Two equations derived from first principles by Brian Josephson that characterize the device, somewhat like an I-V curve:
where:

$I(t)=I_{c}sin(φ(t))dtdφ(t) =ℏ2eV(t) $

- $I_{c}$: Josephson current
- $φ$: the Josephson phase, a function $R→R$ 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