Kof's Rules and Complex DC Circuits

Overview

• Some circuits cannot have all resistors reduced to a single equivalent resistance.
• More complex circuits require Kirchhoff's rules to solve for different currents in the circuit.
• Kirchhoff's rules help set up a system of equations to find currents.

Kirchhoff's Rules

Junction Rule

• The current flowing into a junction equals the current flowing out.
• Example: If 10 amps flow in, 10 amps must flow out.
• If the current direction is chosen incorrectly, the math will reflect this by giving a negative current value.

Loop Rule

• For any closed loop, the sum of voltage increases must equal the sum of voltage decreases.
• Voltage increases and decreases occur when crossing batteries and resistors.
• Voltage increases when moving from the negative to positive terminal of a battery and decreases when moving from positive to negative.
• Voltage drops occur across resistors when moving in the same direction as the defined current.
• If encountered opposite to current direction, it results in a voltage increase.

Applying Kirchhoff's Rules

• In complex circuits with multiple batteries and resistors, set up a system with junction and loop rules.
• Define currents (i1, i2, i3) and determine their direction based on the batteries.
• Example setup:
  • Define i1 governed by a 20V battery flowing through the circuit.
  • Define i2 by a 16V battery in another part of the circuit.
  • Define i3 for the remaining part of the circuit.
• Set up junction and loop equations based on the defined currents.

Solving the System of Equations

Example Problem Setup

• Given diagram, find the current through each resistor:
  • Define and label currents properly.
  • Apply Kirchhoff’s junction rule to find the relationship between currents.
  • Apply Kirchhoff’s loop rule to set up equations for voltage drops and increases.

Solving the Equations

• Combine and manipulate equations to solve for currents.
• Example manipulation:
  • Substitute one current equation into another to isolate variables.
  • Use algebraic methods such as linear combinations.
• Solve for each current step-by-step:
  • Example: solve for i1 first, then use that to solve for i2 and i3.
  • Verify solutions by checking voltage drops in the loop rules.

Using Matrix Algebra

• System of equations can be solved with matrices using a calculator.
  • Example: Use TI-83+ to input and solve the matrix.
  • Define the matrix with coefficients and constants from equations.
  • Use calculator functions for reduced row Echelon form to directly solve for each current.

Summary

• Kirchhoff's rules are essential for solving complex circuits.
• Matrix algebra can simplify solving systems of equations.
• Practice setting up and solving systems to master techniques.

Tools Mentioned

• TI-83 calculator for matrix algebra.

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