Index
1. Introduction
2. BGR Overview
3. Works Cited
4. Analysis
5. Comparison1. Introduction
Many blocks in an integrated circuit (IC) require precise voltage or current references to function properly. For example, the comparator in a conventional LDO requires an exact voltage at its input. Bandgap reference (BGR) is the core IC block that serves this purpose; it provides a voltage or current reference that is invariant to temperature change and process corners.
In this post, an overview of BGR circuit structure and working principle is given in the next section. Following the overview, several state-of-the-art articles on BGR are selected, cited, and analyzed. In the last section, a chart which compares the important parameters and features of all the cited works is provided.
2. BGR Overview
The conventional BGR is built around BJTs. It is well known that the current flowing through the collector of a BJT, IC, is expressed as:
IC = IS exp(VBE/VT), where
IS describes the transfer characteristic of the BJT in the forward-active region, typical values of IS are between 10-14A and 10-16A[*];
VBE is the voltage across the base and emitter terminal of the BJT;
VT = kT/q is the thermal voltage, which is 25.85mV at 300K;
The equation can be re-written as
VBE = VT ln(IS/IC).
In reality, both IC and IS vary non-linearly with temperature, thus VBE has a complex relationship to the temperature. However, the overall trend is that VBE decreases as temperature increases in a rather uniform fashion, which is characterized by a complementary-to-absolute-temperature (CTAT) relation. This brings to mind that if we can generate a voltage which is proportional to absolute temperature (PTAT) and add this voltage to VBE with proper ratio, we may obtain a voltage reference that is relatively zero (unchanged) to absolute temperature (ZTAT) as shown in the following figure[*]:
With certain approximations and assumptions, we can eliminate some non-linearities and arrive at a[*]
VOUT = VBE + M*VT = VG0 + VT(γ – α)[1 + ln(T0/T)], where
M is the proportionality factor which is chosen at a specific value to reduce non-linearities;
VG0 is the band-gap voltage of silicon extrapolated to 0K, which has a value of 1.205V;
γ and α are temperature-dependent constants which can be approximated at a certain temperature;
T0 is the temperature at which the temperature coefficient, TCF, of the output is zero.
For example, to achieve zero TCF at 300K, assuming that γ = 3.2 and α = 1,
VOUT|T = T0 = 300K = VG0 + 2.2VT|T0 = 300K = 1.262V,
which is close to the bandgap voltage of silicon, hence the name BGR.
For the ease of analysis, let us assume that VBE of a BJT that takes a fixed current through its collector is linearly CTAT. For the implementation below[*], Q2 is n times Q1, i.e., Q2 is equivalent to n Q1‘s connected in parallel.
Since the amplifier force the same voltage value and takes 0 current at its inputs,
VBE1 – VBE2 = I2R3,
which is to say this configuration conveniently sets the voltage across R3 to be the difference in VBE1 and VBE2. On the other hand,
IE1/IE2 = IC1/IC2 = exp[(VBE1 – VBE2)/VT]/n, or
VBE1 – VBE2 = VT ln(IE1/IE2) = VT ln(nI1/I2).
Since I1R1 = I2R2, we have VBE1 – VBE2 = VTln(nR2/R1). Thus we can express VOUT as:
VOUT = VBE2 + I2(R3 + R2) = VBE2 + (1 + R2/R3)VT ln(nR2/R1).
This expression contains a CTAT term and a PTAT term, where the coefficient to the PTAT term is set by R1, R2, and R3 values.
Traditional BGR circuits mostly follow this idea of combining a CTAT and a PTAT term. In reality, the output of this topology is a curve that concaves down. Advanced BGR circuits contain curvature correction circuitry to further reduce the voltage/current variation on temperature. In Section 3, we cite several state-of-the-art BGR circuits as of 2021 and proceed to analyze each of them in Section 4.
[*] P.R. Gray, “Band-Gap-Referenced Bias Circuits in Bipolar Technology,” in Analysis and design of analog integrated circuits, New York: Wiley, 2011, pp. 315-321.
3. Works Cited
[1]
[2]
[3]
[4]
[5]
[6]
[7]
