Barrie Gilbert Analog Devices - PowerPoint Presentation

Barrie Gilbert Analog Devices B i CMOS and CMOS B AND -G AP R EFERENCES

PLEASE READ THIS BEFORE ATTENDING The following notes are provided mainly for study ahead of the presentation. Accordingly, be well prepared to raise your hand whenever a question is asked of you, the audience. The aim is to provide a genuine learning – not merely a passive listening – experience. It is well-known that we learn mainly by doing. This invariably entails going down many wrong paths before the deepest insights arise, and one’s personally-forged solutions can be adopted.Not all of the following slides will be shown, and more recent material might be presented.

Bandgap Voltage References range from simple, almost casual, low-accuracy cells – sufficient to meet numerous everyday and incidental biasing purposes – to advanced solutions of excellent accuracy. Today’s best designs cannot match the performance of a tightly-temperature-stabilized zener reference (much less that of the Josephen junction voltage standard) but they offer the only solution in modern low-voltage systems. There is an increasing demand for voltage references using CMOS processes at all technology nodes, often operating from sub-bandgap supply voltages at currents in the low microamp range. However, as we shall see, these are fundamentally noisy. Preview

2.5002V 2.4998V A DIGITALLY - TRIMMED CMOS DESIGN A N A LO G DEVICES Preview

THIS EARLY ALL-CMOS DESIGN uses SPNPs extensively.Later BiCMOS designs eliminate the need for the amplifiersand include the means for refined curvature-correction . Preview

FOUR PARTS 1. SOME UNAVOIDABLE BASIC THEORY 2. EARLY IMPLEMENTATIONS OF THE BGR 3. ADVANCED BiCMOS EMBODIMENTS 4. REFERENCES USING STANDARD CMOS

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 Much of the mathematics has been moved to an Appendix

T (K) 0 0 T HE E SSENTIAL F ORM OF VBE (T) 300K ~ 0.75V (I C =100  A) E GE V BE ~1.143V Why?

T HE SILICON SCAFFOLDING 1.22  1010 electrons/cm3 0.543nm

I N T RI N S I C BAND-GAP ENERGY EGO CAN BE VIEWED AS THE ENERGYNEEDED TO BREAK A VALENCE BONDIN A SAMPLE OF PURE SILICON AT T = 0 Electron-Hole Pairs Available for Conduction E GO 0.235 nm VALENCE BOND (Electron Sharing) Electrons are tightly bound to Si Atoms

INTRINSIC CARRIER GENERATION Electrons are tightly bound to Si Atoms No carriers a vailable for c onduction Electrons g iven the t hermal energy t o escape Electron-hole pairs available for conduction Very high r ate of e lectron release ? T = 0 T = 300K T = 600K EQUILIBRIUM

E GO is a property of pure silicon determined usingoptical absorption techniques referenced to T = 0.With the addition of dopants, the band-gap energyis slightly reduced – by a few tens of millivolts.Our starting point needs to be a different quantity:E GE where the little ‘E ’ refers to the ‘Engineering’ ,‘Effective’ or ‘Extrapolated’ band-gap energy of a particular device, obtained by direct measurement E FFECTIVE B AND- G AP E NERGY

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

AT VERY LOW CURRENTS VBE falls to zero at moderate temperatures E GE T (K) 0 0 T Z  100 º C V BE HOW CAN THIS BE? HOW CAN IT NOT BE SO?

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 Common “T-Shapes” Other shapes of practical value include “Super-PTAT ” (varying at a rate greater than PTAT) and “Ultra-ZTAT ” (having a very high robustness in production, including excellent supply rejection and temperature stability, achieved only through painstaking attention to circuit topology, and to numerous, invariably very subtle details)

