2. Consider the series RLC circuit in Figure 5 with C-12nF, L=1mH and Vin(t)=Vpeak u(t). V Figure 5: Circuit 2 (a) Show that the differential equation for Vc(t) is given as d²/di²Vc(t)+(R/L) * (d/dt) * Vc(t) + (1/LC) Vc(t) = (1/LC) + Vin (t) The characteristic equation for this differential equation is s² +2as + wo² = 0. The roots of the characteristic equation are denoted by s₁ and s2 when they are distinct and by so when the roots are repeated. (b) When the roots of the characteristics equation are repeated the response is called critically damped. Determine the value of the resistor R-Ro for the critically damped response. (c) Determine the values of the currents and voltages at t=0, t=0+ and t=0 for all circuit elements. (d) For R>Ro (over damped response): Solve the differential equation and show that the variables Ve(t) and IL (t) are given below: where s₁ and s2 are the real distinct roots of the characteristic equation. V(t) = V peak+ Vpeak S₁-S₂ (Szent-se²²) į₁(t)= 1 Vpeak LS-S (Szent-s₁e²²) Determine VL(t), VR(t), Ic(t) using the above results. Plot VL(t), VR(t), Ic(t) using simulation software for R-3R0. (e) For R-Ro (critically damped response) Show that the variables Ve(t) and IL (t) are given as where so is the only (repeated) root of V(t)-peak+Vpeak (5-1) (t)- the characteristic equation. 1 Determine VL(t), VR(t), Ic(t) using the above results. Plot V(t), VR(t), Ic(t) using simulation software for R-Ro. (f) For R<Ro (underdamped response) Show that the variables Vc(t) and IL(t) are given as where s₁--a+jwa and s2=-a-jwd V(t)-Vpeak-peak 4₂(t)= sin (wat + tan-4) Vakat sin(wt) نارس are the complex conjugate roots of the characteristic equation and w0=1/√(LC) Determine VL(t), VR(t), Ic(t) using the above results. Plot VL(t), VR(t), Ic(t) using circuit simulation software for R=Ro/4. Repeat parts 2.b, 2.c, 2.d, 2.e and 2.f for C= 47nF.