A series RL circuit contains a resistor and an inductor connected to a voltage source, often with a switch that starts or stops the current. It is a basic engineering model for motors, relays, solenoids, filters, and any wiring that has both resistance and inductance. The key idea is that the current cannot change instantly because the inductor creates a back emf that opposes changes in current.
This makes RL circuits useful for understanding transients, switching behavior, and energy storage in magnetic fields.
When a DC source is connected, the current rises exponentially toward its final value V/R instead of jumping there immediately. The speed of this rise is set by the time constant tau = L/R, where larger inductance slows the change and larger resistance speeds it up. When the source is removed, the inductor releases its stored magnetic energy and the current decays exponentially.
Engineers use these equations to predict delays, voltage spikes, power loss, and safe switching requirements in real circuits.
Key Facts
- Time constant for a series RL circuit: tau = L/R
- Final steady current after connecting a DC source: I_final = V/R
- Current rise after closing the switch: I(t) = (V/R)(1 - e^(-t/tau))
- Current decay after disconnecting the source: I(t) = I0 e^(-t/tau)
- Inductor voltage is proportional to rate of current change: v_L = L dI/dt
- Magnetic energy stored in an inductor: U = 1/2 L I^2
Vocabulary
- RL circuit
- An RL circuit is an electrical circuit that contains resistance R and inductance L, often used to study changing current over time.
- Inductor
- An inductor is a coil or circuit element that stores energy in a magnetic field and opposes changes in current.
- Time constant
- The time constant tau is the characteristic time for the current in an RL circuit to move about 63 percent of the way toward its final value during a rise.
- Transient response
- Transient response is the temporary behavior of voltage and current immediately after a circuit is switched or disturbed.
- Back emf
- Back emf is the induced voltage across an inductor that opposes the change in current that produced it.
Common Mistakes to Avoid
- Assuming the current changes instantly, which is wrong because an inductor resists any sudden change in current through back emf.
- Using tau = R/L instead of tau = L/R, which gives the wrong units and reverses the effect of resistance and inductance on the circuit speed.
- Forgetting that the final DC current is V/R, which is wrong because the inductor behaves like an ideal short circuit after a long time with steady current.
- Treating the inductor voltage as constant during the transient, which is wrong because v_L = L dI/dt and changes as the current curve becomes less steep.
Practice Questions
- 1 A 12 V battery is connected in series with a 6 ohm resistor and a 3 H inductor. Find the time constant and the final steady current.
- 2 For an RL circuit with R = 20 ohm and L = 5 H, the initial current before decay is 2.0 A. Find the current after 0.50 s when the source is disconnected.
- 3 A relay coil with a large inductance is switched off suddenly. Explain why a voltage spike can appear across the switch and how a diode or snubber circuit helps protect the switch.