The outputs of all gates in a circuit are assumed to be observable under the highly observable condition, which is mainly based on the use of E-beam testers. When using the E-beam tester, it is desirable that the test set for a circuit is small and the test vectors in the test set can be applied in a successive and repetitive manner. For a combinational circuit, these requirements can be satisfied by modifying the circuit into a k-UCP circuit, which needs only a small number of tests for diagnosis. For a sequential circuit, however, even if the combinational portion has been modified into a k-UCP circuit, it is impossible that the test vectors for the combinational portion can always be applied in a successive and repetitive manner because of the existence of feedback loops. To solve this problem, the concept of k-UCP scan circuits is proposed in this paper. It is shown that the test vectors for the combinational portion in a k-UCP scan circuit can be applied in a successive and repetitive manner through a specially constructed scan-path. An efficient method of modifying a sequential circuit into a k-UCP scan circuit is also presented.