How OAuth and portable data can revolutionize your web app - Chris Messina
Ch04s
30
Chapter 4 Problem Solutions 4.1 (a) ( ) ( ) 2 2 2 0.5 2 0.040 0.5 3.125 μ ω ⎛ ⎞⎛ ⎞ = = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ = ⇒ = ⎜ ⎟ ⎝ ⎠ n ox m n D D C W g KI I L W L L (b) ( ) ( )( )( ) 2 2 2 0.5 0.04 3.125 0.8 2.80 V n ox D GS TN GS GS C W I V V L V V μ ⎛ ⎞⎛ ⎞ = − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ = − ⇒ = 4.2 (a) ( ) ( ) 2 2 2 2 50 20 100 0.3125 2 p ox m p D D C W g KI I L W W L L μ ⎛ ⎞ ⎛ ⎞ = = ⎜ ⎟⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ = ⇒ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (b) ( ) ( )( )( ) 2 2 2 100 20 0.3125 1.2 5.2 V p ox D SG TP SG SG C W I V V L V V μ ⎛ ⎞ ⎛ ⎞ = + ⎜ ⎟⎜ ⎟ ⎝ ⎠ ⎝ ⎠ = − ⇒ = 4.3 ( )( ) ( ) () [ ] [ ] ( ) ( ) 2 1 1 2 2 1 1 10 1 3.4 1 3.0 1 5 3.4 1 5 3.0 1 10 3.4 3.0 3 10 3.4 5 0.0308 5 12.5 k 0.4 D n GS TN DS DS D D DS DS o D I K V V V V I I V V r I λ λ λ λ λ λ λ λ λ = − + + + = ⇒ = + + + = + − = ⋅ − ⋅ ⇒ = Δ = = = Ω Δ 4.4 ( )( ) 0 0 1 1 1 0.833 mA 0.012 100 D D D r I I I r λ λ = = = ⇒ = 4.5 ( )( ) () () ( ) ( ) ( ) 2 1 1 2 2 1 1 1 1 1 2 0.20 0.22 1 4 0.20 1 4 0.22 1 2 0.8 0.44 0.22 0.20 0.0556 D n GS TN DS DS D D DS I K V V V V I I V V λ λ λ λ λ λ λ λ λ − = − + + = + + = + + = + − = − =
-
Upload
bilal-sarwar -
Category
Technology
-
view
356 -
download
0
Transcript of Ch04s
Chapter 4 Problem Solutions 4.1 (a)
( ) ( )
2 22
0.5 2 0.040 0.5 3.125
μ
ω
⎛ ⎞⎛ ⎞= = ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⎛ ⎞= ⇒ =⎜ ⎟⎝ ⎠
n oxm n D D
C Wg K I IL
WL L
(b)
( )
( )( )( )
2
2
2
0.5 0.04 3.125 0.8 2.80 V
n oxD GS TN
GS GS
C WI V VL
V V
μ⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
= − ⇒ =
4.2
(a)
( ) ( )2
2 22
50 20 100 0.31252
p oxm p D D
C Wg K I IL
W WL L
μ⎛ ⎞⎛ ⎞= = ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⎛ ⎞ ⎛ ⎞= ⇒ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
(b) ( )
( )( )( )
2
2
2
100 20 0.3125 1.2 5.2 V
p oxD SG TP
SG SG
C WI V VL
V V
μ⎛ ⎞⎛ ⎞= +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
= − ⇒ =
4.3
( ) ( )( )( )
[ ] [ ]( )( )
2
11
2 2
11 101 3.4
1 3.0 1 53.4 1 5 3.0 1 103.4 3.0 3 10 3.4 5 0.0308
5 12.5 k0.4
D n GS TN DS
DSD
D DS
DSo
D
I K V V V
VII V
VrI
λλλ
λ λλ λ
λ λ
= − +++
= ⇒ =+ +
+ = +− = ⋅ − ⋅ ⇒ =
Δ= = = Ω
Δ
4.4
( )( )
0
0
1
1 1 0.833 mA0.012 100
D
D D
rI
I Ir
λ
λ
=
= = ⇒ =
4.5
( ) ( )
( )( )
( ) ( )( )
2
11
2 2
1
1111 20.20
0.22 1 4
0.20 1 4 0.22 1 20.8 0.44 0.22 0.20
0.0556
D n GS TN DS
DSD
D DS
I K V V VVI
I V
V
λλλλλ
λ λλλ −
= − ++
=++
=+
+ = +− = −
=
2 100 K0.02
DSo o
D
Vr rI
Δ= = ⇒ =
Δ
4.6 (a)
(i) ( )( )
1 1 1000 K0.02 0.05o
D
rIλ
= = =
(ii) ( )( )
1 100 K0.02 0.5or = =
(b)
(i) 1 0.001 mA 1.0 A1000
DSD
o
VIr
μΔΔ = = = =
1.0 2%50
D
D
II
Δ= ⇒
(ii) 1 0.01 10 A100
10 2%500
DSD
o
D
D
VIr
II
μΔΔ = = = ⇒
Δ = ⇒
4.7
( )( )
1.0 mA1 1 100 K
0.01 1
D
oD
I
rIλ
=
= = =
4.8
( ) ( )( )0 || 1 50 ||10 8.33v m D vA g r R A= − = − ⇒ = − 4.9
a. 10 6 8 k0.5
− −= = ⇒ = ΩDD D SQD D
DQ
V VR R
I
( )
( ) ( )
2
2
For 2 V, for example,
2
0.5 0.030 2 0.8 11.6
μ=
⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞= − ⇒ =⎜ ⎟⎝ ⎠
GSQ
n oxDQ GSQ TN
V
C WI V VL
W WL L
b. ( )
( )( )( )
( )( )
2 2
2 0.030 11.6 2 0.8 0.835 mA/V
1 1 133 k0.015 0.50
m n DQ n GSQ TN
m m
o oDQ
g K I K V V
g g
r rIλ
= = −
= − ⇒ =
= = ⇒ = Ω
c. ( ) ( )( )0 0.835 133 8 6.30v m D vA g r R A= − = − ⇒ = − 4.10
[ ]
[ ]
22 2 2
2
22
sin sin
1sin 1 cos 22
So 1 cos 22
ω ω
ω ω
ω
⎡ ⎤= =⎣ ⎦
= −
= −
n gs n gs n gs
n gsn gs
K v K V t K V t
t t
K VK v t
( )
( )
2
cos 22Ratio of signal at 2 to that at :
2 sin
The coefficient of this expression is then:4
n gs
n GSQ TN gs
gs
GSQ TN
K Vt
K V V V t
V
V V
ωω ω
ω
⋅
− ⋅
−
4.11
( )( )( )( )
0.014
So 0.01 4 3 1 0.08 V
=−
= − ⇒ =
gs
GSQ TN
gs gs
VV V
V V
4.12
( )
( )
( )( ) ( )( )( )
1 2
1 2
50 0.961550 2
0.9615 1 0.9615 50 10 8.01
o m gs o D
gs i i iSi
v m o D v
V g V r R
R RV V V V
R R RA g r R A
= −
⎛ ⎞= ⋅ = ⋅ =⎜ ⎟+ +⎝ ⎠= − = − ⇒ = −
4.13
( )( )0 ||
10 100 || 5 2.1 mA/Vv m D
m m
A g r Rg g
= −− = − ⇒ =
4.14 a.
( )
( )
( )( )( )
( ) ( )( )( )
( )( )( )
( )( )
2
1 2
2
2
2
2
2
200 12 4.8 V200 300
4.8 1 2 4 4
2 7 3.2 0
7 7 4 2 3.22.96 V
2 2
1 2.96 2 0.920 mA
12 0.92 3 2 7.4 V
⎛ ⎞= ⎜ ⎟+⎝ ⎠⎛ ⎞= =⎜ ⎟+⎝ ⎠
−= = −
− = − +
− + =
± −= =
= − ⇒ =
= − += − + ⇒ =
G DD
G
G GSD n GS TN
S
GS GS GS
GS GS
GS
D D
DS DD D D S
DS
RV VR R
V
V VI K V VR
V V V
V V
V
I I
V V I R RV
(b) ( )
1m g D L
om S
g V R RV
g R−
=+
where ( )1 2
1 2
300 200 120 0.9836300 200 2 120 2g i i i i
Si
R RV V V V V
R R R= ⋅ = ⋅ = ⋅ =
+ + +
Then
( )( )
( )( )( )( )( )
( )( )
0.9836 ||1
2 1 2.96 2 1.92 /1.92 0.9836 3 || 3
So 0.5851 1.92 2
m D Lv
m S
m
v v
g R RA
g Rg mA V
A A
−=
+= − =
−= ⇒ = −
+
c.
