Thévenin and Norton equivalent circuits simplify complex networks into single-source models. They're powerful tools for analyzing circuits, making it easier to understand voltage, current, and power relationships.

These techniques are part of a broader toolkit for circuit analysis. By converting networks to equivalent forms, we can solve problems more efficiently and gain insights into circuit behavior.

Thévenin and Norton Equivalent Circuits

Thévenin's Theorem and Equivalent Voltage Source

• States any linear electrical network with voltage and current sources can be replaced by an equivalent circuit consisting of a single voltage source $V_{Th}$ in series with a single series resistance $R_{Th}$
• $V_{Th}$ is the open-circuit voltage at the terminals
• $R_{Th}$ is the input or equivalent resistance at the terminals when the independent sources are turned off
• Useful for analyzing complex circuits by reducing them to a single voltage source and series resistance
• Helps determine the voltage across and current through a load resistance connected to the terminals

Norton's Theorem and Equivalent Current Source

• States any linear electrical network with voltage and current sources can be replaced by an equivalent circuit consisting of a single current source $I_N$ in parallel with a single parallel resistance $R_N$
• $I_N$ is the short-circuit current at the terminals
• $R_N$ is the input or equivalent resistance at the terminals when the independent sources are turned off
• Useful for analyzing complex circuits by reducing them to a single current source and parallel resistance
• Helps determine the current delivered to a load resistance connected to the terminals

Relationship Between Thévenin and Norton Equivalent Circuits

• Thévenin and Norton equivalent circuits are interchangeable
• $V_{Th} = I_N \times R_N$
• $R_{Th} = R_N$
• Can convert between the two using source transformation
• Choice of Thévenin or Norton depends on the type of analysis and the given information about the circuit

Circuit Analysis Techniques

Determining Open-Circuit Voltage

• Open-circuit voltage $V_{oc}$ is the voltage across the terminals of a circuit when no load is connected
• To find $V_{oc}$, remove the load resistance and calculate the voltage across the open terminals
• In a Thévenin equivalent circuit, $V_{oc} = V_{Th}$
• Helps determine the voltage source in the Thévenin equivalent
• Can be measured using a voltmeter connected across the open terminals

Determining Short-Circuit Current

• Short-circuit current $I_{sc}$ is the current that would flow if the terminals were connected together (short-circuited)
• To find $I_{sc}$, replace the load resistance with a short circuit (wire) and calculate the current flowing through it
• In a Norton equivalent circuit, $I_{sc} = I_N$
• Helps determine the current source in the Norton equivalent
• Can be measured using an ammeter connected across the shorted terminals

Source Transformation

• Technique for converting between Thévenin and Norton equivalent circuits
• To convert from Thévenin to Norton:
  1. $I_N = \frac{V_{Th}}{R_{Th}}$
  2. $R_N = R_{Th}$
• To convert from Norton to Thévenin:
  1. $V_{Th} = I_N \times R_N$
  2. $R_{Th} = R_N$
• Allows for flexibility in solving circuit problems by choosing the most convenient representation
• Helps simplify circuit analysis by using the equivalent form that best suits the given information and desired quantities

Power Considerations

Maximum Power Transfer Theorem

• States that for a linear circuit, the maximum power is delivered to the load resistance when it is equal to the Thévenin or Norton equivalent resistance
• $R_{load} = R_{Th} = R_N$ for maximum power transfer
• At maximum power transfer, the power delivered to the load is:
  • $P_{max} = \frac{V_{Th}^2}{4R_{Th}} = \frac{I_N^2 R_N}{4}$
• Efficiency at maximum power transfer is 50% (half of the power is dissipated in the Thévenin or Norton resistance)
• In practice, the load resistance is often chosen to be much larger than the Thévenin or Norton resistance to improve efficiency at the cost of reduced power transfer
• Important consideration in the design of power delivery systems and impedance matching networks