BANDGAP reference BANDGAP REFERENCESBANDGAP REFERENCES

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Bandgap References Lausanne Aug 2013 © Barrie Gilbert
Barrie Gilbert Analog Devices
BiCMOS and CMOS
BAND-GAP REFERENCES

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THIS ALL-CMOS DESIGN
uses SPNPs extensively
However, there is an interesting point
of comparison: Often, many Weston cells
were connected in series to average out
individual errors; and here, a similar
idea is used in a band-gap reference:

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2.5002V
2.4998V
A DIGITALLY TRIMMED ALL-CMOS DESIGN
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Preview

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FOUR PARTS
1. BAND-GAP REFERENCE (BGR) PRINCIPLES
2. VARIOUS IMPLEMENTATIONS OF THE BGR
3. SOME ADVANCED BiCMOS EMBODIMENTS
4. BGRs USING STANDARD CMOS

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PART 1
The “Golden EG” – the Band-Gap Energy of Si
Essential Features of the Base-Emitter Voltage
The Rock-Bottom Model of a Bipolar Transistor
Preliminary and Illustrative Design Exercises*
* Some of the mathematics will be found in the Appendix

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Curvature in VBE(T) – Small but Troublesome
Other Non-Ideal Aspects in the Design Process
Elaborations of the Basic Ideas using BiCMOS
Frequency and Pulse Response; Wideband Noise
C
PART 2

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T (K)
0
0
THE ESSENTIAL FORM OF VBE(T)
300K
~ 0.75V
(IC=100A)
EGE
VBE
~1.143V
Why?

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THE SILICON SCAFFOLDING
1.22  1010 electrons/cm3
0.543nm

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INTRINSIC BAND-GAP ENERGY
EGO CAN BE VIEWED AS THE ENERGY
NEEDED TO BREAK A VALENCE BOND
IN A SAMPLE OF PURE SILICON AT T = 0
Electrons
Bound to
Si Atoms
Electron-Hole
Pairs Available
for Conduction
EGO
VALENCE BOND
(Electron Sharing)

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EGO is a property of pure silicon determined using
optical absorption techniques referenced to T = 0.
After the addition of dopants, the band-gap energy
falls with temperature, by a few tens of millivolts.
Thus, our starting point will be a different quantity:
EGE where the little ‘E ’ refers to the ‘Engineering’,
‘Effective’ or ‘Extrapolated’ band-gap energy of a
particular device, obtained by direct measurement
EFFECTIVE BAND-GAP ENERGY

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A SIMPLE EXPLANATION
of why VBE falls linearly with temperature
EGE
T (K)
0
0
TZ
VBE
THIS MUCH ENERGY IS PROVIDED
BY HEAT (kT)
: THE REST MUST BE
PROVIDED BY THE VOLTAGE BIAS
VBE APPLIED TO THE JUNCTION
T
PTAT
CTAT

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AT VERY LOW CURRENTS
VBE falls to zero at moderate temperatures
EGE
T (K)
0
0 TZ  100ºC
VBE
HOW CAN
THIS BE?
HOW CAN IT
NOT BE SO?

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The Temperature-Shaping of voltages and currents
in IC design is always of the greatest significance.
PTAT: Proportional To Absolute Temperature
The “natural language” of bipolar transistors
CTAT: Complementary To Absolute Temperature
The fundamental shape of VBE(T)
ZTAT: Zero sensitivity To Absolute Temperature
Other shapes of practical value include “Super-PTAT ” (varies at
a rate greater than PTAT) and “Ultra-ZTAT ” (having a very high
robustness in production, including excellent supply rejection and
temperature stability, achieved through painstaking attention to
circuit topology, and to numerous, invariably subtle details)
Common “T-Shapes”

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THE ESSENTIAL PRINCIPLE
OF THE BAND-GAP REFERENCE

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T (K)
0
0
THE ESSENTIAL PRINCIPLE
300K
~ 0.75V
~ 0.39V
+ VPT
= VBE
VSUM
EGE
VBE
VPT
~1.143V

