# Kirchhoff's Current Law Conservation of charge is basic physics principle behind KCL, equation $\sum_{k}i_{k}=0$ >[!definition] Kirchoff's Current Law >The sum of all currents leaving a node is equal to zero ![[Pasted image 20260128173705.png|center|600]] - Nodes occur on straight paths - Essential nodes occur @ junctions where $\geq3$ elements are connected - Nonessential node is a point of circuit where $\leq$ 2 elements are connected - At junction, current is divided evenly. I.e. assume junction w/ 3 elements, all of equal resistance -> current would be divided by 3 passing through each resistor # Kirchhoff's Voltage Law - Conservation of energy is basic physics principle behind KVL, equation $\sum_{k}v_{k}=0$ >[!definition] Kirchhoff's Voltage Law >The sum of voltages around any loop is equal to zero ## Apply KVL consistently every time 1. Start from lowest left corner of each loop 2. Move clockwise around a loop 3. Use first voltage we encounter ## Resistors in Series ![[Pasted image 20260128174522.png|center|700]] $\text{Apply KVL around the loop:}-v+iR_{1}+iR_{2}+\dots iR_{n}=0\equiv-v+iR_{eq}=0$$-v+i \sum_{k}R_{k}=0, i= \frac{v}{\sum_{k}R_{k}}$ $\text{Voltage division: }v_{j}=iR_{j}=\left( \frac{v}{\sum_{k}R_{k}} \right) \to v_{j}=v \frac{R_{j}}{\sum_{k}R_{k}}$ $\text{The equivalent resistance of resistors in parallel is smaller than the smaller resistance}$ ## Resistors in Parallel ![[Pasted image 20260128174917.png|center|700]] $\text{Apply KCL at top node:}-i+\frac{v}{R_{1}}+\frac{v}{R_{2}}+\dots+\frac{v}{R_{n}}=0=-i+v(G_{1}+G_{2}+\dots G_{n})$ $\text{Equivalent to} -i+\frac{v}{R_{eq}}=0\to G_{eq}=G_{1}+G_{2}+\dots+G_{n}=\sum_{k}G_{k}$ $\text{Current division: }i_{j}=vG_{j}=\left( \frac{i}{\sum_kG_{k}} \right)G_{j}\to i_{j}=i \frac{G_{j}}{\sum_{k}G_{k}}$ $\text{For identical resistors in parallel, $G_{eq}=NG\sim R_{eq}=\frac{R}{N}$}$