24 bits at 5 V full scale: \(\frac{5 V}{2^{24}} \approx\) 298 nV, the noise of 50 Ω in 96 kHz
\[SNR = 6.02 \cdot B + 1.76 \text{ dB} = 146 \text{ dB at } B = 24\]
The best DAC of 2005: 120 dB, about 19.6 bits
| What is discrete | Noise it gives |
|---|---|
| Charge, \(q\) | Shot noise, \(2qI\) |
| Energy per mode, \(hf\) | Thermal noise, \(4kTR\) |
| Localized states, traps | Telegraph noise, then \(1/f\) |
\[\overline{x(t)} = \lim_{T\to\infty} \frac{1}{T}\int^{+T/2}_{-T/2}{ x(t) dt} \tag{1}\]
\[\overline{x^2(t)} =\lim_{T\to\infty} \frac{1}{T}\int^{+T/2}_{-T/2}{ x^2(t) dt} \tag{2}\]
\[\sigma^2 = \overline{x^2(t)} - \overline{x(t)}^2 \tag{3}\]
\[R_x(\tau) = \overline{x(t)x(t + \tau)}\]
\[P_{av} = \lim_{T\to\infty} \frac{1}{T} \int^{+T/2}_{-T/2} x^2(t) dt \tag{4}\]
\[P_{av} = \frac{1}{N}\sum_{i=0}^N x^2(i) \tag{5}\]
\[S_x(f) = 2\int_{-\infty}^{\infty}{R_x(\tau)e^{-j2\pi f \tau}d\tau} \tag{6}\]
\[\overline{x^2(t)} = \int_{0}^{\infty}{S_x(f)df} \tag{7}\]
\[S_y(f) = S_x(f)\vert H(f)\vert ^2 \tag{8}\]
\[f_x = \frac{\pi f_0}{2} = \frac{1}{4 R C}\]
\[R_x(\tau) = \overline{x^2(t)}\,\delta(\tau) \quad\Rightarrow\quad S_x(f) = 2\,\overline{x^2(t)}\]
White: the same power density at every frequency
\[v_{tot}^2(t) = (v_1(t) + v_2(t))^2 = v_1^2(t) + 2v_1(t)v_2(t) + v_2^2(t)\]
\[\begin{aligned} \overline{e_{tot}^2} &= \lim_{T\to\infty} \frac{1}{T}\int^{+T/2}_{-T/2}{ v_{tot}^2(t) dt} \\ &= \lim_{T\to\infty} \frac{1}{T}\int^{+T/2}_{-T/2}{ v_1^2(t) dt} \\ &+ \lim_{T\to\infty} \frac{1}{T}\int^{+T/2}_{-T/2}{ v_2^2(t) dt} \\ &+ \lim_{T\to\infty} \frac{1}{T}\int^{+T/2}_{-T/2}{ 2v_1(t)v_2(t) dt} \\ &= \overline{e_{1}^2} + \overline{e_{2}^2} + \lim_{T\to\infty} \frac{1}{T}\int^{+T/2}_{-T/2}{ 2v_1(t)v_2(t) dt} \end{aligned}\]
\[\overline{e_{tot}^2} = \overline{e_{1}^2} + \overline{e_{2}^2}\]
Uncorrelated sources: the cross term averages to zero, and the powers add
\[\begin{aligned} SNR &= 10 \log\left(\frac{Signal\:power}{Noise\:power}\right)\\ &= 10 \log\left(\frac{\overline{v_{sig}^2}}{\overline{e_{n}^2}}\right)\\ &= 20 \log\left(\frac{v_{rms}}{\sqrt{\overline{e_{n}^2}}}\right) \end{aligned}\]
\[F = \frac{\overline{v_o^2}}{source\:contribution\:to\:\overline{v_o^2}}\]
\[NF = 10 \log(F)\]
\[F = \frac{SNR_{input}}{SNR_{output}}\]
\[F = 1 + F_1 - 1 + \frac{F_2 -1}{G_{1}} + \frac{F_3-1}{G_{1}G_{2}} + ....\]
\[S_{x1}(f) = \int_{-\infty}^{\infty}{R_{x1}(\tau)e^{-j\omega\tau}d\tau}\]
