EECS 105 Fall 1998Lecture 28
Frequency Response of Transistor Amplifiers
■ Simplest case: CE short-circuit current gain Ai(jω) as a function of frequency
Kirchhoff’s current law at the output node:
Kirchhoff’s current law at the input node:
■ Solving for Vπ at the input node:
Cπ
Cµ
gmVπrπ Vπ
+
−
Iin
Io
Io gmVπ Vπ jωCµ–=
Is
VπZπ------ VπjωCµ+= where Zπ rπ
1jωCπ-------------
=
VπIs
1 Zπ⁄( ) jωCµ+--------------------------------------=
28
EECS 105 Fall 1998Lecture 28
Short-Circuit Gain Frequency Response
■ Substituting Vπ into the output node equation--
■ Substituting for Zπ and simplifying --
Current gain has one pole:
and one zero
Io
Is----
gmZπ 1jωCµ
gm-------------–
1 jωCµZπ+------------------------------------------=
Io
Is----
gmrπ 1jωCµ
gm-------------–
1 jωrπ Cπ Cµ+( )+-----------------------------------------------
βo 1jωCµ
gm-------------–
1 jωrπ Cπ Cµ+( )+----------------------------------------------- βo
1 jωωz------–
1 jωωp------+
------------------= = =
ωp rπ Cπ Cµ+( )( ) 1–=
ωz gm1–Cµ( )
1–ωp»=
EECS 105 Fall 1998Lecture 28
Bode Plot of Short-Circuit Current Gain
■ Note low frequency magnitude of gain is βo
■ Frequency at which current gain is reduced to 0 dB is defined as the transition frequency ωT. Neglecting the zero,
1
ω
IoIin
1
0
−45
−90
−135
−180
IoIin
∠
ω
(a)
log scale
log scale
(b)
Cµ Cπ + Cµ
rπ(Cπ + Cµ)
1rπ(Cπ + Cµ)
gmCπ + Cµ
gm Cµ
gm Cµ
βo = gmrπ
ωT
gm
Cπ Cµ+( )-------------------------=
EECS 105 Fall 1998Lecture 28
Transition Frequency of the Bipolar Transistor
■ Dependence of transition time τT = ωT-1 on the bias collector current IC:
■ If the collector current is increased enough to make the second term negligible, then the minimum τT is the base transit time, τF.. In practice, the ωT decreases at very high values of IC due to other effects and the minimum τT may not be achieved.
■ Numerical values of fT = (1/2π)ωT range from 10 MHz for lateral pnp’s to 10 GHz for oxide-isolated npn’s
Note that the small-signal model is not valid above fT (due to distributed effects in the base) and the zero in the current gain is not observed
τT1
ωT-------
Cπ Cµ+
gm--------------------
gmτF CjE Cµ+ +
gm------------------------------------------= = =
τT τF
CjE Cµ+
gm----------------------
+ τF
Vth
IC-------- CjE Cµ+( )+= =
EECS 105 Fall 1998Lecture 28
Common-Source Current Gain
■ CS amplifier has a non-infinite input impedance for ω > 0 and we can measure its small-signal current gain.