0.92
7.4 12
AC load line
Slope �
�1
3��3 � 2�1
3.5 K�
( )( )
1i3.5 k
i 0.92 mA 0.92 3.5 3.22 6.44 V peak-to-peak
D ds
D ds
v
v
−Δ = ⋅ ΔΩ
Δ = ⇒ Δ = = ⇒
4.15 a.
( )( )( )
( )( )
( )( )
2
2
2
1 2 1
11
22
2
3 mA 3 0.5 1.5 V
3 2 2 3.225 V3.225 1.5 4.725 V
1
14.725 200 15 635 k
635 200 292 k635
= ⇒ = = =
= −
= − ⇒ == + = + =
⎛ ⎞= = ⋅ ⋅⎜ ⎟+⎝ ⎠
= ⇒ = Ω
= ⇒ = Ω+
D S DQ SQ
DQ n GS TN
GS GS
G GS S
G DD in DD
I V I R
I K V V
V VV V V
RV V R VR R R
RR
R RR
b. ( )
( )( )( )( )
( )( )
||1
2 2 3.225 2 4.90 mA / V4.90 2 || 5
2.031 4.90 0.5
m D Lv
m S
m
v v
g R RA
g Rg
A A
−=
+= − =
−= ⇒ = −
+
4.16
( )
( )( ) ( )( )
1 2
1 2
From Problem 4.14: 0.920 mA7.4 V1.92 mA/V
||||||
200 || 3001.92 3 || 3 2.88 0.9836200 || 300 2
2.83
D
DS
m
v m D LSi
v
IVg
R RA g R RR R R
A
===
⎛ ⎞= − ⎜ ⎟+⎝ ⎠
⎛ ⎞= − = −⎜ ⎟+⎝ ⎠= −
AC load line
Slope � �1
3��3�1
1.5 K
0.92
7.4 12
�
( ) ( )1 , 0.92 mA 0.92 1.5 1.381.5 k
2.76 V peak-to-peak output voltage swing
−Δ = ⋅ Δ Δ = ⇒ Δ = =Ω
⇒
D DS D DSi v i v
4.17 (a)
15 2 7.5 Kv m D
D D
A g RR R
= −− = − ⇒ =
(b)
( )( )( )
12 7.5
5 1 K1 2
m Dv
m S
SS
g RA
g R
RR
−=
+−
− = ⇒ =+
4.18
(a) 1
m Dv
m S
g RA
g R−
=+
(1) ( )
81 1
m D
m
g Rg
−− =
+
(2) 16 m Dg R− = −
Then ( )
168 1 mA/V1 1
16 K
mm
D
gg
R
= ⇒ =+
=
(b) ( )( )
( )1 16
101 1
0.6 K
vS
S
AR
R
−= − =
+=
4.19 a.
IDQ
VDS(sat) VDSQ
AC load line
Slope � �1
5 K
( )( )
( )Output Voltage Swing sat
1 1So 5 5
1 5 (10) 15 k
1.2 2 0.4 mA
DSQ DQ D GSQ
DSQ DS
DQ D GSQ GSQ TN
DQ D TN
D DQ DS DQ D TN
DQ DQ
DQ DQ DQ Q
V V I R VV V
V I R V V V
V I R V
I I V V I R Vk k
I I
I I I I
+
+
+
+
= − − −= −
⎡ ⎤= − + − −⎣ ⎦= − +
⎡ ⎤Δ = = ⋅ Δ = − +⎣ ⎦Ω Ω
⎡ ⎤= − +⎣ ⎦Ω= − = ⇒ = =
b. ( )( )
( )( )( )
( )( )
0
0
2 2 0.5 0.4 0.8944 /
1 1 250 k0.01 0.4|| ||
0.8944 10 ||10 || 250 4.38
m n DQ
DQ
v m D L
v
g K I mA V
rI
A g R R rA
λ
= = =
= = = Ω
= −= − ⇒ = −
4.20 (a)
( )( )
( )
( ) ( )( )( )
2
20.1 0.85 0.8
1.143 V1.143 5
38.6 K0.1
1 500 K0.02 0.1
1 0.1
40 500 37.04 10 K 1040 500 37.04
13.7 K
DQ n GSQ TN
GSQ
GSQ
S S
I D L o o
D L o
D DD L o
D D
D
I K V V
V
V
R R
V R R r r
R R r
R RR R rR R
R
= −
= −
=− − −
= ⇒ =
Δ = Δ = =
=
= = ⇒ =+ +
=
(b) ( )( )2 2 0.85 0.1
0.583 mA/V500 K
m n D
m
o
g K I
gr
= =
==
(c) ( )
( )( ) ( )( )0.583 13.7 40 500 0.583 10
5.83
v m D L o
v
A g R R r
A
= −
= − = −
= −
4.21 a.
( )( )( )
2
22 4 1
0.293 V 0.293 V (2) 0.146 k
D n GS TN
GS
GS S D S S S
I K V V
V
V V I R R R
= −
= − −
= − ⇒ = = = ⇒ = Ω
6 0.293 6.29310 6.293 1.85 k
2
D DS S
D D
V V V
R R
= + = + =−= ⇒ = Ω
b. ( )
( )( )( )( )( )( )( )
122 4 0.293 1 5.66 mA/V
5.66 1.85 22.98
1 5.66 0.146
m D Lv
m S
m n GS TN
m
v v
g R RA
g Rg K V Vg
A A
−=
+= −= − + =
−= ⇒ = −
+
c.
�1 1.107 K2 mA
6 10
�
�1 1.85��2 � 0.146
AC load line
Slope �
( )( ) ( )( )0
i
1.85 || 2 2 1.85 || 2 1.92 V1.92 0.645 0.645 V2.98
D
i
v i
v V
Δ = Δ = =
Δ = = ⇒ =
4.22 a.
( )( )
( )
( )
2
2
2
2
2
2.5 5 22.5
22.52
2.5 2.52 2
2.5 2.54 12 2
2 2.5 2.542 2
2 1.54 2.5
2
4 4 6 2.25
DSQ DD DQ D S
DQ S
DQS
GS SDQ n GS TN GS DQ S
S S
Sn TN
S S
S
S S
S S
S S
S
S
S
V V I R RI R
IR
V RI K V V V I RR R
RK V
R R
RR R
R RR R
RR
R
= − += − +
=+
− −= − = ⇒ = − =
+
⎡ ⎤−− =⎢ ⎥+ +⎣ ⎦
⎡ ⎤− =⎢ ⎥+ +⎣ ⎦
⎡ ⎤+ −=⎢ ⎥+ +⎣ ⎦
−=
+
− +( ) ( )
( ) ( )( )( )
2
2
2
2.5 2
9 26.5 11 0
26.5 26.5 4 9 112 9
0.5 k or 2.44 k
S S
S S
S
S
R R
R R
R
R
= +
− + =
± −=
= Ω Ω
But 2.44 k 1.37S GSR V= Ω ⇒ = − Cut off. 0.5 k , 1.0 mAS DQR I⇒ = Ω =
b. ( )
( )( )( )( )
( ) ( )
||1
2 2 4 1 4 /
4 2 || 21.33
1 4 0.5
m D Lv
m S
m n DQ
v v
g R RA
g R
g K I mA V
A A
−=
+
= = =
−= ⇒ = −
+
4.23 a.
( )5 55 6 10 5
DQ S SDQ DQ D
DQ S DQ
I R V I RI R I
= + + −= + + −
1. 410
5
DQS
S SDQ DQ D SGQ
IR
V V I R V
=+
= + − =
2. ( )1 10DQ SGQI V+ =
3. ( )22DQ p SGQI K V= −
4Choose 10 k 0.20 mA20S DQR I= Ω ⇒ = =
2 2
1 (0.2)(10) 3 V0.20 (3 2) 0.20 /
SGQ
P P
VK K mA V
= + == − ⇒ =
b. ( )( )
( )( )( ) ( )( )
20.20 3 2 0.20 mA
2 2 0.2 0.2 0.4 /
|| 0.4 10 ||10 2.0
DQ
m P DQ
v m D L v
I
g K I mA V
A g R R A
= − =
= = =
= − = − ⇒ = −
c.
