General energy relations for Josephson junctions

The Josephson junction acts as a nonlinear lossless inductor and can be applied for mixing and parametric amplification in the microwave region. Since an alternating current can be generated in the junction by an applied dc voltage, a dc power flow must be taken into consideration. Energy relations for the Josephson junction are derived which are similar to the Manley-Rowe equations but with an additional term for the dc power. These equations show the possibility of realizing dc-pumped parametric amplifiers.