Spring 2018
Equivalent Resistance
Use Series, Parallel, and Ohm's law to find Equivalent Resistance
Learn It!
Pre-Requisite Knowledge
Goal
Find the equivalent resistance.
This group of resistors will act like one equivalent resistor. What is its resistance?
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Part 1
Redraw The Circuit
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1A
Circuit transformation
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Now our circuit is nicely transformed. After doing this a few times, you will be able to do all this in your head. It just takes a little practice.
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Are the legal moves 'real'
Part 2
Finding Equivalent Resistance
Use parallel and series formulas to decompose circuits one piece at a time.
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2A
Reduce two series resistors on the right
\[R_{eq1} = R_1 + R_2 = 4+2 = 6 \]
Now our circuit is nicely transformed. After doing this a few times, you will be able to do all this in your head. It just takes a little practice.
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2B
Reduce two parallel resistors on the left
\[R_{eq2} = \dfrac{1}{\dfrac{1}{R_1} + {\dfrac{1}{R_2}}} = \dfrac {1} {\dfrac{1}{20} + \dfrac{1}{30}} = 12 \]
Apply the parallel resistance formula! Note that our answer is reasonable: a parallel configuration reduces resistance.
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2C
Reduce the series resistors on the left
\[R_{eq3} = 8 +12 = 20 \]
Apply the series equivalent resistance formula.
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2D
Combine three parallel resistors
\(R_{eq} = \dfrac{1}{\dfrac{1}{20} + \dfrac{1}{10} + \dfrac{1}{6}} = 3.157 \Omega \)
\[R_{eq} = \dfrac{1}{\dfrac{1}{20} + \dfrac{1}{10} + \dfrac{1}{6}} = 3.157 \Omega \]
Apply the parallel resistance formula to three resistors. We have succeeded! The whole mess we started with behaves as though it were a single resistor with a resistance of about \(3.1 \Omega \).
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Can all circuits be broken down with series and parallel?