We report fabrication and modeling of self-defined gated filed-emission devices based on encapsulated vertically aligned carbon nanotubes grown on Si substrates with a PECVD method. The electrical characteristics of such devices have been theoretically modeled using an expanded Fowler-Nordheim tunneling effect. Devices fabricated here, resemble vacuum tubes where CNTs act as cathode. They show a saturation behavior for large distances between anode and cathode electrodes, making them suitable for transistor applications. The physical properties of the CNTs were investigated using SEM, evidencing the evolution of vertical CNTs with a tip-growth mechanism. The triode structure proposed in this paper uses CNT as the field-emission electrode (cathode), the surrounding Cr as the gate and a second Si substrate as the anode electrode. The emission current from the nanotube tips is significantly controlled by applying a negative voltage between CNT and the gate. The effect of the anode-cathode and gate-cathode voltages on the emission current has been experimentally observed and theoretically modeled using an expanded F-N effect.