Carefully annotating voltages and currents on yourschematics while also indicating their T-Shape is auseful habit: it conveys important biasing ideas forboth immediate and future reference. PTAT: 100uAP 123mVP etcSuper-PTAT: 100uAsP 321mVsP etcCTAT: 100uAC 750mVC etc ZTAT: 100uAZ 500mVZ etc Ultra-ZTAT: 100uAZZ 1.250VZZ etc The special importance of the PTAT designation was first presented in my 1976 JSSC paper entitled A Monolithic Voltage-to-Frequency Converter . Some years later, Paul Brokaw began using ‘ CTAT’, and shortly thereafter, I added ZTAT and various other T-Shape designations as the needs arose. Common “T-Shapes”

THE FUNDAMENTAL PRINCIPLESOF THE BAND-GAP REFERENCE

T (K) 0 0 300K ~ 0.75V ~ 0.39V + V PT = V BE V SUM E GE V BE V PT ~1.143V T HE E SSENTIAL I DEA

E GE 00300K~27 C T N 1.143V T I N T HE E FFECT OF I C on V BE V BN D V BE = k T /q  some factor which must be dimensionless I C PTAT I N is a “normalizing” value for the general I C at a “normalizing” temperature of T N (It’s simple geometry)

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

Actual PNP transistor with LE=100um, WE=2um, EG=1.13, XTI=4.03, IS = 2.644e-16. IC = 10pA, 100pA, 1nA and 10nA. To optimally illustrate this characteristic, the 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 º C V BE 1.14 10pAZ 100pAZ 1nAZ 10nAZ 125  C -55  C 30C Negative V BE ! These are not the current levels at which band-gap references operate; but there are IC products where the peculiarity of a VBE  0 is important .

V BE 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 On-chip currents are never exact (because resistors are uncertain) Your process is not the same as “my” process

… SO WE MUST IDEALIZE Assume – just for now - that for an ideal BJT: Its effective band-gap is E GE = 1.200V Its VBE is 750mV at IC = 100A and 300K (which can be expressed as a certain saturation current, I S, also at 300K) DC beta and Early voltages are very high (let’s just say a million) BJT resistances are very low (let’s say zero, for now) …. and our on-chip resistors have zero TCR

The entered SPICE parameters for this ideal BJT might thus be: EG = 1.2 Volts IS = 2.5E-17 Amps BF = BR = 1E6 VAF = VAR = 1E6 Although this is a huge concession to simplification you’ll find – perhaps surprisingly – that even with these extremely simplified values, all of the essential ideas about band-gap references can be clearly revealed. . (All the rest take on their default values)

T HE MAGICAL VBE

IC = IS(T) exp VBE / VT VBE = VT log (IC / IS(T) ) THE MAGICAL V BE V T = kT/q I C V BE + - I S (T)     a voltage - controlled current - source I S (T) is determined by the base doping Its transconductance ( g m ) is I C / V T 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”) The BJT is:

IC = IS(T) exp VBE / VT VBE = VT log (IC / IS(T) ) THE MAGICAL V BE V T = kT/q I C V BE + - I S (T)     a voltage - controlled current - source I S (T) is determined by the base doping Its transconductance ( g m ) is I C / V T and is independent of - The material (Si, Ge , SiGe , GaAs …) The device scaling (its dimensions) The BJT is: Its transconductance ( g m ) is I C / V T and is independent of - The material (Si, Ge , SiGe , GaAs …) The device scaling (its dimensions) Stated differently, r e = V T / I C Note carefully that the r e is not a resistance but simply a derivative having the dimension of resistance

V BE (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 I C (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 I S (T)

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 n i(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 k T a - E GO (Later, we ’ ll note that this exponent determines the curvature in V BE ) n i 2 (T) = 32( p m e m h ) 3 T 3 exp exp

The Saturation Current IS(T) In practice IS(T) cannot be accurately calculated - or even measured - for utilizing in the basic expression VBE = kT/q log I C / IS (T). Instead, the V BE 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.

