The QED Lagrangian — everything in electronics follows from this
\[\mathcal{L} = \bar{\psi}[i \hbar c \gamma^\mu\partial_\mu - mc^2]\psi - q[\bar{\psi} \gamma^\mu \psi] A_\mu - \frac{1}{16 \pi}F_{\mu\nu}F^{\mu\nu}\]
Schrödinger equation
\[i\hbar \frac{d}{dt} \psi(r,t) = \widehat{H} \psi(r,t)\]
Probability density of a particle
\[P = \vert \psi(r,t)\vert ^2 \text{ , } \psi(r,t) = A e^{i(kr - \omega t)}\]
Heisenberg uncertainty
\[\sigma_x \sigma_p \ge \frac{\hbar}{2} \text{ , } \Delta E \Delta t > \frac{h}{2\pi}\]
Fermi-Dirac distribution, and its Boltzmann tail
\[f(E) = \frac{1}{e^{(E - E_F)/kT} + 1} \approx e^{(E_F - E)/kT}\]
Maxwell's equations
\[\oint_{\partial \Omega} \mathbf{E} \cdot d\mathbf{S} = \frac{1}{\epsilon_0} \iiint_{V} \rho\cdot dV \text{ , } \oint_{\partial \Omega} \mathbf{B} \cdot d\mathbf{S} = 0\]
\[\oint_{\partial \Sigma} \mathbf{E} \cdot d\mathbf{\ell} = - \frac{d}{dt}\iint_\Sigma \mathbf{B}\cdot d\mathbf{S}\]
\[\oint_{\partial \Sigma} \mathbf{B} \cdot d\mathbf{\ell} = \mu_0\left(\iint_\Sigma \mathbf{J} \cdot d\mathbf{S} + \epsilon_0 \frac{d}{dt}\iint_\Sigma\mathbf{E} \cdot d\mathbf{S} \right)\]
Force on a charge
\[\vec{F} = q\vec{E}\]
Density of electrons in the conduction band
\[n = \int_{E_C}^{\infty} N(E) f(E) dE\]
Effective density of states
\[N_c = 2 \left[\frac{2 \pi k T m_n^*}{h^2}\right]^{3/2} \text{ , } N_v = 2 \left[\frac{2 \pi k T m_p^*}{h^2}\right]^{3/2}\]
Intrinsic carrier concentration
\[n_i = \sqrt{N_c N_v}\, e^{-E_g/(2 k T)}\]
Doped silicon (mass action)
\[n_n = N_D \text{ , } p_n = \frac{n_i^2}{N_D} \text{ ; } p_p = N_A \text{ , } n_p = \frac{n_i^2}{N_A}\]
Drift current
\[\vec{J} = q n \mu \vec{E}\]
Diffusion current
\[J = -q D_n \frac{d\rho}{dx}\]
Thermal voltage
\[V_T = \frac{kT}{q} \approx 25.85\text{ mV @ 300 K}\]
Built-in voltage of a pn junction
\[\Phi_0 = V_T \ln\left(\frac{N_A N_D}{n_i^2}\right)\]
Depletion width
\[W = \sqrt{\frac{2 \varepsilon_{si} (\Phi_0 + V_R)}{q} \cdot \frac{N_A + N_D}{N_A N_D}}\]
Diode equation
\[I_D = I_S\left(e^{V_D/V_T} - 1\right) \text{ , } I_S = q A n_i^2 \left( \frac{D_n}{L_n N_A} + \frac{D_p}{L_p N_D}\right)\]
Forward voltage temperature dependence
\[V_D = \frac{kT}{q}(\ell - 3 \ln T) + V_G \Rightarrow \frac{dV_D}{dT} = \frac{k}{q}\left(\ell - 3\ln T - 3\right)\]
Difference of two diode voltages (PTAT)
\[V_{D1} - V_{D2} = V_T \ln N\]
Generation (leakage) current of a reverse biased junction
\[I_{gen} = \frac{q A n_i W}{\tau_g}\]
Weak inversion
\[I_{D} = I_{D0} \frac{W}{L} e^{V_{eff}/n V_T} \text{ if } V_{DS} > 3V_T\]
