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Network lifetime (NL) is a critical metric in the design of energy-constrained wireless sensor networks (WSNs). In this paper, we investigate a joint optimal design of the physical, medium access control (MAC) and routing layers to maximize NL of a multiple-sources and single-sink (MSSS) WSN with energy constraints. The problem of NL maximization (NLM) can be formulated as a mixed integer-convex optimization problem with adoption of time division multiple access (TDMA) technique. When the integer constraints are relaxed to take real values, the problem can be transformed into a convex problem and the solution achieves the upper bounds. We provide an analytical framework for the relaxed NLM problem of a WSN in general planar topology. We first restrict the topologies to the planar networks on a small scale, including triangle and regular quadrangle topologies. In this special case, we employ the Karush-Kuhn-Tucker (KKT) optimality conditions to derive analytical expressions of the globally optimal NL, which take the influence of data rate, link access and routing into account. To handle larger scale planar networks, an iterative algorithm is proposed using the D&C approach. Numerical results illustrate that the proposed algorithm can be extended to the large planar case and its performance is close to globally optimal performance.