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

T (K)0 VBE = 0 EGE and IS are obtained from VBE (T) E GE 1.143V (typically) V BE (T) 200 400 300 I C1 I C2 > I C1 …… using several values of collector current I C3 < I C1 ; polynomial regression is then used to extract EG as well as XTI SPICE PARAMETERS WILL BE SHOWN IN BROWN IN SLIDES * * *

- V BEN I N V CB V=0 I=0 V BEN = k T I N log q I S (T ) V BE = k T I C log q I S (T ) I C - V BE T HE E FFECT OF I C on V BE D V BE = - = k T I C log q I S (T ) k T I N log q I S (T ) D V BE = k T I C log q I N + -

D V BE is PTAT and close to 60mV/decade at T = 300K 300K 0 0 I C ten times greater E GE 1.14V I C ten times smaller V BE 0 -60mV +60mV T T HE E FFECT OF I C on V BE

The FACTOR H T = T N H 1 V BEN 0.73 1.33 T = -55  C 27  C 126  C T N H Bandgap References Lausanne Aug 2013 © Barrie Gilbert A N A LO G DEVICES

35 B UILDING a COMPLETE VBE V BE = EGE - H (V BEN - EGE ) H provides a useful mnemonic. By having a value of1 at TN, it avoids the need to write, and interpret, thecomposite factor T/T N in every equation. ROOT LINEAR - in - T TERM Bandgap References Lausanne Aug 2013 © Barrie Gilbert A N A LO G DEVICES

36 V BE = EGE - H (EGE - VBEN ) + H VTN log (IC /IN) where V TN = k T N / q 25.85mV at TN = 300K There is at least one more term to add, in order to model the curvature in V BE ( T) as discussed more fully in the Appendix. Note that fundamentally correct circuit modification can be used to eliminate this curvature. ROOT LINEAR - in - T TERM LOG - in - I C TERM B UILDING a COMPLETE VBE Bandgap References Lausanne Aug 2013 © Barrie Gilbert A N A LO G DEVICES

37 1.143V T (K) 0 0 CURVATURE IN V BE (T) E GE V BE ( T) 200 400 300 Curvature is here greatly exaggerated Bandgap References Lausanne Aug 2013 © Barrie Gilbert A N A LO G DEVICES

38 ALTHOUGH THE CURVATURE IN VBE(T) IS FAIRLY SMALL, IT SIGNIFICANTLY AFFECTSTHE DESIGN OF BANDGAP REFERENCES &MUST BE INCLUDED IN THE FULL THEORY. CURVATURE IN V BE (T) ITS MAGNITUDE IS - ( h k T /q ) log (T/TN) = - hVTN log where the factor h is typically between 3.5 and 5.0 ALTHOUGH THE CURVATURE IN V BE (T) IS FAIRLY SMALL, IT SIGNIFICANTLY AFFECTSTHE DESIGN OF BANDGAP REFERENCES & MUST BE INCLUDED IN THE FULL THEORY. H H A N A LO GDEVICES Bandgap References Lausanne Aug 2013 © Barrie Gilbert

The raw function H log H Error in V BE (H) over wide range mV H

40 90 80 70 60 50 40 30 20 10 0 -10 -20 -30 TEMPERATURE, º C V BE ( T) - End fit forced at T = -30°C and 90°C ~ 1.5mV 57 º C 63 º C 27 º C TYPICAL CURVATURE IN V BE (T) Bandgap References Lausanne Aug 2013 © Barrie Gilbert A N A LO G DEVICES

41 B UILDING a COMPLETE VBE V BE = EGE - H (E GE- VBEN ) + VTN H log (I C /I N) ROOT LINEAR - in - T TERM LOG - in - I C TERM VTN = k TN / q (25.85mV at 300K), H = T/ TN - h V TN H log H CURVATURE TERM A N A LO G DEVICES Bandgap References Lausanne Aug 2013 © Barrie Gilbert

THE LITTLE MATTER OF BJT NOISE

Ideal Noise of the VBE (T, IC) IC + - E B C The ‘shot noise’ unavoidably a ssociated with the collector current is  2qI C per root-hertz The ‘ r e ’ is simply kT / qI C Thus, neglecting the BJT’s alpha, the NSD of the base-emitter voltage is kT / qI C   2qI C = kT  2/ qI C This is the noise spectral d ensity (NSD)