2 2
4Choose 20 k 0.133 mA30
1 (0.133)(10) 2.33 V0.133 (2.33 2) 1.22 /
2 (1.22)(0.133) 0.806 mA/V(0.806)(10 10) 4.03
S DQ
SGQ
p p
m
v v
R I
VK K mA V
gA A
= Ω ⇒ = =
= + == − ⇒ =
= == − ⇒ = −
A larger gain can be achieved. 4.24 (a)
( )( )
( )
2
20.25 0.8 0.5
1.059 V3 1.059 7.76 K
0.251.059 1.5 0.441 V
0.441 310.2 K
0.25
DQ p SGQ TP
SGQ
SGQ
S S
D S SDQ
D D
I K V V
V
V
R R
V V V
R R
= +
= −
=−= ⇒ =
= − = − = −− − −
= ⇒ =
(b)
( )( )( )
( )( )2 2 0.8 0.25 0.8944 mA/V
0.8944 10.2 || 21.50
v m D L
m p DQ
v
v
A g R R
g K I
AA
= −
= = =
= −= −
(c) ( ) ( )|| 0.25 10.2 || 2 0.418
So 0.836 peak-to-peakO D L
O
V I R RV
Δ = Δ = =Δ =
4.25
( )
( )( )
( )
( )
( )( )
2
2
2
2
2.2 2 6 0.202 /
6 0.202 2.8 2.65
188 10 1.33 k 1.33
6
1
12.21
11 2.2 1.33
2.21 2.93 2.21
3.93 2.2
DQ n GSQ TN
m n DQ
n n
TN TN
DSQ DQ S D
S D S D
m D Lv
m S
D
D
D
DD
D
I K V V
g K I
K K mA V
V V V
V I R R
R R R R
g R RA
g R
RR
RRRR
= −
=
= ⇒ =
= − ⇒ = −
= − +1 −+ = = Ω ⇒ = −
= −+
⎛ ⎞⋅− ⎜ ⎟+⎝ ⎠− =+ −
+ − =+
−( )( )
( ) ( )( )( )( )( )
( )( )
2
2
2
11 1
22
2
1 2.23.93 1.73 2.2 2.22.2 0.47 3.93 0
0.47 0.47 4 2.2 3.931.23 k , 0.10 k
2 2.22.8 6 0.1 3.4 V
1 1 100 18 3.4 529 k
529 100 123 k529
D D D
D D D
D D
D D S
G GS S
G in DD
R R RR R R
R R
R R R
V V V
V R V RR RR RR
+ =+ − =
+ − =
− + += ⇒ = Ω = Ω
= + = + =
= ⋅ ⋅ = = ⇒ = Ω
= ⇒ = Ω+
4.26 a.
VSD(sat) VSDQ V1
( )( ) ( )
( ) ( ) ( )
( )
( )
1
1
2
9 ,
29
29 1.5 1.5
23.75 9
Set 0.1 2 k3.75 9 2 2.625 mA
SG SD SG TP
SDSDQ SD
SG SG TPSG TP
SG
SDQ SG SG DQ D
DQ p SG TP
D L
DQ DQ
V V V sat V VV V sat
V V sat
V V VV V
V
V V V I R
I K V VR R
I I
= + = +−
= +
+ − += + +
+= + −
= + = + −
= += = Ω= − ⇒ =
b. ( )( )
( )( )0
2 2 2 2.625 4.58 /
1 1 38.1 k0.01 2.625
m p DQ
DQ
g K I mA V
rIλ
= = =
= = = Ω
Open circuit. ( )
( )0||
4.58 2 || 38.1 8.70v m D
v v
A g R rA A
= −= − ⇒ = −
c. With RL ( )4.58 2 || 20 || 38.1 7.947 Change 8.66%v vA A= − ⇒ = − ⇒ =
4.27 (a)
( )( )
( )
2
20.5 0.25 0.8
0.614 10 0.614 18.8 K
0.50.614 3 2.386 V
2.386 1015.2 K
0.5
DQ p SGQ TP
SGQ
SGQ S
S S
D S SDQ
D D
I K V V
V
V V V
R R
V V V
R R
= +
= +
= =−= ⇒ =
= − = − = −− − −
= ⇒ =
(b)
( )( )( )( )
2 2 0.25 0.5 0.7071 mA/V
0.7071 15.210.7
v m D
m p DQ
v
v
A g R
g K I
AA
= −
= = =
= −= −
4.28
( )( )
( )( )10 20 1 10 k
v m D L
DSQ DD DQ S D
S D S D
A g R R
V V I R RR R R R
= −
= − += − + ⇒ + = Ω
Let 8 k , 2 kD SR R= Ω = Ω
( )( ) 2
10 8 20
1.75 / 2 2 1 0.766 /v m
m n DQ n n
A g
g mA V K I K K mA V
= − = −
= = = ⇒ =
( )( )( ) ( )
( )( )
2 2
11 1
22
2
1 2 2 V
1 0.766 2 3.14 V3.14 2 5.14
1 1 200 20 5.14 778 k
778 200 269 k778
S DQ S
DQ n GS TN GS GS
G GS S
G in DD
V I R
I K V V V VV V V
V R V RR RR RR
= = =
= − ⇒ = − ⇒ == + = + =
= ⋅ ⋅ ⇒ = ⇒ = Ω
= ⇒ = Ω+
4.29
( )( )( )( )
( )( )
( )( )
( )( )
0
0
0 0
0
0
0 0 0
4 500.995
1 1 4 50
1 1 50 0.249 k4
4 50 2.51 1 4 50 2.5
4 2.380.905
1 4 2.38
1 1 50 2.5 0.226 k4
mv v
m
om
m Sv
m S
v
Sm
g rA A
g r
R r Rg
g r RA
g r R
A
R r R Rg
= = ⇒ =+ +
= = ⇒ = Ω
= =+ +
= ⇒ =+
= = ⇒ = Ω
4.30
( )( )1
0.98 491
m L ov
m L o
m om o
m o
g R rA
g R r
g r g rg r
=+
= ⇒ =+
( )( )
( )( )
( )( )( )( )
Also 0.491
1
0.49
49 1 490.491 49 1 50
L om
m L o L o
m L o L om
L o
m L o
L o m L o
o o
R rgg R r R r
g R r R rgR r
g R rR r g R r
r r
⎛ ⎞⎜ ⎟+⎝ ⎠= =
+ ⎛ ⎞+ ⎜ ⎟+⎝ ⎠
=+ +
= =+ + +
50 K
0.98 mA/Vo
m
r
g
=
=
4.31 (a)
( )( )( )( )2 25
1 1 2 250.981 1 25 0.5 || 25
20.49 K
m ov
m o
v
o om
o
g rA
g rA
R rg
R
= =+ +
=
= = =
=
(b)
( )( )
( )( )
( )( )
|| 2 25 || 2 2 1.8521 || 1 2 25 || 2 1 2 1.8520.787
m o Lv
m o L
v
g r RA
g r RA
= = =+ + +
=
4.32
( )( )
( )( )( )
( )( )
|| 5 10 ||100 5 9.091 || 1 5 10 ||100 1 5 9.090.978
1 1 10 ||100 0.2 || 9.09095
0.196 K
m L ov
m L o
v
o L om
o
g R rA
g R rA
R R rg
R
= = =+ + +
=
= = =
=
4.33 a.
( )
( ) ( )
( )( )( )
( ) ( )( )( )
( )( )( )( )
2
1 2
2
2
2
2
2
396 10 2.42 V396 1240
10 2.42
7.58 2 4 4 4
8 31 24.4 0
31 31 4 8 24.42.78 V
2 8
2 2.78 2 1.21 mA
10 10 1.21 4 5
⎛ ⎞ ⎛ ⎞= = ⇒ =⎜ ⎟ ⎜ ⎟+ +⎝ ⎠⎝ ⎠− +
= = +
− = − +
− + =
± −= ⇒ =
= − ⇒ =
= − = − ⇒ =
G DD G
SGDQ p SG TP
S
SG SG SG
SG SG
SG SG
DQ DQ
SDQ DQ S SDQ
RV V VR R
VI K V V
R
V V V
V V
V V
I I
V I R V .16 V
b.
