Applications of Linear Differential Equations to Electric Circuits

  • Prerequisites :

Differential Equations of First Order and First Degree

Linear Differential Equations of Higher Order


 

There are 3 basic components of an electric circuit, where a change in voltage is possible. They are:

1) Resistance ({R}), voltage drop = {i R}, we saw this in Ohm’s law.

2) Capacitance ({C}), voltage drop =  {\frac {1}{C} \times \int i dt}

3) Inductance ({L}), voltage drop = {L \times \frac {di}{dt}}

Note that the electric current {i} is the rate of flow of charge {q}, hence, {i = \frac {dq}{dt}}.

To solve a differential equation, we use the Kirchoff’s voltage law, which states that the sum of all the voltages around a loop is equal to zero.

The general circuit will consist of all 3 elements, {R,L} and {C} as well as the voltage source, {E}. The D.E. will be

{L \frac {di}{dt} + i R + \frac {1}{C} \int i dt = E sin (\omega_a t)}

Expressing in terms of the amount of charge, {q},

{L \frac {d^2q}{dt^2} + R \frac {dq}{dt} + \frac {q}{C} = E sin (\omega_a t)}

{\omega_a} is the frequency of applied voltage.

Resonance is a special condition in {R-L-C} circuits, when the imaginary parts of impedances due to inductor and capacitor cancel each other. It occurs when the applied frequency {\omega_a} becomes equal to the natural frequency, given by

{\omega_n = \frac {1}{\sqrt {LC}}}

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