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EGE
0
0
300K
~27C
TN
1.143V
T
IN
THE EFFECT OF IC on VBE
VBN
DVBE = kT/q
 some factor
which must be
dimensionless
IC
PTAT
IN is a “normalizing”
value for the general
IC at a “normalizing”
temperature of TN

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Ideal VBE(T,IC) in Current Steps
T (K)
VBE = 0
EGE
VBE(T)
mV
0
540
480
420
360
300
240
180
120
60
600
660
720
780
1080
1020
960
900
1140
840
1mA
100A
10mA
100mA
The slight curvature in VBE is ignored here
1A
10A
1A
1nA
1pA
1fA
335K 407K 518K 712.5K 814K 950K
SLOPE = 3.4mV/K
0.4mV/K
1mV/K

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Actual PNP transistor having LE=100um, WE=2um, EG=1.13, XTI=4.03, IS = 2.644e-16
IC = 10pA, 100pA, 1nA and 10nA. To optimally illustrate the effect, VBC was adjusted
to 52mVZ (10pA), 83mVZ (100pA), 110mVZ (1nA) and 150mVZ (10nA)
An Actual VBE vs. Temperature
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
-240 -200 -160 -120 -80 -40 0 40 80 120 160 200 240
ºC
VBE
1.14
10pAZ
100pAZ
1nAZ
10nAZ
125C
-55C 30C
Negative VBE!
These are not current levels at
which band-gap references operate;
but there are IC products where the
peculiarity of a VBE  0 is important
.

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VBE IS NOT AN ABSOLUTE
The VBE of a BJT is subject to many influences.
 Doping profiles vary from lot to lot
 Photolithography determines size
 Numerous details affect precise value
Furthermore…
 On-chip currents are not exact
(because resistors are uncertain)
Your process is not the same as “my” process

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WE MUST STANDARDIZE
Let’s assume for now this ideal BJT:
 Its effective band-gap is EGE = 1.200V
 Its VBE is 750mV at IC = 100A and 300K
which can be expressed as a certain
saturation current, IS, also at 300K
 DC beta and Early voltages are very high
 BJT resistances are very low
…. and
Our biasing resistors have zero TCR

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Thus, the SPICE parameters for this BJT are:
 EG = 1.2 Volts
 IS = 2.5E-17 Amps
 BF = BR = 1E6
 VAF = VAR = 1E6
This is a huge concession to simplification; but
surprisingly, even with these extremely basic
values, the essential ideas about band-gap
references can be clearly demonstrated. .

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STEPS TOWARD THE
BROKAW REFERENCE

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A TEST CELL
1 emitter
1 by 16um
15 units
of Q1
70mV for
IC2 = 20µA
IC2
IC1
kT/q log(15) =
70mV at 300K

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100A = 100mVP of
DVBE divided by 1k
IC
(A)
48:1
IC1
IC2
A 1-unit and15-unit transistor with a 3.5k resistor
<>i(q1,c) <>i(q2,c)
1e-11
1e-10
1e-9
1e-8
1e-7
1e-6
1e-5
1e-4
.001
.01
350 400 450 500 550 600 650 700 750 800 850
vb, x1e-3
VB (mV)
750mV
100A
708.4mV
20A
IC1
IC2
70mV/3.5k
= 20A
20A 20A
T = 27ºC
The VBE at the intersection is
25.85mV log (100µA/20µA)
= 41.6mV below 750mV
The ratio IC2/IC1 is
15 at low currents
in this example

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<>v(q1_c,q2_c)
-100
-80
-60
-40
-20
0
20
40
60
80
100
x1e-3
350 400 450 500 550 600 650 700 750 800 850
vb, x1e-3
Note the balance point in IC1 and IC2
VB (mV)
VCC
(mV)
708.4mV
VCC

