How do you perform nodal analysis for circuits with voltage and current sources?

Identify the Nodes: Identify all the essential nodes in the circuit. A node is a point where two or more circuit elements are connected together.

Choose a Reference Node (Ground): Select one of the nodes as the reference or ground node. Assign a voltage of 0V to this node. Usually, it is convenient to choose the node connected to the negative terminal of the voltage sources.

Assign Node Voltages: For the remaining nodes (excluding the reference node), assign variables (usually voltages) to represent their potentials relative to the reference node.

Apply KCL at Each Node: Write down the KCL equation for each non-reference node. For each node, sum up all the currents flowing into that node and set it equal to zero. Consider the currents due to both voltage sources and resistors connected to the node.

Express Currents in Terms of Node Voltages: Express all the currents in the KCL equations in terms of the node voltages using Ohm's law (I = V/R) and the relationships for other circuit elements (e.g., current sources).

Solve the Equations: You will end up with a set of equations that represent the circuit's behavior. Typically, you'll have as many equations as there are non-reference nodes. Use algebraic manipulation to solve for the node voltages.

Determine Other Unknowns: Once you have the node voltages, you can use Ohm's law and other relevant relationships to find other unknowns like branch currents or power dissipation in resistors.

Check the Results: Double-check your solutions and ensure they make sense. Verify that the current directions and voltage polarities are consistent with your initial assumptions.

It's important to note that in some cases, a node voltage may turn out to be negative, which only indicates that the polarity of that voltage is opposite to the assumed direction. It doesn't have a physical significance other than indicating that the actual polarity is reversed compared to the assumed one.

Overall, nodal analysis is a systematic method to analyze complex circuits and is often preferred due to its efficiency and accuracy. However, for very large circuits, computer-aided circuit analysis tools like SPICE may be used to perform the analysis.