Kirchhoff’s Current Law (KCL)

Kirchhoff’s Current Law (KCL) is another statement about the law of conservation of energy within a circuit, affirming the conservation of charge flow. It states that the sum of currents entering a junction is equal to the sum of currents leaving the junction. Mathematically we can sum up this idea as

\[ \Sigma I_{in} = \Sigma I_{out} \]

We read the above literally as the sum of currents going in equals the sum of currents going out of the junction. This idea of Kirchhoff’s is also called The Junction Rule. If we have a number of conducting wires going into a common point or junction (which in electronics is also called a node), Kirchhoff’s Current Law / Junction Rule states that whatever current goes into a node, must equal to the current leaving the node. This is due to the conservation of energy: charges cannot be created or destroyed, so they must be going somewhere.

\[ \mbox{sum of all currents enter = sum of all currents leaving} \]

If we consider the diagram to the right of 5-wires connecting to a common point, $P$, a junction, we know that the sum of all currents entering the node must equal the sum of all currents leaving the node. Each wire is labelled as $I_1$, $I_2$, $I_3$, $I_4$, and $I_5$ — current 1, current 2, etc. — and also has the arrows by the resistor symbol which show the direction of current. We can see that both $I_2$ and $I_5$ have current entering the junction, while $I_1$, $I_3$, and $I_4$ all have current leaving the junction. So

\[ I_2 + I_5 = I_1 + I_3 + I_4 \]

Equations of Kirchhoff’s Current Law

In order to perform calculations with Kirchhoff’s Current Law, we need to establish our frame of reference. We must choose our positive and negative polarity conventions, however abstract they may be. Consider that the wires carrying currents towards the node are positive (+) and the wires carrying current away from that common junction are negative (-).

This allows us to state the Junction Rule in another way: the algebraic sum of all currents entering and leaving a junction is zero, because all energy must be conserved. The term ‘algebraic sum’ refers to the fact that some values are positive and some negative. Those leaving are negative (-) and those entering are positive (+), which will cancel once we add them together.

This is expressed in the following equation

\[ \Sigma I = 0 \]

which means that the sum of all current (entering and leaving a junction) is zero. So we can rewrite the initial expression we wrote for the 5-wire circuit junction as

\[ \Sigma I = -I_1 + I_2 – I_3 – I_4 + I_5 \]

The $I_2$ and $I_5$ are positive because the direction of current is towards the junction. $I_1$, $I_1$ and $I_1$ are negative because their direction of current is away from the node. Note that we could have achieved the same result by simply balancing the first expression that we wrote with algebra.

\[ I_2 + I_5 = I_1 + I_3 + I_4 \]

\[ I_2 + I_5 – I_1 – I_3 – I_4 = 0 \]

Applying the Junction Rule (KCL)

Looking at the diagram to the right, how would you write the equation for $I_1$, the only current carrying wire that is entering the junction? After $I_1$ reaches the junction it branches off into four separate wires each carrying a unique current that depends on the value of the specific resistor on that wire. We know all of those are negative because they are leaving the node, $P$, so

\[ \Sigma I = I_1 – I_2 – I_3 – I_4 – I_5 = 0 \]

which we can then manipulate algebraically to solve for $I_1$ as below

\[ I_1 = I_2 + I_3 + I_4 + I_5 \]

[latexpage]