Question

Problem 1 An RLC circuit (as shown in page 79 in the textbook) has a resistor of Rohms, an inductor with an inductance of 2 henries, and a capacitor with a capacitance of 0.5 farads. A battery is connected to the circuit giving a voltage V(1) = 2 cos(31) in volts) where 1 is given in seconds. 1) Write the differential equation satisfied by the charge Q(1) (in Coulombs) on the capacitor at time t. Answer: ii) Let $R=1$. Use Matlab command (dsolve) to find a formula for the steady periodic solution to the differential equation in (1). Include your formula and your code in your script. % Write your code here. iii) Verify your formula of the steady solution graphically. Plot your steady solution together with some of the solutions of the differential equation in (ii) using multiple initial conditions. All your graphs must be in the same figure. % Write your code here. iv) For which values of R is the vibration of the function (t) underdamped? Plot solutions to the differential equation with values near the threshold value R=R, at which the underdampened motion occurs to prove your answer graphically % Write your code here.

Answer 1

if parallel RLC circuit is given ,

since be series hic circuit pie figure com page 79) is mising am as assuming it to R Applying K VL in circuit mm Vs ist) Vs td VR + V + Vc where Vs ballert battery emf = 2cos (34) voltage across Resistor across inductor Vį → voltage voltage R ilt] Ldi at Vc - Cucos capacitor - < fidt 9cos 3t R ilt) + L di at both sides wrt t Differenting R di L di dt? į Also (1) di فر -2x3 sin3t = - 6 gin3t = dt + + at R d² t) + Ld² Ut) dt² + dut) cdt (Answer) at3

Just apply KCL i.e current rule instead of voltage rule

Find current across each component . finally substitute the current with charge

Thanks !! Happy to help :)