Do not use C as a variable, since it is a built-in symbol in the system. We put X =WU2U0a - WU2U0b
. In this problem we have a special case when X
goes through 0 abruptly - see fig. Thus, all solvers fail on this problem. Fortunately, we can approximately determine the root of the equation as the minimum point X
. It is possible that this solution makes sense for technical applications in trigger generators.
p = 9; Za = I*w*L + 1/(I*w*C1 + 2/Z0);
Zb = 1/(I*w*C1) + 1/(1/(I*w*L) + 2/Z0);
Zl = Re[1/(1/Za + 1/Zb)] - I*Im[1/(1/Za + 1/Zb)];
Zal = 1/(1/Zl + 1/Za);
Zbl = 1/(1/Zl + 1/Zb);
WU2U0a = Arg[Zbl/2/(1/(1/(Zbl + I*w*L) + I*w*C1) + Z0/2)];
WU2U0b = Arg[-Zal/
2/(Z0/2 + 1/(1/(1/(Zal + 1/(I*w*C1))) + (1/(I*w*L))))];
Z0 = 44 - I*15; w = Rationalize[2*Pi*868*^6, 10^-p]; L = 18*^-10;
X = WU2U0a - WU2U0b;
Block[{$MinPrecision = p, $MaxPrecision = p},
NMinimize[{X, 0 <= C1 < 5*10^-11}, C1, WorkingPrecision -> p]]
{-2.96652668, {C1 -> 2.77937850*10^-11}}
{Plot[X, {C1, 2.77`30*10^-11, 2.79`30*10^-11}],
Plot[{WU2U0a, WU2U0b}, {C1, 2.77`30*10^-11, 2.79`30*10^-11},
PlotLegends -> "Expressions"]}