We present a bifurcation analysis of the weak electrolyte model for the electrohydrodynamic convection in a planar layer of nematic liquid crystals. The method developed provides a systematic investigation of nonlinear physical mechanisms generating the wave patterns observable near onset of electroconvection. The linear stability problem is solved analytically for the velocity and electric potential. Globally coupled Ginzburg Landau amplitude equations are used for the weakly nonlinear analysis near threshold. A rich variety of patterns, like traveling waves and rectangles, standing rectangles and rolls, alternating waves and more complex spatiotemporal structure, is predicted at Hopf bifurcation, and compared with patterns observed experimentally.