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Transistor Models & Characteristics Reference Sheet

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NMOS and T-model:
+ VGD
-
D
ID
+
IG
VDS
G
VGS
S
Characteristics in Active Region :
𝐼𝐷 =
+
-
-
1
π‘Š
𝑉𝐷𝑆
µπ‘› π‘π‘œπ‘₯ (𝑉𝐺𝑆 − 𝑉𝑑 )2 (1 +
)
2
𝐿
𝑉𝐴
𝐼𝐺 = 0,
𝑉𝐴 𝑉𝐴′ 𝐿
π‘Ÿπ‘œ =
=
𝐼𝐷
𝐼𝐷
𝑅𝑖𝑛 = ∞,
𝑉𝐺𝑆 − 𝑉𝑑 = 𝑉𝑂𝑉 ,
π‘”π‘š =
𝑉𝐷𝑆 ≥ 𝑉𝑂𝑉 ,
2𝐼𝐷
π‘Š
= µπ‘› π‘π‘œπ‘₯ (𝑉𝐺𝑆 − 𝑉𝑑 )
π‘‰π‘œπ‘£
𝐿
π‘”π‘š = √2𝐼𝐷 µπ‘› π‘π‘œπ‘₯ π‘Š/𝐿
π‘π‘œπ‘₯
πœ€π‘œπ‘₯
=
,
π‘‘π‘œπ‘₯
𝑓𝑇 =
𝑅𝐿
𝑅𝐿 +π‘…π‘œ
RC
CS with Source Resistance:
π‘”π‘š
πΊπ‘š =
1 + (π‘”π‘š + π‘”π‘šπ‘ )𝑅𝑠
R1
RB
π‘–π‘œ = πΊπ‘š 𝑣𝑖 = πΊπ‘š 𝑣𝑔
RE
MOS Resistances as seen from pins:
𝑅1 = π‘Ÿπ‘œ + (1 + π‘”π‘š ′π‘Ÿπ‘œ )(𝑅𝐸 //(π‘Ÿπœ‹ + 𝑅𝐡 ))
′
π‘”π‘š
=
RD
R1
+
𝑅2 =
D
ro
gmvgs
R2
R3
CG: π΄π‘‰π‘œ = 1 + (π‘”π‘š + π‘”π‘šπ‘ )π‘Ÿπ‘œ
cgd
R2
In tr i n s ic G a i n a n d T r a n s i ti o n F r e qu e n c y :
𝐴0 =
2𝑉𝐴
,
π‘‰π‘œπ‘£
𝑓𝑇 =
2πœ‹(𝑐𝑔𝑠 +𝑐𝑔𝑑 )
Body Effect in MOSFETS:
G
vgs
gmbvbs
gmvgs
ro
S
π‘”π‘šπ‘
= 𝝌 = 𝑓(𝑉𝑠𝑏 )
π‘”π‘š
𝑅2 =
𝑅1 = π‘Ÿπ‘œ + [1 + (π‘”π‘š + π‘”π‘šπ‘ )π‘Ÿπ‘œ ]𝑅𝑠
-
IC
IB
CD (Source Follower):
+
VCE
-
VDD
+
VBE
gmvbe
=βie
ib
B
+
vbe
ro
-
-
𝑅3 ≅ π‘Ÿπœ‹ + (𝛽 + 1)
α
re= g
m
ie
E
B
vbe
Rout
vo
RS
1
,
π‘…π‘œπ‘’π‘‘ = π‘…π‘œ //𝑅𝑆
π‘”π‘š + π‘”π‘šπ‘
π‘”π‘š π‘Ÿπ‘œ
π‘”π‘š
=
≅
1 + (π‘”π‘š + π‘”π‘šπ‘ )π‘Ÿπ‘œ π‘”π‘š + π‘”π‘šπ‘
π‘…π‘œ =
π΄π‘£π‘œ
𝐴𝑣 = π΄π‘£π‘œ
C
+
𝑅𝑠
