Exercises: 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).

State whether the following statement is true or false, justifying your answer: "A transformer can increase electrical power." Explain what quantity a step-up transformer actually increases, what quantity it decreases in the same proportion (in the ideal case), and why the power never increases.

The core of a transformer is made of stacked magnetic laminations insulated from each other, not solid iron. Explain what eddy currents are, how laminations reduce them, and what would happen to the efficiency and temperature of the device with a solid core.

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 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.

List three appliances around you that contain a transformer (laptop charger, old halogen light cooker hood, soldering iron) and indicate for each one whether it steps up or steps down the voltage and what approximate voltage it delivers. Comment on why modern switch-mode power supplies are much lighter than traditional $50\ \text{Hz}$ transformers.

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.

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 $230\ \text{V}$.

Explain what an autotransformer is: how its winding differs from that of a normal transformer (a single coil with an intermediate tap), what advantages it has (less copper and iron, cheaper), and what safety drawback makes it unacceptable for powering, for example, a child's electric train (absence of galvanic isolation between input and output).

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.

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.

Based on the nameplate of the single-phase distribution transformer shown in the figure:

1. a) An industrial facility plans to connect loads to the secondary with a total active power of $42\ \text{kW}$ and an average power factor $\cos\varphi = 0,70$. Prove by calculating the demanded apparent power and the rated currents whether this transformer can supply said installation continuously without overloading.
2. b) If the maximum allowable current density in the copper windings is $3,2\ \text{A}/\text{mm}^2$, calculate the minimum cross-section that the conductor of each winding must have and justify the huge disparity between the two based on the principle of conservation of power.
3. c) Predict what would happen if you connected the primary of this transformer to a $20\,000\ \text{V}$ grid but at $60\ \text{Hz}$ instead of the design $50\ \text{Hz}$: analyze how it would affect the maximum magnetic flux in the core and reason whether there would be a danger of saturation or overheating in the iron.

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.

A single-phase laboratory transformer ($230\ \text{V} / 12\ \text{V}$, $115\ \text{VA}$, $50\ \text{Hz}$) has an internal ohmic resistance in its primary winding of $4,6\ \Omega$. A student makes the mistake of supplying the primary winding with a direct current power supply regulated to $230\ \text{V}$.

1. a) Predict what voltage will be induced across the secondary terminals both at the initial moment the switch closes and a few seconds later in steady state. Physically justify the difference based on Faraday-Lenz's law.
2. b) Calculate the current drawn by the primary in direct current and compare it quantitatively with its rated current in alternating current at full load. What destructive thermal effect is triggered?
3. c) Justify why a transformer cannot operate on direct current and explain what electromagnetic property of alternating current allows the no-load alternating current at $230\ \text{V}$ RMS to be merely a few milliamperes.

During the inspection of a single-phase distribution transformer ($10\ \text{kVA}$, $15\,000\ \text{V} / 400\ \text{V}$), the maintenance team notices an abnormally loud acoustic hum and excessive heating of the tank, even when the transformer is operating practically under no-load conditions. Three possible causes are considered: loosening or breakdown of the insulating varnish of the magnetic core laminations, an internal short-circuit between several adjacent turns of a winding, or severely obstructed external ventilation.

1. a) Explain the physical mechanism by which the first two anomalies cause additional energy losses and temperature rise, identifying whether they predominantly affect copper losses or iron losses (eddy currents or localized Joule effect).
2. b) Design an electrical diagnostic protocol to identify the actual cause: what standardized tests (no-load test or short-circuit test) or measurements would you carry out, and what quantities should you record to distinguish between a core fault and a winding fault?
3. c) If in the no-load test at rated voltage the wattmeter measures core losses $P_0$ that are four times higher than the catalog values, but the no-load current $I_0$ barely increases by $10\,\%$ and the no-load voltage ratio remains exact, issue a justified technical assessment determining which of the anomalies is responsible.

Design the power supply installation for an outdoor illuminated sign composed of extra-low voltage LED strips that consume a total of $180\ \text{W}$ at $12\ \text{V}$ with $\cos\varphi = 0,90$, powered from the $230\ \text{V}$ ($50\ \text{Hz}$) single-phase mains.

1. a) Determine the minimum apparent power demanded by the load and select the most suitable standard commercial transformer rating (choose between $160\ \text{VA}$, $250\ \text{VA}$, $400\ \text{VA}$, or $500\ \text{VA}$), ensuring a reserve margin of at least $20\,\%$ above the required power.
2. b) With the chosen commercial transformer, calculate the rated primary current, the rated secondary current, and the transformation ratio $m$.
3. c) Select the rated fuse rating to protect the primary winding against short circuits and overloads, justifying whether it should have a fast-acting (gG) or slow/time-delay curve taking into account the transient inrush current (inrush current), and explain what galvanic isolation requirements this transformer must meet according to the REBT to be classified as a SELV safety transformer.

When connecting the primary of a single-phase transformer to the $230\ \text{V}$ mains with the secondary open-circuited (without any connected load), an ammeter detects a very small current ($I_0 = 0,15\ \text{A}$) and the winding suffers no damage at all, even though the resistance of the copper wire is barely $2\ \Omega$.

1. a) Quantitatively demonstrate what current would result if only the ohmic resistance acted against a voltage of $230\ \text{V}$, and deduce what electrical quantity dependent on the winding inductance actually limits the current in alternating current operation.
2. b) The active power recorded by a wattmeter under no-load conditions is $6,5\ \text{W}$. Decompose the no-load current $I_0$ into its core-loss active component ($I_{Fe}$) and its magnetizing reactive component ($I_\mu$), and calculate the no-load power factor $\cos\varphi_0$.
3. c) Justify the physical destination of those $6,5\ \text{W}$ continuously consumed by the transformer and interpret why electric utilities penalize prolonged no-load operation of large transformers in industrial customers.

An advertisement promotes a device based on a step-up transformer with the following claim: “Increase the electrical energy of your mains tenfold: input $230\ \text{V}$ and obtain $2\,300\ \text{V}$ while maintaining the same current to power ten times more machinery at no extra cost”.

1. a) Rigorously refute this claim by applying the first law of thermodynamics, and formulate the active and apparent power balance equation that governs an ideal transformer.
2. b) If the device is a real transformer with a turns ratio $N_2 / N_1 = 10$, designed to supply $4,6\ \text{kVA}$ with an efficiency $\eta = 95\,\%$ at full load with unity power factor: calculate the current that will actually flow through the secondary when the rated $20\ \text{A}$ flows through the primary at $230\ \text{V}$.
3. c) Determine the power dissipated as heat inside the machine under these conditions, and conclude what electromagnetic parameter is the only one that the transformer truly steps up, and at the expense of what physical quantity it necessarily achieves this.