Tolerance effects in submicroampere current generator design

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Tolerance effects in submicroampere current generator design by B. L. Hart. and R. W. J. Barkerit

It is shown that recently proposed monolithic low current generators employing the principle of 'power-law current division" require some form of on-chip trimming if a close definition of output current is required.

1. Introduction The design of monolithic microampere and submicroampere current generators using a low total value of circuit resistance has recently attracted some attention and circuits based on this technique, termed here 'power-law current division', have been described./2 Subsequently, a variation of this scheme was considered and a claim of superior accuracy made? The object of this note is to show that for a truly monolithic arrangement, using realistic component tolerances, on-chip trimming is necessary if the output current is to be defined to within 10% of a chosen nominal value.

Qt(Q.,), I, is a process-dependent saturation current. Similarly, I¢2= I~= al,exp (V,E,_/VT) . . . . . . . . . . . . . . . . . . . . . . .

(2)

But, VBe2=VnEI--[I2+(Iola2)JR,

(3)

in which,

a2=B2/(l+f12).

Combining (I), (2) and (3) yields,

Io=a lt ~-( llfl,)-(lolI,fl2)~ ~exp(--IoR,la2Vx)] Lexp(- I~R,/VT)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Output

2.

Analysis The basic scheme of Reference 3, shown in Fig. 1, is specifically referred to in this note, but the approach used in the tolerance analysis is also applicable to other previously published schemes. ~.2 In Fig. 1, the emitter area of Q2 is greater than that of QI by a factor 'a'. Both devices operate in the forwardactive mode under low-level injection conditions but at collector currents well above leakage values. Initially it should be noted that, with the modern planar transistors used in monolithic systems, it is the collectorcurrent rather than the emitter current which is exponentially related to the base-emitter voltage, as defined in eqn. (1) below? This distinction is particularly important at the low collector current levels which are of immediate concern. For Fig. 1, the current range considered and/3t>> 1, Appendix 1 shows that,

Id:I~exp(VBE,/VT)

--~ I,

El--(l/~,)--(Io/I,~2)l ...(1)

where VT is the thermal voltage (= kT/q). In eqn. (1),/31 (f12) is the common-emitter direct current gain of +NE London Polytechnic, Dagenham, UK. --Trent Polytechnic, Nottingham, UK.

22

................

(4)

V÷

IoIo

I AreaRatio V+

I1

~ B t:l : ~ E2 12(PTAT)VTd.LogebN o q2

¢ Fig. 1 Submicroampere current generator employing power-law current division. Ideally (lo/lt) = a/bha. where n~ (RdR_,). Suppose the current, 12, is derived 'on-chip' and is proportional to absolute temperature (PTAT). It can be expressed in the form, Iz=(VT d/R2) loge b . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(5)

where d and b are numbers corresponding to emitter area ratios. IfX& ~ - ( 1 / B 0 - ( I o / I t f l 2 ) ;

~xp(-loRt/c~zVx)]

..(6,

MICROELECTRONICSJOURNALVol. 13 No. 2 © 1982 Mackintosh Publications Ltd., Luton.

then substitution of (5) and (6) into (4) and rearranging gives, I o = ( h I 0 a b-"a=(hIt)al b "a . . . . . . . . . . . . . . . . . . . . . .

(7)

where n ~ ( R d R 2 ) . Equation (7) describes a 'power-law current division' process. We assume initially that ,kit is known precisely (this assumption is not strictly valid and is dealt with again later). This facilitates an examination of the effect of tolerances of a, b, n, d, alone, on the definition oflo. It would be unduly pessimistic to consider a worst-case tolerance analysis; consequently, a statistical approach is adopted. It is well-known, for uncorrelated variables, that the square of the fractional standard error of the product equals the suna of the squares of the fractional errors of the factors forming the product. 5 Hence from (7),

70)

+

(ndlogob) 2 (~-~), . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(8)

3. Discussion In eqn. (8), a, b, n, d, as indicated previously are all area ratios, and it is unrealistic to expect a standard deviation of less than 1% in area-ratio matching with standard production techniques. For a matched transistor pair this corresponds to a base-emitter offset voltage of about 0.25 mV assuming VT=25 mV. Accordingly, we substitute into (8), (Sa/a) = (Sb/b) = ((Sn/n)= ( S d / d ) = 0.01

. . . . . . . (9)

Then, (8Io/!o) = 0.01 [ l + ( n d ) z { l+2(Ioge b)2}] '/1 . . . . . . (10) For operation at I o < < 1/xA a normal component choice would be n _> 4, b=2, d=2. Thus, ""(8Io/Io)x 100~11% . . . . . . . . . . . . . . . . . . . . . . . . . . .

h-~ [1-(1//3,) - (I,,/I\,/32)] . . . . . . . . . . . . . . . . . . .

