To start: I am really a huge beginner in Josephson junction. I need to understand the electrical equations associated to it, I don't need to go in the superconducting physics and I would like to stick on this if possible on the answers.
My question is the following:
The current, voltage relationship for a Josephson junction are:
$$I=I_c \sin(\delta)$$ $$V=\frac{\hbar}{2e} \frac{d \delta}{d t} $$
From this, we deduce the inductance associated to the junction:
$$V * \frac{1}{\frac{dI}{dt}} = L = \frac{\hbar}{2 e I_c \cos(\delta)} $$
When we represent electrically a josephson junction, we say that it is an inductance in parallel to a capacitance.
My question:
Here, as the current is the total one (thus going through the inductance and the capacitance in the electrical circuit). For me it means that we can model the josephson junction with a unique inductance (no capacitance) that has the value calculated here.
Then, why do we say that it is modelled with a capacitance in parallel to an inductance ? It is in contradiction from the equations above for me. Indeed in those equations everything behave as if it was an inductance that depends on $\delta$ only.
I insist on the fact that I am looking on an explanation based on the electrical equations. I have no problem to consider that for some reason physically we expect to have a capacitance somewhere, but from the electrical equations such behavior is not showing up from my understanding.
