The behavior of particles is written in the mathematics of quantum mechanics

\(\psi(x,t) = Ae^{j(kx - \omega t)}\) _{Probability amplitude of a particle}

\(\frac{1}{2 m} \frac{\hbar}{j^2} \frac{\partial^2}{\partial^2 x}\psi(x,t) + U(x)\psi(x,t) = -\frac{\hbar}{j}\frac{\partial}{\partial t} \psi(x,t)\) _{Time evolution of the energy of a particle[^2]}

\(\frac{n_n}{n_p} = \frac{e^{(E_{p} - \mu) / kT} + 1}{e^{(E_{n} - \mu) / kT} + 1}\) _{Relates the average number of fermions in thermal equlilibrium to the energy of a single-particle state[^3]}

[^2]: Schrödinger equation

[^3]: Fermi-Dirac statistics

The world is analog and is written in the mathematics of calculus [^1]

\(\oint_{\partial \Omega} \mathbf{E} \cdot d\mathbf{S} = \frac{1}{\epsilon_0} \iiint_{V} \rho \cdot dV\)
_{Relates net electric flux to net enclosed electric charge}

\(\oint_{\partial \Omega} \mathbf{B} \cdot d\mathbf{S} = 0\) _{Relates net magnetic flux to net enclosed magnetic charge}

\(\oint_{\partial \Sigma} \mathbf{E} \cdot d\mathbf{\ell} = - \frac{d}{dt}\iint_\Sigma \mathbf{B} \cdot d\mathbf{S}\) _{Relates induced electric field to changing magnetic flux}

\(\oint_{\partial \Sigma} \mathbf{B} \cdot d\mathbf{\ell} = \mu_0\left( \iint_\Sigma \mathbf{J} \cdot d\mathbf{S} + \epsilon_0 \frac{d}{dt}\iint_\Sigma \mathbf{E} \cdot d\mathbf{S} \right)\) _{Relates induced magnetic field to changing electric flux and to current}

[^1]: Maxwell's equations

The abstract digital world is written in the mathematics of boolean algebra[^4]

\(1 = \text{True}\), \(0 = \text{False}\)

All digital processing can be made with the NOT(A AND B) function!

[^4]: Boolean algebra

People that make digital circuits can easily reuse the work of others

People that make analog circuits can learn from others, but need to deal with the real world on their own

Should we do as much as possible in the abstract digital world?

First order filter

1. any line with a coefficient is a multiplier
2. any box output is a multiplication of the coefficient and the input
3. any sum, well, sum all inputs
4. be aware of gremlins (a sudden -+ swap)

\[H(s) =\frac{V_o(s)}{V_i(s)} = \frac{ k_1 s + k_0 }{s + w_o}\]

Second order filter

Bi-quadratic is a general purpose second order filter.

\[H(s) = \frac{k_2 s^2 + k_1 s + k_0}{s^2 + \frac{\omega_0}{Q} s + \omega_o^2}\]

\[V_o = \frac{I_o}{s C} = \frac{\omega_{ti}}{s} V_i\]

\[\omega_{ti} = \frac{G_m}{C}\]

\[s C V_o = G_m Vi\]

\[H(s) = \frac{V_o}{V_i} = \frac{G_m}{sC}\]

\[H(s) = \frac{V_o}{V_i} = -\frac{G_m}{sC}\]

\[H(s) = \frac{ k_1 s + k_0 }{s + w_o}\]

\[H(s) = \frac{s \frac{C_x}{C_a + C_x} + \frac{G_{m1}}{C_a + C_x}}{s + \frac{G_{m2}}{C_a + C_x}}\]

\[H(s) = \frac{k_2 s^2 + k_1 s + k_0}{s^2 + \frac{\omega_0}{Q} s + \omega_o^2}\]

\[H(s) = \frac{ s^2\frac{C_X}{C_X + C_B} + s\frac{G_{m5}}{C_X + C_B} + \frac{G_{m2}G_{m4}}{C_A(C_X + C_B)}} {s^2 + s\frac{G_{m3}}{C_X + C_B} + \frac{G_{m1}G_{m2}}{C_A(C_X + C_B)} }\]

General purpose first order filter

\[H(s) = \frac{ k_1 s + k_0 }{s + w_o}\]

\[H(s) = \frac{ -\frac{C_1}{C_2}s -\frac{G_1}{ C_2}}{s + \frac{G_2}{ C_2}}\]

General purpose biquad

\[H(s) = \frac{k_2 s^2 + k_1 s + k_0}{s^2 + \frac{\omega_0}{Q} s + \omega_o^2}\]

\[H(s) = \frac{\left[ \frac{C_1}{C_B}s^2 + \frac{G_2}{C_B}s + (\frac{G_1G_3}{C_A C_B})\right]}{\left[ s^2 + \frac{G_5}{C_B}s + \frac{G_3 G_4}{C_A C_B}\right]}\]

The OTA is not ideal

\[H(s) \approx \frac{A_0}{(1 + s A_o R C)(1 + \frac{s}{w_{ta}})}\]

where \(A_0\) is the gain of the amplifier, and \(\omega_{ta}\) is the unity-gain frequency.