A SIMPLE DISTRIBUTED RGC MODEL OF MOSFET FOR PRE-PINCH OFF REGION

The differential equation describing the small signal behavior of a MOSFET channel is 
derived. Based on the analogy of the channel to distributed transmission lines, which 
has been very well established in literature, an entirely new RGC line model of 
MOSFET is presented. The element values of the line are determined by equivalence 
to a general distributed transmission line and subsequently the model is lumped into a 
single section in two possible Π and T representations. The postulated model 
considerably simplifies the study of the properties and behavior of MOSFET structures 
and can be suitably utilized in analysis and Computer Aided Design.


INTRODUCTION
The transmission line approach to model microelectronic circuit components has been commonly used to evaluate transient and fre- quency responses.This theory has been applied to a considerably large number of cases for thin film and diffusion resistors, capacitors and conductors and the undesirable interaction between different components of integrated circuits.The transmission lines have also been successfully used in simulation of active microelectronic ele- ments; in particular, the field effect devices.Also, any integrated circuit employing FETs may be considered as consisting of transmis- sion lines and additional lumped elements.
There have been numerous attempts to derive a distributed equivalent circuit model.These include those by Hoffmann1'2, Kocprgak3, McNutt et al4, Popor & Bickart 5 and Lonngren6.A great deal of work on this topic has been reported by  who has made the extrapolation to charge transfer devices5.We have proceeded from an entirely different method to arrive at such a representation. 55 TEORETICAL ANALYSIS AND DERIVATION Fig. 1 shows an n-channel MOSFET which the source and substrate short circuited to the ground.A voltage Vds consisting of a d.c.bias Vds is applied between the drain and the source.ID is the output drain current.

Now if
IEox" permittivity of the oxide tox" oxide thickness Z" is the MOSFET width in the direction transverse to the current flow U" total gate channel potential at any point x in the channel.

IEox Z tox
gate channel capacitance per unit length of the channel.Then Cox U mobile channel charge per unit length of the channel.Now if mobility of the carriers (electrons) in the channel, then 0u /b--velocity of the carries in the channel.
Hence =I (1)   channel current at any point x in the channel.Assuming that we can express the gate potential and current I as the sum of a.c. and d.c.components, i.e., U(x, t) v(x) + u(x)e jt  (3) ignored Ignoring the second order term in the a.c.component under small signal approximation and equating the time independent and time dependent terms on both the sides we get and Now considering an incrementat section of length Ax of the channel Again, substituting for U + I from equations (2) + (3) in (6) we get From ( 5) we have i Cox [u(x) dv(x) du(x)] d---+v(x)" dx 1 Differentiating we get.

di(x)
dx Cox [u(x) d2 + +v comparing with equation ( 7) we get d e jou + u(x.xV(X / u(x.v(x which can be written as u=0 where a denotes a derivative v.r.t.x.This is the standard differential equation controlling the operation of MOS transistor.Now for the general transmission line structure shown in Fig. 2 as the characteristic equation where u(x) is the small signal line voltage at any point x.
Equation ( 9) is exactly analogous to equation (8) obtained for the MOS channel.Hence by comparison we get, z- v Hv -YZ (11)   from (10) we get upon integrating lnZ which is purely real and hence a pure series of resistances, r, per unit length.
Also from (11) Y can be expressed as jo v" tovZ vZ tt)V V" ku vZ jO)V V"V g + joc where VttV g= k as the shunt conductance per unit length and v as the shunt capacitance per unit length.

PROPOSED MODEL FOR MOSFET
Hence, the resulting small signal distributed RGC line model of the MOSFET intrinsic portion can be represented as shown in Fig. 3.
To determine the values of the elements as functions of distance, we need to know the voltage distribution v as a function of x.
Since the d.c.current at any point x is given by I # Coxv(X) Ov and is constant throughout the channel independent of x, we have"   (VG Vt-VD)[ln(VG VT-VD- 1)] we can lump the distributed model derived above and obtain single T or H analog model which can be represented as shown in Fig. 4.

FIGURE 3
FIGURE 3 Proposed distributed RGC line model of MOSFETs.