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Two equations derived <from first principles> by <Brian Josephson> that characterize the device, somewhat like an <I-V curve>:
$$
I(t) = I_c \sin (\varphi (t)) \\
\dv{\varphi(t)}{t} = \frac{2 e V(t)}{\hbar}
$$
where:
* $I_c$: <Josephson current>
* $\varphi$: the <Josephson phase>, a function $\R \to \R$ defined by the second equation plus initial conditions
* $V(t)$: input voltage of the system
* $I(t)$: current across the junction, determined by the input voltage

Note how these equations are not a typical <I-V curve>, as they are not an instantaneous dependency between voltage and current: the history of the voltage matters! Or in other words, the system has an internal state, represented by the <Josephson phase> at a given point in time.

To understand them better, it is important to look at some important cases separately:
* <AC Josephson effect>: V is a fixed <DC voltage>


 Josephson equations