A slight rearrangement for memorizing isSVBE = kT /q  2q/IC A more compact form for memorizing is SVBE = 13.5 pA / IC at T = 300K, with IC expressed in amps For example when I C = 1µA, SVBE = 14.63nV (per  Hz) Ideal Noise of the V BE (T, IC )

Actual Noise of a VBE (T, IC)Q1 is a practical transistor model havinga single 1µm-wide by 50µm-long emitterfor which the base resistance is 47.5

BLUE: IDEAL GRN: 50µ/1µRED: 5µ/1µ LARGE-SIGNALBETAV/Hzf = 10kHz

IC = 1mAIC = 1µAV/Hz V rms?

SRES = 129 pV   R All you need to know (for now)about resistor (Johnson) noise at T = 300K with R expressed in ohms For example, when R = 50 SRES = 912nV (per Hz )It increases with the square-root of temperature

IC = 1mAIC = 1µAV/Hz V rmsThis excess noiseis attributable to anrbb of 47.5 ohms Total SRES =SQRT ( (463pV)2 + (889pV) 2 ) = 1.002 nV

STEPS TOWARD THE BROKAW REFERENCE

A TEST CELL 1 emitter 1 by 16um 15 units of Q1 70mV for I C2 = 20 µ A I C2 I C1 kT /q log(15) = 70mV at 300K

100  A = 100mVP ofVBE divided by 1k I C (A) 48:1 I C1 I C2 A 1-unit and15-unit transistor with a 3.5k  resistor V B (mV) 750mV 100  A 708.4mV 20  A I C1 I C2 70mV/3.5k  = 20  A 20  A 20  A T = 27 º C The V BE at the intersection is 25.85mV log (100µA/20µA) = 41.6mV below 750mV The ratio I C2 /I C1 is 15 at low currents in this example

Note the balance point in I C1 and I C2 V B (mV) V CC (mV) 708.4mV V CC

V B (V) VCC(mV) T = 150 º C T = 50 º C T = -50 º C Now add R2 and check the balance at T extremes V CC 1.2V Constant value for V B for all temperatures

FINISHING UP…. I C1 I C2 V BE V PT Maintain I C1 = I C2 by control of V B R1 R2

This is the essential form of the well-known Brokaw Band-gap Voltage Reference. It ranks among the most elegant of all analog circuits and is the most readily understood. It has the further advantage of being analytic – that is, the output voltage, the output impedance and the output noise spectral density and other key aspects of performance can all be stated in closed form .

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

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

End of Part 1The several slides that follow (but will not be discussed) explain how to design the Brokaw Bandgap reference for minimum variation in output around a chosen temperature .

A PPENDIX COMPLETE SOLUTIONFOR ESTABLISHING ATEMPERATURE-STABLE WORKING-POINT IN ABROKAW BANDGAP

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.

VBN IS THE MEASURED VALUE OF VBE AT A CHOSEN ‘NORMALIZING’ COLLECTOR CURRENT IC = IN, AND AT A TEMPERATURE T = TN. TO CLARIFY THE MATH, WE WILL USE THE SIMPLIFYING FACTOR H = T/TN. VTN IS THE FIXED THERMAL VOLTAGE kTN/q (THAT IS, THE VALUE OF kT /q AT H = 1). THE CURVATURE TERM, DENOTED BY  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 (TYPICALLY IN THE RANGE 3.5 TO 5) AND IT IS DETERMINED BY CURVE-FITTING VBE(T,IC) . IT CORRESPONDS TO THE SPICE PARAMETER XTI.