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<>v(q1_c,q2_c) -1:<>v(q1_c,q2_c) -2:<>v(q1_c,q2_c)
-120
-100
-80
-60
-40
-20
0
20
40
60
80
100
120
x1e-3
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 1.1 1.2 1.3 1.4
vb
VB (V)
VCC
(mV)
T = 150ºC
T = 50ºC
T = -50ºC
Now add R2 and check the balance at T extremes
VCC
1.2V
Constant value for VB
for all temperatures

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FINISHING UP….
IC1 IC2
VBE
VPT
Maintain IC1 = IC2
by control of VB
R1
R2

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q1_b q1_b -1:q1_b
1.195
1.196
1.197
1.198
1.199
1.2
1.201
1.202
1.203
1.204
1.205
-40 -20 0 20 40 60 80 100 120 140 160 180
tdegc
q1_b q1_b -1:q1_b
1.195
1.196
1.197
1.198
1.199
1.2
1.201
1.202
1.203
1.204
1.205
-40 -20 0 20 40 60 80 100 120 140 160 180
tdegc
q1_b q1_b -1:q1_b
1.195
1.196
1.197
1.198
1.199
1.2
1.201
1.202
1.203
1.204
1.205
-40 -20 0 20 40 60 80 100 120 140 160 180
tdegc
NICE, BUT
HOW DO WE
CHOOSE THE
VALUES FOR
R1 AND R2?
Close the loop to force the balance point
T ºC
SIMULATED OUTPUT

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A useful (and general) transformation:
sum the PTAT and CTAT quantities in
the current mode
MOVE R2
TO THIS
POSITION
TO GENERATE
LOAD-INDEPENDENT
OUTPUT CURRENT

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THE MAGICAL
VBE

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IC = IS(T)exp VBE / VT VBE = VT log (IC / IS(T))
THE MAGICAL VBE
a voltage-controlled current-source
IS(T)is determined by the base doping
Its transconductance (gm) is IC /VT
and is independent of -
The material (Si, Ge, SiGe, GaAs…)
The device scaling (its dimensions)
The base current is purely incidental
(it is best viewed simply as a “defect”)
VT =
kT/q
IC
VBE
+ -
IS (T)




The BJT is:

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VBE (T,IC ) =
kT IC
log
q IS(T)
This fundamental and widely used formulation
is reliable over huge temperature ranges (-150C to
250C) and collector current IC (typically from 1pA to
1mA) although contact resistances and other effects
cause VBE to exceed this value at higher currents; and
variations in collector-base voltage VCB significantly
alter the VBE. Unlike CMOS, there is no “body effect”.
?
But what is this strange quantity IS(T)

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The Saturation Current IS(T)
If the amazing VBE can be called the ‘heart’ of the
bipolar transistor, then IS(T) must surely be its
‘soul’! This extremely tiny quantity arises from a
complex fundamental expression for ni(T), the
intrinsic carrier concentration – which is the
number of free holes and electrons in a unit
volume generated solely by the thermal energy in
an unbiased semiconductor:
k
h k kT
a -EGO
(Later, we’ll note that this exponent determines the curvature in VBE )
ni
2(T) = 32(p memh)3 T3 exp exp

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The Saturation Current IS(T)
In practice IS(T) cannot be accurately calculated
- or even measured, directly - for use in the
basic expression VBE = kT/q log IC /IS(T).
Instead, the VBE of a representative transistor is
measured at a known temperature and current;
then a developed formulation for VBE is used for
calculating its value at other operating points.