■ Analysis is similar to CE case; result is
■ Transition frequency for the MOSFET is
+
−
IingmVgs
Cgd
Cgs
Io
Vgs
Io
I in------
gm 1jωCgd
gm----------------–
jω Cgs Cgd+( )--------------------------------------gm
ω Cgs Cgd+( )----------------------------------≈=
ωT
gm
Cgs Cgd+-------------------------≈
EECS 105 Fall 1998Lecture 28
Transition Frequency of the MOSFET
■ Substitution of gate-source capacitance and transconductance:
and
■ The transition time is the inverse of ωT and can be written as the average time for electrons to drift from source to drain
velocity saturation causes τT to decrease linearly with L; however,submicron MOSFETs have transition frequencies that are approaching those for oxide-isolated BJTs
Cgs23---WLCox= Cgd» gm
WL-----µnCox VGS VTn–( )=
ωT
gm
Cgs---------
WL-----µnCox VGS VTn–( )
23---WLCox
---------------------------------------------------- 32---µn
VGS VTn–( )L
------------------------------ L= =≈
τTL
µn23---
VGS VTn–( )L
------------------------------
---------------------------------------------- L
vdr
----------= =
EECS 105 Fall 1998Lecture 28
Frequency Response of Voltage Amplifiers
■ Common-emitter amplifier:
Procedure: substitute small-signal model and perform phasor analysis
+
−
V +
V−
RS
RL
iSUP
iOUT
Vs
VBIAS
vOUT+−
+−
EECS 105 Fall 1998Lecture 28
Brute Force Phasor Analysis
■ “Exact” analysis: transform into Norton form at input to facilitate nodal analysis
Note that Ccs is omitted, along with rb
Details: see Section 10.4
Factor (approximately!) into one high-frequency zero and two poles ω1 << ω2
Vs rπ
RSCµ
CπVπ
+
−
VoutRL
+
−
gmVπ ro roc+−
Is R'in
R'in = RS rπ R'out = ro roc RL
R'out
Cµ
CπVπ
+
−
Vout
+
−
gmVπ
(a)
(b)
ω11–
rπ RS( ) Cπ 1 gmro roc RL+( )Cµ+( ) ro roc RL( )Cµ+=
ω21–
ro roc RL( ) rπ RS( )CµCπrπ RS( ) Cπ 1 gmro roc RL+( )Cµ+( ) ro roc RL( )Cµ+
-------------------------------------------------------------------------------------------------------------------------------------------------=
EECS 105 Fall 1998Lecture 28
The Miller Approximation
■ The “exact” analysis is not particularly helpful for gaining insight into the frequency response ... consider the effect of Cµ on the input only
neglect the feedforward current Iµ in comparison with gm Vπ ... a good approximation
It = (Vt - Vo) / Zµ
Vo = - gmVt RL / (RL + Rout’) = AvCµ Vt
where AvCµ is the low frequency voltage gain across Cµ
It = Vt (1- Av) / Zµ
Zeff = Vt / It = Zµ / (1 - Av)
is the Miller capacitor
Cµ
gmVt R'out = ro roc RL Vout
+
−
Vt
It
+−
Zeff1
jωCµ------------- 1
1 AvCµ–
--------------------- 1
jω Cµ 1 AvCµ–( )( )-------------------------------------------- 1
jωCM---------------= = =
CM 1 AvCµ–( )Cµ=
EECS 105 Fall 1998Lecture 28
Generalized Miller Approximation
■ An impedance Z connected across an amplifier with voltage gain AvZ canbe replaced by an impedance to ground ... multiplied by (1-AvZ)
■ Common-emitter and common-source:
AvZ = large and negative for Cµ or Cgd --> capacitance at the input is magnified
■ Common-collector and common-drain:
--> capacitance at the input due to Cπ or Cgs is greatly reduced
Vi
+
−Vo
+
−
Vi
+
−Vo
+
−
AvoAvo
Z
Zeff
Zeff = Z /(1 − Avo)
AvZ 1≈
EECS 105 Fall 1998Lecture 28
Voltage Gain vs. Frequency for CE AmplifierUsing the Miller Approximation
■ The Miller capacitance is lumped together with Cπ, which results in a single-pole low-pass RC filter at the input
Transfer function has one pole and no zero after Miller approximation:
from the exact analysis (final term is missing)
Vs rπ
RS
CMCπ
CM = Cµ(1 + gmR'out)
R'outVπ
+
−
Vout
+
−
gmVπ+−
ω3dB1–
rπ RS( ) Cπ CM+( )=
ω3dB1–
rπ RS( ) Cπ 1 gmro roc RL+( )Cµ+[ ]=
ω3dB1– ω1
1–≈ Rout′Cµ
EECS 105 Fall 1998Lecture 28
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