Overview
This lecture explains how to find the Thevenin equivalent circuit for a given network using node analysis and parallel/series resistor calculations.
Problem Setup & Simplification
- Identify and ignore components not affecting the open-circuit (here, the 1 kΩ resistor is ignored).
- Replace the load resistor with a wire to analyze the open-circuit voltage.
- Assign and label circuit nodes; ground is at 0 V, supply node at 3 V, and define unknown node as ( v_x ).
Open-Circuit Voltage (Vth) Calculation
- Open-circuit voltage is equivalent to the voltage ( v_x ) at the node of interest.
- Set up node analysis at ( v_x ), summing currents leaving/arriving at the node.
- Combine all current expressions: ( 7 + \frac{v_x - 3}{2} + \frac{v_x}{5} = 0 ), using milliamps (mA) for current and kilo-ohms (kΩ) for resistance.
- Multiply both sides by 10 to clear denominators, simplify to ( 7v_x = -55 ), so ( v_x = -7.86 ) V.
Using Equation Solvers
- Online equation solvers or graphing calculators can assist in solving node-voltage equations, ensuring accuracy and saving time.
Thevenin Resistance (Rth) Calculation
- To find ( R_{th} ), turn off the independent voltage source (replace with short circuit) and independent current source (replace with open circuit).
- The remaining two resistors, 2 kΩ and 5 kΩ, are in parallel: ( \frac{2 \times 5}{2 + 5} = 1.43 ) kΩ (rounded).
- Add ignored 1 kΩ resistor in series: ( 1 + 1.43 = 2.43 ) kΩ.
Thevenin Equivalent Circuit
- The Thevenin equivalent voltage (( V_{th} )) is (-7.86) V.
- The Thevenin equivalent resistance (( R_{th} )) is (2.43) kΩ.
Key Terms & Definitions
- Thevenin Equivalent — A simplified circuit with one voltage source and one resistor that behaves identically at terminals of interest.
- Node Analysis — Technique using Kirchhoff’s current law to solve for voltages at circuit nodes.
- Parallel Resistance — For resistors ( R_1 ) and ( R_2 ), ( R_{eq} = \frac{R_1 R_2}{R_1 + R_2} ).
- Open Circuit Voltage — Voltage measured at the open terminals where the load resistor is removed.
Action Items / Next Steps
- Complete assigned homework problems involving Thevenin equivalents.
- Practice node analysis and solving parallel and series resistor networks.