Transconductance. (Saturated) MOSFET Small-Signal Model. The small-signal drain current due to v gs is therefore given by

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1 11 (Saturated) MOSFET Small-Signal Model Transconductance Concept: find an equivalent circuit which interrelates the incremental changes in i D v GS v DS etc. for the MOSFET in saturation The small-signal current due to is therefore given by = g m. v GS = V GS i D = I D -- we want to find = (?) D i D = I D We have the functional dependence of the total current in saturation: i D = µ n C ox (W/2L) (v GS - V Tn ) 2 (1 λ n v DS ) = i D (v GS v DS ) Solution: do a Taylor expansion around the DC operating point (also called the quiescent point or point) defined by the DC voltages (V GS V DS ): V GS = 3 V G S B V DS = 4 V 2 i D 1 i D i D = I D ( v vgs gs ) -- ( v 2 2 gs ) 2 v GS If the small-signal voltage is really small then we can neglect all everything past the linear term -- i D (µa) v GS = V GS v GS = V GS = 3 V g m = / 100 i D i D = I D ( v vgs gs ) = I D g m V DS (V) where the partial derivative is defined as the transconductance g m.

2 Another View of g m Transconductance (cont.) * Plot the current as a function of the - voltage so that the slope can be identified with the transconductance: Evaluating the partial derivative: D i D = I D g m = µ n C W ox ---- ( VGS V L Tn )( 1 λ n V DS ) V GS = 3 V 600 G S B V DS = 4 V i D (v GS V DS = 4 V) In order to find a simple expression that highlights the dependence of g m on the DC current we neglect the (usually) small error in writing: W 2I g m 2µ n C ox ---- D = ID = L V GS V Tn i D g m = / For typical values (W/L) = 10 I D = 100 µa and µ n C ox = 50 µav -2 we find that (µa) g m = 320 µav -1 = 0.32 ms v GS = V GS = 3 V v GS = V GS v GS (V)

3 (Partial) Small-Signal Circuit Model Output Conductance/Resistance How do we make a circuit which expresses = g m? Since the current is not across its controlling voltage we need a voltage-controlled current : We can also find the change in current due to an increment in the voltage: i D W g o = = µ v n C ox ( VGS V DS 2L Tn ) 2 λ n λ n I D The output resistance is the inverse of the output conductance g m 1 r o = ---- = g o λ n I D The (partial) small-signal circuit model with r o added looks like: = g m (1/r o )v ds g m ro v ds

4 MOSFET Capacitances in Saturation fringe electric field lines n n C sb q N (v GS ) C depletion overlap L D overlap L db D region Complete Small-Signal Model All these capacitances are patched onto the small-signal circuit schematic containing g m and r o... g mb is open-circuited for EECS 105 since v bs = 0 V. C gd C gs C gb g m g mb v bs r o v bs bulk C sb C db In saturation the - capacitance contains two terms one due to the channel charge s dependence on v GS [(2/3)WLC ox ] and one due to the overlap of and (WC ov where C ov is the overlap capacitance in ff per µm of width) 2 C gs = -- WLC 3 ox WC ov In addition there is the small but very important - capacitance (just the overlap capacitance C gd = C ov ) There are depletion capacitances between the and bulk (C db ) and between and bulk (C sb ). Finally the extension of the over the field oxide leads to a small -bulk capacitance C gb.

5 p-channel MOSFETs p-channel MOSFET Models Structure is complementary to the n-channel MOSFET In a CMOS technology one or the other type of MOSFET is built into a well -- a deep diffused region -- so that there are electrically isolated bulk regions in the same substrate DC current in the three operating regions: -I D > 0 I D = 0 A ( VSG V T ) I D = µ p C ox ( W L) [ V SG V Tp ( V SD 2) ]( 1 λ p V SD )V SD ( VSG V Tp V SD V SG V Tp ) I D = µ p C ox ( W ( 2L) )( V SG V Tp ) 2 ( 1 λ p V SD ) ( VSG V Tp V SD V SG V Tp ) (a) A (b) n-channel MOSFET p-channel MOSFET common bulk contact for isolated bulk contact with all n-channel MOSFETs p-channel MOSFET (to ground or to the supply) shorted to p n n p p n p-type substrate n well A The threshold voltage with back effect is given by: V Tp = V TOp γ p (( V SB 2φ n ) 2φ n ) Numerical values: µ p C ox is a measured parameter. Typical value: µ p C ox = 25 µav µmV 1 λ p L V Tp = -0.7 to -1.0 V which should be approximately -V Tn for a well-controlled CMOS process

6 p-channel MOSFET small-signal model the is the highest potential and is located at the top of the schematic v sg C gs g m v sg g mb v sb r o v sb C gd C gb C sb bulk C db