Kirchoff's Current Law

Kirchoff's Current Law (KCL) is a circuit law related to currents entering and exiting a junction.

It states that the the total current entering a junction is equal to the total current exiting the junction. This is a consequence of the conservation of charge where charge can neither be created nor destroyed. This is mathematically described as:

As a simple example if 1A and 2A branch currents enter a junction then a current of 3A must leave the junction as shown below:

Current directions are assigned to each branch and a negative current result means that the actual current travels in the opposite direction.

We can use the circuit below to illustrate how KCL is used:

To apply KCL we assign currents to each branch and then derive equations at each junction. In this case we just have a single junction connecting Vs to R1 and R2:

Applying KCL at the junction marked A we have Is entering and I2 and I3 exiting:

We can then replace each of the branch currents in terms of the voltage across each resistor and their resistance via Ohm's Law:

Which is equivalent to the result we would expect for the above circuit by Ohm's Law as the total effective resistance is the sum of the reciprocal of the resistances for a parallel circuit, i.e. 1/Rt = 1/R1 + 1/R2... Which is equal to the above result as Is = Vs*(1/Rt) in the above :)

To solve more complex circuits we can use both KVL and KCL which we will see in an example.

Example:

Q1. Use KCL to determine the currents in the circuit below:

Let's begin by labelling all of the branch currents:
Now we can write the KCL equation:
Then we can substitute in the value of Is and replace I1,I2 and I3 in terms of voltage and resistance via Ohm's Law:
As we now know the voltage across each resistor we can compute all of the currents one by one with Ohm's Law:

Q2. Use KVL and KCL to determine all of the currents in the circuit below:

The first step is to redraw the circuit with nodes, KVL loops and all currents and voltages assigned:
In the above we have shown 2 KVL loops from Vs: One from node A to B and then C through R3 and another from node A to B and then C through R2. We can use these and a KCL equation at junction B to write 3 equations to determine the 3 unknown currents. Note: For problems which will involve several equations it helps to subsitute in values and simplify the equations as we go as this will help us identify relationships between equations to simplify the mathematics. KVL for loop 1:
KVL for loop 2:

We can now substitute this back into [1]:

KCL at junction B:
We can now substitute in Is and I3 to solve for I2:

Questions:

Q1. Use KCL to determine the currents in the circuit below:

Answer

Is = 8.333mA, IR2=4.17mA, IR3=4.167mA

Q2. Use KVL and KCL to determine the currents in the circuit below:

Answer

Is = 1.358mA, IR2=1.241mA, IR3=116.704uA

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