notes/Resources/electricity/formulas.md

## Ohms Law 

*Solve for voltage:* 

```latex
\displaystyle V = I*R
```

*Solve for resistance:* 

```latex
\displaystyle R = \frac{V}{I}
```

*Solve for current* 
```latex
\displaystyle I = \frac{V}{R}
```

## Resistors in Series 

```latex
R = R1 + R2 + R3 ...
```

## Resistors in Parallel 

```latex
\frac{1}{R} = \frac{1}{R1} + \frac{1}{R2} + \frac{1}{R3} ... \\
\textit{}\\
\textit{For two resistors in parallel:}\\
\textit{}\\
R = \frac{R1 * R2}{R1 + R2}
```

***Tip:***
If resistors of the same value are in parallel the total resistance is a single resistor divided by the amount if resistors.

## Voltage Divider

```latex
V_{out} = V_{in}(\frac{R_{1}}{R_1+R_2})
```

## Thevenin’s Theorem 
States that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single voltage source and series resistance connected to a load.

## Conservation of Charge (First Law) 

All current entering a node must also leave that node

```latex
\sum{I_{IN}} = \sum{I_{OUT}}
```
**Example:**
![](Resources/electricity/assets/kirchhoffs-law-1.svg)

For this circuit kirchhoffs law states that:

```latex
\displaystyle i1 = i2 + i3 + i4
```

## Conservation of Energy (Second Law)
All the potential differences around the loop must sum to zero.

```latex
\displaystyle \sum{V} = 0
```

## Capacitors in Series

```latex
\displaystyle \frac{1}{C_{t}} = \frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{3}} ...
```

## Impedance in a Circuit 
```latex
Z = \sqrt{R^2 + X^2} \\
\textit{}\\
X = X_{L} - X_{C} \\ 
```

## Capacitive Reactance 

```latex
\displaystyle X_{c} = \frac{1}{2 \pi fC}
```

## Inductive Reactance 
```latex
\displaystyle X_{l} = 2\pi fL
```

## Analog Filters

## Cutoff Frequency for RC Filters 
```latex
\displaystyle f_{c} = \frac{1}{2\pi RC}
```

## Cutoff Frequency for RL Filters
```latex
\displaystyle f_{c} = \frac{R}{2\pi L}
```

## Cutoff Frequency for multiple Low Pass Filters
```latex
\displaystyle f_{(-3db)} = f_{c}\sqrt{2^{(\frac{1}{n})}-1}
```

Where $n$ = Number if **identical** filters

## Resonance Frequency for RLC Low Pass Filter
```latex
\displaystyle f_{o} = \frac{1}{2\pi \sqrt{LC}}
```

## Center Frequency with Fc and Fh
```latex
f_{c} = \sqrt{f_{h}*f_{l}}
```

## Filter Response for RC Filters 
```latex
V_{out} = V_{in}(\frac{X_c}{\sqrt{R_{1}^2+X_{c}^2}})
```

## Cutoff Frequency $\pi$ Topology Filter
When the two capacitors have the same capacitance, it can be calculated like this:
```latex
\displaystyle f_c = \frac{1}{4\pi\sqrt{LC}}
```
## Angular Frequency ($\omega$) 
```latex
\omega = 2\pi f = \frac{2\pi}{T}
```

## RLC Series Response

This is basically Ohms Law:

```latex
\displaystyle V = IZ
```

Where $Z$ is the impedance:

```latex
Z = \sqrt{R^2 + (X_L - X_C)^2}
```

$X_L$ = Reactive Inductance
$X_C$ = Reactive Capacativw 

## Current through a transistor

```latex
\displaystyle I_{EQ} = \frac{V_{BB}-{V_{BE}}}{\frac{R_B}{(\beta+1)}+R_E}
```

## Gain Bandwidth Product 
```latex
GBP = A_V * f_c
```

```latex
\displaystyle f_c = \frac{GBP}{A_V}
```

## Bandwidth of Multiple OpAmps

Where $n$ = number of stages
and $BW$ = Bandwidth of single op-amp

```latex
BW_E = BW\sqrt{2^\frac{1}{n}-1}
```

## Power lost in a Resistor
```latex
P = IV = I^2R = \frac{V^2}{R}
```