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Modeling The Bipolar Transistor For Bandgap Reference Simulations

Michael O. Peralta , Medtronic , December 8, 2010

• It is common to model bipolar transistors by curve fitting measured vs

simulated data for "ic vs vbe", "beta vs vbe", etc.

• Bandgap circuit designs can still have significant differences between the

actual measured circuit behavior and simulation.

• This usually happens because the measured vs simulated temperature

behavior of BEV∆ vs T and VBE vs T is not directly optimized (curve fit).

• Here we will show how to include BEV∆ vs T and VBE vs T behaviors so that

the bjt model properly emphasizes the temperature fit for bandgap circuits.

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VBE Difference At Different Collector Currents

The equations for the same physical bipolar transistor in low level current injection at two collector currents at different VBE can be very accurately described as:

TBE VV

SC eII /1

1⋅= 1-1

TBE VV

SC eII /2

2⋅= 1-2

where q

TkVT

⋅= 1-3

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Select IC2 and IC1 such that their ratio is exactly m

meI

ITBEBE VVV

C

C == − /)(

1

2 12

“exactly” 1-4

Define 12 BEBEBE VVV −=∆ 1-5

Then using equations 1-3 and 1-4 we see that,

)(ln)(ln12 mq

kTmVVVV TBEBEBE ==−=∆ 1-6

As an example we will select nAIC 11 = and nAIC 102 = . Since we selected

mI

I

C

C =1

2 we see that 112 10 CCC IImI == 1-7

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Using Linear Interpolation To Assure That m = 10 Exactly

• When measuring gummel (icvb) curves, VBE is incremented in steps and IC

current is measured.

• This means that hitting IC1 and IC2 at exactly 1nA and 10nA is very unlikely.

• Hence we will use interpolation to find the VBE’s that correspond to exactly 1nA

and 10nA.

• Because of the exponential relationship between VBE and IC we will use values

of ln(IC) and VBE in our linear interpolation scheme.

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Figure 1. ln(IC) , VBE Interpolation

• As depicted in Figure 1 above, to determine VBE1 we find the IC1 just below 1nA

which we label IC1A (the corresponding VBE is labeled VBE1A); and the IC1 equal

to or greater than 1nA is labeled IC1B (the corresponding VBE is labeled VBE1B).

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• From these pairs -- (ln(IC1A),VBE1A) and (ln(IC1B),VBE1B) -- we use linear

interpolation to get VBE1 which corresponds to exactly IC1 = 1nA.

• For the IC2 = 10nA case we use the same linear interpolation scheme to obtain

the VBE2 value corresponding to exactly IC2 = 10nA.

Using equation 1-6 we then obtain

)(ln)1()10( 12 mq

TknAatVnAatVV BEBEBE =−=∆ 1-8

• Notice that both VBE1 and VBE2 are for the same transistor.

• Since the measurement sweep (depicted in Figure 1) is rapid and operated at

low power (1nA and 10nA), then VBE1 and VBE2 are practically at the same

temperature, T.

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• By contrast, in a bandgap circuit application, the different currents are created

by two physically separate transistors -- which can be affected by device-to-

device mismatch.

• At another temperature, 'T , we keep exactly the same ratio between 2'CI and

1'CI as we did for temperature T -- that is mII CC =12 / (Eq. 1-4):

me

I

ITBE VV

C

C == ∆ /'

1

2

''

1-9

)(ln'

)(ln'''' 12 mq

TkmVVVV TBEBEBE ==−=∆ 1-10

• Again we use linear interpolation at the new temperature, 'T , to get 1'BEV and

2'BEV at exactly 1nA and 10nA respectively, thereby assuring that we have

exactly m = 10.

