Problem-solving strategies:
Resistor Networks

If you have already read this introduction, click here.

Let's review the rules for resistors in series and resistors in parallel:

Series resistors:

• are connected together end-to-end in one circuit branch, with no intervening resistors or junctions. In the diagram above, R₂ is in series with R₃, but R₁ is not in series with R₄ (they are in separate branches).
• share the same amount of current. In the diagram above, resistors 4 & 5 & 6 all carry branch current i₄₅₆.
• are additive. You can replace series resistors with a single equivalent resistor with value equal to the sum of the individual resistances.
• are found in the same branch between two junctions, with the sum of voltages across the individual resistors equal to the voltage across the entire branch. In the diagram, V_AB=V₂+V₃.

Parallel resistors:

• are connected together at both ends simultaneously in separate branches, with no intervening resistors. In the diagram above, the branch containing resistors 2 and 3 is in parallel with the branch containing resistors 4 & 5 & 6, and they can be combined after simplification.
• share the same amount of voltage. In the diagram, the voltage V_AB is across both parallel branches.
• are inverse-additive. You can replace parallel resistors with a single equivalent resistor with value equal to the inverse of the sum of inverses. The equivalent resistor always is smaller in value than the smallest of the combined resistors.
• carry additive currents. In the diagram, i_batt=i₂₃+i₄₅₆.

This is the standard recipe for simplification of a complex resistor network:

1. Identify the branches. The network above has three branches: the first containing the battery and R₁, a second with R₂ and R₃, and a third with R₄ & R₅ & R₆.
2. Combine any series resistors in the branches. The network shown has two branches that require series simplification.
3. Combine any simplified branches that are in parallel. The two branches between junctions A and B will be combined in this way eventually.
4. Pick up any left over series resistors. Resistor 1 will not be combined until the last step.

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