Bandgap Reference

Posted on Edited on In

image-20250729004456566

Subthreshold Conduction

By square-law, the Eq \(g_m = \sqrt{2\mu C_{ox}\frac{W}{L}I_D}\), it is possible to obtain a higer transconductance by increasing \(W\) while maintaining \(I_D\) constant. However, if \(W\) increases while \(I_D\) remains constant, then \(V_{GS} \to V_{TH}\) and device enters the subthreshold region. \[ I_D = I_0\exp \frac{V_{GS}}{\xi V_T} \]

where \(I_0\) is proportional to \(W/L\), \(\xi \gt 1\) is a nonideality factor, and \(V_T = kT/q\)

As a result, the transconductance in subthreshold region is \[ g_m = \frac{I_D}{\xi V_T} \]

which is \(g_m \propto I_D\)

image-20240627230726326

image-20240627230744044

PTAT with subthreshold MOS

MOS working in the weak inversion region ("subthreshold conduction") have the similar characteristics to BJTs and diodes, since the effect of diffusion current becomes more significant than that of drift current

image-20240803193343915

image-20240803195500321

image-20240803200129592

Hongprasit, Saweth, Worawat Sa-ngiamvibool and Apinan Aurasopon. "Design of Bandgap Core and Startup Circuits for All CMOS Bandgap Voltage Reference." Przegląd Elektrotechniczny (2012): 277-280.

Curvature Compensation

VBE

image-20250728233542026

In advanced node, N4P, \(V_{BE}\) is about -1.45mV/K

Assuming \(I_C\) is constant

image-20250728233112550

image-20250728233350355

image-20250728233839563

Assuming \(I_C\) is PTAT, \(I_C = (V_T \ln n) / R_3\)

image-20250728233317599

image-20250729002704253

The first-order linear temperature dependence term of \(V_{BE}\) can be eliminated with IPTAT. \(V_T(\eta - \theta)\ln)T/T_r\) is the high-order nonlinear temperature-dependent term of \(V_{BE}\), which requires high-order curvature compensation

G. Zhu, Y. Yang and Q. Zhang, "A 4.6-ppm/°C High-Order Curvature Compensated Bandgap Reference for BMIC," in IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 66, no. 9, pp. 1492-1496, Sept. 2019 [https://sci-hub.se/10.1109/TCSII.2018.2889808]

X. Fu, D. M. Colombo, Y. Yin and K. El-Sankary, "Low Noise, High PSRR, High-Order Piecewise Curvature Compensated CMOS Bandgap Reference," in IEEE Access, vol. 10, pp. 110970-110982, 2022 [https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=9923910]

image-20240903234720200

image-20250728225624247

Tutorials | 08012023 | 1.2.1 Bandgap Voltage Regular [https://youtu.be/dz067SOX0XQ&t=6362]

temperature coefficient

The parameter that shows the dependence of the reference voltage on temperature variation is called the temperature coefficient and is defined as: \[ TC_F=\frac{1}{V_{\text{REF}}}\left[ \frac{V_{\text{max}}-V_{\text{min}}}{T_{\text{max}}-T_{\text{min}}} \right]\times10^6\;ppm/^oC \]

Choice of n

image-20221117002714125

classic bandgap reference

bg.drawio

\[ V_{bg} = \frac{\Delta V_{be}}{R_1} (R_1+R_2) + V_{be2} = \frac{\Delta V_{be}}{R_1} R_2 + V_{be1} \]

\[ V_{bg} = \left(\frac{\Delta V_{be}}{R_1} + \frac{V_{be1}}{R_2}\right)R_3 = \left(\frac{\Delta V_{be}}{R_1} R_2 + V_{be1}\right)\frac{R_3}{R_2} \]

OTA offset effect

bg_ota_vos.drawio

\[\begin{align} V_{be1} &= \frac{kT}{q}\ln(\frac{I_{e1}}{I_{ss}}) \\ V_{be2} &= \frac{kT}{q}\ln(\frac{I_{e2}}{nI_{ss}}) \end{align}\]

Here, we assume \(I_e = I_c\)

Hence,

\[\begin{align} \Delta V_{be} &= \frac{kT}{q}\ln(n\frac{I_{e1}}{I_{e2}}) \\ &= \frac{kT}{q}\ln(n) + \frac{kT}{q}\ln(\frac{I_{e1}}{I_{e2}}) \\ &= \Delta V_{be,0} + \frac{kT}{q}\ln(\frac{I_{e1}}{I_{e2}}) \end{align}\]

Therefore,

\[\begin{align} V_{bg} &= \frac{\Delta V_{be}+V_{os}}{R_2}(R_1+R_2) + V_{be2} \\ &= \alpha \Delta V_{be,0} + \alpha \frac{kT}{q}\ln(\frac{I_{e1}}{I_{e2}}) + \alpha V_{os} + \frac{kT}{q}\ln(\frac{I_{e2}}{nI_{ss}}) \\ &= \alpha \Delta V_{be,0} + \alpha \frac{kT}{q}\ln(\frac{I_{e1}}{I_{e2}}) + \alpha V_{os} + \frac{kT}{q}\ln(\frac{I_{e2,0}}{nI_{ss}})+\frac{kT}{q}\ln(\frac{I_{e2}}{I_{e2,0}}) \end{align}\]

