RLC 4-16-2020

The state variables are the inductor currents and the capacitor voltages. Derive expressions for the time derivatives of the state variables in terms of the state variables and the forcing function.

Basic equations:

$V_{L1} = L_1 \dot{I}_{L1} \\ V_{L2} = L_2 \dot{I}_{L2} \\ I_{C1} = C_1 \dot{V}_{C1} \\ I_{C2} = C_2 \dot{V}_{C2}$

Re-writing the left-hand sides in terms of state variables and the source voltage:

$V_S - R_2 I_{L1} - V_{C2} = L_1 \dot{I}_{L1} \\ V_{C2} = L_2 \dot{I}_{L2} \\ \frac{V_S - V_{C1}}{R_1} = C_1 \dot{V}_{C1} \\ I_{L1} - I_{L2} - \frac{V_{C2}}{R_3} = C_2 \dot{V}_{C2}$

And the source current is:

$I_S = I_{C1} + I_{L1} = \frac{V_S - V_{C1}}{R_1} + I_{L1}$

Then we numerically integrate, as usual. By inspection, the currents at $t = 0$ and $t = \infty$ are $10$ . The minimum current is $\approx 6.366$ . Simulation code is attached.

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import math

# Parameters

VS = 10.0

L1 = 2.0
L2 = 2.0

R1 = 1.0
R2 = 1.0
R3 = 1.0

C1 = 1.0
C2 = 1.0

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

# Initialize Simulation

dt = 10.0**(-6.0)
t = 0.0
count = 0

IL1 = 0.0
IL2 = 0.0

VC1 = 0.0
VC2 = 0.0

IL1d = (VS - R2*IL1 - VC2)/L1
IL2d = VC2/L2

VC1d = (VS - VC1)/(R1*C1)
VC2d = (IL1 - IL2 - VC2/R3)/C2

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

# Numerical Integration

ISmin = 9999999.0

while t <= 40.0:

    IL1 = IL1 + IL1d * dt
    IL2 = IL2 + IL2d * dt

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

    IL1d = (VS - R2*IL1 - VC2)/L1
    IL2d = VC2/L2

    VC1d = (VS - VC1)/(R1*C1)
    VC2d = (IL1 - IL2 - VC2/R3)/C2

    IS = (VS-VC1)/R1 + IL1

    if IS < ISmin:
        ISmin = IS

    t = t + dt
    count = count + 1

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

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

print ""
print dt
print ""
print ISmin
print (ISmin*10.0*10.0)

#>>> 

#0.0001

#6.36650497537
#636.650497537
#>>> ================================ RESTART ================================
#>>> 

#1e-05

#6.36656156622
#636.656156622
#>>> ================================ RESTART ================================
#>>> 

#1e-06

#6.36656722509
#636.656722509
#>>>