Circuits and MOS Small-Signal Models Prof. Ali M. Niknejad Prof....
Transcript of Circuits and MOS Small-Signal Models Prof. Ali M. Niknejad Prof....
Department of EECS University of California, Berkeley
EECS 105 Spring 2017, Module 3
Prof. Ali M. Niknejad Prof. Rikky Muller
Circuits and MOS Small-Signal Models
EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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Announcements
l HW 7 is last HW before Midterm l Next HW will be posted before Spring Break and
due Friday after Spring Break (2 weeks to complete)
l Next week: Midterm Review (Tuesday), Midterm (Thursday)
l Syllabus after Spring Break will be updated
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EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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A Simple Circuit: An MOS Amplifier
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DSI
GSV
vgs
DR DDV
vGS =VGS + vgs
ov
Input signal
Output signal
Supply “Rail”
EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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Selecting the Output Bias Point l The bias voltage VGS is selected so that the output is
mid-rail (between VDD and ground) l For gain, the transistor is biased in saturation l Constraint on the DC drain current:
l All the resistor current flows into transistor:
l Must ensure that this gives a self-consistent solution (transistor is biased in saturation)
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DD o DD DSR
D D
V V V VIR R− −
= =
,R DS satI I=
DS GS TV V V> −
EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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Finding the Input Bias Voltage l Ignoring the output impedance
l Typical numbers: W = 40 µm, L = 2 µm, RD = 25kΩ, µnCox = 100 µA/V2, VTn = 1 V, VDD = 5 V
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2,
1 ( )2DS sat n ox GS Tn
WI C V VLµ= −
2,
1 ( )2 2D
DDR DS sat n ox GS Tn
D
V WI I C V VR L
µ= = = −
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5V µA 1100µA 20 100 ( 1)50k V 2 GSV= = ⋅ ⋅ −
Ω
.1= (VGS −1V )2 VGS =1.32V VGS −VT = .32V <VDS = 2.5Vü
EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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Applying the Small-Signal Voltage
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Approach I. Just use vGS in the equation for the total drain current iD and find vo
vGS =VGS + vgs
ˆ coss sv v tω=
vO =VDD − RDiDS =VDD − RDµnCoxWL12(VGS + vgs −VT )
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Note: Neglecting charge storage effects. Ignoring device output impedance.
EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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Solving for the Output Voltage vO
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vO =VDD − RDiDS =VDD − RDµnCoxWL12(VGS + vgs −VT )
2
vO =VDD − RDiDS =VDD − RDµnCoxWL12(VGS −VT )
2 1+vgs
VGS −VT
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
DSI
vO =VDD − RDIDS 1+vgs
VGS −VT
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
2DDV
EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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Small-Signal Case l Linearize the output voltage for the s.s. case l Expand (1 + x)2 = 1 + 2x + x2 … last term can be
dropped when x << 1
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Neglect
vO ≈VDD − RDIDS 1+2vgs
VGS −VT
⎛
⎝⎜⎜
⎞
⎠⎟⎟
1+2vgs
VGS −VT
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
=1+2vgs
VGS −VT+
vgsVGS −VT
⎛
⎝⎜⎜
⎞
⎠⎟⎟
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EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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Linearized Output Voltage For this case, the total output voltage is: The small-signal output voltage:
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vO ≈VDD − IDRDS 1+2vgs
VGS −VT
⎛
⎝⎜⎜
⎞
⎠⎟⎟
vO ≈ (VDD − IDRDS )−2IDRDSvgsVGS −VT
=VDD2−VDDvgsVGS −VT
“DC” Small-signal output
vo ≈ −2IDRDSvgsVGS −VT
= −VDDvgsVGS −VT
Avvgs
Voltage gain
EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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Plot of Output Waveform (Gain!)
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Numbers: VDD / (VGS – VT) = 5/ 0.32 = 16
output
input
mV
EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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There is a Better Way! l What’s missing: didn’t include device output
impedance or charge storage effects (must solve non-linear differential equations…)
l Approach II: Solve problem in two steps. – 1) DC voltages and currents (ignore small signals
sources): set bias point of the MOSFET ... we had to do this to pick VGS already
– 2) Substitute the small-signal model of the MOSFET and the small-signal models of the other circuit elements …
l This constitutes small-signal analysis
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EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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Total Small Signal Current
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( )DS DS dsi t I i= +
DS DSds gs ds
gs ds
i ii v vv v∂ ∂
= +∂ ∂
1ds m gs ds
o
i g v vr
= +
Transconductance Conductance
EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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Changing One Variable at a Time
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Assumption: VDS > VDS,SAT = VGS – VTn (square law)
GSV
/DSI k
3VDSV =1VTV =
Square Law Saturation
Region
Linear Triode Region
Slope of Tangent: Incremental current increase
EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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The Transconductance gm
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Defined as the change in drain current due to a change in the gate-source voltage, with everything else constant
, ,
( )(1 )GS DS GS DS
D Dm ox GS T DS
GS GSV V V V
i i Wg C V V Vv v L
µ λΔ ∂
= = = − +Δ ∂
2, ( ) (1 )
2ox
DS sat GS T DSCWI V V V
Lµ
λ= − +
( )m ox GS TWg C V VL
µ= −
0≈
2 2DSm ox ox DS
ox
IW Wg C C IWL LCL
µ µµ
= =
2( )
DSm
GS T
IgV V
=−
Gate Bias
Drain Current Bias
Drain Current Bias and Gate Bias
EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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Output Resistance ro
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Defined as the inverse of the change in drain current due to a change in the drain-source voltage, with everything else constant
Non-Zero Slope
DSVδ
DSIδ
IDS / K
VDS (V )
EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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Evaluating ro
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1
,GS DS
Do
DS V V
irv
−⎛ ⎞∂⎜ ⎟=⎜ ⎟∂⎝ ⎠
2( ) (1 )2ox
D GS T DSCWi V V V
Lµ
λ= − +
02
1
( )2ox
GS T
r CW V VLµ
λ=
−
01
DS
rIλ
≈
EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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Three-Terminal Small-Signal Model
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ids = gmvgs +
1ro
vds
EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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P-Channel MOSFET
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Everything is flipped around compared to an NMOS. Currents are negative of NMOS.
EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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Square-Law PMOS Characteristics
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/DSI k
VDS (V)
VGS = −2V
VGS = −3V
VGS = −4V
VGS = −1V
EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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Right Side Up!
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VSG = 2V
VSG = 3V
VSG = 4V
VSG = 1V
ISD / k
VSD (V)
EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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Small-Signal PMOS Model
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EE 105 Spring 2017 Prof. A. M. Niknejad Prof. Rikky Muller
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What we’re ignoring for now…
l The fourth terminal of the MOSFET is the body of the transistor… it can act like a second gate, or “back gate”
l The junctions of the transistor form pn-junction diodes with body, this introduces parasitic capacitance
l Modern transistors are very short channel lengths, and there are many “short channel” effects that we’re ignoring, most prominently velocity saturation
l Subthreshold conduction l Capacitance!
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