( )( )
( )( )( )
( )( )
( )( )
0
0
0
2 2 2 1.21 3.11 /
1 1 41.3 k0.02 1.21
1
3.11 4 4 41.30.856
1 3.11 4 4 41.3
m p DQ
DQ
m S Lv
m S L
v
g K I mA V
rI
g R R rA
g R R r
A
λ
= = =
= = = Ω
=+
= ⇒ =+
�
�R1��R2
RS
I0
Ii
Vi
gmVsgVsg
��
r0
RL
( )
( )( )
00
0
0
1 2
1
Sm sg
S L
isg
m S L
i i
R rI g V
R r R
VV
g R R r
V I R R
⎛ ⎞= − ⎜ ⎟⎜ ⎟+⎝ ⎠
−=
+
=
( )( )
( )( )( )( )
( )( )( )( )
1 2 00
001
3.11 396 1240 4 41.34 41.3 41 3.11 4 4 41.3
3.11 300 3.647 64.21 3.11 1.908 3.647 4
m Si
i S Lm S L
v
g R R R rIA
I R r Rg R R r
A
= = ⋅++
= ⋅++
= ⋅ ⇒ =+ +
0 0 01 1 4 41.3 0.295 k
3.11Sm
R R r Rg
= = ⇒ = Ω
4.34
( )( )
( )( )( )
( )( )
( )
0
0
0
0 0 0
2 2 1 1 2 /
1 1 100 k0.01 1
2 100 40.885
1 1 2 100 4
1 1 100 0.498 k2
m n Q
Q
m Lv v
m L
m
g K I mA V
rI
g r RA A
g r R
R r Rg
λ
= = =
= = = Ω
= = ⇒ =+ +
= = ⇒ = Ω
4.35 a.
( )( )
( )
( ) ( )
40.95
1 1 40.95 4 1 0.95 4.75 mA/V
122
4.75 2 0.030 4 47.0
mm Lv
m L m
m m
m n ox Q
gg RA
g R gg g
Wg C IL
W WL L
μ
= ⇒ =+ +
= − ⇒ =
⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⎛ ⎞= ⇒ =⎜ ⎟⎝ ⎠
b.
( )( )
122
4.75 2 0.030 60 3.13 mA
m n ox Q
Q Q
Wg C IL
I I
μ⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
= ⇒ =
4.36
( )( )
( )
( )( )
( )( )
2
2
0
5 5 2 1 V 1 V5 1 5 1.2 k
5
2 2 5 5 10 /
1 1 20 k0.01 5
DQ n GS TN
GS GS S GS
SDQ S S
S
m n DQ
DQ
I K V V
V V V VV
I R RR
g K I mA V
rIλ
= −
= + ⇒ = − ⇒ = − =− − += ⇒ = ⇒ = Ω
= = =
= = = Ω
( )( )
( )( )( )( )
0
01
10 20 1.2 20.878
1 10 20 1.2 2
m S Lv
m S L
v
g r R RA
g r R R
A
=+
= ⇒ =+
0 01 1|| || || 20 ||1.2 91.9
10S om
R r R Rg
= = ⇒ = Ω
4.37
( )( )
( )
0 0
100.90
1 1 100.90 10 1 0.90 0.90 mA/V
1 1 10 1 k0.90
mm Sv
m S m
m m
Sm
gg RAg R g
g g
R R Rg
= ⇒ =+ +
= − ⇒ =
= = ⇒ = Ω
( )( )
( )( )( )( )
With :|| 0.90 10 || 2
0.601 || 1 0.90 10 || 2
L
m S Lv v
m S L
Rg R R
A Ag R R
= = ⇒ =+ +
4.38
01
Sm
R Rg
=
Output resistance determined primarily by gm
( )( )
( )
( )
( )( )
( )( )
2
2
1Set 0.2 k 5 mA/V
2 5 2 4 1.56 mA
1.56 4 21.38 V, 1.38 V
1.38 54.09 k
1.565 4.09 2
0.8701 1 5 4.09 2
mm
m n DQ DQ DQ
DQ n GS TN
GS
GS S GS
S S
m S Lv v
m S L
gg
g K I I I
I K V V
VV V V
R R
g R RA A
g R R
= Ω ⇒ =
= ⇒ = ⇒ =
= −
= += − = − =
− −= ⇒ = Ω
= = ⇒ =+ +
4.39 a. ( ) ( )2 2 5 5 10 /
1 1 100 10
m p DQ
o om
g K I mA V
R Rg
= = =
= = ⇒ = Ω
b. Open-circuit gain 0
01m
vm
g rAg r
=+
But 0r = ∞ so 1.0vA =
With RL:
( )
110
0.50 0.50 10 1 0.5 0.1 k1 10
m Lv
m L
LL L
L
g RA
g RR
R RR
=+
= ⇒ = − ⇒ = Ω+
4.40
( )
0
0
0
1
min1
D DQS L
DQ S LDQ S L
S L
DQ S
S
L
i I vR R
I R Rv I R R
R RI R
vRR
−Δ = = ⋅ Δ
⋅Δ = − ⋅ = −
+
= −+
( )( )
( ) ( )( )
( ) ( )
0
1
1
min 1
m S Lv
im S L
DQ S L m S Li
m S L
DQi m S L
m
g R R vA
vg R R
I R R g R Rv
g R R
Iv g R R
g
= =+
⎡ ⎤− +⎣ ⎦=
⎡ ⎤= − +⎣ ⎦
4.41 (a)
( )3 1.5 0.25 6 KDD DSQ DQ S
S S
V V I RR R
= += + ⇒ =
( )( )
2
2
1.5 V
0.25 0.5 0.4
1.107 V
S
DQ n GSQ TN
GSQ
GSQ
V
I K V V
V
V
=
= −
= −
=
( )( )
2
1 2 1
1 21
1.107 1.5 2.607 V
1
12.607 300 3 345.2 K 2291 K
G GSQ S
G DD L DD
V V V
RV V R VR R R
R RR
= + + + =
⎛ ⎞= = − −⎜ ⎟+⎝ ⎠
= ⇒ = ⇒ =
(b)
( )( )
( )( )( )( )
( )
2 2 0.5 0.25 0.7071 mA/V1
0.7071 60.809
1 0.7071 6
1 1 6 1.414 || 60.7071
1.14 K
m Sv m n DQ
m S
v v
o Sm
o
g RA g K Ig R
A A
R Rg
R
= = = =+
= ⇒ =+
= = =
=
4.42
( )( )5 420
1 110 10 0.2 ||105
0.196 K
v m D
v
im
i
A g RA
Rg
R
= + ==
= = =
=
4.43 a.
( )
( )( )( )
( ) ( )( )( )
( )( )( )( )
2
2
2
2
2
55
5 10 3 2 1
30 59 25 0
59 59 4 30 251.35 V
2 30
3 1.35 1 0.365 mA
10 0.365 5 10 4.53 V
GS DQ S
GSDQ n GS TN
S
GS GS GS
GS GS
GS GS
DQ DQ
DSQ DSQ
V I RVI K V V
R
V V V
V V
V V
I I
V V
+ =−
= = −
− = − +
− + =
± −= ⇒ =
= − ⇒ =
= − + ⇒ =
b. ( )( )
( )( )0 0
2 2 3 0.365 2.093 /
1 10 0.365
m n DQ m
DQ
g K I g mA V
r rIλ
= = ⇒ =
= = ⇒ = ∞
c. ( ) ( )( )2.093 5 4 4.65v m D L vA g R R A= = ⇒ = 4.44 a.
( )( )( )
( )( )( )
2
20.75 0.5 1 2.225 V5 2.2255 3.70 k
0.7510
6 10 0.75 3.70 1.63 k
DQ p SG TP
SG SG
DQ S SG S S
SDQ DQ S D
D D
I K V V
V V
I R V R R
V I R RR R
= +
= − ⇒ =−= + ⇒ = ⇒ = Ω
= − += − + ⇒ = Ω
b.
( )( )
1
2 2 0.5 0.75 1.225 /
1 0.816 k1.225
1.63 k
im
m p DQ
i i
o D o
Rg
g K I mA V
R R
R R R
=
= = =
= ⇒ = Ω
= ⇒ = Ω
c.
