I am trying to work out how to match a current source coupled with a capacitor to a 50 Ω load for maximum power transfer. Specifically I have a 6pf PIN photodiode which driven by a 2.5 ghz optical signal. At 2.5 Ghz the PIN diode has an impedance of -10j Ωs.
$$\begin{align} Z_c &= \frac{1}{jwC}\\ &= \frac{1}{j2\pi f C} \\ &= \frac{1}{j2{\pi}2.5 \cdot 10^9 \cdot 6 \cdot 10^{-12}}\\ &\approx -10j ~ \Omega \end{align}$$
simulate this circuit – Schematic created using CircuitLab
I have seen impedance matching but usually they are matching a non reactive source impedance, e.g. 50 Ω, to some complex load. In this case the source has a purely complex impedance of -10j and I want to match this to 50 Ω by using some transformer method (Single stub, 1/4 wave or combination or what ever). One conceptual problem I have is that that the load and sources should be complex conjugates of each other and If I manage this then I would end up with a purely inductive impedance and the circuit load and source would be a simple LC resonator and no power transfer would occur ? I haven't done this for quite a while and would appreciate some guidance.
I have seen some approaches which match a mixed complex load with real and imaginary parts, but not a matching a purely complex generator to a purely real load, ie 50 Ω.