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Combinations of Resistors
Resistors do not occur in isolation.  They are almost always part of a larger circuit, and frequently that larger circuit contains many resistors. It is often the case that resistors occur in combinations that repeat.
Goals

 
Series Combinations of Resistors :

Two elements are said to be in series whenever the same current physically flows through both of the elements. The critical point is that the same current flows through both resistors when two are in series.  The particular configuration does not matter.  The only thing that matters is that exactly the same current flows through both resistors.  Current flows into one element, through the element, out of the element into the other element, through the second element and out of the second element.  No part of the current that flows through one resistor "escapes" and none is added.  This figure shows several different ways that two resistors in series might appear as part of a larger circuit diagram.

The current through the two resistors is the same.The voltage across the series combination is given by:Vs= Va + VbThe voltages across the two resistors are given by Ohm's Law:Va = Is RaVb = Is RbWe can combine all of these relations, and when we do that we find the following:
Vs= Va + VbVs= Is Ra  + Is RbVs= Is (Ra  + Rb)Vs= Is RseriesHere, we take Rseries to be the series equivalent of the two resistors in series, and the expression for Rseries is:

Rseries = Ra  + Rb
 

Parallel Resistors
The voltage across the two resistors is the same.The current through the parallel combination is given by:Ip= Ia + IbThe currents through the two resistors are given by Ohm's Law:Ia = Vp /RaIb = Vp /RbWe can combine all of these relations, and when we do that we find the following:
Ip= Ia + IbIp= Vp /Ra + Vp /RbIp= Vp[ 1/Ra + 1/Rb]Ip= Vp/RparallelHere, we take Rparallel to be the parallel equivalent of the two resistors in parallel, and the expression for Rparallel is:
                                                                         

1/Rparallel = 1/Ra  + 1/Rb 
 

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