Figure The series RLC circuit.

The current j (in A – amperes), is a function of time t. The resistance R (in ? – ohms), the inductance L (in H – henrys), and the capacitance C (in F – farads) are all positive and are assumed to be known constants. The applied voltage V (in V – volts) is a given function of time. Another physical quantity that enters the discussion is the total charge q (in C – coulombs) on the capacitor at time t. The relation between charge q and current j is given by

j = dq

dt . (1)

The flow of current in the circuit is governed by Kirchhoff ’s Voltage Law: In a closed circuit, the applied voltage is equal to the algebraic sum of the voltages across the elements in the rest of the circuit.

According to the elementary electric circuit theory, we know that

• The voltage across the inductor is L dj

dt .

• The voltage across the resistor is Rj.

• The voltage across the capacitor is 1

C q.

Hence, by Kirchhoff ’s Voltage Law stated above, we have the following:

L dj

dt + Rj +

1

C q = V (t). (2)

Substituting for j from (1) into (2), we obtain the following second-order equation for q:

L d2q

dt2 + R

dq

dt +

1

C q = V (t) . (3)

(Can you see that!? You DO NOT need to check this