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Node
vs1:=(1+I)/6;
vs2:=0.5 + I*sqrt(3)/2;
numeric::solve([(vs1-va)/5 - va - (va + vs2)/(sqrt(3)*I) = 0],[va])
Mesh
vs1:=(1+I)/6;
vs2:=0.5 + I*sqrt(3)/2;
numeric::solve([i1-i2-vs1+5*i1,i2*sqrt(3)*I - vs2 + i2 - i1,i3 = i1-i2],[i1,i2,i3])
Thevenin voltage
vs1:=(1+I)/6;
vs2:=0.5 + I*sqrt(3)/2;
numeric::solve([vth=(vs1+vs2)*sqrt(3)*I/(5 + sqrt(3)*I)-vs2],[vth])
Norton current
vs1:=(1+I)/6;
vs2:=0.5 + I*sqrt(3)/2;
numeric::solve([xn=vs1/5 - vs2/(sqrt(3)*I)],[xn])
calculating rth
1/(1/5 + 1/(sqrt(3)*i)) = 0.5357 + 1.5465*i
checking rth
vth = 0.7479777026 + 0.5491833001*i;
xn = - 0.4666666667 + 0.3220084679*i;
rth = vth/xn
current in thevenin equivalent is
vth = 0.7479777026 + 0.5491833001*i;
rth=0.5357 + 1.5465*i
ixx = vth/(rth + 1)
power is
( 0.7479777026^2 + 0.5491833001^2)/(2* 0.5357)
checking with simulation
period is 2*pi = 6.2832 seconds
for the 1 ohm resistor
va = - 0.4206268848 + 0.06596618565*i =
va(t) = 0.4258 * cos(t + 2.9860) (171 degrees)
xn = - 0.4666666667 + 0.3220084679*i
i(t) = 0.5670 * cos(t+ 2.5376) (145.4 degrees)
