55 RL Circuits

Current Growth and Decay in RL Circuits

In the previous section, we learned that an inductor resists changes in the current flowing through it. Unlike a resistor, which responds immediately to an applied voltage, an inductor cannot allow the current to change instantaneously. Any attempt to increase or decrease the current changes the magnetic field around the inductor, and that changing magnetic field induces an emf that opposes the change, as described by Lenz's law.

This raises an important question: How long does this opposition last? Although the current cannot change instantly, it does eventually reach its final value after the circuit is switched on, and it eventually falls to zero after the circuit is switched off. Understanding how quickly these changes occur is important in many technologies, from electric motors and MRI gradient coils to power supplies and electronic timing circuits.

Figure 55.1 shows a simple resistor-inductor (RL) circuit used to study how current changes with time. In position 1, the battery, resistor, and inductor are connected in series so that current begins to increase. In position 2, the battery is disconnected, allowing the energy stored in the inductor's magnetic field to dissipate through the resistor.

Current Growth When an RL Circuit Is Switched On

Suppose the switch is moved to position 1 at time [latex]t=0[/latex]. Initially, no current flows through the circuit because the inductor strongly opposes any sudden increase in current. As the current slowly increases, the induced emf becomes smaller because the rate of change of current decreases. Eventually, the induced emf becomes nearly zero, and the current reaches its steady-state value

where [latex]V[/latex] is the battery voltage and [latex]R[/latex] is the total resistance of the circuit.

The current does not increase linearly. Instead, it follows an exponential growth given by

This equation has the same mathematical form as the charging of a capacitor in an RC circuit, although the underlying physics is different. In an RL circuit, the delay arises because energy must first build up in the magnetic field surrounding the inductor.

The quantity

is called the characteristic time constant of the RL circuit. Since

the time constant has units of seconds.

The time constant determines how rapidly the current approaches its final value. After one time constant,

meaning the current has reached approximately 63.2% of its final value. After another time constant, it gains 63.2% of the remaining difference. Because the process is exponential, the current never reaches the final value mathematically, but after about five time constants it is more than 99% of the final current, which is sufficiently close for nearly all practical applications.

Figure 55.1(b) illustrates this gradual increase.

The RL Time Constant

The characteristic time constant depends only on the inductance and resistance of the circuit:

• A larger inductance produces a larger time constant because the inductor stores more magnetic energy and offers greater opposition to changes in current.
• A smaller resistance also produces a larger time constant because the final current is larger, requiring more energy to be stored in the magnetic field before the steady state is reached.

Consequently, circuits containing large inductors or small resistances respond more slowly when they are switched on or off.

This behavior is important in many biomedical technologies. For example, MRI systems use large electromagnets with very high inductance. Their currents cannot be changed instantaneously because enormous amounts of magnetic energy must first be stored or removed. Electronic control systems therefore carefully regulate how quickly these currents change to protect both the equipment and the patient.

Current Decay When an RL Circuit Is Switched Off

Now suppose the switch is moved to position 2, disconnecting the battery. Although the external voltage source has been removed, the current does not immediately drop to zero. The magnetic field surrounding the inductor stores energy, and as that magnetic field collapses it induces an emf that drives current in the same direction as before, opposing the decrease in current.

The stored magnetic energy,

is gradually converted into thermal energy in the resistor. Because the rate of energy dissipation depends on [latex]I^2R[/latex], the current decreases rapidly at first and then more slowly as it approaches zero.

The current during this discharge process is described by another exponential equation:

where the same time constant

determines how quickly the current decreases.

After one time constant, the current has fallen to

or approximately 36.8% of its original value. During each additional time constant, the current falls to 36.8% of its previous value. As shown in Figure 55.1(c), the current approaches zero asymptotically and becomes negligibly small after only a few time constants.

This gradual decay explains why devices containing large inductors, such as industrial electromagnets, MRI systems, and electric motors, often continue carrying current briefly after power has been removed. The energy stored in their magnetic fields cannot disappear instantaneously and must instead be released over time.

Current and Voltage in RL Circuits

In summary, whenever the voltage applied to an inductor changes, the current cannot respond immediately. Instead, the current changes gradually because the inductor generates an induced emf that opposes any change in current. In other words, the current lags behind the applied voltage in an RL circuit.

In the next section, we will see how this behavior influences circuits powered by alternating current (AC), where the voltage changes continuously with time. The concepts of inductive and capacitive reactance build directly on the ideas introduced here.