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Using Several Temperatures To Curve Fit Bandgap BEV∆ vs T and VBE vs T

• Say we take measurements at 4 different temperatures ''','',', TTTT then we

would have 4 different sVBE '∆ for measured and corresponding simulated

sVBE '∆ :

Measured ''','',', ____ MBEMBEMBEMBE VVVV ∆∆∆∆ 1-11

Simulated ''','',', ____ SBESBESBESBE VVVV ∆∆∆∆ 1-12

• At this point the simulated sVBE '∆ are curve fit (optimized) to the corresponding

measured sVBE '∆ . Figure 2 illustrates the curve fitting of the measured vs

simulated sVBE '∆ . As seen through Eq. 1-8, the sVBE '∆ are Proportional To

Absolute Temperature (PTAT).

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Figure 2. Curve Fit Measured vs Simulated sVBE '∆

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• When we curve fit the measured vs simulated BEV∆ curve we also curve fit the

icvb (gummel curves), and the other measured vs simulated curves that are

usually curve fit over temperature.

• Several curves are curve fit simultaneously by assigning different "weights" to

each different curve set.

• Since the BEV∆ measured vs simulated curve set has only 4 or 5 data points

(temperatures) we weight that curve by a factor of 100 to 300 compared to the

other typical curves (icvb =forward gummel curves, etc.)

• These other typical curves (icvb etc.) usually contain hundreds of data points

and so they are given a weight of about 1.

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• It is important to note that, in modern simulators (spectre, hspice, etc.), in

addition to the "usual" temperature parameters (xti, xtb, etc.) there are several

parameters that are also given temperature dependencies (usually first and

second order). Refer to Appendix B.

• Among some of the parameters that are given temperature coefficients are:

nf, nr, nc, ne, vaf, var, bf, br, ikf, ikr. This helps immensely in getting a good fit

while keeping the parameters as close to physical as possible.

• As well as curve fitting the measured vs simulated sVBE '∆ we also curve fit the

measured vs simulated VBE's over the temperatures measured (VBE vs T

behavior). This is illustrated in Figure 3 below.

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Figure 3. Curve Fit Measured vs Simulated sVBE '

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• Again, since we will only have 4 or 5 data points (temperatures) we also weight

the VBE measured vs simulated curve set by a factor of 100 to 300 compared

to the other typical curves. Notice that this curve fit is also simultaneously

performed with the BEV∆ curve fit.

• Before performing the simultaneous BEV∆ vs T and VBE vs T curve fits, we

perform curve fits at room temperature to extract ic vs vbe , beta vs vbe,

Forward Early Voltage (VAF), Reverse Early Voltage (VAR), and others as

deemed appropriate. This gives an extraction "baseline" for parameters that do

not need to be co-extracted with the BEV∆ vs T and VBE vs T curve fitting.

• It also prepares the way so that the BEV∆ vs T and VBE vs T curve fits

converge faster to the final extraction results.

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Advantage of Bandgap Centric Curve Fit Approach

• The advantage of simultaneously curve fitting the BEV∆ vs T and VBE vs T

curves with the typical (icvb, beta, etc.) curves, assures that the parameters

that affect bandgap circuit behavior is emphasized. Without this approach, it is

not likely that the model parameters that affect bandgap circuit behavior will be

optimized for bandgap circuits.

• The advantage of the Bandgap Centric approach is illustrated by comparing

Figures 4,5,6 (not bandgap centric) against Figures 7,8,9 (bandgap centric).

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Figure 4. ∆VBE Curve Fit Without Bandgap Centric Approach

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Figure 5. VBE(meas) - VBE(sim) Curve Fit Without Bandgap Centric Approach

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Figure 6. IE vs vb Curve Fit Without Bandgap Centric Approach

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Figure 7. ∆VBE Curve Fit With Bandgap Centric Approach

Compare this with Figure 4 which does not use the bandgap centric approach.

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Figure 8. VBE(meas) - VBE(sim) Curve Fit With Bandgap Centric Approach

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Figure 9. IE vs vb Curve Fit With Bandgap Centric Approach

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• It is important to notice that the IE vs VB curve fits (and the other typical

modeling curves) may turn out good with or without the bandgap centric

curve fit approach -- compare Figures 6 and 9.

• However, to get good curve fits for bandgap circuit applications we need

good curve fits for the BEV∆ vs T and VBE vs T curves -- as we can see

by comparing Figures 4 and 5 (not bandgap centric) against Figures 7 and 8

(bandgap centric).

