Mesh Current Analysis With Current Source

Let's dive into the world of mesh current analysis, particularly when dealing with circuits containing current sources. This technique, also known as loop analysis, offers a systematic approach to solving for unknown currents in complex circuits. When current sources are involved, the process requires a slightly different approach compared to circuits with only voltage sources. This article will guide you through the theory, steps, and practical considerations of mesh current analysis with current sources, ensuring you gain a solid understanding of this essential circuit analysis method.

Mesh Current Analysis: A Foundation

Mesh current analysis is a powerful technique rooted in Kirchhoff's Voltage Law (KVL). KVL states that the sum of all voltages around any closed loop in a circuit must equal zero. In mesh analysis, we assign a hypothetical "mesh current" to each independent loop in the circuit. These mesh currents are then used to express the voltage drops across resistors in terms of the assigned currents. By applying KVL to each mesh, we create a system of equations that can be solved to find the unknown mesh currents. Once the mesh currents are known, we can easily determine the actual branch currents and voltages throughout the circuit.

Why Use Mesh Current Analysis?

• Systematic Approach: Mesh analysis provides a structured method for solving complex circuits.
• Reduced Complexity: It often reduces the number of equations needed compared to other methods like nodal analysis, especially in circuits with many series-connected elements.
• Direct Calculation of Currents: It directly calculates the loop currents, which are often the desired unknowns.

The Core Principles of Mesh Current Analysis

Before we delve into the intricacies of handling current sources, let's solidify the fundamental principles of mesh current analysis:

1. Identify Independent Loops: A loop is a closed path in the circuit. An independent loop is one that contains at least one element not included in any other loop.

2. Assign Mesh Currents: Assign a clockwise (or counter-clockwise, but consistently) mesh current to each independent loop. Conventionally, clockwise is preferred.

3. Apply Kirchhoff's Voltage Law (KVL): For each mesh, apply KVL. Express the voltage drops across resistors in terms of the assigned mesh currents. Remember the sign convention: a voltage drop is positive if the mesh current enters the positive terminal of the resistor.

4. Formulate Equations: Write down the KVL equation for each mesh. This will result in a system of linear equations.

5. Solve the System of Equations: Solve the system of equations to find the values of the mesh currents. Methods for solving include substitution, Cramer's rule, or matrix algebra.

6. Determine Branch Currents: Once the mesh currents are known, the branch currents (the actual currents flowing through each element) can be easily determined. If a branch is part of only one mesh, its branch current is equal to the corresponding mesh current. If a branch is common to two meshes, its branch current is the algebraic sum of the two mesh currents.

Mesh Current Analysis with Current Sources: The Supermesh Technique

The presence of current sources introduces a twist to the standard mesh analysis procedure. When a current source is present in a branch shared by two meshes, we can no longer directly apply KVL to those individual meshes in the traditional way. This is because we don't know the voltage drop across the current source. Instead, we employ a technique called the supermesh.

What is a Supermesh?

A supermesh is a larger mesh created by combining two or more adjacent meshes that share a current source. The supermesh excludes the current source and any elements directly in series with it. Think of it as "bypassing" the current source.

Steps for Mesh Current Analysis with Current Sources (Supermesh):

1. Identify Independent Loops and Assign Mesh Currents: Same as before. Assign clockwise mesh currents to each independent loop.

2. Identify Current Sources Shared by Two Meshes: Locate any current sources situated in branches shared by two meshes.

3. Form Supermeshes: For each current source shared by two meshes, create a supermesh by excluding the current source and any elements in series with it. The supermesh encompasses the remaining elements of the two meshes.

4. Apply KVL to Supermeshes: Apply KVL around each supermesh. When writing the KVL equation for the supermesh, consider all voltage drops in the loop, but exclude the voltage drop across the current source itself (or any elements in series with it that were bypassed). Express voltage drops across resistors in terms of the mesh currents.

5. Formulate Constraint Equations: For each current source forming a supermesh, write a constraint equation that relates the mesh currents to the value of the current source. This equation states that the difference between the two mesh currents is equal to the value of the current source. The sign depends on the direction of the current source relative to the assigned mesh currents. If the current source direction aligns with the direction of a mesh current, that current is considered positive; otherwise, it's negative.

6. Apply KVL to Any Remaining Standard Meshes: If there are any meshes that do not share a current source with another mesh, apply KVL to those meshes as usual.

7. Solve the System of Equations: Solve the system of equations (KVL equations from supermeshes and standard meshes, plus the constraint equations) to find the values of the mesh currents.

