# An analytical MOS transistor model valid in all regions of operation and dedicated to low-voltage and low-current applications

DOI: 10.1007/BF01239381

- Cite this article as:
- Enz, C.C., Krummenacher, F. & Vittoz, E.A. Analog Integr Circ Sig Process (1995) 8: 83. doi:10.1007/BF01239381

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## Abstract

A fully analytical MOS transistor model dedicated to the design and analysis of low-voltage, low-current analog circuits is presented. All the large- and small-signal variables, namely the currents, the transconductances, the intrinsic capacitances, the non-quasi-static transadmittances and the thermal noise are continuous in all regions of operation, including weak inversion, moderate inversion, strong inversion, conduction and saturation. The same approach is used to derive all the equations of the model: the weak and strong inversion asymptotes are first derived, then the variables of interest are normalized and linked using an appropriate interpolation function. The model exploits the inherent symmetry of the device by referring all the voltages to the local substrate. It is shown that the inversion charge*Q*_{inv}^{′} is controlled by the voltage difference*V*_{P} − V_{ch}, where*V*_{ch} is the channel voltage, defined as the difference between the quasi-Fermi potentials of the carriers. The pinch-off voltage*V*_{P} is defined as the particular value of*V*_{ch} such that the inversion charge is zero for a given gate voltage. It depends only on the gate voltage and can be interpreted as the equivalent effect of the gate voltage referred to the channel. The various modes of operation of the transistor are then presented in terms of voltages*V*_{P} − V_{S} and*V*_{P} − V_{D}. Using the charge sheet model with the assumption of constant doping in the channel, the drain current*I*_{D} is derived and expressed as the difference between a forward component*I*_{F} and a reverse component*I*_{R}. Each of these is proportional to a function of*V*_{P} − V_{S}, respectively*V*_{P} − V_{D}, through a specific current*I*_{S}. This function is exponential in weak inversion and quadratic in strong inversion. The current in the moderate inversion region is then modelled by using an appropriate interpolation function resulting in a continuous expression valid from weak to strong inversion. A quasi-static small-signal model including the transconductances and the intrinsic capacitances is obtained from an accurate evaluation of the total charges stored on the gate and in the channel. The transconductances and the intrinsic capacitances are modelled in moderate inversion using the same interpolation function and without any additional parameters. This small-signal model is then extended to higher frequencies by replacing the transconductances by first order transadmittances obtained from a non-quasi-static calculation. All these transadmittances have the same characteristic time constant which depends on the bias condition in a continuous manner. To complete the model, a general expression for the thermal noise valid in all regions of operation is derived. This model has been successfully implemented in several computer simulation programs and has only 9 physical parameters, 3 fine tuning fitting coefficients and 2 additional temperature parameters.