notes/Resources/electricity/formulas.md

---
cards-deck: electricity
---

## Ohms Law 

*Solve for voltage:* 
#card

$\displaystyle V = I*R$
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*Solve for resistance:* 
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$\displaystyle R = \frac{V}{I}$
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*Solve for current* 
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$\displaystyle I = \frac{V}{R}$
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## Resistors in Series 
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$R = R1 + R2 + R3 ...$
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## Resistors in Parallel 
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$$
\begin{flalign}
&\frac{1}{R} = \frac{1}{R1} + \frac{1}{R2} + \frac{1}{R3} ... &\\
\\
&\textit{For two resistors in parallel:} &\\
\\
&R = \frac{R1 * R2}{R1 + R2} &\\\
\end{flalign}
$$
***Tip:***
If resistors of the same value are in parallel the total resistance is a single resistor divided by the amount if resistors.
## Voltage Divider
#card 
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$V_{out} = V_{in}(\frac{R_{1}}{R_1+R_2})$

## 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) 
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All current entering a node must also leave that node
$$
\begin{flalign}
\sum{I_{IN}} = \sum{I_{OUT}}&&
\end{flalign}
$$
**Example:**
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![](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
#card 

$\displaystyle \frac{1}{C_{t}} = \frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{3}} ...$
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## Impedance in a Circuit 
#card 
$$
\begin{flalign}
&Z = \sqrt{R^2 + X^2} &\\\
\\
&X = X_{L} - X_{C} \\ 
\end{flalign}
$$
## Capacitive Reactance 
#card
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$\displaystyle X_{c} = \frac{1}{2 \pi fC}$

## Inductive Reactance 
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$\displaystyle X_{l} = 2\pi fL$
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## Analog Filters

## Cutoff Frequency for RC Filters 
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$\displaystyle f_{c} = \frac{1}{2\pi RC}$
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## Cutoff Frequency for RL Filters
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$\displaystyle f_{c} = \frac{R}{2\pi L}$
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## 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
#card 

$\displaystyle f_{o} = \frac{1}{2\pi \sqrt{LC}}$
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## Center Frequency with Fc and Fh
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$f_{c} = \sqrt{f_{h}*f_{l}}$
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## Filter Response for RC Filters 
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$V_{out} = V_{in}(\frac{X_c}{\sqrt{R_{1}^2+X_{c}^2}})$
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## Cutoff Frequency $\pi$ Topology Filter
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When the two capacitors have the same capacitance, it can be calculated like this:
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$\displaystyle f_c = \frac{1}{4\pi\sqrt{LC}}$

## Angular Frequency ($\omega$) 
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$\omega = 2\pi f = \frac{2\pi}{T}$
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## RLC Series Response

This is basically Ohms Law:

$\displaystyle V = IZ$

Where $Z$ is the impedance:

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

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

## Current through a transistor
 
$\displaystyle I_{EQ} = \frac{V_{BB}-{V_{BE}}}{\frac{R_B}{(\beta+1)}+R_E}$

## Gain Bandwidth Product 
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$GBP = A_V * f_c$
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$\displaystyle f_c = \frac{GBP}{A_V}$

## Bandwidth of Multiple OpAmps

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

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

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