EE105  Fall 2006 Microelectronic Devices and Circuits. Some Administrative Issues


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1 EE105  Fall 006 Microelectronic evices and Circuits Prof. Jan M. Rabaey Lecture 8: MOS Small Signal Model Some Administrative Issues REIEW Session Next Week Tu Sept 6 6:007:30pm; 060 alley LSB MITERM 1 next week Th will cover everything up to and including MOSCAP Th Sept 8 6:308pm; Sibley Auditorium Some old midterm posted on website 1
2 Overview Last lecture MOS Capacitor; MOS Transistor Intro This lecture MOS Transistor (cntd Small signal model 3 MOSFET: Observed Behavior: I  I / k = 4 nonlinear resistor region constant current I resistor region = 3 = For low values of drain voltage, the device is like a resistor As the voltage is increases, the resistance behaves nonlinearly and the rate of increase of current slows Eventually the current stops growing and remains essentially constant (current source 4
3 Linear Region Current p+ NMOS > S n+ n+ ptype G Inversion layer channel 100m If the gate is biased above threshold, the surface is inverted This inverted region forms a channel that connects the drain and gate If a drain voltage is applied positive, electrons will flow from source to drain x y 5 MOSFET Linear Region The current in this channel is given by I = Wv Q y N The charge proportional to the voltage applied across the oxide over threshold Q = C ( N ox I = Wv C ( y ox If the channel is uniform density, only drift current flows v L y = μne y Ey = W I = ncox( > L μ 100m 6 3
4 MOSFET: ariable Resistor Notice that in the linear region, the current is proportional to the voltage I = W C ( L μ n ox Can define a voltagedependent resistor R eq 1 L L = = = R ( I μ C ( W W n ox This is a nice variable resistor, electronically tunable! 7 Finding I = f (, Approximate inversion charge Q N (y: drain is higher than the source less charge at drain end of channel 8 4
5 Inversion Charge at Source/rain ( Q Q ( y = 0 + Q ( y = L / N N N Q N ( y = 0 = Cox ( Q N ( y = L = C ox ( G G = 9 Average Inversion Charge Source End rain End Cox( T + Cox( G T QN ( y Cox ( T + Cox ( T QN ( y Cox ( T CoxS QN( y = Cox( T Charge at drain end is lower since field is lower Simple approximation: In reality we should integrate the total charge minus the bulk depletion charge across the channel 10 5
6 rift elocity and rain Current Longchannel assumption: use mobility to find v Substituting: μn vy ( = μney ( μn( Δ/ Δ y = L I = WvQN Wμ Cox( T L W I C ( L μ ox T Inverted Parabolas 11 SquareLaw Characteristics TRIOE REGION Boundary: what is I,SAT? SATURATION REGION 1 6
7 The Saturation Region When >, there isn t any inversion charge at the drain according to our simplistic model Why do curves flatten out? 13 SquareLaw Current in Saturation Current stays at maximum (where = =,SAT W I = C ( L μ ox T W T I, sat Cox( T ( T L μ = W μcox I, sat = ( T L Measurement: I increases slightly with increasing model with linear fudge factor W μc I L ox, sat= ( T(1 +λ 14 7
8 Pinching the MOS Transistors epletion Region > S G p+ n+ n+ ptype NMOS PinchOff Point When >,sat, the channel is pinched off at drain end (hence the name pinchoff region rain mobile charge goes to zero (region is depleted, the remaining electric field is dropped across this highfield depletion region As the drain voltage is increases further, the pinch off point moves back towards source Channel Length Modulation: The effective channel length is thus reduced higher I 15 ShortChannel MOSFET Model Channel (inversion charge: neglect reduction at drain elocity saturation defines,sat =E sat L = constant rain current: v sat / μ n I = WvQ = W ( v [ C (, SAT N sat ox E sat = 10 4 /cm, L = 0.1 μm,sat = 0.1! ], I = v WC ( (1 + λ, SAT sat ox n 16 8
9 Linear I Characteristics: shortchannel MOSFET 17 I versus 6 x Resistive =.5 Saturation = x =.5 =.0 I (A 3 =  T = 1.5 I (A 1 = = = ( Long Channel ( Short Channel 18 9
10 A Simple Circuit: An MOS Amplifier v in Input signal R Supply Rail v O = O + v o v = + v s v s i = I + i ds Output signal v o 19 Large Signal Analysis OUT = = I R Load Line 0 10
11 Sweep Input oltage v IN + MOSFET is saturated high slope v O = v MOSFET is triode low slope 0 (negative supply, SS = 0 v IN = v 1 Finding the Input Bias oltage Hand calculation: use I,SAT I W μ C = L n ox, SAT ( Typical numbers: W = 40 μm, L = μm, R = 5 kω μ n C ox = 100 μa/, = 1, = 5 I 1 = = = I, SAT = ( 1 R Want ~.5: I,SAT = /R = 0.1mA, =
12 Applying the SmallSignal oltage Approach 1. Just use v IN in the equation for the total drain current i and find v OUT v = v = + v v s IN ( t = cos( ωt sm s Result: v OUT = R i = R W μnc L ox ( + v s 3 Solving for the Output oltage v OUT v OUT = R W μ ( 1 ncox vs + L I v OUT = R I 1+ vs 4 1
13 Output oltage v O = v t 0 v IN = v t 5 SmallSignal Case Linearize the output voltage for the smallsignal case: Expand (1 + x = 1 + x + x last term can be dropped when x << vs v s = 1+ v s + vs 6 13
14 Linearized Output oltage For this case, the total output voltage is v O = = ( R I = O R + v o I vs 1+ RI The output voltage is the sum of the C voltage O and the smallsignal voltage v o. RI vo = v RI A = s = A v s v s 7 Plot of Output Waveform (Gain! Numbers: / ( T = 5/ 0.3 = 16 output input m 8 14
15 There is a Better Way! What s missing: didn t include device output impedance or charge storage effects (must solve nonlinear differential equations Approach. o problem in two steps. C voltages and currents (ignore small signals sources: set bias point of the MOSFET... we had to do this to pick already Substitute the smallsignal model of the MOSFET and the smallsignal models of the other circuit elements This constitutes smallsignal analysis 9 An MOS Amplifier Input signal R Supply Rail v o v s I v = + vs Output signal 30 15
16 SmallSignal Analysis Step 1. Find C Bias ignore smallsignal source I,Q,BIAS 31 Which Operating Region? TRIOE = = 3 3 SAT OFF 3 16
17 I / k Changing One ariable at a Time Square Law Saturation Region = 3 = 1 T Linear Triode Region Slope of Tangent: Incremental current increase Assumption: >,SAT = (square law 33 The Transconductance g m efined as the change in drain current due to a change in the gatesource voltage, with everything else constant W μc I L ox, sat = ( T (1 +λ Δi i W g = = = μc ( (1 + λ m ox T Δv v L,, 0 W gm = μcox ( T L Gate Bias W I W g = μc = μc I L W μc L ox L m ox ox g m I = ( T rain Current Bias rain Current Bias and Gate Bias 34 17
18 Output Resistance r o efined as the inverse of the change in drain current due to a change in the drainsource voltage, with everything else constant NonZero Slope δ I δ 35 Evaluating r o W μc i L ox = ( T (1 +λ o = v, r i r0 = W μc L ox 1 1 ( λ 1 r0 λi T 36 18
19 Total Small Signal Current i ( t = I + i ds i i i = v + v ds gs ds vgs vds 1 i = g v + v r ds m gs ds o Transconductance Conductance 37 Putting Together a Circuit Model 1 i = g v + v r ds m gs ds o 38 19
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