π‘”π‘š 𝑅𝑠
≅
π‘…π‘œ +𝑅𝑠 1 + (π‘”π‘š + π‘”π‘šπ‘ )𝑅𝑠
gmvbe
rπ
Emitter degeneration :
π‘”π‘š
π‘”π‘š,𝑒𝑓𝑓 =
1 + π‘”π‘š 𝑅𝐸
ro
πΊπ‘Žπ‘–π‘› = −π‘”π‘š,𝑒𝑓𝑓 π‘…π‘œπ‘’π‘‘
E
B
Forward Active Region:
𝑣𝐢𝐸
𝑖𝐢 = 𝐼𝑆 𝑒 𝑣𝐡𝐸/𝑉𝑇 (1 +
),
𝑖𝐡 = 𝑖𝐢 /𝛽
𝑉𝐴
𝐼𝐢
,
𝑉𝑇
π‘Ÿπ‘œ =
𝑉𝐴
,
𝐼𝐢
𝐴0 =
𝑅𝑖𝑛 π‘€π‘–π‘‘β„Ž π‘’π‘šπ‘–π‘‘π‘‘π‘’π‘Ÿ π‘”π‘Ÿπ‘œπ‘’π‘›π‘‘π‘’π‘‘ π‘Ÿπœ‹ =
High Frequency Model:
π‘Ÿπ‘œ 𝑅𝐸
π‘Ÿπ‘œ + 𝑅𝐢 + 𝑅𝐸
𝑅3 ≅ π‘Ÿπœ‹ + (𝛽 + 1)𝑅𝐸 π‘€β„Žπ‘’π‘› π‘Ÿπ‘œ = ∞
-
π‘”π‘š =
π‘Ÿπœ‹ + 𝑅𝐡
𝛽+1
𝑅𝑐
(𝛽 + 1)
𝑅3 = (𝛽 + 1) (π‘Ÿπ‘’ + 𝑅𝐸
)
π‘Ÿπ‘œ + 𝑅𝐢 + 𝑅𝐸
C
+ VBC
𝑅𝐢
π‘”π‘š π‘Ÿπ‘œ
π‘Ÿπ‘œ +
NPN, Hybrid-П and T Model :
𝐼𝑓 𝑉𝑠𝑏 = 0 → π‘”π‘šπ‘ = 0
RO
(π‘Ÿπœ‹ + 𝑅𝐡 )
π‘€β„Žπ‘’π‘› π‘Ÿπ‘œ = ∞
(𝛽 + 1)
𝐼𝑓 𝑅𝐢 = 0 → 𝑅2 ≅
1
𝑅𝐷
𝑅2 ≅
+
π‘”π‘š + π‘”π‘šπ‘ 1 + (π‘”π‘š + π‘”π‘šπ‘ )π‘Ÿπ‘œ
π‘’π‘ π‘’π‘Žπ‘™π‘™π‘¦ 0.1 < 𝝌 < 0.3
vi
(π‘Ÿπ‘œ + 𝑅𝐢 )(π‘Ÿπœ‹ + 𝑅𝐡 )
(𝛽 + 1)π‘Ÿπ‘œ
𝐼𝑓 𝑅𝐡 = 0 → 𝑅2 ≅ π‘Ÿπ‘’ +
π‘Ÿπ‘œ + 𝑅𝐷
𝑅2 =
1 + (π‘”π‘š + π‘”π‘šπ‘ )π‘Ÿπ‘œ
D
+
𝑅2 ≅
RS
π‘”π‘š
1
1
𝑅
+ 𝐡
π‘”π‘š 𝛽
(π‘Ÿπ‘œ + 𝑅𝐢 )(π‘Ÿπœ‹ + 𝑅𝐡 )
(𝛽 + 1)π‘Ÿπ‘œ + π‘Ÿπœ‹ + 𝑅𝐡 + 𝑅𝐢
S
π‘”π‘š
2πœ‹(π‘πœ‹ +𝑐µ )
BJT Resistances as seen from pins:
𝑅𝐿
𝐴𝑉 = π΄π‘‰π‘œ
𝑅𝐿 + π‘…π‘œ
High Frequency Model:
ro
gmvbe
Transition Frequency:
CB: π΄π‘‰π‘œ = 1 + π‘”π‘š π‘Ÿπ‘œ
πœ€π‘œπ‘₯ = 3.9 × πœ€0 ,
vgs cgs
cπ
-
πœ€0 = 8.854 × 10−12 𝐹/π‘š
G
rπ
E
RL vout
AVovi
π‘‰π‘œπ‘’π‘‘ = 𝐴𝑉 𝑉𝑖 = π΄π‘‰π‘œ 𝑉𝑖
C
-
+
vi
cµ
B+
+
vbe
+
1
gm
-
rb
B
Ro
ro
+
vgs
-
π‘”π‘š
1
𝑖𝑓 𝑅𝑠 ≫
