The Bandgap Reference

1 A CirCuit for All SeASonSBehzad Razavi IEEE SOLID-STATE CIRCUITS MAGAZINE Summer 2016 9 SSince its inception in the late 1960s, the Bandgap circuit has served as an essential component in most inte-grated circuits. This simple, robust idea provides a temperature-indepen-dent (TI) voltage and a proportional-to-absolute-temperature (PTAT) current . In this article, we study the principles of Bandgap circuit Brief HistorySemiconductor technology does not directly offer any electric quantity that is nominally independent of the ambi-ent temperature.

2 Thus, temperature independence has generally been envi-sioned in the form of combining two phenomena that have opposite tem-perature coefficients (TCs). For exam-ple, a resistor with a low TC can be constructed by placing in series, with proper weighting factors, a positive-TC resistor and a negative-TC this idea to voltage quan-tities proved more difficult. It had been long recognized that the voltage across a forward-biased diode has a negative TC (if its bias current does not change much with temperature). However, a positive-TC voltage was missing. In 1964, Hilbiber of Fairchild Semiconductor observed that two di-ode stacks biased at different current densities can provide a TI voltage [1].

3 In 1965, Widlar, from the same com-pany, more explicitly showed that the base-emitter voltages of two transis-tors biased at different current densi-ties had a PTAT difference [2] and in 1971 introduced the first Bandgap circuit (Figure 1) [3]. This was fol-lowed by another topology presented by Brokaw in 1974 (Figure 2) [4] and many others rise of CMOS technology in the 1970s posed the question of whether a stable volt -age Reference could be created without the use of bipolar devices [5]. However, it was observed that the high-threshold mismatch of MOS transistors leads to significant error and drift in such references.

4 Subsequent work there-fore focused on the native bipolar transistor available in standard CMOS processes [6], [7]. Figure 3 shows an example similar to Brokaw s cell [8], ex-cept that the current -measuring resis-tors are moved from the collectors to the emitters because the former must be tied to .VDD (Early CMOS technologies used an n-substrate and hence accom-modated a vertical npn transistor.) A useful feature of this topology is that the op amp does not drive resistors and can thus maintain a high loop Basic IdeaAs mentioned in the pre-vious section, the volt -age across a pn junction and hence the base-emitter voltage , ,VBE of a bipolar transistor exhibit a negative TC.

5 It can be shown that, for a constant collector current , ()/,TVTVmVEq4 TgBEBE22=-+- (1)where T is the absolute temperature, /,/ , and Eg the Bandgap Digital Object Identifier Bandgap ReferenceDate of publication: 2 September 2016 VREF = VBE +R2R3 VBEIV+ VBE VBEQ1Q2Q3R3600R1600R26 KGroundR2R3 VBEF igure 1: Widlar s Bandgap useful feature of this topology is that the op amp does not drive resistors and can thus maintain a high loop Summer 2016 IEEE SOLID-STATE CIRCUITS MAGAZINE + R1R2R3Q2Q1 AnAXYVoutFigure 5: t he bipolar Bandgap circuit proposed by With typical current den-sities, (/), yielding a TC of about mV/K at room wish to create a voltage with a TC equal to + mV/K.

6 As shown in Figure 4, we bias two identical diode-connected bipolar transistors at differ-ent current levels, obtaining different current densities and VVV12 BEBED=- (2) lnlnVInIVIITSTS00=- (3) .lnVnT= (4)The same result holds if the bias cur-rents are equal but ,InISS21= , if Q2 consists of n parallel units, each identical to .Q1 We postulate that lnVVVnREFBET=+ (5)can have a zero TC if n is chosen prop-erly. That is, we must calculate the necessary value of n and find a circuit that adds VBE and .lnVnT The tem-perature coefficient of VD is equal to (/).

7 Lnlnkqnn0 087mV/K.+=+ To reach + mV/K, we require lnn..172 and, therefore, ,295107# an impractically large scaling factor for the bias currents or the term Bandgap Reference can be appreciated by calculating /VTREF22 from (5), setting it to zero, finding lnVnT from the result, and substitut-ing in (5) ().VqEmV4gTREF=++ (6)This expression suggests that the value of VREF extrapolated to T0K= is equal to the Bandgap voltage , / .EqgThe problem of .n295107#= can be mitigated if lnVnT is somehow amplified before it is added to .VBE A compact implementation of this idea is shown in Figure 5 [9], where V1BE trav-els through a noninverting amplifier with a gain of /RR123+ (if )gRm213%- and V2BE sees a gain of /RR23- (if ).

8 GRm111%- That is, VVRRVRR1323212outBEBE=+-cm (7) ().VVVRR32112 BEBEBE=+- (8)One can prove that this relation holds even if we do not assume gRm213%- and .gRm111%- We select RR12= so that Q1 and Q2 carry equal currents, and an emitter area ratio of n to one, thereby obtaining lnVVVnT12 BEBE-= for the voltage sustained by .R3 It fol-lows that lnVVRRVnT321outBE=+ (9) .lnVRRVn1T322BE=++cm (10)We can now choose (/)lnRRV1T23+ .n172. with a moderate value for n, , in the range of 10 flaw in our results is that the col-lector current is assumed constant in (1) but it is PTAT in this circuit.

9 Fortu-nately, this issue is easily remedied by reducing the factor of to about standard CMOS technologies, the circuit of Figure 5 faces three issues. R = RV+V + AVOUTQ2Q18AR2R1 VGO + (m 1) KToqR1R2J1J2 KTqin2J1J2 KTqinVBEF igure 2: Brokaw s Bandgap +Figure 3: t he CMoS Bandgap circuit proposed by Gregorian et V+ Figure 4: the generation of a PtAt voltage . IEEE SOLID-STATE CIRCUITS MAGAZINE Summer 2016 11 First, a CMOS op amp directly driv-ing the feedback resistors must deal with gain-power-stabil-ity trade-offs.

10 Second, it is difficult to real-ize bipolar transistors whose collectors are not grounded; the vertical structure available in typical CMOS processes utilizes the p-substrate as the collector. Third, the op amp offset is amplified by a factor of /RR123+ as it appears in t,Vou causing error and temperature drift. For example, n10= translates to /..RR17523.+To address these issues, we first recognize that resistors R1 and R2 in Figure 5 provide equal bias cur-rents, a function that can be real-ized by controlled current sources. In the spirit of Figure 3, we arrive at the topology depicted in Fig-ure 6(a).