From 4739a5c7a1c2c2431952f9dd11f5631f7677778e Mon Sep 17 00:00:00 2001
From: eggy <danielchen04@hotmail.ca>
Date: Wed, 22 Mar 2023 21:44:16 -0400
Subject: [PATCH] ece140: add inductors, opamps, capacitors

so scufffed
---
 docs/1b/ece140.md | 99 +++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 99 insertions(+)

diff --git a/docs/1b/ece140.md b/docs/1b/ece140.md
index a2442f6..22edb5e 100644
--- a/docs/1b/ece140.md
+++ b/docs/1b/ece140.md
@@ -198,3 +198,102 @@ $$
 To maximise the power transferred from the circuit to the load, $R_L$ should be equal to $R_{Th}$.
 
 $$P_L=v_Li_L$$
+
+## Operational amplifiers
+
+The entire op-amp follows KCL. The output current is the sum of all input currents (the two inputs and V+, V-).
+
+Where $\Delta V$ is the difference between the two inputs, and $A$ is the gain of the opamp:
+
+$$\boxed{V_{out}=A\Delta V}$$
+
+Output voltage is limited by the maximum/minimum of the power supply $V_cc$.
+
+If the output is fed directly into the inverting input (as a **voltage follower**), the gain is ignored and results in $V_{out}=\Delta V$.
+
+An **ideal opamp** has no input current and equal voltages entering the opamp.
+
+$$
+\boxed{i_1=i_2=0} \\
+\boxed{v_1=v_2}
+$$
+
+**Inverting amplifiers** feed their input back and return negative voltage.
+
+$$V_{out}=-\frac{R_f}{R_i}V_{in}$$
+
+<img src="https://upload.wikimedia.org/wikipedia/commons/4/41/Op-Amp_Inverting_Amplifier.svg" width=700>(Source: Wikimedia Commons)</img>
+
+**Non-inverting amplifiers** moves the voltage source to the non-inverting terminal.
+
+$$v_o=\left(1+\frac{R_f}{R_i}v_i\right)$$
+
+<img src="https://upload.wikimedia.org/wikipedia/commons/6/66/Operational_amplifier_noninverting.svg" width=700>(Source: Wikimedia Commons)</img>
+
+**Voltage followers** have either $R_f=0$ or $R_i=\infty$, so:
+
+$$v_o=v_i$$
+
+<img src="https://upload.wikimedia.org/wikipedia/commons/f/f7/Op-Amp_Unity-Gain_Buffer.svg" width=700>(Source: Wikimedia Commons)</img>
+
+A **summing amplifier** splits an inverting amplifier's input into multiple voltage sources in series with resistances, all parallelised into the opamp:
+
+$$v_o=-R_f\left(\frac{V_1}{R_1}+\frac{V_2}{R_2}+\frac{V_3}{R_3}\right)$$
+
+<img src="https://upload.wikimedia.org/wikipedia/commons/3/3e/Op-Amp_Summing_Amplifier.svg" width=700>(Source: Wikimedia Commons)</img>
+
+A **difference amplifier** is funky. To ensure that output is zero when inputs are equal, $\frac{R_1}{R_2}=\frac{R_3}{R_4}$.
+
+$$v_o=\frac{R_2}{R_1}(v_2-v_1)$$
+
+<img src="https://upload.wikimedia.org/wikipedia/commons/a/a2/Op-Amp_Differential_Amplifier.svg" width=700>(Source: Wikimedia Commons)</img>
+
+
+## Capacitors
+
+Capacitors are open circuits in DC that store energy in electric fields. Capacitance is measured in **farads** ($\pu{1 F = 1 C/V}$).
+
+Where $A$ is the cross-section area of the wire, $\epsilon$ is the permittivity of the dielectric, and $d$ is the distance between plates:
+
+$$C=\frac{\epsilon A}{d}$$
+
+Capacitors charge only when power is positive ($VI>0$).
+
+For linear capacitors:
+
+$$i=C\frac{dv}{dt}$$
+
+$$v(t)=\frac{1}{C}\int^t_{t_0}i(t)dt+v(t_0)$$
+
+The energy in a capacitor can be interconverted.
+
+$$U=\frac 1 2 CV^2$$
+
+Capacitor rules are the opposite of resistor rules.
+
+- In parallel: $C_{eq} = C_1 + C_2 + ...$
+- In series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$
+
+## Inductors
+
+Inductors store energy in their magnetic field. Inductance is measured in **henrys** ($\pu{1 H = 1 V\cdot S/A}$). An ideal inductor has zero resistance and capacitance
+
+Where $L$ is the inductance (opposition of charge flow):
+
+$$V=L\frac{di}{dt}$$
+
+Inductor rules are the same as resistor rules.
+
+### Selenoids
+
+Selenoids have an inductance based on their cross sectional area $A$, number of coils $N$, length $\ell$, and core permeability $\mu$:
+
+$$L=\frac{N^2\mu A}{\ell}$$
+
+Where $i(t_0)$ is the total current for $-\infty<t<t_0$
+
+$$i=\frac 1 L\int^t_{t_0}v(t)dt + i(t_0)$$
+
+Much like capacitors, inductors have energy now based on current.
+
+$$U=\frac 1 2 Li^2$$