The transmission electrode technique has been recently proposed as a versatile method to obtain various types of liquid-crystal (LC) lenses. In this work, an equivalent electric circuit and new analytical expressions based on this technique are developed. In addition, novel electrode shapes are proposed in order to generate different phase profiles. The analytical expressions depend on manufacturing parameters that have been optimized by using the least squares method. Thanks to the proposed design equations and the associated optimization, the feasibility of engineering any kind of aspheric LC lenses is demonstrated, which is key to obtain aberration-free lenses. The results are compared to numerical simulations validating the proposed equations. This novel technique, in combination with the proposed design equations, opens a new path for the design and fabrication of LC lenses and even other types of adaptive-focus lenses based on voltage control.