\[S_{x2}(f) = 2\int_{-\infty}^{\infty}{R_{x2}(\tau)e^{-j\omega\tau}d\tau}\]
\[S_{x2}(f) = 2S_{x1}(f)\]
\[\overline{x_2^2(t)} = \int_0^{\infty}{S_{x2}(f)df}\]
\[\overline{x_1^2(t)} = 2\int_0^{\infty}{S_{x1}(f)df}\]
\(S_{x1}\): two-sided, signal processing books. \(S_{x2}\): one-sided, noise books
\[S_{th}(f) = 4kTR\]
\[S_{th}(f) = 2kTR\]
\[\mathfrak{M} (\Delta) = \overline{F(t)F(t + \Delta)}\]
\[I(\theta) = \int_0^{T}{\mathfrak{M}(\Delta) \cos ( \pi \frac{\Delta}{\theta})d\Delta}\]
\[S_V(f) = 4R\frac{hf}{e^{hf/kT}-1} \approx 4kTR \quad \text{if } hf \ll kT\]
\[\frac{kT}{h} = 6.25 \text{ THz at 300 K}\]
\[\overline{i_n^2} = 2qI\,\Delta f\]
In weak inversion, with \(\gamma = n/2\) and \(g_m = I_D/nV_T\)
\[4kT\gamma g_m = 4kT\frac{n}{2}\frac{I_D}{nV_T} = 2qI_D\]
\[i\hbar\frac{\partial}{\partial t}\Psi(\vec{r},t) = \left[-\frac{\hbar^2}{2m}\nabla^2 + V(\vec{r})\right]\Psi(\vec{r},t)\]
Stationary states, \(\Psi = \psi(\vec{r})e^{-iEt/\hbar}\)
\[-\frac{\hbar^2}{2m}\nabla^2\psi + V\psi = E\psi\]
\[f(E) = \frac{1}{1 + e^{(E - E_F)/kT}}\]
\[n = \int_{E_C}^{\infty} N(E) f(E)\, dE\]
\[\psi_{ch}(x) \propto e^{-\kappa x} \text{ in the oxide}, \quad \kappa = \frac{\sqrt{2 m_{ox}(\Phi_B - E)}}{\hbar}\]
\[\Phi_B = 3.1 \text{ eV}, \quad m_{ox} \approx 0.5 m_0 \Rightarrow \kappa \approx 6.4 \text{ nm}^{-1}\]
\[\frac{1}{\tau} = \frac{2\pi}{\hbar}\left\vert\langle\psi_T\vert H'\vert\psi_{ch}\rangle\right\vert^2 \rho(E)\]
\[\left\vert\langle\psi_T\vert H'\vert\psi_{ch}\rangle\right\vert^2 \propto e^{-2\kappa x_T}\]
\[\tau = \tau_0 e^{x_T/\lambda}, \quad \lambda = \frac{1}{2\kappa} \approx 0.08 \text{ nm}\]
| Trap depth | \(\tau\) (\(\tau_0\) = 0.1 ns) |
|---|---|
| 0 nm | 0.1 ns |
| 1 nm | 40 µs |
| 2 nm | 13 s |
| 3 nm | 2 months |
\[\frac{1}{\tau_c} \propto e^{-x_T/\lambda}\, e^{-E_B/kT}\, n_s\]
\[\frac{\tau_c}{\tau_e} = \frac{1}{g}e^{(E_T - E_F)/kT}, \quad g = 2\]
\[\frac{\partial \ln(\tau_c/\tau_e)}{\partial V_{GS}} = -\frac{q}{kT}\frac{x_T}{t_{ox}}\]
\[R_x(\tau) = \Delta I^2 \frac{\tau_c\tau_e}{(\tau_c + \tau_e)^2} e^{-\vert\tau\vert/\tau_0}, \quad \frac{1}{\tau_0} = \frac{1}{\tau_c} + \frac{1}{\tau_e}\]
\[S_x(f) = 4\int_0^\infty R_x(\tau)\cos(\omega\tau)d\tau = \frac{4\,\Delta I^2}{(\tau_c+\tau_e)\left[\left(\frac{1}{\tau_c}+\frac{1}{\tau_e}\right)^2 + (2\pi f)^2\right]}\]
\[\Delta V_T = \frac{q}{C_{ox}WL}\left(1 - \frac{x_T}{t_{ox}}\right), \quad \Delta I_D = g_m \Delta V_T\]
