14.6 Rlc series circuits  (Page 7/4)

When the switch is closed in the RLC circuit    of [link] (a), the capacitor begins to discharge and electromagnetic energy is dissipated by the resistor at a rate $i^{2}R$ . With U given by [link] , we have

where i and q are time-dependent functions. This reduces to

This equation is analogous to

which is the equation of motion for a damped mass-spring system (you first encountered this equation in Oscillations ). As we saw in that chapter, it can be shown that the solution to this differential equation takes three forms, depending on whether the angular frequency of the undamped spring is greater than, equal to, or less than b /2 m . Therefore, the result can be underdamped $(\sqrt{k\text{/}m}>b\text{/}2m)$ , critically damped $(\sqrt{k\text{/}m}=b\text{/}2m)$ , or overdamped $(\sqrt{k\text{/}m}<b\text{/}2m)$ . By analogy, the solution q ( t ) to the RLC differential equation has the same feature. Here we look only at the case of under-damping. By replacing m by L , b by R , k by 1/ C , and x by q in [link] , and assuming $\sqrt{1\text{/}LC}>R\text{/}2L$ , we obtain

where the angular frequency of the oscillations is given by

This underdamped solution is shown in [link] (b). Notice that the amplitude of the oscillations decreases as energy is dissipated in the resistor. [link] can be confirmed experimentally by measuring the voltage across the capacitor as a function of time. This voltage, multiplied by the capacitance of the capacitor, then gives q ( t ).

Summary

• The underdamped solution for the capacitor charge in an RLC circuit is
  
  $q(t)=q_0{e}^{\text{−}Rt\text{/}2L}\text{cos}(\omega \text{′}t+\varphi ).$
• The angular frequency given in the underdamped solution for the RLC circuit is
  
  ${\omega }^{\prime }=\sqrt{\frac{1}{LC}-{\left(\frac{R}{2L}\right)}^2}.$

Key equations