PHYS345 PHYS345 Class 3

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Multiloop Circuits

Kirchhoff's Current Law (KCL) - also known as the junction rule:
At any junction point in a circuit where the current can divide, the sum of the currents into the junction must equal the sum of the currents out of the junction.
Kirchhoff's Voltage Law (KVL) - also known as the loop rule:
When any closed circuit loop is traversed, the algebraic sum of the changes in voltage must equal zero.

Apply the loop rule by starting at any point in circuit, going around the loop considering all changes in voltage, until returning to the original point. The following sign conventions must be obeyed:

  1. For a resistor crossed in the same direction as the current, the change in voltage is - iR.
  2. For a resistor crossed in the opposite direction of the current, the change in voltage is + iR.
  3. If a source of emf is crossed in the same direction as the emf, the change in voltage is +.
  4. If a source of emf is crossed in the opposite direction of the emf, the change in voltage is -.

Example, going from point A to B

[circuit] [sign convention]

      Concept Check: Series and Parallel Combinations

Simple Circuit Example

Real Batteries and Battery Testers

Ideal Voltage and Current Sources

      Concept Check: Current Sources

One Method of Solving Multiloop Circuits (from PHYS208)
- Assigning Branch Currents and Applying KVL and KCL

  1. Replace any combination of resistors in series or parallel with their equivalent resistance. Label resistors and emfs with symbols.
  2. Choose a direction for the current in each branch of the circuit, and label the currents in a circuit diagram. Add plus signs to indicate the high votlage sides of each emf and resistor.
  3. Apply the junction rule to each junction where the current divides.
  4. In a circuit containing n interior loops, apply the rule to any n loops.
  5. Solve the equations to obtain the values of the unknowns.
  6. Check your results by assigning a voltage of zero to one point in the circuit (the ground) and use the values of the currents found to determine the voltage at other points in the circuit.
Adapted from Tipler, "Physics for Scientists and Engineers"

Detailed Multiloop Circuit Example

A Simpler Multiloop Circuit Example

Relevant Online Resources
Linear Algebra Solver for those pesky simultaneous equations...


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Last updated Sept. 9, 1998.
Copyright George Watson, Univ. of Delaware, 1998.