π‘”π‘š + π‘”π‘šπ‘
π‘”π‘š + π‘”π‘šπ‘
gmvgs
ig=0
-
+
𝐴𝑣 ≅
𝑉𝐴
𝑉𝑇
𝛽
π‘”π‘š
C
+
vbe
gmvbe
rπ
E
Vsig
B
RE
RE
C
+
vb
Rin
gm,eff vb
Rout
-
Wilson Current Mirror:
ro
π‘…π‘œ =
π›½π‘Ÿπ‘œ
(𝐡𝐽𝑇)
2
OCTC method:
π‘…π‘œ ≅ π‘”π‘š3 π‘Ÿπ‘œ3 π‘Ÿπ‘œ2 (𝑀𝑂𝑆)
𝜏𝐻 = ∑ 𝑅𝑖 𝐢𝑖 → 𝑓𝐻 ≅
Widlar Current Mirror:
πΌπ‘Ÿπ‘’π‘“
πΌπ‘œ 𝑅𝐸 = 𝑉𝑇 𝑙𝑛
πΌπ‘œ
𝑅𝑔𝑑 = 𝑅𝐿 ′ + πΊπ‘š 𝑅𝑠𝑖𝑔 𝑅𝐿 ′ + 𝑅𝑠𝑖𝑔
+
vi
−𝑅𝐷 π›₯π‘”π‘š
(
)
2𝑅𝑆𝑆 π‘”π‘š
𝐡𝐽𝑇: π΄π‘π‘š =
R1
vx
−𝑅𝐢 π›₯𝑅𝐢
(
)
2𝑅𝐸𝐸 𝑅𝐢
cx
v1
π‘ π‘π‘š
)
2π‘”π‘š3
𝐴𝑑 =
𝑠𝑐
(1 + 𝑠𝑐𝐿 π‘…π‘œ ) (1 + π‘š )
π‘”π‘š3
π‘”π‘š π‘…π‘œ (1 +
↓
v1
c1
(𝑀𝑂𝑆)
π›₯𝑅𝐢 2
π›₯𝐼𝑆 2
π‘‰π‘œπ‘  = 𝑉𝑇 √(
) + ( ) (𝐡𝐽𝑇)
𝑅𝐢
𝐼𝑆
𝐴𝑑 ≅
v2
(1-V2/V1)cx c2
(1-V1/V2)cx
π‘”π‘š π‘…π‘œ
π‘”π‘š3
π‘“π‘œπ‘Ÿ πœ” β‰ͺ
1 + 𝑠𝑐𝐿 π‘…π‘œ
π‘π‘š
Low Frequency Response:
πœ”π‘,1 =
Diff. Amp. Input Offset Current:
π›₯𝛽
πΌπ‘œπ‘  = 𝐼𝐡 ( )
𝛽
(Ci’s are bypass/coupling capacitances)
look like MOS high freq model:
MOS:
Rsig
𝑠
∏ (1 +
)
𝑀𝑧,𝑖
π‘Šβ„Žπ‘’π‘› πœ”π» (𝑠) =
𝑖𝑠 π‘˜π‘›π‘œπ‘€π‘›
𝑠
∏ (1 +
)
𝑀𝑝,𝑗
rπ
πœ”πΏ = max(πœ”π‘,𝑖 )
cµ
rx
Vsig
Usually bypass capacitance pole is 10
times larger than others
cπ
BJT:
πœ”πΏ = πœ”π‘,1 + πœ”π‘,2 + πœ”π‘,3
↓
𝐼𝑓 πœ”π‘,1 β‰ͺ πœ”π‘,𝑗 & πœ”π‘,1 β‰ͺ πœ”π‘§,𝑖
𝑒π‘₯: πœ”π‘,3 = 0.8πœ”πΏ & πœ”π‘,1 = πœ”π‘,2 = 0.1πœ”πΏ
cµ
Rsig’
πœ”π‘,1 𝑖𝑠 π‘‘β„Žπ‘’ π‘‘π‘œπ‘šπ‘–π‘›π‘Žπ‘›π‘‘ π‘π‘œπ‘™π‘’ π‘Žπ‘›π‘‘ πœ”π» = πœ”π‘,1
Two port parameters:
1
πœ”π‘§,𝑖 2
Vsig’
cπ
𝑉
𝐼
[ 1] = 𝑍 [ 1]
𝑉2
𝐼2
π‘Œ = 𝑍 −1
𝑉
𝐼