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Measurement of VBE(T,IC)
In BJT modeling, the default collector bias is
VCB = 0, that is, VCE = VBE. Collector current
IC is forced by an electrometer-grade op amp
-VBE
IC
VCB
DUT
+ -
High-Accuracy
Current Source
1pA to 10mA
High-Accuracy
DC Voltmeter
1.00000 V FS
V=0
I=0
A

E
B
C

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T (K)
0
VBE = 0
EGE and IS are obtained from VBE(T)
EGE 1.143V (typically)
VBE(T)
200 400
300
IC1
IC2 > IC1
……using several values
of collector current
IC3 < IC1
* SPICE PARAMETERS USE
BROWN TEXT IN SLIDES
;
polynomial regression
is then used to extract
EG * as well as XTI *

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-VBEN
IN
VCB
+ -
V=0 I=0
VBEN =
kT IN
log
q IS(T)
VBE =
kT IC
log
q IS(T)
IC
-VBE
kT IC
log
q IN
DVBE = - =
kT IC
log
q IS(T)
kT IN
log =
q IS(T)
VBE = VBEN + kT IC
log
q IN

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DVBE is PTAT and close to
60mV/decade at T = 300K
300K
0
0
IC ten times greater
EGE 1.14V
IC ten times smaller
VBE
0
-60mV
+60mV
T

The FACTOR H
T
=
TN
H
1
VBEN
0.73 1.33
T = -55C 27C 126C
TN H
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BUILDING a COMPLETE VBE
VBE = EGE
- H (VBEN- EGE )
H provides a useful mnemonic. By having a value of
1 at TN, it avoids the need to write, and interpret, the
composite factor T/TN in every equation.
ROOT
LINEAR-in-T TERM
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VBE = EGE
- H (EGE- VBEN)
+ H VTN log (IC/IN)
where VTN = kTN /q 25.85mV at TN = 300K
There is at least one more term to add, in order to
model the curvature in VBE(T). This is explained in
the Appendix. A simple, fundamentally correct, circuit
modification can be used to eliminate this curvature.
ROOT
LINEAR-in-T TERM
LOG-in-IC TERM
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1.143V
T (K)
0
0
CURVATURE IN VBE(T)
EGE
VBE(T)
200 400
300
Curvature is
here greatly
exaggerated
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90
80
70
60
50
40
30
20
10
0
-10
-20
-30
TEMPERATURE, ºC
VBE(T) - End fit at T = -30°C and 90°C
~ 1.5mV
57ºC
63ºC
27ºC
CURVATURE IN VBE(T)
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ALTHOUGH THE CURVATURE IN VBE(T) IS
FAIRLY SMALL, IT SIGNIFICANTLY AFFECTS
THE DESIGN OF BANDGAP REFERENCES &
MUST BE INCLUDED IN THE FULL THEORY.
CURVATURE IN VBE(T)
ITS MAGNITUDE IS
- (hkT/q) log (T/TN) = - hVTN log
where the factor h is typically between 3.5 and 5.0
ALTHOUGH THE CURVATURE IN VBE(T) IS
FAIRLY SMALL, IT SIGNIFICANTLY AFFECTS
THE DESIGN OF BANDGAP REFERENCES &
MUST BE INCLUDED IN THE FULL THEORY.
H H
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H*(log(H))
-.4
-.2
0
.2
.4
.6
.8
1
1.2
1.4
-4*25.95m*H*(log(H))
-150
-125
-100
-75
-50
-25
0
25
50
x1e-3
0 .2 .4 .6 .8 1 1.2 1.4 1.6 1.8 2
h, from 0.001 to 2
The raw function
Error in VBE(H)
over entire range
mV
H

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VBE = EGE
- H (EGE- VBEN)
+ VTN H log (IC/IN)
ROOT
LINEAR-in-T TERM
LOG-in-IC TERM
VTN = kTN/q (25.85mV at 300K), H = T/TN
- hVTN H log H CURVATURE TERM
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EXAMPLES OF
BGR DESIGN
IN BiCMOS

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APPENDIX
COMPLETE SOLUTION
FOR ACHIEVING A
TEMPERATURE-STABLE
WORKING-POINT

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A FEW REMINDERS
EGE IS THE “EXTRAPOLATED” BAND-GAP
VOLTAGE; IT IS SLIGHTLY LOWER THAN
THE INTRINSIC SILICON VALUE, EGO AND
DETERMINED IN PRACTICE BY CURVE-
FITTING TO MEASURED VBE(T,IC) DATA.
IN MOST SIMULATORS, SUCH AS SPICE,
THIS VERY IMPORTANT QUANTITY IS
REPRESENTED BY THE PARAMETER EG,
TYPICALLY 1.143V FOR A MODERN BJT.

