# Biasing MOSFET Amplifier ```Biasing MOSFET Amplifier
Biasing with resistors
Adjust the values of resistors and MOSFET parameters to put the
MOSFET in an operating Point such as:
Biasing with constant current source
Current source:
ID2 is constant,
Its value adjusted by the choice of R to the desired
value:
1
𝑊
2
𝐿 1
Q1 saturation ⟹ 𝐼𝐷1 = 𝑘𝑛′
𝑉𝐺𝑆 − 𝑉𝑡
2
ID1 is the current through R ⟹
𝐼𝐷1 = 𝐼𝑅𝐸𝐹 =
𝑉𝐷𝐷 + 𝑉𝑆𝑆 − 𝑉𝐺𝑆
𝑅
The current 𝐼𝑅𝐸𝐹 is the reference current
1
𝑊
2
𝐿 2
If Q2 is in saturation ⟹ 𝐼𝐷2 = 𝑘𝑛′
Since Q2 has same VGS as Q1 then
𝐼𝐷2 = 𝐼𝑅𝐸𝐹
𝑊 𝐿
𝑊 𝐿
2
1
𝑉𝐺𝑆 − 𝑉𝑡
2
Small signal operation and models
Simple conceptual MOSFET circuit:
 DC VGS applied on the Gate-Source for biasing
 A small signal vgs applied: the ac signal
The DC Bias Point: Assume small signal vgs is zero
 vGS = VGS + vgs = VGS
only DC
1
 𝑖𝐷 = 𝐼𝐷 = 2 𝑘𝑛′
𝑊
𝐿
𝑉𝐺𝑆 − 𝑉𝑡
2
only DC
 𝑣𝐷 = 𝑉𝐷 = 𝑉𝐷𝐷 − 𝑅𝐷 𝐼𝐷
only DC
To ensure proper amplifier operation MOSFET must be saturated:
⇒ 𝑉𝐷 &gt; 𝑉𝐺𝑆 − 𝑉𝑡
𝑉𝐷 has to be half way between VDD and 𝑉𝐺𝑆 − 𝑉𝑡
Signal current in the drain:
 vGS = VGS + vgs
1
 𝑖𝐷 = 2 𝑘𝑛′
𝑖𝐷 =
𝑊
𝐿
𝑣𝐺𝑆 − 𝑉𝑡
2
1 ′ 𝑊
𝑘
𝑉 − 𝑉𝑡
2 𝑛 𝐿 𝐺𝑆
1
𝑊
2
𝐿
= 𝑘𝑛′
2
+ 𝑘𝑛′
𝑉𝐺𝑆 + 𝑣𝑔𝑠 − 𝑉𝑡
2
𝑊
1 𝑊 2
𝑉𝐺𝑆 − 𝑉𝑡 𝑣𝑔𝑠 + 𝑘𝑛′
𝑣
𝐿
2
𝐿 𝑔𝑠
1. First term is the DC component ID,
2. Second term has a linear dependence with the signal vgs
3. Third term has nonlinear dependence on vgs, this term is undesirable it
represent nonlinear distortion:
 To reduce this distortion we need to keep the signal small such
1
𝑊 2
𝑊
that 𝑘𝑛′ 𝑣𝑔𝑠
≪ 𝑘𝑛′
𝑉𝐺𝑆 − 𝑉𝑡 𝑣𝑔𝑠
2
Resulting in
𝐿
𝐿
𝑣𝑔𝑠 ≪ 2 𝑉𝐺𝑆 − 𝑉𝑡
or 𝑣𝑔𝑠 ≪ 2𝑉𝑂𝑉
This is the small signal approximation
𝑖𝐷 ≅ 𝐼𝐷 + 𝑖𝑑
Where
𝑊
𝑖𝑑 = 𝑘𝑛′
𝑉𝐺𝑆 − 𝑉𝑡 𝑣𝑔𝑠
𝐿
the small signal component of the drain current;
it is proportional to the small signal 𝑣𝑔𝑠
Small signal parameter
The parameter that relates 𝑖𝑑 to 𝑣𝑔𝑠 : is the MOSFET transconductance 𝒈𝒎
𝑔𝑚 =
Graphical Interpretation:
𝑔𝑚 =
𝜕𝑖𝐷
𝜕𝑣𝑔𝑠
𝑣𝐺𝑆 =𝑉𝐺𝑆
𝑖𝑑
𝑊
𝑊
= 𝑘𝑛′
𝑉𝐺𝑆 − 𝑉𝑡 = 𝑘𝑛′
𝑉
𝑣𝑔𝑠
𝐿
𝐿 𝑂𝑉
iD
Slope = gm
iD Approximate
iD Exacte
id
Q
iD
t
IDQ
VGSQ
Vt
vGS
vgs
t
Voltage Gain
𝑣𝐷
𝑣𝐷
𝑣𝐷
𝑣𝐷
= 𝑉𝐷𝐷 − 𝑅𝐷 𝑖𝐷
= 𝑉𝐷𝐷 − 𝑅𝐷 (𝐼𝐷 + 𝑖𝑑 )
= (𝑉𝐷𝐷 − 𝑅𝐷 𝐼𝐷 ) − 𝑅𝐷 𝑖𝑑
= 𝑉𝐷 + 𝑣𝑑
Where
𝑣𝑑 = −𝑖𝑑 𝑅𝐷 = −𝑔𝑚 𝑣𝑔𝑠 𝑅𝐷
𝐴𝑣 ≡
The voltage gain is:
𝑣𝑑
𝑣𝑔𝑠
= −𝑔𝑚 𝑅𝐷
The minus sign indicate that 𝑣𝑑 is 180 &ordm; out of phase with 𝑣𝑔𝑠
vgs
vd
t
Small Signal Equivalent Circuit:
𝑖𝑑 = 𝑔𝑚 𝑣𝑔𝑠
𝑉𝐴
𝑟𝑜 =
𝐼𝐷
1 𝑊
𝐼𝐷 = 𝑘𝑛′
𝑉 − 𝑉𝑡
2
𝐿 𝐺𝑆
example
G
D
gmvgs
vgs
2
S
ro
Low frequency Equivalent circuit
At low frequency no capacitive effect.
G
D
gmvgs
vgs
At high frequency charging and discharging of
the different capacitors have to be taken into
consideration
ro
S
G
The figure shows the different capacitors in a
MOSFET and where they originate:
 Cgs and Cgd : Oxide capacitance
 Csb and Cdb : Junction Capacitance
S
D
Cgd
Cgs
n+
Csb
n+
p type substrate
B
Cdb
Complete high frequency small
signal Equivalent circuit:
Cgd
g mbbs
g m  gs
G
 gs
ro
Cgs
Cdb
bs
S
D
Csb
B
Body and Source connected and body
effect neglected
Cgd
g m  gs
G
 gs
D
ro
Cgs
S
Cdb
Cdb can be neglected to give the simplified
high frequency small signal Equivalent
circuit
Cgd
g m  gs
G
 gs
ro
Cgs
S
D
```