RL Circuit

Inductor's and Inductance

Inductors are usually tightly wound coils of wire induce a changing magnetic flux when a changing current passes through them. In fact, any wire induces a magnetic field due to a change of current, however winding wire into a coil greatly amplifies the strength of this field. The size of the magnetic flux \(\Phi_B(t)\) produced by a wire for some current \(I\) is dependent on its geometry and this effect is quantified by the Inductance of the wire \(L\)\[L=\frac{\Phi_B}{I}.\]

When a current first begins to flow through an inductor, the increasing current causes in magnetic flux. This change in magnetic flux induces an electromotive force (EMF) across the inductor, thanks to Faraday's Law of Induction. This EMF, \(\mathcal{E}\), is equal to the magnitude of the rate of change of flux \(\Phi_B(t)\):\[\mathcal{E}=-\frac{\mathrm{d}\Phi_B(t)}{\mathrm{d}t}\]

Combining Faraday's Law with the definition of inductance, shows us that the size of the induced EMF across the inductor depends on its inductance:\[\mathcal{E}=-L\frac{\mathrm{d}I(t)}{\mathrm{d}t}\]

According to Lenz's Law, the induced EMF is directed such that it opposes the change in magnetic flux that created it. This means the EMF induced across the inductor opposes the EMF of the battery, hence resisting the change in current.

Differential equation for LR Circuit

Using what we know about inductors, let's apply Kirchoff's Loop rule for potential difference to Figure 1. Starting at the battery and moving clockwise summing the potentials, we find\[V+I(t)R-L\frac{\mathrm{d}I(t)}{\mathrm{d}t}=0.\]

So, the inductor will clearly have an effect on how the current reaches its maximum value when the switch is first turned on. Solving this equation will tell us how the current increases just after the switch is turned on.

Current in RL Circuits

As we have found, the current is defined by the following differential equation.\[\begin{align}V+I(t)R-L\frac{\mathrm{d}I(t)}{\mathrm{d}t}&=0\\\frac{\mathrm{d}I(t)}{\mathrm{d}t}&=\frac{I(t)R}{L}-V\end{align}\]

It's clear that this differential equation can be solved by a function of the form:\[I(t)=\frac{V}{R}\left(1-\mathrm{e}^{-\frac{Rt}{L}}\right).\]

So, what does this equation tell us about current in an LR circuit?

Fig. 2 - The current in an LR circuit increases as an inverse exponential until it reaches its maximum value as defined by the voltage and resistance of the circuit. The rate at which it does this is defined by the inductance of the inductor.

Figure 2 shows a plot of the current in an LR circuit just after the circuit is switched on. It shows the current quickly increasing, but with the rate of this increase slowing as time goes on as the current tends towards its maximum value of \(I=\frac{V}{R}\) asymptotically.

Let's try and understand this physically. When the switch is first turned on, current begins to flow around the circuit, increasing at a rate that would be expected if there were no inductors in the circuit. However, this change in current through induces a changing magnetic flux in the inductor, which in turn induces an EMF that opposes the current. This EMF slows the increase in current over time, however the smaller increase in current reduces the size of the induced EMF and so the current tends towards a steady maximum value as the induced EMF tends towards zero.

Time Constant of an RL Circuit

As we can see in Figure 2, the rate at which an RL circuit reaches its maximum current, is defined by the inductance of the inductor. The larger the inductance, the greater the opposing EMF will be, and the longer it will take for the current to reach its maximum value.

The resistance of the circuit also appears in the exponential term of the current, but in the numerator of the fraction. So, we see that the larger the resistance of the circuit, the faster the current will reach its maximum value. However, this maximum current, defined as \(I=\frac{V}{R}\), will be lower. This rate of current increase is quantified using the Time-Constant of the circuit:\[\begin{align}\tau&=\frac{L}{R}\\\implies\,I(t)&=\frac{V}{R}\left(1-\mathrm{e}^{-\frac{t}{\tau}}\right).\end{align}\]

We can see that at \(t=\tau\) the current will be\[\begin{align}I(\tau)&=\frac{V}{R}\left(1-e^{-1}\right)\\&\approx0.63\frac{V}{R}.\end{align}\]

So, the time constant defines how long it will take an RL circuit to reach \(0.63\) times it maximum value. Note how this time constant depends only on the resistance and inductance of the circuit, no matter the voltage of the battery the rate at which LR circuits with equal values of \(L\) and \(R\) reach their maximum current is the same.

RL Circuit Discharging

Having looked at how an RL circuit behaves just after the switch is turned on, it is also interesting to look at what happens just after the switch is turned off. When we switch the circuit, there is again a change in magnetic flux in the inductor, causing the circuit to behave quite differently from a usual Ohmic circuit. An RL circuit discharging is essentially the same as the circuit charging, but in reverse. When the switch is first opened, the reduction in charge induces a changing magnetic flux in the inductor once again, but this time because the changing magnetic flux acts in the opposite direction as there it is caused by a decreasing current. Hence, this time the induced EMF acts to maintain the current, keeping the current flowing for longer than would be seen in an Ohmic circuit.

Fig. 3 - Graph showing how the current changes in an LR circuit after the switch is opened.

RL Circuits Application

Having got to grips with the characteristic features of an RL circuit and how they work, let's look at how they are used in modern technology. RL circuits are very common in applications using AC (Alternating Current). In AC circuits, the current is constantly changing, so there is always and EMF induced across the inductor. One use of an RL circuit in AC circuits is as a frequency filter, by configuring the circuit such that signals (or alternating currents) at a certain frequency are prevented from flowing by the strong counteracting EMF. This is vital to radio communication, where being able to choose the frequency of signals allows clear communication over a certain channel.