11.4 RL Circuits
LEARNING OBJECTIVES
- Analyze circuits that have an inductor and resistor in series
- Describe how current and voltage exponentially grow or decay based on the initial conditions
A circuit with resistance and self-inductance is known as an circuit. Figure 11.4.1(a) shows an circuit consisting of a resistor, an inductor, a constant source of emf, and switches and . When is closed, the circuit is equivalent to a single-loop circuit consisting of a resistor and an inductor connected across a source of emf (Figure 11.4.1(b)). When is opened and is closed, the circuit becomes a single-loop circuit with only a resistor and an inductor (Figure 11.4.1(c)).
(Figure 11.4.1)
We first consider the circuit of Figure 11.4.1(b). Once is closed and is open, the source of emf produces a current in the circuit. If there were no self-inductance in the circuit, the current would rise immediately to a steady value of . However, from Faraday’s law, the increasing current produces an emf across the inductor. In accordance with Lenz’s law, the induced emf counteracts the increase in the current and is directed as shown in the figure. As a result, starts at zero and increases asymptotically to its final value.
Applying Kirchhoff’s loop rule to this circuit, we obtain
(11.4.1)
where
is the inductive time constant of the circuit.
The current is plotted in Figure 11.4.2(a). It starts at zero, and as , approaches asymptotically. The induced emf is directly proportional to , or the slope of the curve. Hence, while at its greatest immediately after the switches are thrown, the induced emf decreases to zero with time as the current approaches its final value of . The circuit then becomes equivalent to a resistor connected across a source of emf.
(Figure 11.4.2)
The energy stored in the magnetic field of an inductor is
(11.4.4)
Thus, as the current approaches the maximum current , the stored energy in the inductor increases from zero and asymptotically approaches a maximum of .
The time constant tells us how rapidly the current increases to its final value. At , the current in the circuit is, from Equation 11.4.2,
which is of the final value . The smaller the inductive time constant , the more rapidly the current approaches .
We can find the time dependence of the induced voltage across the inductor in this circuit by using and Equation 11.4.2:
(11.4.6)
The magnitude of this function is plotted in Figure 11.4.2(b). The greatest value of is ; it occurs when is greatest, which is immediately after is closed and is opened. In the approach to steady state, decreases to zero. As a result, the voltage across the inductor also vanishes as .
The time constant also tells us how quickly the induced voltage decays. At , the magnitude of the induced voltage is
The voltage across the inductor therefore drops to about of its initial value after one time constant. The shorter the time constant , the more rapidly the voltage decreases.
After enough time has elapsed so that the