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VBN IS THE MEASURED VALUE OF VBE AT SOME
CHOSEN ‘NORMALIZING’ COLLECTOR CURRENT IC =
IN, AND AT A TEMPERATURE T = TN. TO CLARIFY THE
MATH, WE WILL USE THE FACTOR H = T/TN.
VTN IS THE FIXED THERMAL VOLTAGE kTN/q (THAT IS,
kT/q AT H = 1).
THE CURVATURE TERM, DENOTED BY h IN THESE
EQUATIONS, IS THE EXPONENT OF TEMPERATURE IN
THE EXPRESSION FOR ni2(T). IDEALLY IT IS EXACTLY
3, BUT IN PRACTICE IT IS HIGHER (USUALLY IN THE
RANGE 3.5 TO 5) AND IT IS DETERMINED BY CURVE-
FITTING VBE(T,IC). IT CORRESPONDS TO THE SPICE
PARAMETER XTI.

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VBE(H,IC) = EGE - H [(EGE-VBN) - VTN {log (IC/IN) - h log H }]
IN ITS MOST COMPACT FORM (AND STRICTLY, FOR VBC = 0)
ROOT LINEAR-in-T LOG-in-IC CURVATURE
TERM TERM TERM
However, in refining bandgap references it is necessary to
also account for several other effects not included here,
such as the excess VBE caused by IE and IB flowing in the
emitter and base resistances (IC ree´/a + IC rbb´/b) and the
reduction in VBE due to VAF when VCB > 0 (= VBCVTN /VAF).
WE MUST START WITH VBE

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VBE´ = VBE IC{ + }
IC
VCB
+ -
V=0 I=0
EFFECT of ree´ & rbb´ on VBE
A =

ree´
rbb´
IC
a
IC/b
VBE ree´ rbb´
a b
Example:
For ree´= 10, rbb´= 100, b =100 and IC = 100A,
the excess is 1.11mV, or roughly 0.1% of a 1.2V VREF
101A
100A
10
100
1A
+ 1.11mV

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IC
+ -
V=0 I=0
EFFECT of VBC on VBE due to VAF
A =

E C
n n
p
VBE
V
BC
=
0
VBE
VBE
VCB
Example:
For VAF= 25, T = 300K, the reduction is 1.034mV/V,
or more than 0.2% of a 1.2V VREF when VCB = 2.4V
VBE´
V
BC
=
0
VBE´ = VBE-
VTNVCB/VAF
NOTE:
Slope of
charge in
base is fixed
by the current

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AS THE NEXT STEP, NOTE THAT THE COLLECTOR CURRENT IN
MOST BANDGAP REFERENCES WILL BE PTAT, AS A NATURAL
OUTCOME OF THE DESIGN AND OPERATING CONDITIONS.
SO LET’S WRITE
IC = l H IN
THE FACTOR l DENOTES THE RATIO OF THE ACTUAL IC TO
THE NORMALIZING CURRENT IN. IT MAY JUST HAPPEN TO BE
UNITY BUT IT CAN BE MADE TO BE SO, SIMPLY BY CHOOSING
TO MEASURE VBN AT IC. BUT THAT HAS USUALLY ALREADY
BEEN DONE DURING THE PROCESS CHARACTERIZATION.
BUILDING THE BAND-GAP SOLUTION

