The transient response from a transmitter–receiver coil system inside a multi-layer cylindrical conductive configuration is obtained. The particular set-up applies to well logging as well as to eddy current tube testing. In this work, a number of improvements are presented to existing models for an efficient calculation of the induced voltage. These include: domain truncation, novel treatment of arbitrary number of layers in order to avoid computational overflows and efficient time response calculation. The latter is based on a combination of Laplace inversion techniques for short- and long-time transient responses. This article is part of the theme issue ‘Advanced electromagnetic non-destructive evaluation and smart monitoring’.