[ 1] = 𝐻 [ 1 ]
𝐼2
𝑉2
𝐺 = 𝐻 −1
Transfer Function of a Two Stage Amplifier
π‘‰π‘œ
πΊπ‘š1 (πΊπ‘š2 − 𝑠𝐢𝑐 )𝑅1 𝑅2
=
𝑉𝑖𝑑 1 + 𝑠[𝐢1 𝑅1 + 𝐢2 𝑅2 + 𝐢𝑐 (πΊπ‘š2 𝑅1 𝑅2 + 𝑅1 + 𝑅2 )] + 𝑠 2 [𝐢1 𝐢2 + 𝐢𝑐 (𝐢1 + 𝐢2 )]𝑅1 𝑅2
Approximated Locations of Poles when a Dominant Pole Exists
πœ”′𝑃1 ≅
1
𝑅𝑖 𝐢𝑖
BJT high freq model can be simplified to
High Frequency Response:
πœ”π‘,𝑗
1 + 𝑠𝑅𝑆𝑆 𝐢𝑆𝑆
𝑠
(1 +
) (… )
πœ”π‘,1
Active Loaded:
c2
2
πœ”π» 2
π΄π‘π‘š (𝑠) = π΄π‘π‘š (0)
v2
c1
π‘‰π‘œπ‘£ π›₯𝑅𝐷 2
π‘‰π‘œπ‘£ π›₯(π‘Š ⁄𝐿)
π‘‰π‘œπ‘  = √(
) +(
) + (π›₯𝑉𝑑 )2
2 𝑅𝐷
2 π‘Š ⁄𝐿
−2∑
2
Resistively loaded:
Miller Method:
Diff. Amp. Input Offset Voltage:
1
𝑅𝑠 + 𝑅𝑠𝑖𝑔
π‘Ÿπ‘œ
1 + π‘”π‘š 𝑅𝑠 (
)
π‘Ÿπ‘œ + 𝑅𝐿
Diff. Amp.
𝑣π‘₯
= 𝑅1 + π‘”π‘š 𝑅1 𝑅2 + 𝑅2
𝑖π‘₯
−1
𝑀𝑂𝑆: π΄π‘π‘š =
2π‘”π‘š3 𝑅𝑆𝑆
−π‘Ÿπ‘œ4
𝐡𝐽𝑇: π΄π‘π‘š =
𝛽3 𝑅𝐸𝐸
=∑
𝑅𝑔𝑠 ≅
R2
𝜏𝐻 = 𝑅𝑔𝑠 𝐢𝑔𝑠 + 𝑅𝑔𝑑 𝐢𝑔𝑑 + 𝑅𝐿 ′𝐢𝐿
load transistor):
1
g mv i
-
-Active load (Q3 diode connected, Q4
𝑒𝑙𝑠𝑒
𝑅𝐿 ′ = 𝑅𝐿 //π‘Ÿπ‘œ
ix
−𝑅𝐷 π›₯𝑅𝐷
=
(
)
2𝑅𝑆𝑆 𝑅𝐷
π‘œπ‘Ÿ π΄π‘π‘š =
CS with source resistance:
π‘”π‘š
πΊπ‘š ≅
& 𝑅𝑠 ≅ π‘Ÿπ‘œ (1 + π‘”π‘š 𝑅𝑠 )
1 + π‘”π‘š 𝑅𝑠
𝜏𝐻 = 𝑅𝑔𝑠 𝐢𝑔𝑠 + 𝑅𝑔𝑑 𝐢𝑔𝑑 + 𝑅𝐿 𝐢𝐿
-Resistive load, differential output:
𝑉𝑠𝑖𝑔 π‘Ÿπœ‹
π‘Ÿπœ‹ + π‘Ÿπ‘₯ +𝑅𝑠𝑖𝑔
𝑅𝑠𝑖𝑔 ′ = π‘Ÿπœ‹ //(π‘Ÿπ‘₯ +𝑅𝑠𝑖𝑔 )
1
2πœ‹πœπ»
For CS:
Differential Amplifier CM Gain:
𝑀𝑂𝑆: π΄π‘π‘š
𝑉𝑠𝑖𝑔 ′ =
1
πΊπ‘š2 𝑅2 𝐢𝑓 𝑅1
πœ”′𝑃2 ≅
πΊπ‘š2 𝐢𝑓
𝐢1 𝐢2 +𝐢𝑓 (𝐢1 +𝐢2 )
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