4. mos small signal modelaries.ucsd.edu/najmabadi/class/ece102/11-f/notes/ece102_f11-lecset... ·...
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4. MOS Small Signal Model
Reading: Sedra & Smith Sec. 5.5 (S&S 5th Ed: Sec. 4.6)
ECE 102, Fall 2011, F. Najmabadi
Circuit response to signal is different
Recall:
o Response of the circuit to small signal is different than response of the circuit to Bias + signal.
o iv characteristics of the circuit elements in response to the signal is different than iv characteristics of the circuit elements in response to the Bias + signal
• Circuit looks different when signal is considered!
f(∙) aAA xXx += aAA yYy += )( AA xfy =Signal + Bias
)( AA XfY =f(∙) AX AYBias (signal = 0)
g(∙)
ax aySignal=
(Signal + Bias) –(Bias) )( aa xgy =
“Signal-only” circuit is different!
Signal only = (Bias + Signal) - Bias
RD: vrd ird = id
MOS: vgs, id, vds
No signal here!
RD: VRD IRD = ID
MOS: VGS, ID, VDS
VDD: VDD Bias
RD: vRD = VRD + vrd iRD = iD = ID + id
MOS: vGS = VGS + vgs vDS = VDS + vds iD = ID + id
VDD: VDD
Bias & Signal
Signal Model for linear circuit elements Independent voltage source (e.g., VDD)
o No signal: effectively grounded
Independent current source o No signal: effectively open circuit (Careful about current mirrors as they
are NOT “ideal” current sources, channel width modulation was ignored!)
Resistors, capacitors, inductors o Remain the same:
Dependent sources o Remain the same with the control parameter related to the signal!
Non-linear Elements: o Different!
iR = IR + ir vR = VR + vr = RIR + vr vR = R iR = R (IR + ir ) = RIR + R ir vr = R ir
−
×=
−
×
=
−
+=−=
1exp1expexp
expexp :Signal
T
dD
T
d
T
Dsd
T
Ds
T
dDsDDd
nVvI
nVv
nVVIi
nVVI
nVvVIIii
Diodes: signal response is non-linear but can be linearized when signal is small
VD
ID
vd
id ?
vd
id
R = nVT/ID
=+
T
DsD nV
vIi exp :Signal Bias
=
T
DsD nV
VII exp :Bias
vD
iD
DT
D
T
dDd
T
d
T
d
T
d
T
d
T
d
T
d
vnVI
nVvIi
nVv
nVv
nVv
nVv
nVv
nVv
=
−
+×≈
+≈
<<
+
+
+=
11
1exp :1 If
.... !2
11exp :Exapnsion SeriesTaylor 2
Formal derivation of small signal model
( ) ( ) ...!2
)()()( 2)2(
)1( +−⋅+−⋅+= AAA
AAAA XxXfXxXfXf
...!2
)()()( 2)2(
)1( +⋅+⋅+= aA
aAA xXfxXfXf
aAAaAA xXfYxXfXf ⋅+=⋅+≈ )()()( )1()1(
2)2(
)1(
!2)()( a
AaA xXfxXf ⋅>>⋅
)()(2 )2(
)1(
A
Aa Xf
Xfx ⋅<<
Small signal means:
aAaa xXfxgy ⋅== )()( )1(
f(∙) aAA xXx += aAA yYy += )( AA xfy =Signal + Bias
)( AA XfY =f(∙) AX AYBias (signal = 0)
g(∙)
Ax AySignal=
(Signal + Bias) –(Bias) )( aa xgy =
)( AAaA xfyyY ==+(Taylor Series Expansion)
Derivation of diode small signal model
−⋅= 1T
DnVv
SD eIi
−⋅= 1)( TnV
x
S eIxf
−⋅== 1)( T
DnVV
SDD eIVfI
dT
SDd
T
nVV
Sd
VxT
nVx
SdDd v
nVIIv
nVeIv
nVeIvVfi
T
D
D
T
⋅
+=⋅
⋅
=⋅
⋅
=⋅=
=
)()1(
dT
Dd
T
SDd v
nVI
vnV
IIi ⋅
≈⋅
+=
D
dd r
vi =D
TD I
nVr ≈vd
id
R = nVT/ID
vD
iD
For Small Signals!
