Question

4) Objective: Use what we've done for mass-spring systems to derive results for a driven LRC circuit a) Show that for a series LRC circuit, the maximum potential across the capacitor occurs at a frequency o=0,Ji-2013, where os?=te and Q - Ore b) Show that for this circuit, the maximum potential across the inductor occurs at frequency or V 29.

Answer 1

It has been assumed that the series RLC circuit is being energized through an AC source. Please observe the following images:

WN Consider the following circailt where Vinav sinut using voltage division rice. Vin t clvc ve = cu loc Vin L8 for some & 4R²+ (col - 1 z = No Let (RWC)? (W2-1)==k² 1 w TR wet+ (W/LED IR WC? Wt 17. cuac then Nela 1 / 2 for vcl to be maximum , k has to be minimum. Maximum potential will be stored in capacitor when Melis maximum since Energy Stored in capacitor = To find minimum value of k' we differentiate 'k with respect to co' and equate it to zero. dk :o dw

o R and lo= usla u tal- then att then - d[rwc) tw2c-13] =0 > Q Rwc +Qcwłc-1)(2WLC) -0 =) Q WC [CR + 2W7C-1)2 ) - 0 w=o or CR72 (WZc-1) L=0 ano tution GR = 2L (1-WELC) - ORTHWic -) CR-1--W1c a) w.pce -O Not a solution putting ② and ③ in Two wo TT 120 l This is the frequenas at which k is minimum and Hence potential in capacitor will be maximum.

Onn (b) considering the following circuit, where Vin= Vo sinut. Agains rmm I. IL y Vitsl= Vowl Vo w? LC Vole R?+CWL-ue) Foc w tw26-1 TRO*+(<citet R TC w2 maximum potential will be stored in inductor M, then I when IV, I is maximum. Because will be maximum when Mis minimum Now suppose dm-0 de → d (8+1kC-1} 1-0 7-a855 +2(LC-) 63) - 0 > 2 [ROP+ acc-)) do a3 = U

→ Wigg or (RO + [LC-5)=0 Not feasible since the = Wo, LC = 1 662 Ako zaz-RC cas derived earlier) putting these in ② & Et luz co LC - - - - Vic-lec? » W- FL - ORC? This is the frequency at which maximum energy is stord in the inductor. C rare