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+// Example 9.12

+// Step Response with variable damping

+V_s1=0; // Voltage source value for t<0

+V_s2=30;//Voltage source value for t>0 

+L=0.1;

+C=1/640;

+omega_0=sqrt(1/(L*C));

+v_C_aft=0; // v_C(0^+)=0;

+v_C_aft_d=0; // v_C'(0^+)=0;

+V_ss=30;

+// for Overdamped Response

+// Let

+R=34;

+alpha=R/(2*L);

+p1=-alpha+sqrt(alpha^2-omega_0^2)

+p2=-alpha-sqrt(alpha^2-omega_0^2)

+P=[1,1;p1,p2];// coefficients of A's matrix

+V=[v_C_aft-V_ss;v_C_aft_d];// initial conditions and excitations

+A=P\V;

+A_1=A(1);

+A_2=A(2);

+t=0:0.001:0.5

+v_C=V_ss+A_1*exp(p1*t)+A_2*exp(p2*t);// t>0

+// for Underdamped Response

+// Let

+R1=5;

+alpha1=R1/(2*L);

+p3=-alpha1+sqrt(alpha1^2-omega_0^2);

+p4=-alpha1-sqrt(alpha1^2-omega_0^2);

+omega_d=sqrt(omega_0^2-alpha1^2);

+P1=[1,1;p3,p4];

+V1=[v_C_aft-V_ss;v_C_aft_d];

+A1=P1\V1

+A_3=A1(1);

+v_C1=V_ss+2*abs(A_3)*exp(-alpha1*t).*cos(omega_d*t+atan(imag(A_3),real(A_3)));

+// for Critically Damped Response

+R2=sqrt(6400/25);

+alpha2=R2/(2*L);

+A_4=v_C_aft-V_ss;

+A_5=v_C_aft_d+alpha2*A_4;

+v_C2=V_ss+A_4*exp(-alpha2*t)+A_5*t.*exp(-alpha2*t);

+plot(t,v_C,t,v_C1,t,v_C2)

+xlabel('t')

+ylabel('v_c(t)')

+title('Step Response with variable damping')

+h1=legend(['Overdamped';'Underdamped';'Critically damped'])