Overview
This lecture introduces nodal analysis, a technique for finding unknown voltages in electrical circuits using Kirchhoff’s Current Law (KCL) and Ohm’s Law, demonstrated step-by-step with example problems.
Introduction to Nodal Analysis
- Nodal analysis uses KCL to solve for unknown voltages at circuit nodes.
- Primarily used to find node voltages, but can also determine currents.
- Circuits must have a chosen ground (reference) node, typically set at zero volts.
Example 1: Finding Node Voltages
- Identify all nodes in the circuit and label them (e.g., A, B, C).
- Assign ground (0 V) to the most logical node, often at the negative side of a power supply.
- The voltage at a node can be determined by considering drops across power supplies (e.g., VA = 50 V if across a 50 V source).
- To find VX, set up VX = VB − VC (if VC is ground, VX = VB).
- Apply KCL at node B: sum of currents leaving node equals current entering.
- Express current through resistor as (Voltage at one end − Voltage at other end) / Resistance.
- Combine all currents leaving node into a single equation and solve for the unknown voltage.
Example 2: Parallel Voltages and Solving for V1, V2
- Components in parallel have the same voltage drop.
- If a resistor is parallel with a voltage source (e.g., a 10 Ω resistor with a 20 V source), the voltage across the resistor equals the source (V2 = 20 V).
- Apply KCL at a chosen node for the unknown voltage (e.g., VA for V1).
- Set up and solve the equation 3 = VA/5 + (VA − VB)/20, substituting known voltages.
- Solve for VA (V1), and use previously found node voltages as needed.
Calculating Power in Resistors
- Power dissipated by a resistor: P = V² / R, where V is the voltage across the resistor.
- The sign of the calculated voltage does not affect power, since voltage is squared.
- Use node voltages to determine voltage across resistors (e.g., VA − VB).
Key Terms & Definitions
- Node — A point in a circuit where two or more elements are connected.
- Nodal Analysis — A method using KCL and Ohm’s Law to find node voltages.
- Ground (Reference Node) — Node chosen as 0 volts to serve as a voltage reference.
- Kirchhoff’s Current Law (KCL) — The sum of currents entering a node equals the sum leaving it.
- Ohm's Law — V = IR, where V is voltage, I is current, and R is resistance.
- Power Dissipation — The rate at which a resistor converts electrical energy to heat, calculated as P = V²/R.
Action Items / Next Steps
- Practice nodal analysis with homework problems, including finding node voltages and resistor power.
- Review textbook sections on KCL, nodal analysis, and power calculations.