BJT Differential Amplifiers (6/24/00)
Page 1
3.4 - BJT DIFFERENTIAL AMPLIFIERS
INTRODUCTION
Objective
The objective of this presentation is:
1.) Define and characterize the differential amplifier
2.) Show the large-signal and small-signal performance
3.) Show alternate implementations of the differential amplifier
Outline
• Characterization and definitions
• Large-signal transconductance
• Large-signal voltage transfer
• Small-signal performance
• Other characteristics of the differential amplifier
• Summary
ECE 4430 - Analog Integrated Circuits and Systems
 P.E. Allen, 2000
BJT Differential Amplifiers (6/24/00)
Page 2
CHARACTERIZATION AND DEFINITIONS
What is a Differential Amplifier?
A differential amplifier is an amplifier that amplifies the difference between two voltages and rejects
the average or common mode value of the two voltages.
Symbol for a differential amplifier:
+
+
v1
-
+
v2
-
-
+
vOUT
Fig. 5.2-1A
Differential and common mode voltages:
v1 and v2 are called single-ended voltages. They are voltages referenced to ac ground.
The differential-mode input voltage, vID, is the voltage difference between v1 and v2.
The common-mode input voltage, vIC, is the average value of v1 and v2 .
v 1+v 2
∴
v ID = v 1 - v 2 and v IC = 2
⇒ v1 = vIC + 0.5vID and v2 = vIC - 0.5vID
vID
2
vIC
vID
2
v 1 + v 2 
v OUT = A VD v ID ± A VC v IC = A VD (v 1 - v 2) ± A VC  2 
+
-
+
vOUT
Fig. 5.2-1B
where
AVD = differential-mode voltage gain
AVC = common-mode voltage gain
ECE 4430 - Analog Integrated Circuits and Systems
 P.E. Allen, 2000
BJT Differential Amplifiers (6/24/00)
Page 3
Differential Amplifier Definitions
• Common mode rejection rato (CMRR)
A VD
CMRR = A 
 VC 
CMRR is a measure of how well the differential amplifier rejects the common-mode input voltage in
favor of the differential-input voltage.
• Input common-mode range (ICMR)
The input common-mode range is the range of common-mode voltages over which the differential
amplifier continues to sense and amplify the difference signal with the same gain.
Typically, the ICMR is defined by the common-mode voltage range over which all MOSFETs
remain in the saturation region and all BJTs remain in the active region.
• Output offset voltage (VOS(out))
The output offset voltage is the voltage which appears at the output of the differential amplifier
when the input terminals are connected together.
• Input offset voltage (VOS(in) = VOS)
The input offset voltage is equal to the output offset voltage divided by the differential voltage gain.
V OS (out)
V OS =
A VD
ECE 4430 - Analog Integrated Circuits and Systems
 P.E. Allen, 2000
BJT Differential Amplifiers (6/24/00)
Page 4
LARGE-SIGNAL TRANSCONDUCTANCE
Transconductance Characteristic of the Differential Amplifier
Consider the following NPN-BJT differential amplifier (sometimes called an emitter-coupled pair):
iC1
vI1
iC2
Q1
+
vBE1
-
Q2
+
vBE2
-
vI2
IEE
VEE
Large-Signal Analysis:
1.) Input loop eq.:
2.) Foward-active region:
3.) If IS1 = IS2 then
BJTDA01
vI1 - vBE1 + vBE2 - vI2 = vI1 - vI2 - vBE1 + vBE2 = vID - vBE1 + vBE2 = 0
 iC1
 iC2
vBE1= Vt ln 
and
v
=
V
ln
BE2
t I 
I S1 
 S2 
iC1
 vI1-vI2
 vID
=
exp

 = exp

iC2
 Vt 
 Vt 
4.) Nodal current equation at the emitters:
5.) Combining the above equations gives: iC1 =
ECE 4430 - Analog Integrated Circuits and Systems
1
-(iE1+iE2) = IEE = α (iC1 + iC2)
F
αFIEE
 -vID
1 + exp  V 
 t 
and
iC2 =
αFIEE
 vID
1 + exp  V 
 t
 P.E. Allen, 2000
BJT Differential Amplifiers (6/24/00)
Page 5
Transconductance Characteristic of the Differential Amplifier - Continued
Plotting the collect current as a function of vID:
1
iC2
iC1
0.8
iC
αIEE
0.6
0.4
0.2
0
-5
-4
-3
Transconductance:
∂iC1
gm1 = ∂v Q| =

ID
-2
-1
0
vID
Vt
αFIEE
αFIEE IC1
= 4V = 2V
 -vID  2
t
t
 1+exp
 V t

 V t 
-αFIEE
-αFIEE -IC2
∂iC2 |
= 4V = 2V
gm2 = ∂v Q =

 vID  2
ID
t
t
 1+exp
 V t

 V t 
ECE 4430 - Analog Integrated Circuits and Systems
1
2
3
4
5
BJTDA00
when VID = 0
when VID = 0
 P.E. Allen, 2000
BJT Differential Amplifiers (6/24/00)
Page 6
VOLTAGE TRANSFER CHARACTERISTICS
Large-Signal Voltage Transfer Function
VCC
Assume load resistors as shown:
RC
iC1
Q1
+
vBE1
-
vi1
RC
+vOD +
+
vO1
vO1
-
iC2
Q2
+
vBE2
-
vi2
IEE
VEE
∴ vOD = vO1-vO2 = VCC-iC1RC - VCC + iC2RC
= RC(iC2-iC1)
Substituting in the previous expressions and
using hyperbolic trig identities gives
 -vID
vOD = αFIEERC tanh

