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RL Circuit - transient response

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.

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