The inverse eigenvalue problem is an important method for designing the circuit or the mass-spring system. The paper considers the following circuit design problem: given some information of the inductance matrix $M$, resistance matrix $C$, and capacitance matrix $K$, determine the rest data such that the system has the prescribed frequency. We transform the problem into a quadratic inverse eigenvalue problem under double complex eigenvalues. The existence and the expression of the solution are derived by solving the quadratic character determinant equations. We present the algorithm and numerical examples, which show that the obtained result is correct.