 2V t 
Illustration →
1
0.5
vOD
αFIEERC
0
-0.5
-1
ECE 4430 - Analog Integrated Circuits and Systems
BJTDA03
-5 -4
-3
-2
-1
0
vID
Vt
1
2
3
4
5
BJTDA04
 P.E. Allen, 2000
BJT Differential Amplifiers (6/24/00)
Page 7
Emitter Degeneration of the BJT Differential Amplifier
Increases the range over which the emitter-coupled pair behaves as a linear amplifier with lower gain at
the cost of lower gain.
VCC
RC
RC
+vOD -
iC1
vi1
vOD
αFIEERC
1
iC2
Q2
+
vBE1
-
+
vBE2
-
RE
Increasing
RE
0.5
-25 -20 -15 -10 -5
vi2
RE
5 10
-0.5
RE=0
15 20 25
vID
Vt
-1
BJTDA05
IEE
VEE
We know that,
 v ID - (R E iD /α F ) 
 v ID - (R E iD /α F ) 



iD = iC1 - iC2 = αFIEE tanh
≈ αFIEE 


2Vt
2Vt
Solving for iD gives,
∴
diD
gm(DC) = dv =
ID
αFIEEvID
iD = iC1 - iC2 = 2V + R I
t
E EE
(αFIEE)
(αFIEE)/2Vt
=
2V t + R E I E E 1+ R E IEE /2V t
ECE 4430 - Analog Integrated Circuits and Systems
 P.E. Allen, 2000
BJT Differential Amplifiers (6/24/00)
Page 8
SMALL-SIGNAL PERFORMANCE
Differential and Common-mode Small-Signal Performance
The small-signal performance of a differential amplifier can be separated into a differential mode and
common mode analysis. This separation allows us to take advantage of the following simplifications.
VCC
Half-Circuit Concept:
VCC
RC
ic1
VCC
RC
RC
ic1
vi1
+
vbe1
Q1
+vod +
+
vo1
vo2
-
IEE
VEE
RC
VEE
ic2
Q2
vid
al
i
t
n
e
2
er
Diff nalysis
A
e
Mod
+
vbe2
-
+
vbe1
Q1
ic2
Q2
-
VCC
vic
+
vbe1
IEE
2
VEE
vid
2
RC
ic1
REE
+
vbe2
-
VCC
RC
vi2
C
Mo omm
de A on
nal
ysis
+vod +
+
vo1
vo2
-
Q1
+vod +
+
vo1
vo2
-
2REE
VEE
ic2
Q2
+
vbe2
IEE
2
2REE
VEE VEE
vic
BJTDA06
Note: The half-circuit concept is valid as long as the resistance seen looking into each emitter is the same.
ECE 4430 - Analog Integrated Circuits and Systems
 P.E. Allen, 2000
BJT Differential Amplifiers (6/24/00)
Page 9
SMALL-SIGNAL PERFORMANCE
Differential and Common-mode Small-Signal Performance
The small-signal performance of a differential amplifier can be separated into a differential mode and
common mode analysis. This separation allows us to take advantage of the following simplifications.
VCC
Half-Circuit Concept:
VCC
RC
ic1
VCC
RC
RC
ic1
vi1
+
vbe1
Q1
+vod +
+
vo1
vo2
-
IEE
VEE
RC
VEE
ic2
Q2
vid
al
i
t
n
e
2
er
Diff nalysis
A
e
Mod
+
vbe2
-
+
vbe1
Q1
ic2
Q2
-
VCC
vic
+
vbe1
IEE
2
VEE
vid
2
RC
ic1
REE
+
vbe2
-
VCC
RC
vi2
C
Mo omm
de A on
nal
ysis
+vod +
+
vo1
vo2
-
Q1
+vod +
+
vo1
vo2
-
2REE
VEE
ic2
Q2
+
vbe2
IEE
2
2REE
VEE VEE
vic
BJTDA06
Note: The half-circuit concept is valid as long as the resistance seen looking into each emitter is the same.
ECE 4430 - Analog Integrated Circuits and Systems
 P.E. Allen, 2000
BJT Differential Amplifiers (6/24/00)
Page 10
Small-Signal, Common Mode Performance of the BJT Differential Amplifier
Circuit and small-signal model:
VCC
Rin
RC
+
ic1
+
vbe1
vic
IEE
2
VEE
Rout
Q1
+
vo1
-
rπ
vic
+
vπ
-
+
gmvπ
ro
RC
vout
2REE
2REE
VEE
BJTDA08
Half-circuit performance:
βο12REE 
vo1
βo1RC

