Branch Current Analysis

Introduction

Branch Current Analysis is one of the most fundamental and systematic methods used in electrical circuit analysis to determine the current flowing through each branch of a network. In many practical electrical circuits, especially those containing multiple loops and sources, it becomes difficult or even impossible to simplify the circuit using simple series and parallel combinations. In such cases, advanced methods like Branch Current Analysis provide a reliable approach to solving the circuit. This method is based on the direct application of basic electrical laws such as Ohm’s Law, Kirchhoff’s Current Law (KCL), and Kirchhoff’s Voltage Law (KVL). By assigning a current to each branch and forming equations based on these laws, it becomes possible to calculate unknown currents and voltages in the circuit accurately. Branch Current Analysis is especially useful when dealing with complex networks where no elements can be easily combined. It provides a clear understanding of how current flows in different parts of a circuit and helps engineers design and analyze electrical systems effectively.

What is a Branch in a Circuit?

A branch in an electrical circuit is defined as a path that connects two nodes and contains one or more elements such as resistors, voltage sources, or current sources. All the components in a branch carry the same current because they are connected in series. For example, if a resistor and a voltage source are connected in series between two nodes, they form a single branch, and the current flowing through both elements will be the same. Each branch in a circuit is assigned a separate current when applying the Branch Current Analysis method. Understanding branches is important because the number of unknown currents in the circuit depends on the number of branches. Each branch introduces a new current variable that needs to be determined.

Why do we need to apply Branch current analysis?

Before examining the details of the first important method of analysis, let us examine the network in [Fig. 1], to be sure that you understand the need for these special methods. Initially, it may appear that we can use the reduce and return approach to work our way back to the source $E_1$ and calculate the source current $I_{s1}$. Unfortunately, however, the series elements $R_3$ and $E_2$ cannot be combined because they are different types of elements. A further examination of the network reveals that there are no two like elements that are in series or parallel. No combination of elements can be performed, and it is clear that another approach must be defined. It should be noted that the network of [Fig. 1] can be solved if we convert each voltage source to a current source and then combine parallel current sources. However, if a specific quantity of the original network is required, it would require working back using the information determined from the source conversion. Further, there will be complex networks for which source conversions will not permit a solution, so it is important to understand the methods to be described in this chapter. The first approach to be introduced is called branch-current analysis because we will define and solve for the currents of each branch of the network. At this point it is important that we are able to identify the branch currents of the network. In general, In [Fig. 1], the source $E1$ and the resistor $R1$ are in series and have the same current, so the two elements define a branch of the network. It is the same for the series combination of the source $E_2$ and resistor $R3$. The branch with the resistor $R2$ has a current different from the other two and, therefore, defines a third branch. The result is three distinct branch currents in the network of [Fig. 1] that need to be determined. Experience shows that the best way to introduce the branch-current method is to take the series of steps listed here.

Basic Principles Used

Branch Current Analysis relies on three important electrical laws:

Ohm’s Law

Ohm’s Law relates voltage, current, and resistance in a circuit. It is given by: $$ V = I R $$ This law is used to express voltage drops across resistors in terms of branch currents.

Kirchhoff’s Current Law (KCL)

Kirchhoff’s Current Law states that the algebraic sum of currents entering and leaving a node is zero. Mathematically: $$ \sum I_{in} = \sum I_{out} $$ This law ensures conservation of charge in a circuit.

Kirchhoff’s Voltage Law (KVL)

Kirchhoff’s Voltage Law states that the sum of all voltages around a closed loop is zero: $$ \sum V = 0 $$ This law ensures conservation of energy in the circuit. By combining these three laws, we can form equations that describe the behavior of the entire circuit.

Branch-Current Analysis Procedure

• Assign a distinct current of arbitrary direction to each branch of the network.
• Indicate the polarities for each resistor as determined by the assumed current direction.
• Apply Kirchhoff's voltage law around each closed, independent loop of the network.
  The best way to determine how many times Kirchhoff's voltage law has to be applied is to determine the number of "windows" in the network. For networks with three windows, as shown in [Fig. 2], three applications of Kirchhoff's voltage law are required, and so on.
  

  Fig. 2: Determining the number of independent closed loops.
• Apply Kirchhoff's current law at the minimum number of nodes that will include all the branch currents of the network.
  The minimum number is one less than the number of independent nodes of the network. For the purposes of this analysis, a node is a junction of two or more branches, where a branch is any combination of series elements. [Fig. 3] defines the number of applications of Kirchhoff's current law for each configuration in [Fig. 2].
  

  Fig. 3: Determining the number of applications of Kirchhoff's current law required.
• Solve the resulting simultaneous linear equations for assumed branch currents.

Advantages of Branch Current Analysis

Branch Current Analysis offers several benefits in circuit analysis:

• Applicable to all types of circuits, including complex networks
• Provides direct calculation of branch currents
• Based on fundamental laws, making it conceptually simple
• Useful for understanding current distribution in a circuit
• Helps in circuit design and troubleshooting

Although it may involve solving multiple equations, it is a reliable and accurate method for circuit analysis.

Limitations of Branch Current Analysis

Despite its usefulness, the method has some limitations:

• Requires solving many equations for large circuits
• Time-consuming compared to nodal or mesh analysis
• Becomes complex when the number of branches increases

For large networks, other methods like nodal analysis or mesh analysis are often preferred because they reduce the number of equations.

Conclusion

Branch Current Analysis is a powerful and fundamental method used to analyze electrical circuits. It provides a systematic approach to determining currents in each branch by applying Kirchhoff’s laws and Ohm’s law. Although the method may involve solving multiple equations, it is highly effective for analyzing circuits that cannot be simplified using basic techniques. It also forms the foundation for understanding more advanced methods like nodal analysis and mesh analysis. For students and engineers, mastering Branch Current Analysis is essential for understanding circuit behavior, designing electrical systems, and solving real-world engineering problems.