The objective of transmission network expansion planning is to find the optimal strategy that balances the investment and the operating costs, considering all generation and transmission constraints. Attempts to address this problem in a tractable manner have led researchers to develop different convex relaxations and approximations. Due to the constant power grid evolution, new and improved approximation models are required to successfully handle the upcoming challenges. In this paper, we present a comprehensive approach to handle this highly complex problem both tractably and accurately. The model is based on a convex polar second-order Taylor expansion approximation of the AC power flows where both the voltage magnitudes and angles are quadratically constrained. The proposed approach achieves high accuracy due to the elimination of constraint relaxation errors, as determined by the presolve, which can occur due to the convexification process. The model demonstrated superior accuracy and similar computation times as the existing approximation models. In comparison to the exact formulations, our model shows similar accuracy while improving the computation time.