(12)

But, (I./Ii) = ( a / b " a ) , s o p u t t i n g a : 1 and using the previous values for n (=4) and b (=2) gives, X= ~-(llBi)-(l1256fl,.)]. Typically, /3,>100 and /3z> 10, even at low currents (see below). Thus cqn. (12) simplifies to, Relating this error component to the error implied in eqn. (ll), the small departure o f h from a value of unity due to non-unity a2 (or al) causes little overall error in Io. For this reason, a base-current compensation scheme for Q2, such as that proposed in ref. 3, is unnecessary and many actually cause error. One reason why this error may not have been apparent could be the fact that /3z normally holds up well at low currents. Samples of CA 4046 have exhibited a 13of one-half the value at Ic=0.5 mA when operated at Ic=20 nA. 6 From the foregoing, it is apparent that if base-current compensation is to be used, it is the base current of Q i which needs to be considered rather than that of Qz. a suitable scheme is shown in Fig. 2, in which the presence of the emitter follower, Q3, forces the condition, Icl= I j, (as shown in Appendix B ). This scheme also facilitates the generation of a 'start-up' current, IK, for the complementary PTAT current source, comprising the components in the dotted rectangle, 7 that supplies I2. A development of the circuit of Fig. 2 results in a cascode output circuit for Io. Summarising, a simple statistical analysis shows that on-chip resistor trimming is required in the design of submicroampere current generators, ira close definition of output current is required in fully monolithic circuits employing 'power-law current division'. This conclusion is supported by previously-published m practical measurements. PTAT CURRENT SOURCE

V.

(ll)

Returning to the term hit, we first consider the It component. For a truly monolithic system, the reference current must be generated on-chip which would normally imply the use of a monolithic resistor. Consequently, It is unlikely to be defined to better than 10%, which would mean an overall standard deviation in Io of some 15% - clearly unacceptable for some applications. Consider now the magnitude of h and its uncertainty. From (6), it is necessary that

Thus, to an accuracy of 1%, we can write,

Q /

iK

I%

I1

IC 1 ...,

Q2 12

a:l

k,,

~RI

Io< (a2 VT/100RI)

if the exponential term is to be within 1% of unity. This condition is not difficult to meet in a practical design but should be checked. (As this error component results from the flow of the emitter current of the output transistor in the ranging resistor, R~, it is useful to note that with the 'shunt' version of the 'power-law division scheme', 2 it is only the base current of the output transistor which flows in the ranging resistor. This feature of the 'shunt' version makes it possible for the circuit to operate over a wider range of output currents).

Hg. 2 Improved version of Fig. I together with PTATcurrcnt source (inside dotted rt~ctangle) supplying 1z. References [ 1] Barker, R. W. J., and Hart, B. L.: 'Novel submicroampere current source design technique for monolithic circuits'. Electronics Letters, 16, pp.609-611, 1980. [2] Barker, R. W. J., and Hart, B. L.: 'Design technique for low-current generators', Electronics Letters. 17, pp.271-273. 1981. 23

Tolerance effects in submicroampere current generator design continued from page 23

[3] Nedungadi, A.: "Accurate submircroampere controlled current source', Electronic Letters, 17, pp.320-323, 1981. [41 Hart, B. L.: "Translinear circuit principle: a reformulation', Electronics Letters, 15, pp.801-803, 1979. 15] Topping, J.: "Errors of observation and their treatment', Chapman and Hall, London, p.81, 1958. [6] McGovern, P. A.: 'Simple square law circuits of wide dynamic range', Proc. IREE (Australia), pp. 119-123, 1975. [7] Van Kessel, Th.J. and Van de Plassche, R. J.: "Integrated linear basic circuits', Philips Technical Review, 32, pp. I-12. 1971.

Appendix A

lo = It I l - ( l / f l O -

A4

Appendix B The emitter current of 03 is given by, The base current of Qs is I3/(133+ 1) where 0005is the common emitter direct current gain of Q3. Current summation at the collector of Qj gives, ~, = ~ ,

Current summation at the collector of Q, gives,

(IdIt~2) ] . . . . . . . . . . . . . . . . . .

+ 1~o/o~033+ ~)} + { |¢,/13, (0003+~)1

B2

.........

I, = Ic,+(1c,/13,)+(I,,/13_,) . . . , . . . . . . . . . . . . . . . . . . .

AI

.'. ,c, ~1 + { 1113,033+ l)~ = El,-1o113.(/33+ 1)1

..... B3

.'. Ic,(l+ 1//3,) = I, - (I,,//3_,) ....................

A2

Ic, ~ 1,

E-I1/13,03~+,)1-I ,,,/I,¢S~(133+l)l]

...B4

•". lcl = II

A3

.............

Using the binomial expansion and assuming 13~>> 1 yields,

24

Equation B4 reduces to lcl I, for the practical cases 13~133>> 1, [3,.00033>>1.