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VBE(H,IC) = EGE - H [(EGE - VBN) - VTN ( log lH - h log H )]
BY SUBSTITUTING l = IC /IN (at H=1) WE HAVE
VBE(H,IC) = EGE - H (EGE - VBN - VTN log l) - (h-1)VTN H logH
THEN
NOW, VBN - VTN log l IS SIMPLY THE VALUE OF VBN AT THE
OPERATING CURRENT IC =lHIN . IN ORDER TO SIMPLIFY THE
MATH, WITHOUT ANY LOSS OF GENERALITY, WE CAN MAKE l
= 1 BY DEFINING VBN AS THE VALUE AT IC RATHER THAN AT IN

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THIS GETS US TO
VBE(H,IC) = EGE - H (EGE - VBN) - (h-1)VTN H logH
WE NOW HAVE TO ADD A PTAT VOLTAGE VPT TO THIS VBE
TO GENERATE THE DESIRED REFERENCE OUTPUT, VSUM:
VSUM = EGE - H (EGE - VBN) - (h-1)VTN H logH + VPT
VSUM = EGE-H (EGE-VBN - VPTN) - (h-1)VTN H logH
THAT IS, + H VPTN

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WE WISH THIS TO BE ZERO, BUT NOTE THAT IT CANNOT BE
COMPLETELY INDEPENDENT OF H. SO WE’LL SOLVE SO AS
TO MAKE THE DERIVATIVE ZERO AT H = 1, where logH = 0.
DIFFERENTIATING WITH RESPECT TO H :
VSUM/H = - (EGE-VBN - VPTN)- (h-1)VTN (1+logH )
VSUM = EGE-H (EGE-VBN - VPTN) - (h-1)VTN H logH
0 = - (EGE-VBN - VPTN) - (h-1)VTN
SO THE REQUIRED VALUE OF THE PTAT VOLTAGE AT T = TN
IS VPTN = (EGE-VBN ) + (h-1)VTN

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NOTING logH = 0 FOR H = 1, WE HAVE ARRIVED AT THE
FORMULATION FOR THE OPTIMUM VALUE OF A STANDARD
BANDGAP REFERENCE OUTPUT, VBG, HAVING ZERO SLOPE
vs TEMPERATURE AT T = TN:
VBG = EGE + (h- 1)VTN
VSUM (H ) = VBN + VPTN =
VBN + (EGE-VBN ) + (h-1)VTN ( 1 - H logH )

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VPTN = EGE - VBN + (h - 1)VTN
1.143V
T
300K, TN
VBN
EGE
FOR EGE = 1.143V, VBN = 0.75V (IS = 2.57E-17A @ T = 300K)
AND h = 3.98, WE FIND VPTN = 0.47V, AND VSUM = 1.220V
VSUM
THE EXCESS OVER
EGE IS (h - 1)VTN
(HERE, 77.03mV) AND
IS INDEPENDENT OF
VBN , HENCE IC

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VCURV = VSUM - VBG
THE SMALL CURVATURE IN VBE, AND THUS IN VSUM,
(THE PTAT VOLTAGE IS ASSUMED TO BE PERFECTLY
LINEAR WITH TEMPERATURE) HAS THE MAGNITUDE
= (h-1) VTN {H ( 1 - logH ) - 1}
WHICH, OVER THE MODERATE RANGE 0.8  H  1.2
(T = -33°C TO +87°C), CAN BE APPROXIMATED BY
VCURV = (1 - h) (VTN /2) (H - 1)2

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T = -55C
Detailed curvature and approximation, for h =3.98
-4
-3.75
-3.5
-3.25
-3
-2.75
-2.5
-2.25
-2
-1.75
-1.5
-1.25
-1
-.75
-.5
-.25
0
.725 .8 .875 .95 1.025 1.1 1.175 1.25 1.325
H, from 0.727 to 1.327
T = 27C
T = -33C T = 87C
T = -55C T = 125C
-4.25
mV Approximation
Full equation
The SPICE
variable XTI

65
ANALOG
DEVICES
A development of the previous idea generates
VOUT = 250mVZ from VSUP > 1.1V @ ~100A
Optional
enable

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