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.
Understanding Engineering: The RL Circuit
At the instant a switch closes, the inductor behaves very differently from an ordinary piece of wire. Before switching, there is no magnetic field from circuit current. Building that field requires energy, so the inductor develops a voltage that resists the attempted increase in current.
At first, nearly all the supply voltage can appear across the inductor. As the current builds, the resistor takes a larger share of the voltage because its voltage depends directly on current.
Eventually the current stops changing. The inductor then has no induced voltage across it in the ideal model, and it acts much like a wire with zero resistance.
The resistor does more than set the final current. It turns electrical energy into heat at every moment that current flows. During the switching period, energy from the source has two destinations.
Some becomes heat in the resistor. Some is placed into the magnetic field around the inductor. When the source is disconnected, that stored field becomes the source of energy for a short time.
The inductor reverses its voltage polarity as needed to keep current moving in the same direction through the circuit. This detail is essential because current direction cannot reverse instantly either. The current first falls to zero, then it can only reverse if another circuit condition drives it that way.
A real inductor has resistance in its wire, so engineers must include this resistance when predicting the final current and heating. Its core material can add further effects. Iron cores can store strong magnetic fields, but they can reach magnetic saturation.
After saturation, increasing current produces a much smaller increase in inductance. The current may then rise faster than a simple constant inductance calculation predicts. Coils near other coils or metal parts can have unwanted coupling.
Long wires have inductance too. This is why even a relay, motor winding, or cable can produce a sharp voltage pulse when a switch opens.
That opening pulse can damage transistor switches, create sparks at mechanical contacts, or interfere with nearby electronics. A diode placed across a DC relay coil gives the current a safe loop after the switch opens. It reduces the voltage spike, though it can make the relay release more slowly because the current decays gently.
Other designs use a resistor and capacitor network, a diode with a zener diode, or a metal oxide varistor when a faster release is needed. Students should carefully mark the assumed current direction and the voltage polarity across each component. Sign mistakes are common in RL work.
It also helps to sketch current values at the start, after one time constant, and long after switching. At one time constant, the change is substantial but incomplete. This makes exponential behavior easier to picture than treating it as an abstract curve.
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.