Rin1 = rπ1 +(1+βo1)2REE, Rout1= ro 1+ 2R +r  + 2REE||rπ1 , and v = - r +(1+β )2R
EE π 1 
ic
π1
o1
EE

Common mode performance:
rπ1+(1+βo1)2REE
βο12REE 
voc
βo1RC

, R oc= ro 1+ 2R +r  + 2REE||rπ1 , and v = - r +(1+β )2R
Ric =
2
EE π 1 
ic
π1
o1
EE

where gm1 = gm2, rπ1 = rπ2 and βo1= βo2.
ECE 4430 - Analog Integrated Circuits and Systems
 P.E. Allen, 2000
BJT Differential Amplifiers (6/24/00)
Page 11
Common Mode Rejection Ratio (CMRR)
The common mode rejection ratio is a measure of the differential amplifier’s ability to reject the common
mode signal and amplify the differential mode signal.
gm1RC
IEEREE
A dm vo1/vid
2
CMRR = 
=
=
≈
g
R
=
m1 EE
βo1RC
2V t
A cm  vo1/vic 
rπ1+(1+βo1)2REE
Thus, the larger the input transconductance or REE, the larger the common mode rejection ratio.
ECE 4430 - Analog Integrated Circuits and Systems
 P.E. Allen, 2000
BJT Differential Amplifiers (6/24/00)
Page 12
OTHER CHARACTERISTICS OF THE DIFFERENTIAL AMPLIFIER
Input Common Mode Voltage Range
The input common mode voltage range (ICMR) is the range of common mode input voltages over which
the differential amplifier amplifies the differential signal without significant change.
Consider the following:
VCC
VCC
+
IEERC
2
IEE
IEE
2
2
Q1
+
+ vCE(sat)
VBE1 -
RC
vic(max)
Q1
Q2
+
vBE2
-
IEE
VBias
RC
RC
vic(min)
RC
IEE
2
+
v
CE(sat)
+
VBE1 -
VBias
Q3
Q3
VEE
Maximum Input Common Mode Voltage:
vic(max) = VCC - 0.5IEERC -vCE1(sat)+VBE1
ECE 4430 - Analog Integrated Circuits and Systems
VEE
IEE
2
Q2
+
vBE2
-
IEE
+
vCE(sat)
BJTDA03
Minimum Input Common Mode Voltage:
vic(min) = VEE+vCE3(sat)+VBE1
 P.E. Allen, 2000
BJT Differential Amplifiers (6/24/00)
Page 13
Slew Rate of the BJT Differential Amplifier
Slew rate is a voltage rate limit due to the fact that the current available to charge a capacitor is constant.
dvC iC
Slew rate = SR = dt = C
where iC and vC are the current through and voltage across a capacitor C.
A differential amplifier that has a capacitive load will experience slew rate which is seen as follows:
VCC
RC
vOD
+
iC1
Cs
vi1<<0
VCC
IEE
iCs 2
RC
iC2 Cs
Co
Q1
+
vBE1
-
Q2
Situation at
vOD ≈ 0 Cs
iC1=0
+
vBE2
-
vi2>>0
vi1<<0
RC
RC
vOD ≈ 0
vC1 + - vC2
Co
iCo
Q1
+
vBE1
-
IEE
Q2
IEE
2
iCs
iC2=IEE
+
vBE2
-
Cs
vi2>>0
IEE
VEE
VEE
BJTDA09A
Note that the current in Q1 is 0 and Q2 is IEE. Therefore, the initial value of iCo is iCo = 0.5IEE - iCs.
iCo
iCo
dvC1
dvOD
If we define the slew rate across Co as SR =
Co , then iCs = 0.5SR·Cs = 2CoC s , i.e. dt = 0.5 dt
∴
SR =
IEE iCs IEE SR·C s
2Co - Co = 2C o - 2Co
Cs  IEE

⇒ SR 1 + 2C  = 2C
o
o

ECE 4430 - Analog Integrated Circuits and Systems
⇒
0.5IEE
IEE
S R = C +0.5C = 2C +C
o
s
o s
 P.E. Allen, 2000
BJT Differential Amplifiers (6/24/00)
•
•
•
•
•
•
Page 14
SUMMARY
A differential amplifier amplifies the difference signal between two voltages and rejects the common
mode signal
The transconductance characteristics of the BJT differential amplifier switches from all of the current in
one side to the other side within ±100mV
The large-signal differential voltage transfer function has the form of hyperbolic tangent
Emitter degeneration increases the range over which the differential amplifier behaves as a linear
amplifier
The half circuit concept is very useful for analyzing the small-signal differential and common mode
performance
The maximum and minimum input common mode range is:
vic(max) = VCC - 0.5IEERC -vCE1(sat)+VBE1
vic(min) = VEE+vCE3(sat)+VBE1
• The differential amplifier has a slew rate limit of IEE/Ceq where Ceq is the capacitance seen to ground
from either collector.
ECE 4430 - Analog Integrated Circuits and Systems
 P.E. Allen, 2000