Linear couplers have been used for busbar protection, but, so far as the authors are aware, detailed consideration of their design and full potentialities has not yet been published. A method of design to secure optimum performance of voltage-summation circuits is considered in detail. Expressions are derived for the internal apparent power of a toroid of any given overall size, together with a rapid trial-and-error method of deducing the appropriate former or air-core size for maximum power output. It is shown that maximum apparent power output depends on former size and, to second order only, on winding space factor. It is determined also by the permeability of air and the conductivity of the winding which are, in general, fixed. Formulae are derived for output voltage per 1 kA in the primary circuit for any given conductor size. They are also obtained for the secondary impedance for specifying a matching relay-coil impedance, or vice versa for designing to a given relay impedance. Appendices give a sample design, test results on stability and operating current of a restricted earth-fault scheme, and a comparison of calculated and measured characteristics.