[ ]
( )( )( ) ( )
0
0
0 0
0 0 0
1/
1.63 3.701.63 2 3.70 0.816
0.368 1.84sin
1.84 2 sin 3.68sin mV
SDi
D L S m
i
i
L
RRi iR R R g
i i
i i i t A
v i R t v t
ω μ
ω ω
⎛ ⎞⎛ ⎞= ⋅⎜ ⎟⎜ ⎟⎜ ⎟+ +⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟+ +⎝ ⎠⎝ ⎠
= = =
= = ⇒ =
4.45 a.
( )( )
( )
2
25 4 2 3.12 V
10 5 12 3.12 0.224 k
DQ n GS TN
GS GS
DQ D DSQ GS
D D
I K V V
V VV I R V V
R R
+
= −
= − ⇒ == + −
= + − ⇒ = Ω
b.
( )( )2 2 4 5 8.944 /
1 1 0.112 k8.94
m n DQ m
i im
g K I g mA V
R Rg
= = ⇒ =
= = ⇒ = Ω
c. ( ) ( )( )8.944 0.224 2 1.80v m D L vA g R R A= = ⇒ = 4.46 a.
( )( )
( )( )( )
2
23 2 2 3.225 V
10 3.225 2.26 k3
2010 20 3 2.26 1.07 k
DQ p SG TP
SG SG
DQ S SG
S S
SDQ DQ S D
D D
I K V V
V VV I R V
R R
V I R RR R
+
= +
= − ⇒ == +
−= ⇒ = Ω
= − += − + ⇒ = Ω
b. ( )
( )( )( )( )2 2 2 3 4.90 /
4.90 1.07 ||10 4.74
v m D L
m p DQ
v v
A g R R
g K I mA V
A A
=
= = =
= ⇒ =
4.47 a.
( )( ) ( ) ( ) ( )
( )( )
( )( )
2
2
/5 / 25 0.5 / 12.5
/
1 30 12.5 375 A/V230 0.5 15 /
Dv D D
L
D n oxD
L
W LA W L W L
W L
WK CL
K A V
μ μ
μ
= = ⇒ = ⇒ =
⎛ ⎞= = =⎜ ⎟⎝ ⎠
= =
Transition point:
( ) ( )
( ) ( )
( ) ( ) ( )
3752 10 2 215
1 5 10 2 2 5 23.33 V and 1.33 V3.33 2 2 2.67 V
2
DGSD TND DD TNL GSD TND
L
GSD GSD
GSD
GSD DSD
GSQ GSQ
Kv V V V v VK
v v
vv v
V V
− = − − −
− = − − −
+ = − + += =
−= + ⇒ =
VDSQ
VGSQ
�GSD
�O
8
2
1.33
3.33
Q-point
b.
( )( )
2
20.375 2.67 2 0.166 mA
8 1.33 1.33 4.67 V2
DQ D GSQ TND
DQ DQ
DSQ DSQ
I K V V
I I
V V
= −
= − ⇒ =
−= + ⇒ =
4.48 (a) Transition point: Load: 10 2 8 OtB DD TNLV V V V= − = − = Driver:
( ) ( ) ( ) ( )( )( ) ( )( )( )
2 2
2 30 0 0
1 1
0.5 0.02 0.1 4 1 0.02 10
D Ot A D Ot A L TNL L DD Ot A
t A t A t A
K v v K V V v
v v v
λ λ⎡ ⎤ ⎡ ⎤+ = − + −⎢ ⎥ ⎣ ⎦⎣ ⎦⎡ ⎤⎡ ⎤+ = + −⎣ ⎦ ⎣ ⎦
We have 3 20 0 00.01 0.5 0.008 0.48 0t A t A t Av v v+ + − =
Therefore 0 0.963 Vt Av =
Now 08 0.963 0.963 4.48 V
2Q DSQv V−= + = =
( ) ( ) ( ) ( )( )( ) ( )( )( ) ( ) ( )( )
2 2
2
Then 1 1
0.5 1.2 1 0.02 4.48 0.1 4 1 0.02 10 4.48 which yields 2.103 V
D GSD TND D OQ L TNL L DD OQ
GSD GSD
K V V v K V V v
V V
λ λ⎡ ⎤⎡ ⎤− + = − + −⎣ ⎦ ⎣ ⎦⎡ ⎤ ⎡ ⎤− + = + − =⎣ ⎦⎣ ⎦
b.
( ) ( )( ) ( )( )( )
2
2
1
0.5 2.103 1.2 1 0.02 4.48
So 0.444 mA
DQ D GSD TND D OQ
DQ
DQ
I K V V v
I
I
⎡ ⎤= − +⎣ ⎦⎡ ⎤= − +⎣ ⎦
=
λ
c.
( )( )( ) ( )( )
( ) ( )( )
0 0
0 0
1 1 112.6 k0.02 0.444
2 2 0.5 2.103 1.2 0.903 /Then 0.903 112.6 112.6 or 50.8
D LDQ
mD D GSD TND mD
v mD v
r rI
g K V V g mA VA g r D r L A
λ= = = = Ω
= − = − ⇒ == − = − = −
4.49
( )
( )
( ) ( )( )
2
2 2
,0 when 1.5 1.5
0.8 3 1.5 0.356 /
2 2 0.356 3 1.5 0.936
D n GS TN DS GS
D DS TN
n n
Do n DS TN o
DS o
I K V V V VI V V V V
K K mA VdI Ig K V V R kdV R
= − == = ⇒ =
= − ⇒ =
= = = − = − ⇒ = Ω
4.50 a.
( ) ( ) ( )( )
( ) ( )( )
22
2 2
20
0 0
0.5 0 1
0.5
0.5 0.030 10 1
0.5 9 4.92 V0.030
DQ nD GS TND
DQ
DQ nL GSL TNL nL DD O TNL
I K V V
I mA
I K V V K V V V
V
V V
= − = − −
=
= − = − −
= − −
= − ⇒ =
b.
( ) ( )
( )
( )
( )( )
2 2
//
500 4.0830
DD DL
nD i TND nL DD o TNL
nDi TND DD o TNL
nL
nDo DD TNL i TND
nL
o nD Dv
i nL L
v v
I I
K V V K V V V
K V V V V VK
KV V V V VK
W LdV KAdV K W L
A A
=
− = − −
− = − −
= − − −
= = − = −
= − ⇒ = −
4.51 (a)
( ) ( )( )( )
( )( )( )
2 2
2
2
2
0.1 4 1 0.9
0.9 1 1 1.95 V1.95 4 5.95 V
DQ L GSL TNL L DSL TNL
D
DQ D GSD TND
GSD GSD
GG GSD DSL GG
I K V V K V V
I mA
I K V V
V VV V V V
= − = −
= − =
= −
= − ⇒ == + = + ⇒ =
b.
( ) ( )
( )
( )
2 2
1
/ 11 / 1 /
DD DL
D GSD TND L GSL TNL
DGG i o TND o TNL
L
D Do GG i TND TNL
L L
D Lov v
i D L L D
I I
K V V K V V
K V V V V V VK
K KV V V V VK K
K KdVA AdV K K K K
=
− = −
+ − − = −
⎛ ⎞+ = + − +⎜ ⎟⎜ ⎟
⎝ ⎠
= = ⇒ =+ +
(c) From Problem 4.49.
( )
( )( )( )( )
( )( )
( )( )( )( )
12
1 1.67 2 0.1 4 1
2 2 1 0.9 1.90 /
|| 1.90 1.67 || 40.691
1 || 1 1.90 1.67 || 4
LDL DSL TNL
m D DQ
m LD Lv v
m LD L
RK V V
k
g K I mA V
g R RA A
g R R
=−
= = Ω−
= = =
= = ⇒ =+ +
4.52 a. From Problem 4.51.
( )( )
( )( )( )( )
|| 1.90 1.67 ||101 || 1 1.90 1.67 ||100.731
m LD Lv
m LD L
v
g R RA
g R RA
= =+ +
=
b.
