III. Gm-C Filtering - Epublications - Université de Limoges

⎛ I s ⎞
Vout
= Vin
+ U T ln ⎜ ⎟ , (D.31)
⎝ I ⎠
which simplifies equation (D.30).
vout
Since the term is close to 0, it is possible to develop Equation (D.31) with a
RL
I
Taylor series, which gives:
Hence
2
3
3
⎡ v
⎛ ⎞⎤
out 1 ⎛ vout
⎞ 1 ⎛ vout
⎞
⎢
⎜⎛
vout
⎞
v
+ ⎟
⎟ + ⎜
⎟
⎥
in = vout
+ U T − ⎜
o
⎢
⎜ ⎜
⎟
(D.32)
R 2
⎟
L I
⎥
⎣ ⎝ RL
I ⎠ 3 ⎝ RL
I ⎠ ⎝⎝
RL
I ⎠ ⎠⎦
⎛ U ⎞ 1 U 1 U
v in ≈ ⎜ + ⎟v
− v +
(D.33)
⎝
T
T 2
T 3
1 out
2 out v 3 out
RL
I ⎟
⎠ 2 ( R ) 3
L I ( RL
I )
This expression is of the form
v in = avout
2 3
+ bvout
+ cv out
(D.34)
and we are looking for a solution of the type
v out
2 3
= Avin
+ Bvin
+ Cvin
(D.35)
to compute the IIP3 of the emitter-follower stage.
and
in
This leads to:
( ) ( ) ( ) 3
2 3
2 3 2
2 3
Av + Bv + Cv + b Av + Bv + Cv + c Av + Bv Cv
v = a
+
(D.36)
in
in
in
in
in
Limiting to third order terms, we obtain:
3
2 2
v = aAv + aB + bA v + aC + cA v
(D.37)
in
in
( ) ( ) 3
Identifying the two sides of this equation gives a system:
aA = 1
Hence
aC + cA
3 =
1 1 RL
I
A = = =
a U T U T + RL
I
1+
R I
L
in
0
4
- 217 -
in
in
in
in
in
(D.38)
(D.39)
(D.40)
( ) ( ) 4
c U T ⎛ RL
I ⎞ U T RL
I
C = − = − 4
3
a 3 R I
⎜
= −
U T RL
I ⎟
. (D.41)
⎝ + ⎠ 3 U + R I
This leads to:
v
out
L
T
RL
I
2 U T RL
I 3
≈ vin
+ Bvin
−
v 4 in
(D.42)
U + R I
3
T
L
( U + R I )
T
L
L