KCL and KVL independent equations - Solutions - Huaqiang Electronic Network

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The general method of circuit analysis refers to a systematic approach used to solve any electrical circuit, especially when dealing with complex networks where multiple branches are present. In such circuits, the goal is often to determine the current and voltage values across each branch. For a circuit containing six branches and n nodes, there are six branch voltages and six branch currents, totaling 12 variables. However, since each branch has both a current and a voltage, this results in 26 unknowns that need to be solved. To achieve this, we must create 26 independent equations based on the properties of the circuit. Using Kirchhoff's Current Law (KCL), we can write (n - 1) independent node equations for the n nodes in the circuit. These equations ensure that no equation is a linear combination of another. Each new equation must introduce at least one new branch current to maintain independence. However, these (n - 1) equations alone are not sufficient to solve for all six branch currents. Similarly, using Kirchhoff's Voltage Law (KVL), we can generate b - (n - 1) independent loop equations for the closed loops in the circuit. Each of these equations includes at least one new branch voltage, ensuring their independence. But again, these equations alone cannot fully determine the six branch voltages. To resolve this, we combine the KCL and KVL equations with the branch voltage-current relationships, known as the branch V-I equations. These equations describe the behavior of individual components, such as resistors, capacitors, or inductors. By including these, we now have (n - 1) + [b - (n - 1)] + 6 = 26 independent equations, which allow us to solve for all 26 branch variables—currents and voltages. This comprehensive approach, which combines KCL, KVL, and component-specific V-I relationships, is known as the 26-method. It provides a structured and reliable way to analyze even the most complex electrical circuits, making it a fundamental tool in electrical engineering.