MOS iv equations: iD = f(vGS, vDS)
),( AAA yxfz =Signal + Bias f(∙, ∙) aAA xXx +=aAA zZz +=
aAA yYy +=
Bias (signal = 0) f(∙, ∙) AX
AZAY
),( AAA YXfZ =
Signal= (Signal + Bias) –(Bias) g(∙, ∙)
axaz
ay),( aaa yxgz =
),( AAAaA yxfzzZ ==+
...)(),()(),(),(,,
+−⋅∂
∂+−⋅
∂∂
+= AAYX
AAYX
AA Yyy
yxfXxx
yxfYXfAAAA
aYX
aYX
A yy
yxfxx
yxfZAAAA
⋅∂
∂+⋅
∂∂
+≈,,
),(),(
aYX
aYX
a yyfx
xfz
AAAA
⋅∂∂
+⋅∂∂
≈,,
Derivation of MOS small signal model
dsVVDS
gsVVGS
d vvfv
vfi
DSGSDSGS
⋅∂∂
+⋅∂∂
=,,
yvxvyxfi
vvfvVvL
WCi
i
DSGSD
DSGSDStGSoxnD
G
↔↔=
=+−=
=
, with ),(
),()1()( 5.0
0
2 λµ
mOV
D
tGS
DStGSoxn
VVDStGSoxnVVGS
gV
IVV
VVVL
WC
vVvL
WCvf
DSGS
DSGS
≡=−
+−×=
+−×=∂∂
2)(
)1()( 5.02
)1)(( 5.02
2
,,
λµ
λµ
oDS
D
DS
DStGSoxn
VVtGSoxn
VVDS
rVI
V
VVVL
WC
VvL
WCvf
DSGSDSGS
1)1()1(
)1()( 5.0
)( 5.0
2
,
2
,
≡+
=+
+−×=
−×=∂∂
λλ
λ
λµλ
µλ
1 *
2
0
Do
OV
Dm
g
o
dsgsmd
Ir
VIg
irvvgi
⋅=
=
=
+⋅=
λ(* For λ vDS << 1)
)1()( 5.0
0
2DStGSoxnD
G
VVVL
WCI
I
λµ +−=
=
MOS small signal “circuit” model
and 0 o
dsgsmdg r
vvgii +⋅==
Do I
r⋅
≈λ
1
OV
Dm V
Ig ⋅=
2 122>>==
OV
A
OVom V
VV
rgλ
id Statement of KCL Two elements in parallel Input open circuit
PMOS “circuit” small signal model is identical to NMOS
Do I
r⋅
=λ
1
OV
Dm V
Ig ⋅=
2
id
vsg gmvsg
id
=
PMOS small-signal circuit model is identical to NMOS o For PMOS small signal model, id flows into the drain (while iD and ID
flows out of the drain). o For both NMOS and PMOS, while iD ≥ 0 and ID ≥ 0, signal quantities: id,
vgs, and vds , can be negative!
and 0 o
sdsgmdg r
vvgii +⋅==
Body Effect
It has been found that with a few unusual exceptions, body effects can be ignored in the initial design of MOS amplifiers (and we ignore it here).
When body effect included:
We can use Taylor Series Expansion in three variables to get MOS small-signal model:
( )),,()1()( 5.0
|2||2|
2
0,
SBDSGSDStGSoxnD
FSBFtt
vvvfvVvL
WCi
vVV
=+−=
−++=
λµ
φφγ
o Additional element. o Note body is connected to
ground for small signal (because it is connected to the negative terminal of the power supply).
Example 1: Draw the small-signal equivalent of the circuit below (assume capacitors are short for small signal).
IVS → 0 R remains
Ground at the bottom
Example 2: Draw the small-signal equivalent of the circuit below (assume capacitors are short for small signal).
Flip PMOS
IVS → 0 Ground at the bottom (100k || 33k = 24.8 k)
Example 3: Draw the small-signal equivalent of the circuit below (assume capacitors are short for small signal).
ICS → 0 (This makes ICS an open circuit)
IVS → 0