%pylab inline

import seaborn
style.use('seaborn-poster')

import sympy
sympy.init_printing()

rcParams['lines.linewidth'] = 1

V = array([3.3, 5, 12, 120, 240])[:, None]
R = array([4.7e6, 4.7e3, 4700, 470])
I = V/R
P = V*I
P

V = 12
Ri = 2.3  # internal
Rl = array([0.01, 0.1, 1, 10, 100])
R = Ri + Rl
I = V/R
Vl = I * Rl
Pl = Vl * I
Pl

V = 12
Ri = 2.3                        # internal
Rl = linspace(0.001, 20, 1000)  # load
R = Ri + Rl
I = V/R
Vl = I * Rl
Pl = Vl * I
opt = Rl[Pl.argmax()]
subplot(211); tick_params(labelbottom=False); plot(Rl, Pl); ylabel("$P_L$"); axvline(opt); axhline(Pl.max())
subplot(212, sharex=gca()); plot(Rl, I, 'r'); ylabel("$I$"); axhline(I.max()); xlabel("$R_L$")
gca().twinx(); plot(Rl, Vl); ylabel("$V_L$"); legend(['$V_L$'], loc=3)
opt

V, Rl, Ri = sympy.symbols('V R_L R_I')
R = Ri + Rl
I = V/R
Vl = I * Rl
Pl = Vl * I
Pl

Pl_concrete = Pl.subs({V: 12, Ri: 2.3})   # P_L at the concrete values we used above
#sympy.plotting.plot(Pl_concrete, (Rl, 0, 20))
R, I, Vl, Pl, Pl_concrete

Plp = Pl.diff(Rl)   # P_L'
Plp, Plp.simplify(), Pl_concrete.diff().simplify()

c = sympy.Eq((Vl.diff(Rl)/Vl), -(I.diff(Rl)/I))
c

c.simplify()

sympy.Eq(c.lhs * R, c.rhs * R).simplify()

Vl, Vl.diff(Rl).simplify()

I, I.diff(Rl).simplify()

V, Ri, m = sympy.symbols('V R_I m')
Rl = m*Ri / (1 - m)
(Rl / (Ri + Rl)).simplify()

R = Rl + Ri
I = V / R
Vl = I * Rl
Pl = Vl * I
Pl.simplify()