CE Amplifier High Frequency Analysis
August 20, 2014
1
High Frequency Analysis
Models include parasitic capacitances:
C gd
G
D
rd
C gs
S
g m vgs
S
Cµ
B
C
rπ
vπ
rO
Cπ
g m vπ
E
E
small-signal incremental model
PARASITIC CAPS LIMIT GAIN AT HIGH FREQS.
1
Summary of the open-circuit time-constant method
1. Replace all coupling and bypass caps by shorts
2. Select one parasitic cap; call it CH1
3. Replace all other parasitic caps by open circuits
4. Find resistance seen by CH1 ; call it RH1
5. High frequency pole associated with CH1 is
ωH1 =
6. Repeat above steps for each parasitic cap
1
1
CH1 RH1
7. Find equivalent high frequency cutoff
ω H = ∑n
1
1
i=1 ωHi
2
Amplifier analysis
VCC
RC
R1
CC2
CC1
RTH
vS
R2
RE
CE
RL
At high frequencies
RTH
rπ
RB
vS
Cµ
B
vπ
RB=R1||R2
Resistance seen by Cπ
Cπ
C
RLL=RC||RL
g m vπ
E
Rπ = rπ || RB || RT H
Resistance seen by Cµ
itest
vtest
Is this correct?
Find Rµ .
Resistance seen by Cµ
RB||RTH||rπ
vπ
Rµ
g m vπ
RLL=RC||RL
itest
vtest
RB||RTH||rπ
vπ
Rµ
g m vπ
RLL=RC||RL
vπ = itest (RB || RT H || rπ )
Applying KVL on the outher loop yields
vtest = vπ + (itest + gm vπ )RLL
= itest (RB || RT H || rπ )
+(1 + gm (RB || RT H || rπ ))itest RLL
or
vtest
itest
= RB || RT H || rπ + RLL
+gm (RB || RT H || rπ )RLL
Rµ =
2
Miller Theorem
Miller’s theorem provides a basis for a simplified analysis of the CE configuration.
Y=sC
iIN
iOUT
vin
Am
Assume that
Am =
vout
vOU T
≪ −1
vIN
is independent of Y = sC. Use vOU T = Am vIN , vIN = vOU T /Am .
Input:
iIN = Y (vIN − vOU T )
= sC(1 − Am )vIN = sCIN vIN
i.e. from the input C looks like a bigger capacitor C(1 − Am ).
Output:
iOU T = Y (vOU T − vIN )
1
= sC(1 −
)vOU T = sCOU T vOU T
Am
i.e. from the output C looks like a capacitor C(1 −
1
)
Am
≈ C.
vIN
vOUT
Am
CIN
COUT
To apply Miller’s Theorem, make sure that
• Am is negative
• Am is real, i.e. load is resistive
3
Unity Gain Frequency
Device manufacturer specify the unity-gain frequency fT and Cµ . You must calculate Cπ
from these values, so an appropriate relationship is needed.
Due to the presence of the parasitic capacitors, the transistor’s β = ic /ib is a function of
frequency. The unity-gain frequency fT is the frequency at which the transistor’s β = 1.
Cµ
ic
ib
ib
Zb
rπ
ic
vπ
Cπ
ic = gm vπ − vπ sCµ
vπ = ib × Zb
g m vπ
1
1
||
sCπ sCµ
1
= 1
+ sCπ + sCµ
rπ
rπ
=
1 + srπ (Cπ + Cµ )
Zb = rπ ||
β(s) =
ic
ib
gm rπ − srπ Cµ
1 + srπ (Cπ + Cµ )
β0
≈
1 + srπ (Cπ + Cµ )
=
Midband β ≡ β0 = gm rπ
β has a pole at
ωβ =
1
rπ (Cπ + Cµ )
Unity-gain frequency fT
f at which | β(s) |= 1
β02 = 1 +
ωT2
ωβ2
√
ωT = ωβ β02 − 1 ≈ β0 ωβ
Data sheet often specifies fT and Cµ ; Cπ can then be found from above equations.