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RL Circuit - transient response HTML5

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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.

Click on the switch to change the state of the circuit. Drag the sliders to change the values of R and L.

Learning goals

  • To simulate the charge and discharge of an inductor.
  • To know that the response of a circuit of the first order is exponential in nature.
  • To understand the influence of the time constant on the first order circuit.


Learn more

Consider the study of the evolution of the current i(t) for this series RL circuit excited by a DC voltage level E..

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

The left side of this equation includes the terms of…

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