\[n = \frac{C_{ox} + C_{j0}}{C_{ox}} \text{ , } I_{D0} = (n-1)\mu_n C_{ox} V_T^2\]
Strong inversion, define
\[V_{eff} = V_{GS} - V_{tn} \text{ , } \ell = \mu_n C_{ox}\frac{W}{L}\]
\[I_{DS} = \ell \begin{cases} V_{eff} V_{DS} & \text{if }V_{DS} << V_{eff} \\[10pt] V_{eff} V_{DS} - V_{DS}^2/2 & \text{if } V_{DS} < V_{eff} \\[10pt] \frac{1}{2} V_{eff}^2\left[1 + \lambda(V_{DS} - V_{eff})\right] & \text{if } V_{DS} > V_{eff} \\[10pt] \end{cases}\]
Transconductance
\[g_m = \frac{\partial I_{DS}}{\partial V_{GS}} = \ell V_{eff} = \sqrt{2 \ell I_D} = \frac{2 I_D}{V_{eff}} \text{ (strong)} \text{ , } g_m = \frac{I_D}{nV_T} \text{ (weak)}\]
Transconductance efficiency
\[\frac{g_m}{I_D} = \frac{1}{nV_T} \text{ (weak)} \text{ , } \frac{g_m}{I_D} = \frac{2}{V_{eff}} \text{ (strong)}\]
Output conductance and intrinsic gain
\[g_{ds} = \frac{1}{r_{ds}} \approx \lambda I_D \text{ , } A = g_m r_{ds} = \frac{2}{\lambda V_{eff}}\]
Capacitances
\[C_{gs} = \frac{2}{3}WLC_{ox} \text{ (saturation)} \text{ , } C_{gd} = C_{ox} W L_{ov}\]
\[C_{sb} = (A_s + A_{ch}) C_{js} \text{ , } C_{js} = \frac{C_{j0}}{\sqrt{1 + \frac{V_{SB}}{\Phi_0}}}\]
Miller's theorem
\[C_{in} = (1 + A)C \text{ , } C_{out} = \left(1 + \frac{1}{A}\right)C \Rightarrow C_{in} \approx C_{gd}\, g_m r_{ds}\]
Matching (Pelgrom)
\[\sigma^2(\Delta P) = \frac{A_P^2}{WL} + S_P^2 D^2\]
Current mismatch (Kinget)
\[\frac{\sigma_{I_D}^2}{I_D^2} = \frac{1}{WL}\left[\left(\frac{g_m}{I_D}\right)^2 \sigma_{vt}^2 + \frac{\sigma_{\ell}^2}{\ell^2}\right] \text{ , } \sigma_{v_i}^2 = \frac{\sigma_{I_D}^2}{g_m^2}\]
Mean square and power spectral density
\[\overline{x^2(t)} = \int_{0}^{\infty}{S_x(f)df} \text{ , } S_y(f) = S_x(f)\vert H(f)\vert ^2\]
Thermal noise of a resistor
\[S_{th}(f) = 4kTR\]
Sampled (kT/C) noise bandwidth of an RC
\[f_x = \frac{\pi f_0}{2} = \frac{1}{4RC}\]
Uncorrelated sources add in power
\[\overline{e_{tot}^2} = \overline{e_{1}^2} + \overline{e_{2}^2}\]
Signal-to-noise ratio
\[SNR = 10 \log\left(\frac{\overline{v_{sig}^2}}{\overline{e_{n}^2}}\right)\]
Noise factor and Friis' formula
\[F = \frac{SNR_{input}}{SNR_{output}} \text{ , } NF = 10\log(F)\]
\[F = F_1 + \frac{F_2 - 1}{G_1} + \frac{F_3 - 1}{G_1 G_2} + \cdots\]
Common source
\[A = -g_m r_{ds} \text{ , } r_{out} = r_{ds}\]
Common drain (source follower)
\[A = \frac{g_m}{g_m + g_{ds} + g_s} \text{ , } r_{out} \approx \frac{1}{g_m}\]
Common gate
\[A = 1 + g_m r_{ds} \text{ , } r_{in} \approx \frac{1}{g_m}\left(1 + \frac{R_L}{r_{ds}}\right)\]
Source degeneration and cascode output resistance
\[r_{out} = r_{ds2}\left[1 + R_s(g_{m2} + g_{ds2})\right] \approx r_{ds2}\, g_{m2} R_s\]
