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We developed a fully analytical ballistic theory of carbon nanotube field effect transistors enabled by the development of an analytical surface potential capturing the temperature dependence and gate and quantum capacitance electrostatics. The analytical ballistic theory is compared to the experimental results of a ballistic transistor with good agreement. The validated analytical theory enables intuitive circuit design, provides techniques for parameter extraction of the bandgap and surface potential, and elucidates on the device physics of drain optical phonon scattering and its role in reducing the linear conductance and intrinsic gain of the transistor. Furthermore, a threshold voltage definition is proposed reflecting the bandgap-diameter dependence. Projections for key analog and digital performances are discussed.