0
0
1 1 1.67 0.526 ||1.671.90
0.40 k
LDm
R Rg
R
= = =
= Ω
4.53
( )
( )( )
21
1 1 1
85 50 2.125 mA/V2
2 2 2.125 0.1 0.9220
n
m n D
K
g K I
⎛ ⎞= ⇒⎜ ⎟⎝ ⎠
= = =
( )( )
( )( )
11 1
22 2
1 1 200 K0.05 0.1
1 1 133.3 K0.075 0.1
oD
oD
rI
rI
λ
λ
= = =
= = =
( ) ( )( )1 1 2|| 0.922 200 ||133.373.7
v m o o
v
A g r rA
= − = −= −
4.54
( )
( )( )
21
1 1 1
40 50 1.0 mA/V2 2
2 2 1 0.1 0.6325 mA/V
pp
m p D
k wKL
g K I
′ ⎛ ⎞ ⎛ ⎞= = ⇒⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= = =
( )( )
( )( )
11 1
22 2
1 1 133.3 K0.075 0.1
1 1 200 K0.05 0.1
oD
oD
rI
rI
λ
λ
= = =
= = =
( ) ( )( )1 1 2 0.6325 133.3 || 200
50.6v m o o
v
A g r r
A
= − = −
= −
4.55 (a)
( )
( )( )
( )( )
( )( )
21
1 1 1
11 1
22 2
85 50 2.125 mA/V2
2 2 2.125 0.1 0.922 mA/V
1 1 200 K0.05 0.1
1 1 133.3 K0.075 0.1
n
m n D
oD
oD
K
g K I
rI
rI
λ
λ
⎛ ⎞= ⇒⎜ ⎟⎝ ⎠
= = =
= = =
= = =
(b)
( ) ( )( )( )( )
11
11
1
11 1 2
1 1 1.085 K0.922
1.085 0.9560.050 1.085 0.050
0.956 0.922 200 133.3
70.5
im
igs i i i
i
gsv m o o
i
v
Rg
RV V V V
RV
A g r rV
A
= = =
⎛ ⎞ ⎛ ⎞= − = − = −⎜ ⎟ ⎜ ⎟+ +⎝ ⎠⎝ ⎠
= − ⋅ = +
=
(c) 1
1 10.05 0.05 1.135 K0.922i i
m
R Rg
= + = + ⇒ =
(d) 1 2 200 133.7 80 Ko o o oR r r R≈ = ⇒ ≈
4.56 (a)
( )( )( )( )
( )( )
1 1
2 2
1 2
2 2 2 0.1 0.8944 mA/V
2 2 2 0.1 0.8944 mA/V
1 1 100 K0.1 0.1
m n D
m p D
o oD
g K I
g K I
r rIλ
= = =
= = =
= = = =
(b) The small-signal equivalent circuit
gm1Vi
gm2Vsg2
rO1
rO2
Vsg2
�
�
Vi
Vo
�
�
(1) 2 21 2 2
1 2
0sg sg om i m sg
o o
V V Vg V g V
r r−
+ + + =
(2)
( )
22 2
2
2 22 2
2 2
1 1 1
1 1 1 0.8944 0.0331750 100 100
o sgom sg
o o
o sg mo o o
o sg sg o
V VVg V
r r
V V gr r r
V V V V
−+ =
⎛ ⎞ ⎛ ⎞+ = +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞+ = + ⇒ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
(1)
( )
( )
1 2 21 2 2
1 1
1 10.8944 0.03317 0.8944100 100 100
0.8944 0.01 0.03033
44
om i sg m
o o o
oi o
i o
o
i
Vg V V g
r r r
VV V
V VVV
⎛ ⎞+ + + =⎜ ⎟
⎝ ⎠⎛ ⎞+ + + =⎜ ⎟⎝ ⎠
= −
= −
(c) For output resistance, set 0.iV = gm2Vsg2
Vsg2
rO2
rO1
RO
rO
�
�
����
Ix
Vx
(1) 22 2
2
x sgxm sg x
o o
V VVg V I
r r−
+ = +
(2) 2 22 2
1 2
0sg sg xm sg
o o
V V Vg V
r r−
+ + =
(2)
( )
2 21 2 2
2
2
1 1
1 10.8944100 100 100
0.010936
xsg m
o o o
xsg
sg x
VV gr r r
VV
V V
⎛ ⎞+ + =⎜ ⎟
⎝ ⎠⎛ ⎞+ + =⎜ ⎟⎝ ⎠=
(1)
( )
( )
2 22 2
1 1 1
1 1 10.010936 0.894450 100 1000.03 0.0098905
49.7 K
x x sg mo o o
x x x
x x
xo
x
I V V gr r r
I V V
I VVRI
⎛ ⎞ ⎛ ⎞= + − +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞= + − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= −
= =
4.57 a.
( )( )
( )( )
( )( )
( )( )( )
21 1 1 1
21 1
1 1 1
1 1 1
21
1 2 1
11
22
2
1 1 1 1
1 1
0.4 0.1 2 4 V0.4 4 1.6 V
1.6 4 5.6 V
1
15.6 200 10 357 k
357 200 455 k357
4 10 0.4 4
D GS TN
GS GS
S D S
G S GS
G DD in DD
DS DD D S D
D D
I K V V
V VV I RV V V
RV V R VR R R
RR
R RR
V V I R RR R
= −
= − ⇒ == = == + = + =
⎛ ⎞= = ⋅ ⋅⎜ ⎟+⎝ ⎠
= ⇒ = Ω
= ⇒ = Ω+= − +
= − + ⇒ =
( )( )( )
( )
( )( )( )
12
2 2 2 22
2 2
2 1 2
2 2
2 2 2 2
2 2
11 k
10 0.4 11 5.6 V
2 1 2 3.41 V5.6 3.41 9.01
10 9.01 0.495 k2
5 10 2 0.495 2.01 k
D
D SG TP
SG SG
S D SG
S S
SD DD D S D
D D
V
I K V V
V VV V V
R R
V V I R RR R
Ω
= − =
= +
= − ⇒ == + = + =
−= ⇒ = Ω
= − += − + ⇒ = Ω
b.
�
�
R1��R2
V0
Vi Vgs2gm1Vgs1 gm2Vsg2
Vgs1
�
�
��
RD1
RD2
( )( )
( )( )( )( )
( )( )( )( )
0 2 2 2 2 2 1 1 1
1
01 2 1 2
1 1 1
2 2 2
2 2 0.1 0.4 0.4 /
2 2 1 2 2.828 /
0.4 2.828 11 2.01 25.0
m sg D m D m gs D
gs i
v m m D Di
m D
m D
v v
V g V R g R g V R
V VVA g g R RV
g K I mA V
g K I mA V
A A
= =
=
= =
= = =
= = =
= ⇒ =
c.
VSD(sat)
4
5 10
2
( )
( )( ) ( )
2 2 22
2 2 2
sat
sat
SD SG Th
DD D D S
DD p D S SD
V V VV I R RV K R R V
= += − += − +
So ( )( ) ( ) ( )
( ) ( )
( ) ( )( )( )
( )
2
2
1 2.01 0.495 sat sat 02.50 sat sat 10 0
1 1 4 2.501 10sat
2 2.501sat 1.81 V
5 1.81 3.19 Max. output swing = 6.38 V peak-to-peak
SD SD DD
SD SD
SD
SD
V V VV V
V
V
+ + − =+ − =
− ± +=
=− = ⇒
4.58 a.
10VSD2(sat)
10
2.5� 4 mA
( ) ( ) ( ) ( )( )( ) ( ) ( )
( ) ( )
( ) ( )( )( )
( )( )
22 2 2 2 2 2 2 2
22 2
22 2
2
2 2
2 2 2 2
2
1 2 0.5 sat sat 10 02.5 sat sat 10 0
1 1 4 2.5 10sat 1.81 V
2 2.510 1.81 1.81 5.91 V
2
5.91 10 2 0.5
SD DD D D S DD p D S SD
SD SD
SD SD
SD
SDQ SDQ
SDQ DD DQ D S
DQ DQ
V sat V I R R V K R R V satV V
V V
V
V V
V V I R RI I
= − + = − ++ + − =
+ − =
− ± += =
−= + ⇒ =
= − += − + ⇒
( )( )( )
( )( )
( )( )( )( )( )
2
22
2 2 2 22
2 2
1 2 2
1 1
21 1 1 1
21 1
1
1 1 1
1.64 mA10 1.64 0.5 9.18 V
1.64 1 2 3.28 V9.18 3.28 5.90 V
10 5.90 10.25 k0.4
0.4 0.1 2 4 V0.4 1 0.4 V
0.4
S
DQ p SG TP
SG SG
D S SG
D D
DQ n GS TN
GS GS
S
G S GS
V
I K V V
V VV V V
R R
I K V V
V VVV V V
== − =
= +
= − ⇒ == − = − =
−= ⇒ = Ω
= −
= − ⇒ == == + =
( )( )
21
1 2 1
11
22
2
4 4.4 V
1
14.4 200 10 455 k
455 200 357 k455
G DD m DDRV V R V
R R R
RR
R RR
+ =
⎛ ⎞= = ⋅ ⋅⎜ ⎟+⎝ ⎠
= ⋅ ⇒ = Ω
= ⇒ = Ω+
b.
