Advanced Math Consider the RLC circuit shown below a Using

Advanced Math

Consider the RLC circuit shown below a) (**) Using Kirchoff\'s Laws, show that this circuit can be described by the following system of first-order ODEs: dV_1/dt = 1/C_1I_1 dV_2/dt = 1/C_2I_1 - 1/R_2C_2V_2 dI_1/dt = 1/LV_2 - 1/LV_1-R_1/LI_1 where V_1 is the voltage across capacitor C_1 and V_2 is the voltage across capacitor C_2. b) (**) After substituting the given values for R, C, and L, convert this system of ODEs to vector form and determine the eigenvalues and eigenvectors of the system. c) (***) The circuit begins with capacitor C_1 charged to 10V and capacitor C_2 charged to 9V, with no initial current. Using your answer from (b), solve the system of equations for expressions describing V_1(t), V_2(t), and I_1(t). Plot your solutions on the same axes.

Solution

V1 is the voltage across capacitor C1

if Q is the charge across C1 then

V1 = Q/C1

as v1 is varying

we can write it as

dV1/dt = 1/C1(dQ/dt)

dQ/dt = I1 current charging C1 hence we can write the ODE as

dV1/dt = I1/C1

similarly for the case of C2

dV2/dt = I3/c2

I3 = I1-I2

dV2/dt = I1/C2 - I2/C2

I2R2 = V2

hence dV2/dt = I1/C2 - V2/R2C2

algebraic summ of emfs in any closed loop is 0

consider the loop of c1 , C2,R1 and L

let V_L is the emf across L, it will be opposit to the direction of current

V2-V1-I1R1 -V_L =0

V_L = V2-V1-I1R1

as the current through L is changing

V_L = L dI/dt

dI/dt = V2/L -V1/L -I1R1/L

dV1/dt = I1/0.05

dV2/dt = I1/0.25 -V2/2.25

dI1/dt = V2-V1-9I1