# Damped Oscillations In an RLC Circuit

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**Damped Oscillations In an RLC Circuit**

A circuit containing resistance, i,inductance, and capacitance is called an RLC circuit. We shall here discuss only series RLC circuits like that shown in With a resistance R present, the total electromagnetic energy U of the circuit (the sum of

the electric energy and magnetic energy) is no longer constant; instead, it decreases with time as energy is transferred to thermal energy in the resistance. Because of this loss of energy, the oscillations of charge, current, and potential difference continuously decrease in amplitude, and the oscillations are said to be damped. As you will see, they are damped in exactly the same way as those of the block spring oscillator of Section To analyze the oscillations of this circuit, we write an equation for the total electromagnetic energy U in the circuit at any instant. Because the resistance does not store electromagnetic energy.

Now, however, this total energy decreases as energy is transferred to thermal energy. The rate of that transfer is, from E, dU

– = -i2R dt ‘

Fig.13-5 A series Rle circuit. As the •charge contained in the circuit oscillates

back and forth through the resistanc , electromagnetic energy is dissipated asthermal energy. damping (decreasing

the amplitude of) the oscillations. where the minus sign indicates that U decreases. By differentiating Eq. 33-22 with

respect to time and then substituting the result in Eq. 33-23, we obtain dU = Li t!i. + 1dq = –