In opportunistic networks, putting devices in energy-saving mode is crucial to preserve their battery, and hence to increase the lifetime of the network and foster user participation. A popular strategy for energy saving is duty cycling. However, when in energy-saving mode, users cannot communicate with each other. The side effects of duty cycling are twofold. On the one hand, duty cycling may reduce the number of usable contacts for delivering messages, increasing intercontact times, and delays. On the other hand, duty cycling may break long contacts into smaller contacts, thus also reducing the capacity of the opportunistic network. Despite the potential serious effects, the role played by duty cycling in opportunistic networks has been often neglected in the literature. In order to fill this gap, in this article, we propose a general model for deriving the pairwise contact and intercontact times measured when a duty cycling policy is superimposed on the original encounter process determined only by node mobility. The model we propose is general, i.e., not bound to a specific distribution of contact and intercontact times, and very accurate, as we show exploiting two traces of real human mobility for validation. Using this model, we derive several interesting results about the properties of measured contact and intercontact times with duty cycling: their distribution, how their coefficient of variation changes depending on the duty cycle value, and how the duty cycling affects the capacity and delay of an opportunistic network. The applicability of these results is broad, ranging from performance models for opportunistic networks that factor in the duty cycling effect, to the optimisation of the duty cycle to meet a certain target performance.