( )( )( )( )
2
2 2
1 1
1.64 mA
10 1.64 2 0.5 5.90 V
10 0.4 10.25 1 5.50 V
DQ
SDQ SDQ
DSQ DSQ
I
V V
V V
=
= − + ⇒ =
= − + ⇒ =
(c) ( )( )( )( )( )( )( )( )
1 1 1
2 2 2
1 2 1 2
2 2 0.1 0.4 0.4 /
2 2 1 1.64 2.56 /
0.4 2.56 10.25 2 21.0
m n DQ
m p DQ
v m m D D v
g K I mA V
g K I mA V
A g g R R A
= = =
= = =
= = ⇒ =
4.59 a.
( )
( )( )( )
2 2 2 2
22
2 2 2 22
2 2
1 2 2
7 10 60.5 mA
0.5 0.2 0.8 2.38 V3 2.38 5.38
DSQ DD DQ S DQ
DQ
DQ GS TN
GS GS
S S GS
V V I R II
I K V V
V VV V V
= = − = −=
= −
= − ⇒ == + = + =
( )( )( )
11
1
21 1 1 1
21 1
1 1 1
5.38 0.269 mA20
0.269 0.2 0.8 1.96 V5.38 1.96 7.34 V
SDQ
S
DQ GS TN
GS GS
G S GS
VIR
I K V V
V VV V V
= = =
= −
= − ⇒ == + = + =
( )( )
21
1 2 1
11
22
2
1
17.34 400 10 545 k
545 400 1.50 M545
G DD in DDRV V R V
R R R
RR
R RR
⎛ ⎞= = ⋅ ⋅⎜ ⎟+⎝ ⎠
= ⇒ = Ω
= ⇒ = Ω+
b.
( )( )1 2
1 1
0.269 mA, 0.5 mA
10 0.269 20 4.62 VDQ DQ
DSQ DSQ
I I
V V
= =
= − ⇒ =
c.
( )( )( )( )
( )( )( )( )
( )( )( )( )
( )( )
1 1 2 2
1 1 2 2
1 1 1
2 2 2
1 1
2 2 0.2 0.269 0.464 /
2 2 0.2 0.5 0.6325 /
0.464 20 0.6325 61 0.464 20 1 0.6325 60.9027 0.7915
0.714
m S m Sv
m S m S
m DQ
m DQ
v
v
g R g RAg R g R
g K I mA V
g K I mA V
A
A
= ⋅+ +
= = =
= = =
= ⋅+ +
== =
gm1Vgs1 gm2Vgs2
����
RS1 RS2
Vgs1
Vx
Vgs2��
��
Ix
0 22
1 1 6 1.581 6 1.25 k0.6325S o
m
R R Rg
= = = ⇒ = Ω
4.60 (a)
( )
( )( )( )
( ) ( )( )( )
211 1 1 1
2
21 1 1
21 1
2
1 1
10
10 4 10 4 4
40 159 150 0
159 159 4 40 1502.435 V
2 40
GSDQ n GS TN
S
GS GS GS
GS GS
GS GS
VI K V V
R
V V V
V V
V V
−= = −
− = − +
− + =
± −= ⇒ =
( )( )( )( )
( )( )
21 1
1 1
2
2 2
4 2.435 2 0.757 mA
20 0.757 10 12.4 V
Also 0.757 mA
20 0.757 10 5 8.65 V
DQ DQ
DSQ DSQ
DQ
DSQ DSQ
I I
V V
I
V V
= − ⇒ =
= − ⇒ =
=
= − + ⇒ =
(b) ( ) ( )1 2 1 22 2 4 0.757 3.48 /m m DQ m mg g KI g g mA V= = = ⇒ = = c.
Vgs2
gm2Vgs2
gm1Vgs1Vgs1
V0Vi
RS1 RD
RG
RL
RS2
����
�
�
�
�
( )( )( )( )
( ) ( )( )( ) ( ) ( )
( )( ) ( )
0 2 2
2 1 1 2 2 1 2
1 2 1 2
2 2 2 1 2 1 2 1 2
2 2 2 1 2 1 2 1 2 1 1 2
1 1 22
2 1 2 1 1 2
1 20
1
m gs D L
gs m gs m gs S S
i gs gs gs i gs
gs m gs S S m i gs S S
gs m gs S S m gs S s m i S S
m i S Sgs
m S S m S S
m mv
i
V g V R R
V g V g V R R
V V V V V V
V g V R R g V V R R
V g V R R g V R R g V R R
g V R RV
g R R g R R
g gVA
V
= −
= − −
= − ⇒ = +
+ = − +
+ + = −
−=
+ +
= =( )( )
( )( )( ) ( )( )
( )( )
1 2
1 2 1 2
2
1
3.48 10 10 5 22.42
1 3.48 3.48 10 10
S S D L
m m S S
v v
R R R Rg g R R
A A
+ +
= ⇒ =+ +
4.61 a.
( )( )( )
( )
( ) ( )
12
12
1 1
3 31 3
1 2 3
3 mA5 3 1.2 5 1.4 V
3 2 1 2.225 V2.225 1.4 0.825 V
5 0.825 5 82.5 k500
DQ
S DQ S
DQ GS TN
GS GS
G GS S
G
IV I R
I K V V
V VV V V
R RV R
R R R
== − = − = −
= −
= − ⇒ == + = − =
⎛ ⎞ ⎛ ⎞= ⇒ = ⇒ = Ω⎜ ⎟ ⎜ ⎟+ + ⎝ ⎠⎝ ⎠
( ) ( )
1 1 1
2 1
2 3 2 32
1 2 3
2 3 2
1.4 2.5 1.1 V1.1 2.225 3.325 V
5 3.325 5500
332.5 250 k
D S DSQ
G D GS
G
V V VV V V
R R R RV
R R RR R R
= + = − + == + = + =
⎛ ⎞+ +⎛ ⎞= ⇒ =⎜ ⎟ ⎜ ⎟+ + ⎝ ⎠⎝ ⎠+ = ⇒ = Ω
1 1
2 1 2
500 250 82.5 167.5 k
1.1 2.5 3.6 V5 3.6 0.467 k
3
D D DSQ
D D
R R
V V V
R R
= − − ⇒ = Ω
= + = + =−= ⇒ = Ω
b.
( )( )( )( )
1
1 2 2 2 3 4.90 /
4.90 0.467 2.29
v m D
m n DQ
v v
A g R
g K I mA V
A A
= −
= = =
= − ⇒ = −
4.62 a.
( )( )( )
( )( )
( )( )
1 12
1 12
1 1
1 1 1 3 3 3
1 1 1
2 1
2 3
2 3 2
2 1 2
10 5 2 10 0
5 4 1.5 2.618 V2.618 V 0.1 26.2 k
0 3.5 3.5V3.5 2.62 6.12V
0.161.2 k 35 k
S DQ S S
DQ GS TN
GS GS
G GS S
D S DSQ
G D GS
D D DSQ
V I R V
I K V V
V VV V V IR R R
V V VV V V
R RR R R
V V V
= − = − ⇒ =
= −
= − ⇒ == + = = = ⇒ = Ω
= + = + == + = + =
= ++ = Ω ⇒ = Ω
= + =
1 1
3.5 3.5 7.0V10 7 0.6 k
510 6.12 38.8 k
0.1
D DR R
R R
+ =−= ⇒ = Ω
−= ⇒ = Ω
b.
( )( )( )( )
1
1 2 2 4 5 8.944 /
8.944 0.6 5.37
v m D
m n DQ
v v
A g R
g K I mA V
A A
= −
= = =
= − ⇒ = −
4.63 a.
( )
( )
( )
2
2
1
4 6 13
43 1 0.551 V6
6 10 4 1 k
GSDQ DSS
P
GS
GS GS
DSQ DD DQ D
D D
VI IV
V
V V
V V I RR R
⎛ ⎞= −⎜ ⎟
⎝ ⎠
⎛ ⎞= −⎜ ⎟⎜ ⎟−⎝ ⎠
⎡ ⎤= − − ⇒ = −⎢ ⎥
⎣ ⎦= −
= − ⇒ = Ω
b.