Bipolar / diode core
\[V_{BE} = V_T \ln\frac{I_C}{I_S} \text{ (CTAT)} \text{ , } \Delta V_{BE} = V_T \ln N \text{ (PTAT)}\]
Brokaw bandgap output
\[V_{REF} = V_{BE3} + \frac{R_2}{R_3} V_T \ln\frac{R_2}{R_1}\]
Bandgap voltage with curvature
\[V_{BG} = V_{G0} + (m-1)\frac{kT}{q}\ln{\frac{T_0}{T}} + T\left[\frac{k}{q}\ln{\frac{J_2}{J_1}}\frac{2R_2}{R_1} - \frac{V_{G0} - V_{be0}}{T_0}\right]\]
Pole/zero frequency
\[\omega_{p\vert z} \propto \frac{1}{RC} \text{ (Active-RC)} \text{ , } \omega_{p\vert z} \propto \frac{G_m}{C} \text{ (Gm-C)}\]
General biquad
\[H(s) = \frac{\frac{C_1}{C_B}s^2 + \frac{G_2}{C_B}s + \frac{G_1G_3}{C_A C_B}}{s^2 + \frac{G_5}{C_B}s + \frac{G_3 G_4}{C_A C_B}}\]
SC resistance
\[Z_{I} = \frac{1}{C_1 f_\phi}\]
SC gain stage and integrator
\[H(z) = \frac{C_1}{C_2}z^{-1} \text{ , } H(z) = \frac{C_1}{C_2}\frac{z^{-1}}{1 - z^{-1}}\]
First and second order IIR
\[H(z) = \frac{b}{z-a} \text{ , } H(z) = \frac{b z}{z^2 - 2a z + (a^2+b^2)}\]
Quantization noise
\[\overline{e_n^2} = \frac{\Delta^2}{12} \Rightarrow SQNR \approx 6.02B + 1.76 \text{ dB}\]
Oversampling
\[SQNR \approx 6.02B + 1.76 + 10\log(OSR)\]
First order noise shaping
\[Y(z) = STF(z)U(z) + NTF(z)E(z) \text{ , } STF = z^{-1} \text{ , } NTF = 1 - z^{-1}\]
\[SQNR = 6.02B + 1.76 - 5.17 + 30\log(OSR)\]
Figures of merit
\[FOM_W = \frac{P}{2^B f_s} \text{ , } FOM_S = SNDR + 10\log\left(\frac{f_s/2}{P}\right)\]
DAC: a digital number scales a reference
\[V_{out} = D_{in} \times V_{ref}\]
The inductor and capacitor of a switcher
\[I_x(t) = \frac{1}{L}\int{V_x(t)dt} \text{ , } V_o(t) = \frac{1}{C}\int{(I_x(t) - I_o(t))dt}\]
Ideal buck output
\[V_o = V_{in} \times \text{Duty-Cycle}\]
A modulated carrier, and phase versus frequency
\[A_m(t)\cos\left(2\pi f_{c}t + \phi_{m}(t)\right) \text{ , } \phi(t) = 2\pi\int_0^t f(t)dt\]
Loop gain of a charge-pump PLL
\[L(s) = \frac{K_{osc} K_{pd} K_{lp} H_{lp}(s)}{N s}\]
\[K_{osc} = 2\pi\frac{df}{dV_{cntl}} \text{ , } K_{pd} = \frac{I_{cp}}{2\pi} \text{ , } K_{lp}H_{lp}(s) = \frac{1}{s(C_1 + C_2)}\frac{1 + sRC_1}{1 + sR\frac{C_1C_2}{C_1+C_2}}\]
Crystal input impedance
\[Z_{in} \approx \frac{L C_F s^2 + 1}{L C_F C_P s^2 + C_F + C_P}\]
Ring oscillator frequency
\[f = \frac{1}{2 N t_{pd}} \text{ , } t_{pd} \approx RC \Rightarrow f = \frac{\mu_n (VDD-V_{th})}{\frac{4}{3} N L^2}\]
Current starved ring
\[f \approx \frac{I_{control}}{C \frac{VDD}{2} N}\]
Friis transmission (free space)
\[P_{RX} = \frac{P_{TX}}{D^2}\left[\frac{\lambda}{4\pi}\right]^2\]
Receiver sensitivity
\[P_{RX_{sens}} = -174\text{ dBm} + 10\log_{10}(DR) + NF + E_b/N_0\]