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Conclusion

• By directly including the BEV∆ vs T and VBE vs T temperature behaviors in

the modeling curve fit procedures, parameters can be extracted so that

bandgap circuit designs can be simulated with a higher degree of accuracy.

This method does require that the modeling software used have the capability

to properly "weight" the BEV∆ vs T and VBE vs T curves so these curves are

emphasized enough in the midst of the curve fit optimizations.

• The accuracy of the method is enhanced by the fact that for each specific

temperature, both the VBE1 and VBE2 used to determine each point of the

BEV∆ vs T and VBE vs T curves, uses the same transistor. This avoids the

usual device-to-device mismatch between the 1x and 8x (or other ratio) device

sizes which are used in the usual bandgap circuit.

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APPENDIX A: ∆ VBE vs. Temperature

Let us start with the Gummel-Poon equations (H. K. Gummel and H. C. Poon, “An integral charge control model of bipolar transistors,” Bell System Technical Journal, vol. 49, pp. 827-52, 1970) :

( ) ( )

( ) ( )11

11

//

//

−⋅−−⋅−

−⋅−−⋅=

TCBCTRBC

TRBCTFBE

VnVSC

VnV

R

S

VnV

B

SVnV

B

SC

eIeI

eq

Ie

q

II

β A-1

( ) ( )11 // −⋅+−⋅= TEBETFBE VnV

SEVnV

F

SBE eIe

II

β A-2

( ) ( )11 // −⋅+−⋅= TCBCTRBC VnV

SCVnV

R

SBC eIe

II

β A-3

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AR

BE

AF

BCB V

V

V

Vqq ++=≅ 11 { low level injection } A-4

Let's force VCB = 0 in which case we obtain,

( )1/ −⋅= TFBE VnV

B

SC e

q

II

A-5

( ) ( )11 // −⋅+−⋅= TEBETFBE VnV

SEVnV

F

SBE eIe

II

β A-6

0=BCI A-7

AR

BEB V

Vqq +=≅ 11 A-8

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For VBE >> VT and letting IC1 = 1nA, IC2 = 10nA = m IC1 . Notice that IC1 and

IC2 are for the same transistor. (In bandgap reference designs they are for

matched transistors with different device areas.) With this we get for IC1 = 1nA

(with a bias of VBE1),

TFBE VnV

B

SC e

q

II /

11

1⋅= A-9

TEBETFBE VnV

SEVnV

F

SBE eIe

II //

111 ⋅+⋅=

β A-10

AR

BEB V

Vq 1

1 1 += A-11

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For IC2 = 10nA (with a bias of VBE2) we see,

TFBE VnV

B

SC e

q

II /

22

2⋅= A-12

TEBETFBE VnV

SEVnV

F

SBE eIe

II //

222 ⋅+⋅=

β A-13

AR

BEB V

Vq 2

2 1 += A-14

Taking the ratio IC2 / IC1 we obtain,

( )( )

( ) TFBEBE

TFBE

TFBEVnVV

B

BVnV

BS

VnVBS

C

C eq

q

eqI

eqI

I

I /

2

1/

1

/2

1

2 12

1

2

// −⋅=

⋅⋅=

A-15

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Since we chose IC1 = 1nA and IC2 = 10nA = m IC1 we also have,

( ) ( ) TFBEBETFBEBE VnVV

ARBE

ARBEVnVV

B

B

C

C

C

C eVV

VVe

q

qm

I

Im

I

I /

2

1/

2

1

1

1

1

2 1212

/1

/1 −− ⋅++

=⋅==⋅= A-16

Rearranging we get,

( ) TFBEBE VnVV

ARBE

ARBE eVV

VVm /

1

2 12

/1

/1 −=

++

⋅ A-17

Taking the natural log of both sides we obtain,

TF

BEBE

ARBE

ARBE

Vn

VV

VV

VVm

⋅−=

++

⋅ 12

1

2

/1

/1ln

A-18

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Solving this last expression for VBE2 - VBE1 we get,