( )( )
( )( )0 0
2 62 0.5511 1 3.265 mA/V3 3
1 1 25 k0.01 4
DSS GSm m
P P
DQ
I Vg gV V
r rIλ
⎛ ⎞ −⎛ ⎞= − = − ⇒ =⎜ ⎟ ⎜ ⎟− −⎝ ⎠⎝ ⎠
= = ⇒ = Ω
c. ( ) ( ) ( )0 3.265 25 1 3.14v m D vA g r R A= − = − ⇒ = −
4.64
( )
( )
( )( )
( ) ( )
( ) ( ) ( )( )
1 22
2
1 2
2
2
2
2
2
0
1
1 0
2 0.1 0.25 1 0
20.7 1 0
2 2
0.175 1.7 0.7 0
1.7 1.7 4 0.175 0.70.4314 V
2 0.175
2
GS DQ S S
GSDQ DSS
P
GSGS DSS S S
P
GSGS
P
VGS GSGS
GS GS
GS GS
m
V I R R
VI IV
VV I R R
V
VV
V
VV
V V
V V
g
+ + =
⎛ ⎞= −⎜ ⎟
⎝ ⎠
⎛ ⎞+ + − =⎜ ⎟
⎝ ⎠
⎛ ⎞+ + − =⎜ ⎟
⎝ ⎠⎛ ⎞⎜ ⎟+ − + =⎜ ⎟− −⎝ ⎠+ + =
− ± −= ⇒ = −
=( )
( ) ( )( )( )( )
( )( ) ( )
1
00 0
2 2 0.4311 1 1.569 mA/V2 2
1.569 8 43.62
1 1 1.569 0.1
/ 503.62 45.2/ 4
DSS GSm
P P
m D Lv v
m S
L Gi i
i i G i L
I Vg
V V
g R RA A
g R
v Ri v RA A
i v R v R
⎛ ⎞ −⎛ ⎞− = − ⇒ =⎜ ⎟ ⎜ ⎟− −⎝ ⎠⎝ ⎠− −
= = ⇒ = −+ +
⎛ ⎞= = = ⋅ = − ⇒ = −⎜ ⎟⎝ ⎠
4.65
( )( )( )
( ) ( )
2
2
4 mA2
10 V2
10 20 4 2.5 k2 V 4 0.5 k , 2.0 k
1
44 8 1 4.2 1 1.23 V4.2 8
2 1.23
DSSDQ
DDDSQ
DSQ DD DQ S D
S D S D
S DQ S S S D
GSDQ DSS
P
GSGS GS
G S GS
G
II
VV
V V I R RR R R R
V I R R R R
VI IV
V V V
V V V
V
= =
= =
= − += − + ⇒ + = Ω= = = ⇒ = Ω = Ω
⎛ ⎞= −⎜ ⎟
⎝ ⎠
⎛ ⎞ ⎛ ⎞= − ⇒ = − − ⇒ = −⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠
= + = −
( ) ( )2 22 1
1 2
0.77 V 20 20 3.85 k , 96.2 K100
R R R RR R
⎛ ⎞ ⎛ ⎞= = = ⇒ = Ω = Ω⎜ ⎟ ⎜ ⎟+ ⎝ ⎠⎝ ⎠
4.66 a.
( ) ( )
( )( )
2
2
2
1 2 1
11
5 mA2
12 6 V2 2
12 6 1.2 k5
1
55 10 1 5 1 1.464 V5 10
6 1.464 4.536 V
1
14.536 100 12 26
DSSDQ
DDDSQ
S S
GSDQ DSS
P
GSGS GS
G S GS
G DD in DD
II
VV
R R
VI I
V
V V V
V V V
RV V R VR R R
RR
= =
= = =
−= ⇒ = Ω
⎛ ⎞= −⎜ ⎟
⎝ ⎠
⎛ ⎞ ⎛ ⎞= − ⇒ = − − ⇒ = −⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠
= + = − =
⎛ ⎞= = ⋅ ⋅⎜ ⎟+⎝ ⎠
= ⇒ =
22
2
5 k
265 100 161 k265
R RR
Ω
= ⇒ = Ω+
b.
( )( )
( )( )( )
( )( ) ( )
( ) ( )
0
0
0
0 0
2 102 1.461 1 2.83 mA/V5 5
1 1 20 k0.01 5
1
2.83 20 1.2 0.50.495
1 2.83 20 1.2 0.5
1 1 1.2 0.353 1.2 0.273 k2.83
DSS GSm m
P P
DQ
m s Lv
m S L
v v
Sm
I Vg g
V V
rI
g r R RA
g r R R
A A
R R Rg
λ
⎛ ⎞ −⎛ ⎞= − = − ⇒ =⎜ ⎟ ⎜ ⎟− −⎝ ⎠⎝ ⎠
= = = Ω
=+
= ⇒ =+
= = = ⇒ = Ω
4.67 a.
( )
( )
( )( )
( )
( ) ( )( )( )
2
1 2
2
2
2
2
2
110 10 5.5 V110 90
101
10 5.5 2 5 11.75
4.5 10 1 1.143 0.3265
3.265 12.43 5.5 0
12.43 12.43 4 3.265 5.50.
2 3.265
G DD
G GS GSDQ DSS
S P
GSGS
GS GS GS
Gs GS
GS GS
RV VR R
V V VI I
R V
VV
V V V
V V
V V
⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟+ +⎝ ⎠⎝ ⎠
− − ⎛ ⎞= = −⎜ ⎟
⎝ ⎠
⎛ ⎞− + = −⎜ ⎟⎝ ⎠
+ = − +
− + =
± −= ⇒ =
( )
( )( )
2
511 V
0.5112 1 1.00 mA1.75
10 1.00 5 5.0 V
DQ DQ
SDQ SDQ
I I
V V
⎛ ⎞= − ⇒ =⎜ ⎟⎝ ⎠
= − ⇒ =
b. ( )
( )( )
( )( )( )( )
( )( )
( )
00
1 2
2 22 0.5111 1 1.618 mA/V1.75 1.75
1.618 5 100.844
1 1 1.618 5 10
//90 110 49.5 k
49.50.844 4.1810
DSS GSm m
P P
m S Lv v
m S L
L ii v
i i i L
i
i i
I Vg g
V V
g R RA A
g R R
v Ri RA Ai v R R
R R R
A A
⎛ ⎞ ⎛ ⎞= − = − ⇒ =⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠
= = ⇒ =+ +
⎛ ⎞= = = ⋅⎜ ⎟
⎝ ⎠= = = Ω
⎛ ⎞= ⇒ =⎜ ⎟⎝ ⎠
c.
1.0
5.0 10
�1 3.33 K
�
�1 5��10
AC load line
Slope �
( )( )1.0 mA3.33 1.0 3.33 V
d
sd
ivΔ =
= =
Maximum swing in output voltage = 6.66 V peak-to-peak 4.68
( )( )
( )
( )
2
2
1
44 8 1 4 1 1.17 V4 8
7.5 20 4 3.125 k2 82 1.171 1 2.83 mA/V
4 43.125
13 1
GSDQ DSS
P
GSGS GS
SDQ DD DQ S D
S D S D
DSS GSm m
P P
S D
m Dv
m S
m S
VI IV
V V V
V V I R RR R R R
I Vg gV V
R Rg RAg R
g R
⎛ ⎞= −⎜ ⎟
⎝ ⎠⎛ ⎞⎛ ⎞= − ⇒ = − ⇒ =⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= − += − + ⇒ + = Ω
⎛ ⎞ ⎛ ⎞= − = − ⇒ =⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠
= −−
=+
− + = −( )( ) ( )
( )( )
( )2 22 1
1 2
3 1 2.83 3.125 2.83
9.844 2.83 0.9433 2.61 k 0.516 k
20 4 0.516 17.94 V17.94 1.17 16.77 V
20 335 k . 65 k400
m D
D D
D D D S
S S
G S GS
G DD
g RR R
R R R R
V VV V V
R RV V R RR R
+ − =⎡ ⎤⎣ ⎦− = ⇒ = Ω = Ω
= − ⇒ == − = − =
⎛ ⎞ ⎛ ⎞= = ⇒ = Ω = Ω⎜ ⎟ ⎜ ⎟+ ⎝ ⎠⎝ ⎠
