Circuit Resistance Analysis

Parallel and Series Branches

• Current divides at junctions, creating parallel branches.
• Current is the same through elements in series.

Top Branch Resistance

• Two resistors in series ($6 \Omega$ and $1 \Omega$) can be combined: $6 \Omega + 1 \Omega = 7 \Omega$.
• A $6 \Omega$ resistor is in parallel with the combined $6 \Omega$ resistor.
• Equivalent resistance of parallel branches: $R<em>{eq} = (1/R</em>1 + 1/R_2)^{-1}$.
• $R_{eq} = (1/6 + 1/6)^{-1} = 3 \Omega$.
• Combined with a $3 \Omega$ resistor in series: $3 \Omega + 3 \Omega = 6 \Omega$.

Bottom Branch Resistance

• Parallel resistors: $1/4 \Omega$ and $1/12 \Omega$.
• Equivalent resistance: $(1/(1/4) + 1/(1/12))^{-1} = (4 + 12)^{-1} = 3 \Omega$.
• Combined with a $9 \Omega$ resistor in series: $3 \Omega + 9 \Omega = 12 \Omega$.

Total Circuit Resistance

• Two parallel branches with $6 \Omega$ and $12 \Omega$ resistances.
• Equivalent resistance: $(1/6 + 1/12)^{-1} = 4 \Omega$.
• Total resistance: $3 \Omega + 4 \Omega = 7 \Omega$ (series resistances added).