We omit the last part \[\begin{align} V_{bg} &\approx \alpha \Delta V_{be,0} + \alpha \frac{kT}{q}\ln(\frac{I_{e1}}{I_{e2}}) + \alpha V_{os} + \frac{kT}{q}\ln(\frac{I_{e2,0}}{nI_{ss}}) \\ &= \alpha \Delta V_{be,0} + V_{be2,0} + \alpha \left(V_{os} + \frac{kT}{q}\ln(\frac{I_{e1}}{I_{e2}})\right) \\ &= V_{bg,0} + \alpha \left(V_{os} + \frac{kT}{q}\ln(\frac{I_{e1}}{I_{e2}})\right) \end{align}\]

i.e. the bg variation due to OTA offset \[ \Delta V_{bg} \approx \alpha \left(V_{os} + \frac{kT}{q}\ln(\frac{I_{e1}}{I_{e2}})\right) \]

  • \(V_{os} \gt 0\)

    ​ \(I_{e1} \gt I_{e2}\): \(\Delta V_{bg} \gt \alpha V_{os}\)

  • \(V_{os} \lt 0\)

    ​ \(I_{e1} \lt I_{e2}\): \(\Delta V_{bg} \lt \alpha V_{os}\)

OTA with chopper

bg_chop.drawio

bg_chop_shift.drawio

\(I_{e1}\), \(I_{e2}\)

\[\begin{align} V_{ip} &= V_{im} + V_{os} \\ \frac{V_{bg}-V_{ip}}{R_2} &= I_{e2} \\ \frac{V_{bg}-V_{im}}{R_2} &= I_{e1} \\ V_{ip} &= I_{e2}R_1 + V_T\frac{I_{e2}}{nI_S} \\ V_{im} &= V_T\frac{I_{e1}}{I_S} \end{align}\] where \(V_T = \frac{kT}{q}\)

we obtain \[ I_{e1} = \frac{V_T\ln n}{R_1} + V_{os}\left(\frac{1}{R_1} + \frac{1}{R_2} \right) - \frac{1}{R_1}\cdot V_T\ln\left(1- \frac{V_{os}}{R_2I_{e1}} \right) \]

we omit the last part \[\begin{align} I_{e1} &= I_{e0} + V_{os}\left(\frac{1}{R_1} + \frac{1}{R_2} \right) \\ I_{e2} &= I_{e1} - \frac{V_{os}}{R_2} = I_{e0} + \frac{V_{os}}{R_1} \end{align}\] where \(I_{e0} = \frac{\Delta V_{be}}{R_1}\), \(\Delta V_{be}=V_T\ln n\)

That is, both \(I_{e1}\) and \(I_{e2}\) are proportional to \(V_{os}\)

\(I_{e1}\) and \(I_{e2}\) can be expressed as \[\begin{align} I_{e1} &= I_{e0} + V_{os}\left(\frac{1}{R_1} + \frac{1}{2R_2} \right) + \frac{V_{os}}{2R_2} \\ I_{e2} &= I_{e0} + V_{os}\left(\frac{1}{R_1} + \frac{1}{2R_2} \right) - \frac{V_{os}}{2R_2} \end{align}\] i.e., \(\Delta I_{e,cm} = V_{os}\left(\frac{1}{R_1} + \frac{1}{2R_2} \right)\) and \(\Delta I_{e,dif} =\frac{V_{os}}{2R_2}\)

bandgap output voltage is

\[\begin{align} V_{bg} &= V_T \ln \frac{I_{e1}}{I_s} + I_{e1}R_2 \\ &= V_T \ln \frac{I_{e0} + V_{os}\left(\frac{1}{R_1} + \frac{1}{R_2} \right)}{I_s} + I_{e1}R_2 \\ &= V_T \ln \frac{I_{e0} + V_{os}\left(\frac{1}{R_1} + \frac{1}{R_2} \right)}{I_s} + I_{e0}R_2 + V_{os}\frac{R_1+R_2}{R_1} \\ &= I_{e0}R_2 + V_T \ln \frac{I_{e0}}{I_s} + V_T\ln\left(1+\frac{V_{os}\left(\frac{1}{R_1} + \frac{1}{R_2} \right)}{I_{e0}} \right) + V_{os}\frac{R_1+R_2}{R_1} \\ &= V_{bg0} + V_T\ln\left(1+\frac{V_{os}\left(\frac{1}{R_1} + \frac{1}{R_2} \right)}{I_{e0}} \right) + V_{os}\frac{R_1+R_2}{R_1} \end{align}\]

Therefore, the averaged output of bandgap

\[ V_{bg,avg} = V_{bg0} +\frac{1}{2}V_T\ln\left(1-\frac{V_{os}^2\left(\frac{1}{R_1} + \frac{1}{R_2} \right)^2}{I_{e0}^2} \right) \lt V_{bg0} \]

\(V_{bg,avg} \lt V_{bg0}\) due to nonlinearity of BJT

reference

ECEN 607 (ESS) Bandgap Reference: Basics URL:https://people.engr.tamu.edu/s-sanchez/607%20Lect%204%20Bandgap-2009.pdf

CICC 2023 Session 12: Forum: Recent Progress in LDOs and Voltage, Current, and Timing References

  • Jae-Yoon Sim, POSTECH. 12-2: Design of Ultra-low-power Bandgap Reference Circuits
  • Inhee Lee, University of Pittsburgh. 12-3: Sub-μW Non-Bandgap Voltage References