# 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:

**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:

**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

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