++

+⋅=−ARBE

ARBETFBEBE VV

VVmVnVV

/1

/1ln)ln(

1

212 A-19

With VBE1 << VAR

( )[ ])/1()/1(ln)ln( 1212 ARBEARBETFBEBE VVVVmVnVV −⋅++⋅=− A-20

Expanding the expression,

ARBEBEARBEARBEARBEARBE VVVVVVVVVVV 2211212 ///1)/1()/1( ⋅−−+=−⋅+ A-21

and assuming VBE1 << VAR and VBE2 << VAR then we see that,

ARBEARBEARBEARBE VVVVVVVV //1)/1()/1( 1212 −+≅−⋅+ A-22

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Defining x = VBE2 / VAR - VBE1 / VAR we see that |x| << 1 in which case

xx ≅+ )1ln( A-23

and so we have,

( ) ARBEARBEARBEARBE VVVVVVVV //)/1()/1(ln 1212 −=−⋅+ A-24

With this equation A-20 then simplifies to,

[ ]ARBEARBETFBEBE VVVVmVnVV //)ln( 1212 −+⋅=− A-25

or collecting 1 / VAR terms together we get,

( )[ ]ARBEBETFBEBE VVVmVnVV /)ln( 1212 −+⋅=− A-26

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Rearranging we get,

( ) ( ) )ln(/1212 mVnVVVVnVV TFARBEBETFBEBE ⋅=−⋅−− A-27

or ( ) ( ) )ln(/112 mVnVVnVV TFARTFBEBE ⋅=−⋅− A-28

Solving for VBE2 - VBE1 we obtain,

ARTF

TFBEBE

VVn

mVnVV

/1

)ln(12

−⋅⋅=−

A-29

Since VT << VAR then

AR

TF

ARTFV

Vn

VVn

⋅+≅−

1/1

1

A-30

and so we find that,

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⋅+⋅⋅⋅=−AR

TFTFBEBE V

VnmVnVV 1)ln(12 A-31

Making the substitution for VT = k T / q we get,

⋅⋅⋅+⋅⋅⋅⋅=−=∆

AR

FFBEBEBE Vq

Tknm

q

TknVVV 1)ln(12 A-32

Usually in a bandgap reference circuit analysis ∆ VBE = VBE2 - VBE1 is taken as,

)ln(12 mq

TknVVV FBEBEBE ⋅⋅⋅=−=∆ A-33

But here, in equation A-32, we see a correction factor due to VAR.

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APPENDIX B: Some Temperature Expressions for the BJT

Saturation Current Temperature Dependence:

−−

⋅

⋅=

1NOMT

gTBTIT

T

V

EXX

NOMSSSST e

T

TII

B-1

Similar expressions exist for ISE , and ISC .

Forward Beta Temperature Dependence:

TBX

NOMFFT T

TBB

⋅=

B-2

A similar expression exists for BR .

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Forward Ideality Factor Temperature Dependence:

( ) ( ){ }2211 NOMnfNOMnffft TTtTTtnn −+−+⋅= B-3

Similar expressions exist for nr , nc , ne , vaf , var , ikf , ikr and others.

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APPENDIX C: Expressions for PTAT Voltage Generation

Generation of PTAT (Proportional To Absolute Temperature) Voltage:

If two matched transistors, IS1 = IS2 = IS , are biased at collector currents of n IO

and IO (with negligible base currents), as shown in Figure C1, then

⋅−

⋅⋅=−=∆21

12 lnlnS

OT

S

OTBEBEBE I

IV

I

InVVVV

C-1

( ) ( )n

q

kTnVV TBE lnln ⋅=⋅=∆

C-2

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Generation of PTAT Voltage Involving Several or Different Area Transistors:

If the transistors are of different areas or multiples, as shown in Figure C2, then

⋅⋅−

⋅⋅=−=∆S

OT

S

OTBEBEBE Im

IV

I

InVVVV lnln12 C-3

( ) ( )mn

q

kTmnVV TBE ⋅⋅=⋅⋅=∆ lnln

C-4