alternating current: what happens to the magnetic flux in the core when the primary current varies, what is induced in the secondary (Faraday's law), and what would stop happening if it were connected to direct current after the initial moment.

Name the three fundamental parts of a transformer (primary, secondary, and core) and explain the function of each one. Reason why the two coils are not electrically connected to each other and what safety advantage this galvanic isolation provides.

Define the turns ratio, $\frac{V_1}{V_2} = \frac{N_1}{N_2}$, explaining what each symbol represents. State the condition satisfied by the turns of a step-down transformer ($N_1 > N_2$) and those of a step-up transformer, and give an everyday example of each (mobile phone charger, power station output).

The figure shows a transformer with its winding data. a) State, without calculating, whether it is a step-up or step-down transformer and how you deduce this from the drawing. b) Calculate the secondary voltage marked with a question mark. c) Name an everyday device that uses a transformer with a similar ratio.

A transformer has $N_1 = 500$ turns in the primary and $N_2 = 50$ in the secondary, and is powered at $230\ \text{V}$. a) Calculate the turns ratio and the secondary voltage. b) State whether it is step-up or step-down. c) Explain what would happen to the output voltage if it were connected "backwards" (the mains connected to the 50-turn winding).

A circuit needs to be powered at $12\ \text{V}$ from the $230\ \text{V}$ mains, and the primary of the transformer has $920$ turns. a) Calculate the number of turns of the secondary. b) If the load consumes $24\ \text{W}$, calculate the secondary current and, assuming an ideal transformer, that of the primary. c) Comment on why the secondary wire is thicker than the primary wire.

An ideal $230/24\ \text{V}$ transformer powers a load of $60\ \text{W}$. a) Calculate the secondary current. b) Calculate the primary current applying conservation of power ($V_1 \cdot I_1 = V_2 \cdot I_2$). c) Generalise the current ratio as a function of the turns ratio and state it in a single sentence.

The output transformer of a power station steps up from $25\ \text{kV}$ to $400\ \text{kV}$. a) Calculate its turns ratio. b) If the secondary has $8\,000$ turns, calculate the primary turns. c) The current in the $400\ \text{kV}$ line is $125\ \text{A}$: calculate the current on the power station side (ideal transformer) and comment on the values.

A transformer absorbs $1\,200\ \text{W}$ from the mains and delivers $1\,140\ \text{W}$ to the load. a) Calculate its efficiency, $\eta = \frac{P_2}{P_1}$. b) Calculate the power lost and explain into which two "categories" it goes (copper losses in the windings due to the Joule effect and iron losses in the core due to hysteresis and eddy currents). c) Reason why a commercial transformer has efficiencies of $95-99\,\%$ compared to $85-90\,\%$ for a motor.

Distinguish between the losses of an unloaded transformer (secondary off-load: iron losses dominate) and a fully loaded transformer (copper losses dominate, which grow with the square of the current). Reason why maximum efficiency is achieved when both are equal and why utility companies disconnect distribution transformers off-peak.

To supply power to a $12\ \text{V}$ industrial welding line that continuously consumes $720\ \text{W}$, two design alternatives are compared using the $230\ \text{V}$ main grid:

• Option A: reuse two transformers already available connected in cascade ($230\ \text{V} / 24\ \text{V}$ with $\eta_1 = 94\,\%$, and $24\ \text{V} / 12\ \text{V}$ with $\eta_2 = 90\,\%$).

• Option B: purchase a new direct $230\ \text{V} / 12\ \text{V}$ transformer with an efficiency of $96\,\%$ and an acquisition cost of $140\ \text{€}$.

1. a) Calculate the active power drawn from the grid and the overall efficiency for both options.
2. b) The facility operates $2\,500\ \text{h}$ per year and the price of electrical energy is $0,20\ \text{€}/\text{kWh}$. Evaluate the annual financial savings provided by Option B and calculate the payback period of the investment.
3. c) Calculate the power dissipated as heat that the electrical cabinet must evacuate in each case and draw a technical conclusion on why the proliferation of intermediate stages degrades the efficiency and reliability of an installation.

Explain the role of transformers in the transmission of electrical energy: why voltage is stepped up to hundreds of kilovolts to travel long distances (same power with less current: fewer Joule losses, $P_p = R \cdot I^2$) and why it is then stepped down to $230\ \text{V}$ before reaching homes (safety).

The figure shows the transmission chain of electrical energy from the power station to a home, with the two transformers covered. a) State what each covered transformer does (step up or step down, and between which voltages). b) Calculate the turns ratio of each transformer. c) Explain why energy is not transmitted directly at $25\ \text{kV}$ nor at $230\ \text{V}$.

A power plant must evacuate $50\ \text{MW}$ through a $100\ \text{km}$ transmission line whose total resistance is $20\ \Omega$. Technical regulations require that Joule heating losses during transmission do not exceed $3\,\%$ of the generated power. a) Calculate the losses that would occur if the evacuation were carried out directly at the generation voltage of $25\ \text{kV}$, numerically justifying why this transmission is unfeasible without transformation. b) Determine the minimum transmission voltage required to meet the $3\,\%$ loss limit and select the most suitable standardized voltage level from the Spanish transmission grid ($66\ \text{kV}$, $132\ \text{kV}$, $220\ \text{kV}$, or $400\ \text{kV}$). c) With the chosen standardized voltage, calculate the overall transmission efficiency if the step-up and step-down transformers each have an efficiency of $98,5\,\%$, and justify why the voltage is not increased indefinitely to reduce losses even further.