Supermesh Analysis: Tips and Tricks for Finding Mesh Currents

Supermesh Analysis: A Step-by-Step Guide with Solved Examples

Supermesh analysis is a technique that can help you solve complex electric circuits with ease. In this article, you will learn what supermesh analysis is, when to use it, how to apply it, and see some solved examples of supermesh analysis.

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What is Supermesh Analysis?

Definition and Concept

A supermesh is a combination of two meshes that have a current source as a common element. A mesh is a loop in a circuit that does not contain any other loops within it. A current source is a device that provides a constant current to a circuit, regardless of the voltage across it.

Supermesh analysis is a technique that allows you to apply Kirchhoff's Voltage Law (KVL) to a supermesh, instead of applying it to each individual mesh. This way, you can reduce the number of equations you need to solve for the mesh currents.

When to Use Supermesh Analysis?

You should use supermesh analysis when you are trying to solve a circuit with mesh analysis, but there is a current source between two meshes. If you apply mesh analysis directly, you will end up with an equation that has no unknown variables, which is useless. By using supermesh analysis, you can eliminate this equation and replace it with another one that relates the mesh currents and the current source.

How to Apply Supermesh Analysis?

Step 1: Identify the Total Number of Meshes

The first step is to count how many meshes are in the circuit. Remember, a mesh is a loop that does not contain any other loops within it.

Step 2: Assign the Mesh Currents and Check for Supermesh

The next step is to assign a mesh current to each mesh in the circuit. You can choose any direction for the mesh currents, but it is usually convenient to follow the passive sign convention, which means that the mesh current enters the positive terminal of a voltage source and leaves the negative terminal.

Then, you need to check if there is a current source between two meshes. If there is, then you have a supermesh. You should remove the current source from the circuit and treat the two meshes as one supermesh.

Step 3: Develop the KVL Equation for the Supermesh

The third step is to apply KVL to the supermesh. This means that you need to sum up all the voltage drops and rises around the supermesh and set them equal to zero.

However, this equation alone is not enough to solve for the mesh currents. You also need another equation that relates the mesh currents and the current source. This equation can be obtained by applying Kirchhoff's Current Law (KCL) to any node that connects the two meshes.

Step 4: Solve the Equations to Find the Mesh Currents

The final step is to solve the system of equations that you have obtained from the previous steps. You can use any method that you prefer, such as substitution, elimination, or matrix inversion. Once you find the mesh currents, you can use them to find any other quantity of interest in the circuit, such as branch currents, voltages, powers, etc.

Solved Examples of Supermesh Analysis

Example 1: Find the Current I Using Supermesh Analysis

Consider the following circuit:

Solution

Step 1: The total number of meshes is 2.

Step 2: Let us assign mesh currents I1 and I2 for meshes 1 and 2 respectively, as shown in the figure below. As you can see, there is a 2 A current source between the two meshes, so we have a supermesh. We remove the current source from the circuit and treat the two meshes as one supermesh.

Step 3: We apply KVL to the supermesh and get:

-10 + 4(I1 - I2) + 6I1 + 8I2 = 0 ... (1)

We also apply KCL to node 0 and get:

I1 - I2 = 2 ... (2)

Step 4: We solve the system of equations (1) and (2) and get:

I1 = 3 A and I2 = 1 A

The current I is equal to I2, so we have:

I = 1 A

Example 2: Find the Power Delivered by the Voltage Source Using Supermesh Analysis

Consider the following circuit:

Solution

Step 1: The total number of meshes is 3.

Step 2: Let us assign mesh currents Ia, Ib, and Ic for meshes a, b, and c respectively, as shown in the figure below. As you can see, there is a 5 A current source between meshes b and c, so we have a supermesh. We remove the current source from the circuit and treat the two meshes as one supermesh.

Step 3: We apply KVL to mesh a and get:

-20 + (10 + R)Ia - RIb = 0 ... (3)

We apply KVL to the supermesh and get:

-RIa + (R + 10)Ib - 10Ic = 0 ... (4)

We also apply KCL to node x and get:

Ib - Ic = 5 ... (5)

We are given that R = 10 Ω, so we substitute this value in equations (3), (4), and (5).

-20 + (20)Ia - (10)Ib = 0 ... (6)

-10Ia + (20)Ib - (10)Ic = 0 ... (7)

Conclusion

Supermesh analysis is a useful technique that can help you solve complex electric circuits with current sources. By applying KVL to a supermesh and KCL to a node, you can obtain a system of equations that can be solved for the mesh currents. Once you have the mesh currents, you can find any other quantity of interest in the circuit.

Supermesh analysis is based on two fundamental laws of electricity: Kirchhoff's Voltage Law and Kirchhoff's Current Law. These laws state that the sum of voltages around a loop is zero, and the sum of currents at a node is zero. By using these laws, you can analyze any circuit with linear elements.

FAQs

What is the difference between mesh analysis and supermesh analysis?

Mesh analysis is a technique that applies KVL to each mesh in a circuit and solves for the mesh currents. Supermesh analysis is a modification of mesh analysis that applies KVL to a supermesh, which is a combination of two meshes that have a current source as a common element.

How many equations do I need for supermesh analysis?

You need one equation for each mesh in the circuit, plus one equation for each supermesh. The number of equations should be equal to the number of unknown mesh currents.

How do I choose the direction of the mesh currents?

You can choose any direction for the mesh currents, but it is usually convenient to follow the passive sign convention, which means that the mesh current enters the positive terminal of a voltage source and leaves the negative terminal. This way, you can avoid negative signs in your equations.

How do I find the voltage across a current source?

You can find the voltage across a current source by applying KVL to any loop that contains it. Alternatively, you can use Ohm's law and multiply the current source value by the resistance of the branch that it is in.

How do I find the power delivered by a voltage source?

You can find the power delivered by a voltage source by multiplying its voltage value by the current flowing through it. If the current is in the same direction as the voltage, then the power is positive, which means that the voltage source delivers power to the circuit. If the current is in the opposite direction as the voltage, then the power is negative, which means that the voltage source absorbs power from the circuit. 71b2f0854b