RLC 6-14-2020

Source current: $I$

Charge on $C_1$ = $Q_1$

Charge on $C_2$ = $Q_2$

Circuit equations:

$I=I_{R_1} + I_{C_1}$ $I_{R_1}=I_{C_2} + I_{R_2}$ $I_{L}=I_{C_1} + I_{C_2}$

$\dot{Q}_1 = I_{C_1}$ $\dot{Q}_2 = I_{C_2}$

$-V_S + \frac{Q_1}{C_1} + L\dot{I}_L=0$ $I_{R_1}R_1 + \frac{Q_2}{C_2} = \frac{Q_1}{C_1}$ $I_{R_2}R_2 = \frac{Q_2}{C_2} + L\dot{I}_L$

$Q_1(0) = I_L(0) = Q_2(0) =0$

I would have preferred to obtain the closed-form solution for the source current using Laplace transforms but the online calculator yields no result. So I went the usual numerical route. The variation of power provided by the source with time is:

It can be seen that at a certain interval, the power provided by the source becomes negative which indicates that power flows into the source. The maximum value of the power inflow to the source is the minimum value of the function plotted above. The answer is $\approx 12.72$

The state variables are the capacitor voltages and the inductor currents. Write the fundamental equations for these elements:

$V_L = L \dot{I}_L \\ I_{C1} = C_1 \dot{V}_{C1} \\ I_{C2} = C_2 \dot{V}_{C2}$

Now write the left sides in terms of state variables (with some intermediate quantities to aid in calculating):

$V_L = V_S - V_{C1} = L \dot{I}_L \\ I_{R1} = \frac{V_{C1} - V_{C2}}{R_1} \\ I_{R2} = \frac{V_S - V_{C1} + V_{C2}}{R2} \\ I_{C2} = I_{R1} - I_{R2} \\ I_{C1} = I_L - I_{C2} \\ I_{C1} = C_1 \dot{V}_{C1} \\ I_{C2} = C_2 \dot{V}_{C2}$

Solve for the derivatives and numerically integrate. The source current (flowing out of the source) is:

$I_S = I_{C1} + I_{R1}$

The power flowing out of the source is:

$P_S = V_S I_S$

If $P_S > 0$ , power is flowing out of the source. If $P_S < 0$ , power is flowing into the source. So find the largest negative value of $P_S$ , and the absolute value of that is the answer.

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import math

dt = 10.0**(-5.0)

VS = 10.0

R1 = 4.0
R2 = 1.0

C1 = 1.0
C2 = 1.0

L = 1.0

###################################

IL = 0.0

VC1 = 0.0
VC2 = 0.0

###################################

t = 0.0
count = 0

ILd = (VS - VC1)/L

IR1 = (VC1 - VC2)/R1
IR2 = (VS - VC1 + VC2)/R2
IC2 = IR1 - IR2

VC2d = IC2/C2

IC1 = IL - IC2

VC1d = IC1/C1

IS = IC1 + IR1

#print t,IS

###################################

ISmin = 999999999.0

while t <= 100.0:

    IL = IL + ILd*dt

    VC1 = VC1 + VC1d*dt
    VC2 = VC2 + VC2d*dt

    ILd = (VS - VC1)/L

    IR1 = (VC1 - VC2)/R1
    IR2 = (VS - VC1 + VC2)/R2
    IC2 = IR1 - IR2

    VC2d = IC2/C2

    IC1 = IL - IC2

    VC1d = IC1/C1

    IS = IC1 + IR1

    if IS < ISmin:
        ISmin = IS

    t = t + dt
    count = count + 1

    #if count % 10000 == 0:
        #print t,IS

###################################

print ""
print ""
print dt
print ISmin
print (VS*ISmin)

#>>> 
#1e-05
#-1.27205753776
#-12.7205753776
#>>>