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 determine the currents and voltages in each branch of a complex electrical circuit. When dealing with a circuit that has six branches and multiple nodes, there are six branch voltages and six branch currents that need to be calculated—resulting in a total of twelve variables. To solve these, we require twelve independent equations derived from fundamental laws like Kirchhoff's Current Law (KCL) and Kirchhoff's Voltage Law (KVL). When using branch current as the variable, KCL can be applied to list (n-1) independent node equations for n nodes. These equations ensure that each new equation introduces at least one new branch current, making them linearly independent. However, this set of equations alone is not sufficient to solve for all six branch currents. On the other hand, when using branch voltage as the variable, KVL can be used to form b-(n-1) independent loop equations. Each equation includes at least one new branch voltage, ensuring independence. But again, this set is incomplete on its own. To fully solve the problem, we combine both approaches. By including the branch voltage-current relationship equations (VAR), we create a complete system of equations. This results in (n-1) independent KCL equations, b-(n-1) independent KVL equations, and six VAR equations, totaling twelve independent equations. With these, we can accurately calculate all twelve unknowns: six branch currents and six branch voltages. This comprehensive technique is known as the "twelve-variable method," offering a structured and reliable way to analyze complex circuits.