Transformers use changing magnetic flux to transfer electrical energy between circuits without direct electrical contact. This cheat sheet covers ideal transformer ratios, mutual inductance, induced emf, power transfer, and common real-world losses. Students need these relationships to solve alternating current circuit problems and to understand how voltage is stepped up or stepped down in power systems.
The main ideas come from Faraday’s law and the link between changing current and changing magnetic flux. In an ideal transformer, the voltage ratio equals the turns ratio, while the current ratio is the inverse of the turns ratio. Mutual inductance connects one coil’s changing current to the emf induced in another coil, using formulas such as .
Key Facts
- For an ideal transformer, the voltage ratio equals the turns ratio: .
- For an ideal transformer, the current ratio is inverse to the turns ratio: .
- Ideal transformer power is conserved, so and .
- A step-up transformer has and increases voltage according to .
- A step-down transformer has and decreases voltage according to .
- Mutual inductance relates induced emf in one coil to changing current in another: .
- For a coil, self-induced emf is given by .
- Transformer efficiency is .
Vocabulary
- Transformer
- A device that uses electromagnetic induction between coils to change alternating voltage and current levels.
- Primary Coil
- The coil connected to the input voltage source in a transformer.
- Secondary Coil
- The coil connected to the output circuit where an induced voltage appears.
- Turns Ratio
- The ratio of secondary turns to primary turns, written as , which determines the ideal voltage ratio.
- Mutual Inductance
- A measure of how effectively a changing current in one coil induces an emf in another coil.
- Induced emf
- A voltage produced by a changing magnetic flux, often described by Faraday’s law.
Common Mistakes to Avoid
- Reversing the turns ratio, which gives the wrong output voltage because must match for an ideal transformer.
- Using the same ratio for current as for voltage, which is wrong because ideal transformer current follows .
- Applying transformer equations to direct current, which is wrong because a transformer needs changing magnetic flux from alternating or changing current.
- Ignoring the negative sign in , which loses the meaning of Lenz’s law and the opposition to the change causing the emf.
- Assuming real transformers are perfectly efficient, which is wrong because heat, eddy currents, flux leakage, and core losses make .
Practice Questions
- 1 An ideal transformer has turns and turns. If , find .
- 2 An ideal transformer steps down to . If the secondary current is , find the primary current.
- 3 Two coils have mutual inductance . If the current in coil 1 changes at , find the magnitude of the induced emf in coil 2.
- 4 Explain why transformers are useful for long-distance power transmission even though they do not create extra energy.
Understanding Transformers and Mutual Inductance Reference
A transformer works because the same magnetic field passes through both windings. Alternating current in the input winding rises, falls, then reverses direction. This produces a magnetic field in the iron core that changes continuously.
The core gives the magnetic field an easy path, so much more of the field reaches the second winding. As the field through that winding changes, charges in its wire are pushed around the circuit. The induced voltage depends on how rapidly the magnetic field changes.
A faster current change creates a larger induced voltage. This is why an ordinary transformer needs alternating current.
With steady direct current, the field changes only briefly when the circuit is switched on or off. After that moment, there is no continuing induced voltage in the other winding.
The minus sign in the induction law represents Lenz's law. An induced current always creates a magnetic effect that opposes the change that produced it. This is not a detail to ignore.
It explains why energy must be supplied to the input coil. When a load is connected to the output, the induced current in that coil resists the original magnetic change. The source then has to provide more input current to maintain the changing flux.
Energy is not created by the second coil. It is transferred through the magnetic field.
Keep track of which quantity is changing. A large current that stays constant produces no sustained transformer action, while a smaller current changing quickly can produce a strong induced voltage.
Real transformers are not ideal because some energy becomes thermal energy or escapes the intended magnetic path. Resistance in the copper windings causes heating. This loss grows strongly when current is large, so power lines use high voltage and low current for long distance transmission.
Some magnetic flux fails to link both coils. The iron core itself loses energy as its magnetic regions repeatedly reverse. Engineers reduce this loss by making cores from thin insulated layers instead of one solid block.
Thin layers reduce unwanted circulating currents inside the metal. A transformer may hum because the core changes shape slightly as its magnetism changes. The hum is usually linked to the alternating supply frequency.
Transformers can change how a load appears to the source. This is called impedance matching. A suitable turns ratio can let an amplifier deliver power effectively to a speaker, or let equipment draw current at a useful level.
The same principle matters in chargers, doorbells, welding equipment, substations, and audio circuits. When solving problems, first identify the input and output sides clearly. Then decide whether the device raises or lowers voltage.
Check that the current change goes in the opposite direction from the voltage change for an ideal case. Finally, use units carefully.
Voltage is measured in volts, current in amperes, power in watts, inductance in henries, and efficiency is a percentage. These checks catch many common errors before any calculation is finished.