Resistance (R), capacitance (C) and inductance (L) are the basic components of linear circuits. The behavior of a circuit composed of only these elements is modeled by differential equations with constant coefficients.

The study of an RL circuit requires the solution of a differential equation of the first order. For this reason, the system is called a* circuit of the first order*.

For this RL series circuit, the switch can simulate the application of a voltage step (E = 5V) causing the inductor to store energy. When the switch is returned to the zero-input position (E = 0), the inductor releases the stored energy.

A simple mesh equation establishes the law that governs the evolution of the current *i*(t):

**di/dt** **+** **(R/L)i** **=** **E/L **

Solving a differential equation always results in two types of solutions:

- The
**transient**(free) state, solution of the differential equation without a second member:

di/dt + (R/L)i = 0. - The
**steady**state, particular solution of the differential equation with second member:

di/dt + (R/L)i = E/L.

The **response** of the circuit (full solution) is the sum of these two individual solutions:

**i(t)** **=** **E/R** **+** **Ke**^{(-tR/L)}

The solution of a differential equation of the first order is always exponential in nature.