Circuit for Practicing Numerical Solution

The state variables are the capacitor voltage $V_C$ and the inductor current $I_L$ . The basic equations are:

$V_L = L \dot{I}_L \\ I_C = C \dot{V}_C$

Writing the left sides in terms of the state variables results in:

$V_C - R I_L = L \dot{I}_L \\ -I_L = C \dot{V}_C$

The plot of the current is included below, along with simulation code. Note that the damped sinusoidal behavior emerges from some very simple math operations.

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import math

# Component values

R = 0.2
L = 1.0
C = 1.0

dt = 10.0**(-5.0)

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

# Initialize quantities

t = 0.0
count = 0

IL = 0.0
VC = 10.0

ILd = (VC-R*IL)/L
VCd = -IL/C

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

Imax = -99999999.0

while t <= 20.0:

    IL = IL + ILd * dt   # Numerical integration
    VC = VC + VCd * dt

    ILd = (VC-R*IL)/L   # Time derivatives
    VCd = -IL/C

    #if count % 1000 == 0:
        #print t,IL

    if (t > 5.0) and (t < 10.0) and (IL > Imax):  # Constrained max check
        Imax = IL

    t = t + dt
    count = count + 1

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

print ""
print ""
print dt
print Imax

#1e-05
#4.58750547921