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

V BE´ = VBE IC { + } IC V CB + - V=0 I=0 E FFECT of ree ´ & rbb ´ on V BE A =  ree ´ rbb ´ I C a I C / b V BE ree ´ rbb ´ a b Example: For ree ´ = 10 W , rbb ´ = 100 W , b =100 and I C = 100 m A, the excess is 1.11mV, or roughly 0.1% of a 1.2V V REF 101 m A 100 m A 10 W 100 W 1 m A + 1.11mV

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

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 I N. IT MAY JUST HAPPEN TO BE UNITY BUT IT CAN BE MADE TO BE SO, SIMPLY BY CHOOSING TO MEASURE V BN AT IC. BUT THAT HAS USUALLY ALREADY BEEN DONE DURING THE PROCESS CHARACTERIZATION. B UILDING THE B AND- G AP SOLUTION

V BE(H, IC) = EGE - H [(E GE - VBN) - VTN ( log lH - h log H )] BY SUBSTITUTING l = IC / IN (at H=1) WE HAVE V BE ( H , I C ) = E GE - H (EGE - VBN - VTN log l) - (h - 1) VTN H log H THEN NOW, V BN - V TN log l IS SIMPLY THE VALUE OF VBN AT THE OPERATING CURRENT IC = l H IN . TO SIMPLIFY THE MATH, WITHOUT LOSS OF GENERALITY, WE CAN MAKE l = 1 BY USING VBN FOR A CHOSEN VALUE OF IC RATHER THAN AT I N . B UILDING THE B AND- GAP SOLUTION

THIS GETS US TO V BE ( H , I C) = EGE - H ( EGE - VBN ) - ( h - 1) VTN H log H WE NOW HAVE TO ADD A PTAT VOLTAGE V PT TO THIS VBE TO GENERATE THE DESIRED REFERENCE OUTPUT, V SUM : V SUM = E GE - H ( EGE - V BN) - (h - 1 ) VTN H log H + VPT V SUM = E GE - H ( EGE - V BN - VPTN ) - (h - 1) VTN H log H THAT IS, + H V PTN B UILDING THE B AND- G AP S OLUTION

WE WISH THIS TO BE ZERO, BUT NOTE THAT IT CANNOT BE COMPLETELY INDEPENDENT OF H. SO WE’LL SOLVE SO ASTO MAKE THE DERIVATIVE ZERO AT H = 1, where log H = 0 . DIFFERENTIATING WITH RESPECT TO H :  V SUM /  H = - (E GE - VBN - VPTN) - (h - 1) V TN (1+ log H ) V SUM = E GE - H (EGE - VBN - V PTN) - ( h - 1) VTN H log H 0 = - ( E GE - VBN - V PTN) - ( h - 1) VTN SO THE REQUIRED VALUE OF THE PTAT VOLTAGE AT T = T N IS V PTN = ( E GE - VBN ) + ( h - 1) VTN B UILDING THE B AND- G AP SOLUTION

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: V BG = E GE + (h - 1) V TN V SUM ( H ) = V BN + VPTN = VBN + (EGE - VBN ) + (h - 1) VTN ( 1 - H log H ) B UILDING THE B AND- GAP SOLUTION

V PTN = EGE - VBN + (h - 1)VTN 1.143V T 300K, T N V BN E GE FOR E GE = 1.143V , V BN = 0.75V ( I S = 2.57 E - 17 A @ T = 300K) AND h = 3.98, WE FIND V PTN = 0.47V, AND V SUM = 1.220V V SUM THE EXCESS OVER E GE IS ( h - 1) V TN (HERE, 77.03mV) AND IS INDEPENDENT OF V BN , HENCE I C B UILDING THE B AND- G AP S OLUTION 1.220V

V CURV = VSUM - VBG THE SMALL CURVATURE IN VBE, AND THUS IN VSUM, (THE PTAT VOLTAGE IS ASSUMED TO BE PERFECTLYLINEAR WITH TEMPERATURE) HAS THE MAGNITUDE = ( h - 1 ) VTN {H ( 1 - log H ) - 1} WHICH, OVER THE MODERATE RANGE 0.8  H  1.2 (T = -33°C TO +87°C), CAN BE APPROXIMATED BY V CURV = ( 1 - h ) (VTN / 2) ( H - 1 ) 2 B UILDING THE B AND- G AP SOLUTION

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 B UILDING THE B AND- G AP S OLUTION