The research reported in this paper represents an in-depth examination of the finding from an earlier investigation that university students often do not develop a functional understanding of the concept of a complete circuit. Participants in this study included undergraduates in introductory and upper-division physics courses as well as graduate teaching assistants. Although the concept of a complete circuit is covered in the standard undergraduate curriculum, students in all three groups had difficulty in applying this concept to single-loop, resistive circuits. Students frequently did not apply the conservation of current when analyzing circuits containing more than one battery. Certain basic conceptual difficulties spanned all of the populations and even persisted after instruction in upper-division courses that involved analog electronics.