In this paper, we study the energy efficiency (EE) of orthogonal frequency-division multiple access (OFDMA) cellular networks under the 5G requirement of EE enhancement. We aim to present a power allocation scheme maximizing the EE of downlink cellular communications while avoiding numerical methods such as fractional programming. We focus on two EE forms; global EE and weighted sum EE. Therefore, we propose a novel explicit expression of the optimal power allocation related to each subcarrier. We also present the power control with limited power budget or/and minimal transmission rate constraint in both base station and subcarrier perspectives. As a result, we notice the occurrence of some transmission outage events depending on the constraints' parameters. From another side, we extend our study to analyze the effect of the channel state information (CSI) unavailability on our proposed power scheme. In the numerical results, we show that our proposed power control improves the EE, especially at high power budget regime and low minimal rate regime. We also show that having more subcarriers enhances the OFDMA EE. Finally, we show that EE degradation due to CSI unavailability is very small showing the robustness of the proposed scheme against CSI imperfectness.