Kirchhoff's Laws

Kirchhoff’s Laws, combined with Ohm’s Law, form the fundamentals for simple circuit analysis. It uses the conservation of energy, as well as the conservation of charge.

Definition §

$$\text{Current Law (KCL): }\sum _{n=1}^N{I}_n=0⟺\text{Sum of currents entering a node must be 0}$$

$I_n$ is the current entering a node, $A$, and $N$ is the total number of branches connected to $A$.

$$\text{Voltage Law (KVL): }\sum _{i=1}^n{V}_i=0⟺\text{Sum of voltages in a loop must be 0}$$

$V_i$ is the voltage of a component in series to the loop.

[!info ] Convention… again To maintain consistency in Kirchhoff’s law, we need to define a current direction to ensure we have an idea of ‘positive’ flow into an object and ‘negative’ flow out a node. Similarly, we need to define reference voltage polarities. Details below

Kirchhoff’s Voltage Law (KVL) §

Physics defines a#todo Law of Conservastion of Charge, which means the net charge in a system must be the same.

The first rule states that no build up of charge can occur in a junction of a circuit, i.e. the sum of a charge in a junction is always equal to zero. Since current is just time-dependent flow of charge, we can extend this same logic to current

$$\sum i=\sum i_{in}+\sum i_{out}=0$$

Generally, the current going into a circuit is defined to be positive, while current leaving a circuit is defined to be negative.

In simple circuit analysis, we use the definitions of nodes to state this. The sum of currents entering a node must be equal to the sum of currents exiting a node:

#todo Add drawing

Kirchhoff’s Current Law (KCL) §

The second rule states the total change in Electrical Potential Energy in a circuit must be zero. We can use the definition of voltage to extend this:

$$U_f-U_i=0\to \Delta V=0$$

#todo redefine.