Kirchoff's Law: A Practical Guide To Application

Kirchhoff's Law is used to analyse complex electrical circuits to find the voltage and currents. It is especially useful when a circuit has both parallel and series resistance, and Ohm's Law falls short. There are two variations of Kirchhoff's Law: Kirchhoff's Current Law and Kirchhoff's Voltage Law. When combined, they form Kirchhoff's Circuit Law.

Kirchhoff's Current Law states that the amount of current entering a node is equal to the amount of current leaving a node. Kirchhoff's Voltage Law states that in a closed-loop circuit, the total voltage will always equal the sum of all the voltage drops within the loop.

To apply Kirchhoff's Law, follow these steps:

- Label all known voltages and resistances in the circuit.

- Name each branch in the circuit with a current label.

- Find Kirchhoff's Current Law for each node in the circuit.

- Find Kirchhoff's Voltage Law for each independent loop in the circuit.

- Use the equations to find any missing currents or voltages.

Identify the simple loops (meshes)

To identify the simple loops or meshes in a circuit, you must first label each node with a letter of the alphabet. Nodes are the essential points where three or more wires join.

Once this is done, you can identify the simple loops, which are the "window panes" of the circuit. These are loops that do not contain other loops within them. Label each loop with a number.

For example, in Figure 1 below, there are two simple loops or meshes.

! [Figure 1. Circuit to be solved.](https://engineering.louisville.edu/wp-content/uploads/sites/1154/2014/08/Fig1.gif)

Label each node with a letter of the alphabet

When applying Kirchhoff's Laws to solve a circuit, it is important to identify each essential node and label them with a letter of the alphabet. An essential node is where three or more wires join. This node need not be a single point and may stretch across a circuit and go around corners.

For instance, in the circuit shown in Figure 1, there are two essential nodes, labelled Node A and Node B.


Once the nodes are labelled, you can proceed to write a Kirchhoff's Current Law equation for each of the nodes.

Assign a current (showing its direction) in every element

To apply Kirchhoff's laws, the first step is to assign a current (with a direction) to every element in the circuit. Here are some detailed instructions on how to do this:

• Assign a current to each element, ensuring that elements in the same branch have the same current.
• Don't forget to also assign currents to voltage sources.
• You can choose the direction of the current arbitrarily.
• For dependent voltage sources, it is necessary to define a current in the branch.

These steps will allow you to systematically apply Kirchhoff's laws to a circuit. Once you have assigned currents and voltages, you can move on to identifying nodes and loops, and then writing and solving the relevant equations.

Write an Ohm's Law equation for each resistor

Ohm's Law is a formula used to calculate the relationship between voltage, current and resistance in an electrical circuit. The law is named after German physicist Georg Ohm and is written as:

V = I x R or V = IR

Where:

• V = voltage (measured in volts)
• I = current (measured in amps)
• R = resistance (measured in ohms)

Using this formula, we can calculate the unknown value in a circuit when the other two values are known.

Example 1: Calculating Voltage

Description: If the resistance of an electric iron is 50 Ω and a current of 3.2 A flows through the resistance, find the voltage between two points.

Solution:

• V = I x R
• V = 3.2 A x 50 Ω
• V = 160 V

Example 2: Calculating Resistance

Description: An EMF source of 8.0 V is connected to a purely resistive electrical appliance (a lightbulb). An electric current of 2.0 A flows through it. Calculate the resistance offered by the electrical appliance.

Solution:

• R = V / I
• R = 8 V / 2 A
• R = 4 Ω

Example 3: Calculating Current

Description: What is the current in a circuit with a voltage of 12V and resistance of 6Ω?

Solution:

• I = V / R
• I = 12V / 6Ω
• I = 2A

Kirchhoff's Law is used to analyse complex electrical circuits with multiple voltage sources and resistors. It consists of two variations:

• Kirchhoff's Current Law: The sum of all currents entering a junction must equal the sum of all currents leaving the junction.
• Kirchhoff's Voltage Law: The total voltage in a closed-loop circuit is equal to the sum of all voltage drops within the loop.

Ohm's Law is used in conjunction with Kirchhoff's Law to calculate the individual resistances, voltages, or currents in a complex circuit.

Write a Kirchhoff's Current Law equation for each node

Kirchhoff's Current Law (KCL) is a fundamental law used for circuit analysis. It states that the total current entering a circuit junction is equal to the total current leaving the same junction. This is because the current has nowhere else to go, and no charge is lost.

In other words, the algebraic sum of all the currents entering and leaving a junction must be equal to zero. This can be expressed as: Σ IIN = Σ IOUT.

Kirchhoff's Junction Rule, or KCL, can be applied to any number of junctions or nodes, and the sum of the currents both entering and leaving will be the same.

For example, consider a junction where I1 = 3 amperes and I2 = 2 amperes. The total current, IT, leaving the junction will be 3 + 2 = 5 amperes.

If we reverse the current's directions, the resulting equations will still hold true. For instance, I1 = IT - I2 = 5 - 2 = 3 amps, and I2 = IT - I1 = 5 - 3 = 2 amps. Thus, we can consider the currents entering the junction as positive, and those leaving as negative.

Now, let's write Kirchhoff's Current Law equations for each node:

Node A:

At point A, the total current IT enters the circuit from the 12V supply voltage. This current then separates at node A, flowing through resistors R1 and R2. So, the KCL equation for node A is:

I1 = IT

Node B:

Node B is a junction where the current can split into two directions. Some of the current will flow downwards through resistor R1, while the rest will continue through resistor R2 via node C. We can use KCL to find the total current entering node B:

I1 = I2 + I3

Node C:

At node C, the currents flowing through resistors R1 and R2 recombine. We can apply KCL to find the total current exiting node C:

I4 = I1 + I2

Node D:

Node D is another junction where the current can take multiple paths. After passing through node C, the current splits again, flowing through resistors R3, R4, and R5 before recombining at node F. We can use KCL to find the total current entering node D:

I5 = I2 + I3 + I4

Node E:

At node E, the currents flowing through resistors R3, R4, and R5 recombine. We can apply KCL to find the total current exiting node E:

I6 = I3 + I4 + I5

Node F:

Finally, at node F, all the currents from the previous nodes recombine to form the main flow. We can use KCL to confirm that the sum of the currents entering node F is equal to the total current IT:

ΣIN = IT

These Kirchhoff's Current Law equations allow us to analyze the circuit and determine the currents at each node, helping us understand the behavior of the electrical system.

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