Solving Math’s Assignment with Sage

Someone sent me this question:

Solve for the currents in the circult of Figure 2, if E(t)=5H(t-2) and the initial currents are zero. [Hint : Use Lapalce transform to solve this problem.]

So, to solve it, form mesh analysis of two loops. Then, convert them from time domain to complex domain with Laplace transform. Next, solve I1 and I2 with normal algebra. Then only inverse I1 and I2 back to time domain.

Of cause, if you familiar with Sage, you can solve it within 30min (or lesser?).

t = var('t')
s = var('s')
I1 = var('I1')
I2 = var('I2')

E(t) = 5*unit_step(t-2)

E(s) = E(t).laplace(t, s); E(s)
# >> 5*e^(-2*s)/s

equation = [
	-E(s) + I1*20*s + 10*(I1-I2) == 0,
	10*(I2-I1) + I2*30*s + I2*10 == 0 ]

solution = solve(equation, I1, I2); solution
# >> [[I1 == 1/2*(3*s + 2)*e^(-2*s)/(6*s^3 + 7*s^2 + s), I2 == 1/2*e^(-2*s)/(6*s^3 + 7*s^2 + s)]]

# Note that Sage cannot inverse-Laplace time-delay function. So, taking out e^(-2*s)
I1(s) = 1/2*(3*s + 2)/(6*s^3 + 7*s^2 + s)
I2(s) = 1/2/(6*s^3 + 7*s^2 + s)

i1_temp(t) = I1(s).inverse_laplace(s, t); i1_temp
# >>t |--> -1/10*e^(-t) - 9/10*e^(-1/6*t) + 1

i2_temp(t) = I2(s).inverse_laplace(s, t); i2_temp
# >> t |--> 1/10*e^(-t) - 3/5*e^(-1/6*t) + 1/2

# Referring to Table. For G(s)= e^(as)F(s), the inverse is g(t) = f(t-a).
u(t) = unit_step(t)
i1(t) = u(t-2) * ( -1/10*e^(-(t-2)) - 9/10*e^(-1/6*(t-2)) + 1 ) # Answer for i1
i2(t) = u(t-2) * ( 1/10*e^(-(t-2)) - 3/5*e^(-1/6*(t-2)) + 1/2 ) # Answer for i2

p1 = plot(i1(t), 0, 10, color='blue', legend_label='i1(t)')
p2 = plot(i2(t), 0, 10, color='red', legend_label='i2(t)')
show(p1 + p2)

And, the final answers are:

i_1(t) =u(t)\left( \frac{-1}{10}e^{-(t-2)} -\frac{9}{10}e^{-(t-2)/6} +1 \right)
i_2(t) =u(t)\left( \frac{1}{10}e^{-(t-2)} -\frac{3}{5}e^{-(t-2)/6} +\frac{1}{2} \right)

Current I1 and I2 vs time.

Bye SAINT2012

I happened to be the last batch of speakers in SAINT2012 conference. Next year it will be merged with COMPSAC as “New COMPSAC”, which will be held in Kyoto in 2013. The name of “SAINT” will not be used anymore.

Badge in COMPSAC/SAINT2012

Happened to know that I have to teeeaaaach in the coming short-semester. It seems like I am getting far far away from my PhD completion. I blame noone but myself. When will I have the gut to tender resignation letter? When will I have the time to do nothing but my own research?

Ephesus, Turkey.

Visit some old cities of Turkey during the SAINT2012 conference. Tortured by the sunshine, and was amazed at the legacy of Ephesus.

Wondering in Mind

Not really sure since when this idea came to my mind – to stay or to leave?

To see the hero dancing with bull.

Or to learn the Ultraman fighting a raksasa.

I have a dream …