| W x L | \(\Delta V_T\) | \(\Delta I_D/I_D\) at \(g_m/I_D\) = 20 |
|---|---|---|
| 0.42 µm x 0.15 µm | 305 µV | 0.6 % |
| 1 µm x 1 µm | 19 µV | 0.04 % |
| 10 µm x 10 µm | 0.19 µV | 0.0004 % |
sky130 nfet_01v8: \(t_{oxe}\) = 4.148 nm, \(C_{ox}\) = 8.3 fF/µm²
Per decade of time constant
\[N \approx N_t \cdot kT \cdot W L \cdot \lambda \ln 10\]
\(N_t = 4 \times 10^{17}\) cm\(^{-3}\)eV\(^{-1}\) (sky130 nfet), \(kT\) = 26 meV, \(\lambda\) = 0.1 nm
| W x L | Active traps per decade |
|---|---|
| 0.42 µm x 0.15 µm | 0.15 |
| 1 µm x 1 µm | 2.4 |
| 10 µm x 10 µm | 240 |
\[\frac{\Delta t}{t} = \frac{\Delta V_{ref} + V_{os}}{V_{ref}} - \frac{\Delta I}{I} = 1760 \text{ ppm}\]
\[\tau_0 = \left(\frac{1}{3.7 \text{ s}} + \frac{1}{1.4 \text{ s}}\right)^{-1} = 1.0 \text{ s}, \quad f_0 = \frac{1}{2\pi\tau_0} = 0.15 \text{ Hz}\]
| Device | W/L [µm] | \(\Delta V_T\), one electron | \(\Delta t/t\) |
|---|---|---|---|
| Comparator input pair | 4 x 3.2/0.94 | 1.6 µV | 3 ppm |
| Comparator load | 3.2/0.94 | 6.4 µV | 11 ppm |
| Reset switch, off leakage | 1.92/0.22 | 46 µV | < 1 ppm |
| PTAT core | 3.2/0.94 | 6.4 µV | 30 to 90 ppm |
| Ramp current mirror unit | 3.2/0.94 | 6.4 µV | 190 ppm |
| Measured | 1760 ppm |
\(C_{ox}\) = 8.2 fF/µm² (pfet), 8.3 fF/µm² (nfet), from \(t_{oxe}\) in the sky130 model cards; \(nV_T\) = 34 mV
| Source | Step \(\Delta t/t\) vs \(T\) | Step \(\Delta t/t\) vs \(V_{DD}\) |
|---|---|---|
| Comparator offset | constant | \(\propto 1/V_{DD}\) |
| Supply | ruled out | ruled out |
| Weak inversion current | \(\propto 1/T\) or faster | constant |
| Measured | falls, 2500 to 1750 ppm from 5 °C to 45 °C | not measured |
\[\tau = \tau_0 e^{x/\lambda} \Rightarrow dx = \lambda\frac{d\tau}{\tau}\]
\[S(f) \propto \int_{\tau_1}^{\tau_2} \frac{\tau}{1 + \omega^2\tau^2}\frac{d\tau}{\tau} = \frac{1}{\omega}\left[\arctan(\omega\tau)\right]_{\tau_1}^{\tau_2} \approx \frac{\pi}{2\omega}\]
for \(1/\tau_2 \ll \omega \ll 1/\tau_1\)
\[S_{V_G}(f) = \frac{q^2\, kT\, \lambda\, N_t}{W L\, C_{ox}^2\, f}\]
BSIM4, flicker noise model 1, as used by ngspice:
\[S_{I_D} \propto \frac{q^2\, kT\, \mu\, I_D}{10^{10}\, f^{EF} A_{bulk} C_{oxe} L^2} \left[\text{NOIA} \ln(\ldots) + \ldots\right]\]
| sky130 01v8 | \(t_{oxe}\) | NOIA [J\(^{-1}\)m\(^{-3}\)] | \(N_t\) [cm\(^{-3}\)eV\(^{-1}\)] | EF |
|---|---|---|---|---|
| nfet | 4.148 nm | 2.5e42 | \(4.0 \times 10^{17}\) | 0.84 |
| pfet | 4.23 nm | 1.5e42 | \(2.4 \times 10^{17}\) | 1.0 |