To solve a circuit using the loop method, the equations are first written for each loop by applyingKCLKVLOhm’s lawThevenin’s theorem

To solve a circuit using the loop method, we follow a systematic approach. Here's how to do it:

1. Break Down the Problem

Identify the independent loops in the circuit. Label the nodes and assign loop currents to each loop. Determine whether the circuit is series, parallel, or a combination of both as this will help in applying the correct laws.

2. Relevant Concepts

• Kirchhoff's Voltage Law (KVL) states that the sum of the electrical potential differences (voltage) around any closed network is zero.
• Ohm’s Law states that $V = IR$ where $V$ is voltage, $I$ is current, and $R$ is resistance.
• Thevenin’s Theorem allows us to simplify a complex network to a simple equivalent circuit with a single voltage source and a single resistor.

3. Analysis and Detail

1. Apply KVL: For each loop, write down an equation based on KVL. For instance, for loop 1 with voltages $V_1, V_2$ and resistances $R_1, R_2$, the equation would be: $-V_1 + I_1 R_1 + I_2 R_2 = 0$ where $I_1$ and $I_2$ are the currents in the loop.

2. Use Ohm’s Law: Substitute $V$ and $I$ terms from Ohm’s Law into your KVL equations to express all voltages in terms of current and resistance.

3. Thevenin’s Theorem: If you can reduce the circuit, find the equivalent voltage $V_{th}$ and equivalent resistance $R_{th}$. This will help simplify calculations in your loop equations.

4. Solve the Systems of Equations: Use algebra (substitution, elimination) or matrix methods to solve the system of linear equations derived from step 1.

4. Verify and Summarize

• Verify the calculated loop currents by substituting them back into KVL equations to ensure they satisfy all equations.
• Summarize your findings, indicating the notable currents and voltages across the components.

Final Answer

Present the final results for loop currents and any other relevant electrical characteristics, ensuring clarity and correctness in units (Amps, Volts, ohms, etc.).

This method provides a structured way to analyze and solve electrical circuits using the loop method efficiently.