## On electronics . note

### Model

#### Circuit elements on the microscopic level

##### Hierarchy of devices

*ordered hierarchy of viewpoints on devices such as resistors (electrical resistance)*

##### On units

Here I want to work out more natural units for stuff like capacitance, to understand the formulas describing different circuit device behaviors. See also On physical units . note.

Thoughts:

• A common idea is to consider charge $[q_x]$ fundamental. The inverse $\frac{1}{[q_x]}$ may be thought of as node flux.
• Time is a given and the notion of motion is relatively basic too.
• Then current $I$ is charge over time.

$[I]=\dfrac{[q_x]}{[t]}$

• The notion of macroscopic voltage $U$ (units of energy over charge) is probably the least removed one from any of the macroscopic theories, because its constituents are often a starting point for the microscopic theory as well. If energy is set to be frequency

$[U]=\dfrac{1}{[q_x][t]}$

• Very generally, I guess a resistance of an electrical device will be a function mapping the two above concepts, $I=f_G(U)$. In the most basic model is a linear $f_G$ and we write

$I=G\cdot U$

$[G]=[q_x]^2$

(If energy is set to be frequency, $[q_x]^2$ is Siemens $[S]$ (Ohm $[\Omega]$ to the power of $-1$))

and one often uses reciprocal quantities, $G\leftrightarrow\dfrac{1}{R}$ (see Electrical conductance and resistance).

$[R]=\dfrac{1}{[q_x]^2}$

##### Field theory

On a very encompassing level, we should maybe view all those problems as question of the dynamics of charge collections, making for a charge density (making up $q_x$) under external forces (contributing to the voltages in the system). The charges interact, or put differently, the charge density self-interacts.

##### Models for devices
$G = a_R + a_C \dfrac{\partial}{\partial t} + a_L \int{\mathrm d}t$

An capacitor are two ends of a conducting line that hold charges with a some voltage between them. Ideally, i.e. in the simplest case, we double the voltage if we double the charges - thus the number $C:=\frac{q_x}{U}$ is a constant. The factor $\frac{1}{C}$ says how strongly $U$ rises in reaction to a rise of $q_x$. Expressing that rise in terms of the time dependent incoming charge current, $q_x\propto \int^t I(t)$, we find $G{\sim}C\dfrac{\partial}{\partial t}$.

The dependence of $I$ on $U$ for various theories (e.g. models for electrical devices in a circuit) is indeed commonly given as such a differential or integral relation, written down using coefficients called capacitance $C$ and also inductance $L$. They encode time scales

$[C] = [q_x]^2[t]$

$[L] = \dfrac{1}{[q_x]^2}[t]$

so that

$[C\dfrac{\partial}{\partial t}] = [G]$

$[\dfrac{1}{L}\int^t {\mathrm d}t] = [G]$

Microscopically,

$G\propto\dfrac{\mathcal A}{\mathcal l}$

(ratio of geometric quantities of device length and cross section) is a common and intuitive dependency. But it might be better to first consider these laws in a more local form.

#### On the level of circuits

##### Axioms

$\forall j.\ \underset{{U_i\,\in\,\mathrm{Schleife}_j}}{\sum}U_i = 0$

$\forall j.\ \underset{{I_i\,\in\,\mathrm{Knoten}_j}}{\sum}I_i = 0$ ?????