notes/Areas/electricity/formulas.md

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# Ohms Law
Solve for voltage:
$\displaystyle V = \frac{I}{R}$
*Solve for resistance:*
$R = \frac{V}{I}$
_Solve for current_
$$
\begin{flalign}
I & = \frac{V}{R} &
\end{flalign}
$$
# Resistors in Series
$R = R1 + R2 + R3 ...$
# Resistors in Parallel
$$
\begin{flalign}
&\frac{1}{R} = \frac{1}{R1} + \frac{1}{R2} + \frac{1}{R3} ... &\\
\\
&\textit{For two resistors in parallel:} &\\
\\
&R = \frac{R1 * R2}{R1 + R2} &\\\
\end{flalign}
$$
***Tip:***
If resistors of the same value are in parallel the total resistance is a single resistor divided by the amount if resistors.
## Thevenins 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.
# Kirchhoff's Law
## Conservation of Charge (First Law)
All current entering a node must also leave that node
$$
\begin{flalign}
\sum{I_{IN}} = \sum{I_{OUT}}&&
\end{flalign}
$$
**Example:**
![](./assets/kirchhoffs-law-01.svg)
For this circuit kirchhoffs law states that:
$\displaystyle i1 = i2 + i3 + i4$
## Conservation of Energy (Second Law)
All the potential differences around the loop must sum to zero.
$\displaystyle \sum{V} = 0$
## Capacitors in Series
$\displaystyle \frac{1}{C_{t}} = \frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{3}} ...$
### Impedance in a Circuit
$$
\begin{flalign}
&Z = \sqrt{R^2 + X^2} &\\\
\\
&X = X_{L} - X_{C} \\
\end{flalign}
$$
# Capacitive Reactance
$\displaystyle X_{c} = \frac{1}{2 \pi fC}$
# Inductive Reactance
$\displaystyle X_{l} = 2\pi fL$
# Analog Filters
## Cutoff Frequency for RC Filters
$\displaystyle f_{c} = \frac{1}{2\pi RC}$
## Cutoff Frequency for RL Filters
$\displaystyle f_{c} = \frac{R}{2\pi L}$
## Signal Response of an RC/RL Filter
$X_c$ = [[#Capacitive Reactance]] || [[#Inductive Reactance]]
$\displaystyle V_{out} = V_{in}(\frac{X_{c}}{\sqrt{R^2+X_{c}^2}})$
## Cutoff Frequency for multiple Low Pass Filters
$\displaystyle f_{(-3db)} = f_{c}\sqrt{2^{(\frac{1}{n})}-1}$
Where $n$ = Number if **identical** filters
## Resonance Frequency for RLC Low Pass Filter
$\displaystyle f_{o} = \frac{1}{2\pi \sqrt{LC}}$
## Center Frequency with Fc and Fh
$f_{c} = \sqrt{f_{h}*f_{l}}$
## Filter Response for RC Filters
$V_{out} = V_{in}(\frac{X_c}{\sqrt{R_{1}^2+X_{c}^2}})$
## Cutoff Frequency $\pi$ Topology Filter
When the two capacitors have the same capacitance, it can be calculated like this:
$\displaystyle f_c = \frac{1}{4\pi\sqrt{LC}}$
# Voltage Divider
$V_{out} = V_{in}(\frac{R_{1}}{R_1+R_2})$
# Angular Frequency ($\omega$)
$\omega = 2\pi f = \frac{2\pi}{T}$ ^4ad7fc
# RLC Series Response
This is basically Ohms Law:
$\displaystyle V = IZ$
Where $Z$ is the impedance:
$Z = \sqrt{R^2 + (X_L - X_C)^2}$
# Current through a transistor
$\displaystyle I_{EQ} = \frac{V_{BB}-{V_{BE}}}{\frac{R_B}{(\beta+1)}+R_E}$