RL Circuits. Non-instantaneous current, time constant, and magnetic energy

Growth and decay of current in an RL circuit

In an RL circuit, the current does not change instantaneously when the power source is connected or disconnected. The inductor opposes any sudden variation and forces the current to evolve gradually, following an exponential law both in the process of increase and in the process of decrease. This delayed response is the fundamental characteristic of RL circuits and explains many of their practical uses.

Current growth

When a voltage is applied to an RL circuit, the current does not reach its final value immediately: the inductor opposes the change and makes the current rise progressively. During this transient process, part of the energy supplied by the source is stored in the magnetic field of the inductor, which explains the initial resistance to current flow and the exponential shape of the growth.

Current decay

When the power supply is interrupted, the current does not drop abruptly either: the inductor tries to maintain it and generates a voltage that forces its circulation until the stored magnetic energy dissipates. This process also follows an exponential law and is responsible for phenomena such as voltage spikes that appear when opening an inductive circuit. The time constant τ = R·C

The time constant τ = L/R

The evolution of current in an RL circuit is determined by the time constant τ, a quantity which sets the speed at which the circuit responds to changes. Just like in RC circuits, this constant defines the duration of the transient phase and allows us to predict how the current will rise or fall depending on the values of its components.

Physical meaning of τ

The time constant τ = L/R indicates the characteristic time it takes for the current to change appreciably. After an interval equal to τ, the current has covered about 63% of the distance between its initial value and its final value, which allows the speed of the transient process to be described precisely.

How L and R affect the speed of the process

A higher value inductor increases the time constant and makes the current change more slowly, while a larger resistor reduces it and speeds up the transition. The relationship between these two components determines whether the circuit responds quickly or slowly to any variation in applied voltage.

Exponential voltage and current curves

The temporal evolution of an RL circuit can be represented by exponential curves describing how current and voltage change during the processes of growth and decay. These graphs allow direct visualisation of the circuit’s transient phase and understanding of how the time constant sets the speed at which the steady state is reached.

Current growth curve

When the power source is connected, the current increases from zero following a rising exponential curve. The initial slope is at its maximum and decreases as the inductor completes the formation of its magnetic field, gradually approaching the final value set by the resistor.

Current decay curve

When the power source is disconnected, the current decreases following a falling exponential curve. The inductor releases the energy stored in its magnetic field, maintaining the current for a brief interval before it drops almost to zero.

Reading and interpreting the graphs

In both curves, the key point is the time constant τ, which marks the speed of the process: after a time equal to τ, the current has completed approximately 63% of the total change. These graphs allow clear identification of the transient phase, the steady state, and the influence of the values of L and R on the circuit’s response.

Series and parallel inductor connections

Inductors can be combined to obtain an equivalent inductance that simplifies circuit analysis. Just as with resistors and capacitors, these combinations allow adjustment of the circuit’s temporal response, as the total inductance directly influences the time constant τ = L/R. The combination rules are opposite to those for capacitors and reflect how magnetic effects are added or shared when several coils act together.

Inductors in series

When several inductors are connected in series, their inductances add directly , because the same current flow passes through all the coils and the magnetic fields are added:

Leq = L1 + L2 + L3 +…

A greater equivalent inductance makes the circuit respond more slowly to changes in current.

Inductors in parallel

In a parallel connection, the current can split between several paths, so the equivalent inductance decreases . The relationship is obtained by summing the reciprocals of each inductance:

1/Leq = 1/L1 + 1/L2 + 1/L3 + …

and, in the particular case of two inductors, it can be written as:

Leq = (L1 · L2) / (L1 + L2)

A smaller equivalent inductance speeds up the circuit’s response to any variation in applied voltage.

Practical applications of RL circuits

RL circuits are used in a wide variety of systems where it is necessary to control the temporal evolution of current, filter signals, or manage magnetic energy. Their ability to oppose abrupt changes in current makes them essential in low-pass and high-pass filters, where inductance determines which frequencies are attenuated or allowed through. They are also used to generate delays and smooth transients in switching stages, soft starts, and protection against current spikes. In power electronics, inductors store and release energy in a controlled manner, allowing regulation of voltages, limiting of currents, and improvement of efficiency in converters and switched power supplies. Thanks to these properties, RL circuits are key elements both in low-signal electronic applications and in larger-scale power systems.