Inductors are circuit elements that resist changes in current by creating a back emf. This cheat sheet covers how inductors behave, how energy is stored in magnetic fields, and how current changes in RL circuits. Students need these ideas to analyze switches, transients, and real circuits where current does not change instantly.
The most important relationships connect induced voltage, inductance, current, and time. In an RL circuit, the time constant controls how quickly current rises or falls. Current growth follows , while current decay follows .
Key Facts
- The induced voltage across an inductor is , where the negative sign means the inductor opposes changes in current.
- Inductance is measured in henries, with .
- The energy stored in an inductor is .
- For an RL circuit, the time constant is .
- When current grows after a switch is closed, and .
- When current decays after the source is removed, .
- After one time constant during current growth, the current reaches about of its final value.
- After one time constant during current decay, the current drops to about of its initial value.
Vocabulary
- Inductor
- A circuit component that stores energy in a magnetic field and opposes changes in current.
- Inductance
- The property of a component that measures how strongly it opposes changes in current, measured in henries.
- Back emf
- The induced voltage that acts against the change in current that produced it.
- RL circuit
- A circuit that contains resistance and inductance , causing current to change gradually over time.
- Time constant
- The time scale that determines how quickly current grows or decays in an RL circuit.
- Transient response
- The temporary changing behavior of voltage or current before a circuit reaches steady state.
Common Mistakes to Avoid
- Treating current through an inductor as changing instantly is wrong because an inductor resists sudden current changes through back emf.
- Using instead of is wrong because the RL time constant increases with inductance and decreases with resistance.
- Forgetting the negative sign in is wrong because the sign shows that the induced voltage opposes the change in current.
- Using for current growth is wrong because that equation describes decay, not the rise toward .
- Assuming the inductor stores energy as electric potential energy is wrong because an inductor stores energy in a magnetic field.
Practice Questions
- 1 An RL circuit has and . Find the time constant .
- 2 A battery is connected to a series circuit with and . Find and at .
- 3 An inductor with carries a current of . How much energy is stored in its magnetic field?
- 4 Explain why a spark can occur when a switch is opened in a circuit containing an inductor.
Understanding Inductors & RL Circuits Detailed
An inductor is usually a coil of wire. When charge moves through the coil, it produces a magnetic field around and through the turns. A changing magnetic field creates an electric effect that pushes against the change that produced it.
This is a consequence of electromagnetic induction. More turns of wire generally make the effect stronger. A magnetic core, such as iron or ferrite, can concentrate the field and greatly increase the inductance.
The core has limits, however. At high current, it can become magnetically saturated, so further current produces a much smaller increase in magnetic field.
The behavior just after a switch moves is often the hardest part of an RL circuit problem. Before current starts, there is no established magnetic field in the coil. At the instant the source is connected, the inductor acts almost like a break in the circuit because it prevents current from jumping to a large value.
As time passes, its opposing voltage becomes smaller and more current can flow. After a long time with a steady direct current source, the current no longer changes. The inductor then has almost no induced voltage across it and behaves much like an ordinary piece of wire, apart from its small wire resistance.
When the source is disconnected, current in the coil cannot stop immediately. The collapsing magnetic field supplies energy that keeps charge moving in the same direction through any available path. This fact explains why a circuit needs a complete discharge path.
If a switch opens without a safe path, the inductor can produce a very large voltage across the opening gap. A spark may form. Relays, motors, solenoids, and car ignition systems show this effect in real devices.
Engineers often place a diode across a coil in a direct current circuit. The diode gives the current a route during switch off and reduces damaging voltage spikes.
The size of the resistance changes how quickly stored magnetic energy becomes thermal energy in the circuit. A larger resistance causes the current to fade more quickly. A larger inductance makes the change slower because more magnetic field effect is associated with a given current.
The energy stored by a coil grows with the square of current. Doubling the current stores four times as much energy if inductance stays unchanged.
This makes current ratings important. Excess current can heat the wire, saturate a magnetic core, or damage connected components.
When solving circuit questions, begin by marking the time interval being studied. Decide what the current was just before the switch changed, since current through an ideal inductor is continuous. Then consider the first instant after switching and the condition after a long time.
These two limits help check every calculation. Keep current direction consistent when assigning signs to voltages.
On graphs, remember that the curve is steepest at the beginning and gradually levels out. A time constant is a useful timing scale, but the current never reaches its exact final value in a finite mathematical time.