Modeling The Bipolar Transistor For Bandgap Reference - MOS-AK

35 Pages·2010·0.14 MB·English

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
D
behavior of V vs T and V vs T is not directly optimized (curve fit).
BE BE

• D
Here we will show how to include V vs T and V vs T behaviors so that
BE BE
the bjt model properly emphasizes the temperature fit for bandgap circuits.
1
V Difference At Different Collector Currents
BE

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

= (cid:215) V /V
I I e
BE1 T
C1 S 1-1

= (cid:215) V /V
I I e
BE2 T
C2 S 1-2

(cid:215)
k T
=
V
where T 1-3
q

2
Select I and I such that their ratio is exactly m
C2 C1
I
C2 = e(VBE2- VBE1)/VT = m
“exactly” 1-4
I
C1
D = -
V V V
Define 1-5
BE BE2 BE1

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

kT
D = - = =
V V V V ln ( m ) ln ( m )
1-6
BE BE2 BE1 T
q

As an example we will select I =1 nA and I =10 nA. Since we selected
C1 C2
I
C2 = m I = m I = 10 I
we see that 1-7
I C2 C1 C1
C1

3
Using Linear Interpolation To Assure That m = 10 Exactly

•
When measuring gummel (icvb) curves, V is incremented in steps and I
BE C
current is measured.

•
This means that hitting I and I at exactly 1nA and 10nA is very unlikely.
C1 C2

•
Hence we will use interpolation to find the V ’s that correspond to exactly 1nA
BE
and 10nA.

•
Because of the exponential relationship between V and I we will use values
BE C
of ln(I ) and V in our linear interpolation scheme.
C BE

4
Figure 1. ln(I ) , V Interpolation
C BE

•
As depicted in Figure 1 above, to determine V we find the I just below 1nA
BE1 C1
which we label I (the corresponding V is labeled V ); and the I equal
C1A BE BE1A C1
to or greater than 1nA is labeled I (the corresponding V is labeled V ).
C1B BE BE1B
5
•
From these pairs -- (ln(I ),V ) and (ln(I ),V ) -- we use linear
C1A BE1A C1B BE1B
interpolation to get V which corresponds to exactly I = 1nA.
BE1 C1

•
For the I = 10nA case we use the same linear interpolation scheme to obtain
C2
the V value corresponding to exactly I = 10nA.
BE2 C2

Using equation 1-6 we then obtain
k T
D = - =
V V (at 10 nA) V (at 1 nA) ln ( m )
1-8
BE BE2 BE1
q
•
Notice that both V and V are for the same transistor.
BE1 BE2

•
Since the measurement sweep (depicted in Figure 1) is rapid and operated at
low power (1nA and 10nA), then V and V are practically at the same
BE1 BE2
temperature, T.

6
•
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 I' and
C2
I' as we did for temperature T -- that is I / I = m (Eq. 1-4):
C1 C2 C1
I'
C2 = eD V' /V = m
BE T
I' 1-9
C1
k T'
D = - = =
V' V' V' V' ln ( m ) ln ( m )
1-10
BE BE2 BE1 T
q
• Again we use linear interpolation at the new temperature, T', to get V' and
BE1
V' at exactly 1nA and 10nA respectively, thereby assuring that we have
BE2
exactly m = 10.

7
D
Using Several Temperatures To Curve Fit Bandgap V vs T and V vs T
BE BE

•
Say we take measurements at 4 different temperatures T, T', T'', T''' then we
D
would have 4 different V 's for measured and corresponding simulated
BE
D
V 's :
BE

D D D D
V , V ', V '', V '''
Measured 1-11
BE_M BE_M BE_M BE_M
D D D D
V , V ', V '', V '''
Simulated 1-12
BE_S BE_S BE_S BE_S

• At this point the simulated D V 's are curve fit (optimized) to the corresponding
BE
D
measured V 's . Figure 2 illustrates the curve fitting of the measured vs
BE
D D
simulated V 's . As seen through Eq. 1-8, the V 's are Proportional To
BE BE
Absolute Temperature (PTAT).
8
D
Figure 2. Curve Fit Measured vs Simulated V 's
BE

9
• D
When we curve fit the measured vs simulated V curve we also curve fit the
BE
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 D V measured vs simulated curve set has only 4 or 5 data points
BE
(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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