Kirchhoff's Laws In Circuits

Voltage

The net voltage around any loop is 0
- If you take a loop and add all the voltages going in the same direction, and subtract the ones going in the opposite direction, it should be 0
Series >> $V_{T} = V_{1} + V_{2} + V_{3} + \dots$
Parallel >> $V_{T} = V_{1} = V_{2} = V_{3} = \dots$
- Each V is for a path (not a component)

Current

The net current at a node is 0
- If you add all the incoming current, and subtract the outgoing current, it should be 0
Series >> $I_{T} = I_{1} = I_{2} = I_{3} = \dots$
- number of electrons stays the same
Parallel >> $I_{T} = I_{1} + I_{2} + I_{3} + \dots$
- electrons are split between the branches

Resistance

Resistance Formulas

Series >> $R_{T} = R_{1} + R_{2} + R_{3} + \dots$ - Parallel >> $\frac{1}{R_{T}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + \dots$ - For 2 leads: $R_{T} = \frac{R_{1} R_{2}}{R_{1} + R_{2}}$
- Your total resistance will always be lower than your lowest resistance
- You can write $R_{1} ∣∣ R_{2} ∣∣ R_{n}$

derivation

$\begin{aligned} i_{s} & = i_{1} + i_{2} + \dots + i_{n} \\ = \frac{V}{R_{1}} + \frac{V}{R_{2}} + \dots + \frac{V}{R_{n}} \\ = V (\frac{1}{R_{1}} + \frac{1}{R_{2}} + \dots + \frac{1}{R_{n}}) \\ \frac{1}{R_{e q}} & = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \dots + \frac{1}{R_{n}} \end{aligned}$

Power

For all: $P_{T} = P_{1} + P_{2} + P_{3} + \dots$