8. Determine Branch Currents: Once the mesh currents are known, determine the branch currents as before.

Example: A Simple Circuit with a Supermesh

Let's illustrate the supermesh technique with a simple example:

Circuit:

Imagine a circuit with two meshes. Mesh 1 has a resistor R1 and Mesh 2 has a resistor R2. The two meshes are connected by a branch containing a current source I.

Analysis:

1. Identify Loops and Assign Mesh Currents: We have two loops. Assign clockwise mesh currents I1 to Mesh 1 and I2 to Mesh 2.

2. Identify Current Source: The current source I is shared by Mesh 1 and Mesh 2.

3. Form Supermesh: Create a supermesh by excluding the current source I. The supermesh encompasses R1 and R2.

4. Apply KVL to Supermesh: Apply KVL around the supermesh:

   R1 * I1* + R2 * I2* = 0

5. Formulate Constraint Equation: The constraint equation relates the mesh currents to the current source:

   I1 - I2 = I (Assuming I1 flows in the same direction as the current source, and I2 flows in the opposite direction)

6. Solve the System of Equations: We now have two equations and two unknowns (I1 and I2):

   • R1 * I1* + R2 * I2* = 0
   • I1 - I2 = I

   Solve this system (using substitution or other methods) to find the values of I1 and I2.

7. Determine Branch Currents:

   • The current through R1 is I1.
   • The current through R2 is I2.
   • The current through the current source I is determined by the source itself.

Handling Current Sources in an Outer Branch

The supermesh technique applies when the current source is shared between two meshes. What happens if the current source is in an outer branch, meaning it's only part of one mesh? In this case, the situation is much simpler. The mesh current associated with that loop is directly defined by the current source.

Example:

Consider a circuit with two meshes. Mesh 1 contains a resistor R1 and a current source I in series. Mesh 2 contains a resistor R2 and is coupled to Mesh 1 through another element.

Analysis:

1. Identify Loops and Assign Mesh Currents: Assign clockwise mesh currents I1 to Mesh 1 and I2 to Mesh 2.

2. Current Source in Outer Branch: The current source I is in the outer branch of Mesh 1.

3. Directly Define Mesh Current: Since the current source I is in series with Mesh 1, we know immediately that I1 = I.

4. Apply KVL to Remaining Meshes: Now, apply KVL only to Mesh 2. The value of I1 is already known. This will result in a single equation with one unknown (I2), which can be easily solved.

5. Determine Branch Currents: Once I2 is known, you can determine all branch currents in the circuit.

Practical Considerations and Common Mistakes

• Sign Conventions: Consistent sign conventions are crucial. Always use the same sign convention for voltage drops across resistors (positive when the mesh current enters the positive terminal). Also, pay close attention to the direction of current sources when formulating constraint equations.

• Dependent Sources: If the circuit contains dependent current sources (sources whose value depends on a voltage or current elsewhere in the circuit), the analysis becomes slightly more complex. You'll need to express the controlling voltage or current in terms of the mesh currents and substitute that expression into the constraint equation for the dependent source.

• Careless Simplification: Avoid making assumptions or simplifying the circuit prematurely. Follow the steps systematically to minimize errors.

• Checking Your Work: Always check your answers. You can use circuit simulation software (like SPICE) to verify your results. You can also perform a power balance calculation: the total power delivered by the sources must equal the total power dissipated by the resistors.

• Choosing the Right Method: While mesh analysis is powerful, it's not always the best choice. In circuits with many parallel branches, nodal analysis may be more efficient. Consider the circuit's topology and choose the method that minimizes the number of equations you need to solve.

Advanced Topics and Extensions

• Matrix Formulation: For larger, more complex circuits, expressing the mesh analysis equations in matrix form can simplify the solution process.

• Computer-Aided Analysis: Software tools like MATLAB, Python (with libraries like NumPy and SciPy), and dedicated circuit simulators can automate the mesh analysis process, especially for very large circuits.

• Frequency Domain Analysis: Mesh analysis can be extended to AC circuits by working with impedances and phasors instead of resistances and DC currents.

Conclusion

Mesh current analysis is a fundamental and versatile technique for solving electrical circuits. While the presence of current sources requires the application of the supermesh technique, the underlying principles remain the same. By following the steps outlined in this article, paying attention to sign conventions, and practicing with various examples, you can master mesh current analysis and confidently analyze a wide range of circuits. Understanding mesh analysis provides a solid foundation for more advanced circuit analysis techniques and is an invaluable tool for electrical engineers and anyone working with electrical circuits. Remember to practice, review, and utilize the resources available to solidify your understanding and build your skills in circuit analysis. Good luck!