7.3 Analysis of multi-loop circuits

Multi-loop circuits can be tricky, but there are cool ways to break them down. We'll look at mesh currents, node voltages, and superposition to simplify complex circuits. These methods help us solve for currents and voltages in different parts of the circuit.

We'll also learn about Thévenin's and Norton's theorems. These let us replace big, complicated circuits with simpler equivalent ones. Plus, we'll see how to get the most power out of a circuit. It's all about making tough problems easier to handle.

Circuit Analysis Techniques

Mesh Current Method

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• Analyzes planar circuits by assigning mesh currents to each loop
• Applies Kirchhoff's voltage law (KVL) to each mesh to generate a system of equations
• Solves the system of equations to determine the mesh currents
• Calculates branch currents and voltages using the mesh currents
• Suitable for circuits with a small number of meshes and a large number of voltage sources

Node Voltage Method

• Analyzes circuits by assigning node voltages to each non-reference node
• Applies Kirchhoff's current law (KCL) to each node to generate a system of equations
• Solves the system of equations to determine the node voltages
• Calculates branch currents and voltages using the node voltages
• Suitable for circuits with a small number of nodes and a large number of current sources

Superposition Principle

• Analyzes linear circuits by considering the effect of each independent source separately
• Determines the contribution of each independent source to the desired quantity (current or voltage)
• Adds the contributions of all independent sources to obtain the final result
• Applicable to circuits with multiple independent sources
• Simplifies the analysis by breaking down the circuit into simpler sub-circuits (one source active at a time)

Circuit Theorems

Thévenin's Theorem

• Replaces a linear two-terminal network with an equivalent circuit consisting of a voltage source ($V_{Th}$) in series with a resistor ($R_{Th}$)
• Determines the Thévenin equivalent voltage ($V_{Th}$) by calculating the open-circuit voltage across the terminals of interest
• Calculates the Thévenin equivalent resistance ($R_{Th}$) by removing all independent sources and finding the resistance between the terminals
• Simplifies the analysis of complex circuits by focusing on the load connected to the two-terminal network
• Useful for analyzing the behavior of a circuit when the load is varied (maximum power transfer, load matching)

Norton's Theorem

• Replaces a linear two-terminal network with an equivalent circuit consisting of a current source ($I_{N}$) in parallel with a resistor ($R_{N}$)
• Determines the Norton equivalent current ($I_{N}$) by calculating the short-circuit current through the terminals of interest
• Calculates the Norton equivalent resistance ($R_{N}$) by removing all independent sources and finding the resistance between the terminals
• Simplifies the analysis of complex circuits by focusing on the load connected to the two-terminal network
• Useful for analyzing the behavior of a circuit when the load is varied (maximum power transfer, load matching)

Maximum Power Transfer Theorem

• States that a linear two-terminal network delivers maximum power to a load when the load resistance equals the Thévenin/Norton equivalent resistance of the network
• Determines the condition for maximum power transfer by setting the load resistance equal to the Thévenin/Norton equivalent resistance
• Calculates the maximum power delivered to the load using the Thévenin/Norton equivalent circuit
• Applies to both Thévenin and Norton equivalent circuits
• Important in the design of power delivery systems and impedance matching networks (audio amplifiers, antenna systems)