What is Electrical Engineering? Electrical Engineering(EE) involves pragmatically applying abstract electromagnetism models in conjunction with fancy mathematical manipulations to meet design requirements. Any system that involves producing, measuring, or transferring electricity can be analyzed from an EE perspective. There are five branches, all of which deals with electric circuit models(commonly simply referred to as circuits): Communication,Computer,Control, Power; Signal Processing. Therefore circuit theory is an essential required framework for any EE.

Table of Contents

## Circuit Theory

Circuit Theory is a specialized subset of Electromagnetic Field Theory that often permits us to model electric properties without using complex mathematical models. However circuit theory makes three system assumptions:
1. Lumped-Parameter Assumption assumes that all electric effects happens instantly. We make this assumption because electricity travels at the speed of light, so in smaller systems the electric signal movement is very fast.
2. Zero Net Charged Components Assumption assumes that the net charge on every circuit element is zero.
3. No Magnetic Coupling Assumption assumes that there is no magnetic coupling between components in a system. However, no assumption is made on the magnetic coupling WITHIN a component.

### Charges, Voltage, Current, Power

Charges
1. Bipolar-Can be positive or negative
2. Discrete Quantities- Integral multiples of $1.602\cdot 10^{-19}$
3. Electric effects are caused by motion and separation of charges.
1. Separation creates Voltage$.=\frac{dw}{dq}\left(\frac{J}{C}=V\right)$
1. is energy per unit charge
2. Motion creates current$.=\frac{dq}{dt}\left(\frac{C}{s}=A\right)$
1. is rate of charge flow
2. Although current is the flow of discrete electrons, we look at the motion of these electrons from a macroscopic perspective.
Power is energy per unit time$.=\frac{dw}{dt}\left(w=VA=\frac{J}{s}\right)$

### Ideal Simple Circuit Elements

1. Two Terminals (Simple)
2. Mathematically represented as equations of current or voltage (Circuit Elements)
3. Can’t be subdivided into other elements (Simple)

## Frequency Domain

When multiple sinusoidal signals are superimposed on each other and graphed as a function of time, the contributions of each sinusoidal signal can be hard to analyze. Instead, a graph of the different sinusoidal frequencies and phase shifts that make up the overall superimposed signal is sometimes easier to analyze. Watch this video to learn more about the relationship between the frequency and time domain: https://www.youtube.com/watch?v=spUNpyF58BY

### Laplace Transform

Laplace($L$) and Inverse Laplace($L^{-1}$) Transforms are used to switch between the time (t) and frequency(s) domains. Often complex time varying differential equations are easier manipulated in the frequency domain. The Laplace Transform is defined as the integral of the product of the function,$f\left(t\right)$, and $e^{-st}$ from negative infinity to positive infinity, but engineers define the lower limit to zero for practicality. $$L\left\{f\left(t\right)\right\}=\int _{s=0}^{s=\infty }\left(f\left(t\right)\cdot e^{-st}\right)dt\:$$ Bellow are the most useful Laplace Transforms for electrical engineering.
Time Domain(t)Frequency Domain(s)
$1$$\frac{1}{s}$ Proof 1
$t$$\frac{1}{s^2}$ Proof 2
$e^{-\alpha t}$$\frac{1}{s+a}$ Proof 3
$sin\left(\alpha t\right)$$\frac{\alpha }{s^2+\alpha ^2}$ Proof 4
$cos\left(\alpha t\right)$$\frac{s}{s^2+\alpha ^2}$ Proof 5
$te^{-\alpha t}$$\frac{1}{\left(s+a\right)^2}$ Proof 6
$\alpha \cdot f\left(t\right)$$\alpha \cdot F\left(s\right)$ Proof 7
$f_1\left(t\right)+f_2\left(t\right)$$F_1\left(s\right)+F_2\left(s\right)$ Proof 8
$\frac{d}{dt}\left(f\left(t\right)\right)$$s\cdot F\left(s\right)-f\left(0\right)$ Proof 9
$\frac{d^2}{dt^2}\left(f\left(t\right)\right)$$s^2\cdot F\left(s\right)-s\cdot f\left(0\right)-\frac{d}{dt}\left(f\left(0\right)\right)$ Proof 10
$\int _0^tf\left(x\right)dx\:$$\frac{F\left(s\right)}{s}$ Proof 11

### Modeling Switches

The unit step function($u\left(t\right)$) is often used to model switches. $$u\left(t\right)=\begin{pmatrix}1&t>0\\ 0&t<0\end{pmatrix}$$ Technically, the unit step function is not defined at $t=0$, but we will assume linearity between $t=0^-$ and $t=0^+$. $$u\left(t\right)=\begin{pmatrix}1&t>0\\ .5&t=0\\ 0&t<0\end{pmatrix}$$ Engineering Notes Summary