Examples/Req

From PrattWiki
Revision as of 16:37, 1 February 2016 by DukeEgr93 (talk | contribs)
Jump to navigation Jump to search

The following shows two step-by-step examples for find the equivalent resistance between two nodes for the following network:

R 0.png

In particular, this example demonstrates how the same network of resistors might reduce differently depending up which two nodes in the network are of interest. Each figure shows the "before" and "after" of a particular step.

\(R_{ab}\!\)

For the first example, we will find the equivalent resistance between nodes a and b.

  • Start by focusing only on nodes a and b

Rab 1.png

  • Next, combine any resistances that are in series. In this particular case, R8 and R11 are now in series because the wire leading to node d has been eliminated. Replace this combination with Req1:

\(R_{eq1}=R_8+R_{11}\) Rab Req1.png

  • Since no resistors are currently in series, now look for resistors that are in parallel. R7 and Req1 are in parallel; also, R3 and R4 are in parallel. Replace each with their equivalent resistances (named Req2 and Req3, respectively):

\(R_{eq2}=R_{eq1}||R_7=R_7||(R_8+R_{11})~~~~R_{eq3}=R_3||R_4~~~~\) Rab Req23.png

  • Now note that Req2 and Req3 are in series; replace them with Req4:

\(R_{eq4}=R_{eq2}+R_{eq3}=(R_7||(R_8+R_{11}))+(R_3||R_4)\) Rab Req4.png

  • After that step, R6 and Req4 are in parallel; replace them with Req5:

\(R_{eq5}=R_6||R_{eq4}=R_6||((R_7||(R_8+R_{11}))+(R_3||R_4))\) Rab Req5.png

  • Next, note that R2, Req5, and R10 are in series; replace them with Req6:

\(R_{eq6}=R_1+R_{eq5}+R_{10}=R_2+(R_6||((R_7||(R_8+R_{11}))+(R_3||R_4)))+R_{10}\) Rab Req6.png

  • Given that network, R5 and Req6 are in parallel; replace them with Req7:

\(R_{eq7}=R_5||R_{eq6}=R_5||(R_2+(R_6||((R_7||(R_8+R_{11}))+(R_3||R_4)))+R_{10}) \) Rab Req7.png

  • Finally, R1, Req7, and R9 are in series; replace them with Req,ab, the overall equivalent resistance between nodes a and b:

\(R_{eq,ab}=R_1+R_{eq7}+R_9=R_1+(R_5||(R_2+(R_6||((R_7||(R_8+R_{11}))+(R_3||R_4)))+R_{10}))+R_9\) Rab Reqab.png

  • Overall, this yields:

\( R_{eq,ab} = R_1~+~(R_5~||~(R_2~+~(R_6~||~((R_7~||~(R_8~+~R_{11}))+(R_3~||~R_4)))~+~R_{10}))~+~R_9\! \)

or, in other words,

\( {\frac {R_[[:Template:1]]R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:7]]+R_[[:Template:1]]R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:8]]+R_[[:Template:1]]R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:11]]+R_[[:Template:1]]R_[[:Template:2]]R_[[:Template:3]]R_ [[:Template:6]]R_[[:Template:7]]+R_[[:Template:1]]R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:6]]R_[[:Template:8]]+R_[[:Template:1]]R_[[:Template:2]]R_[[:Template:3]] R_[[:Template:6]]R_[[:Template:11]]+R_[[:Template:1]]R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:7]]R_[[:Template:8]]+R_[[:Template:1]]R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:7]]R_[[:Template:11]]+R_[[:Template:1]]R_[[:Template:2]]R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:7]]+R_[[:Template:1]]R_[[:Template:2]]R _[[:Template:4]]R_[[:Template:6]]R_[[:Template:8]]+R_[[:Template:1]]R_[[:Template:2]]R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:11]]+R_[[:Template:1]]R_[[:Template:2]]R_[[:Template:4]]R_[[:Template:7]]R_[[:Template:8]]+R_[[:Template:1]]R_[[:Template:2]]R_[[:Template:4]]R_[[:Template:7]]R_[[:Template:11]]+R_[[:Template:1]]R_ [[:Template:3]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:7]]+R_[[:Template:1]]R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:8]]+R_[[:Template:1]] R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:11]]+R_[[:Template:1]]R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:7]]+R_[[:Template:1]]R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:8]]+R_[[:Template:1]]R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:11]]+R _[[:Template:1]]R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:7]]R_[[:Template:10]]+R_[[:Template:1]]R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:8]]R_[[:Template:10]]+R_[[:Template:1]]R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:10]]R_[[:Template:11]]+R_[[:Template:1]]R_[[:Template:3]]R_[[:Template:5]]R_[[:Template:6]]R _[[:Template:7]]+R_[[:Template:1]]R_[[:Template:3]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:8]]+R_[[:Template:1]]R_[[:Template:3]]R_[[:Template:5]]R_{{6} }R_[[:Template:11]]+R_[[:Template:1]]R_[[:Template:3]]R_[[:Template:5]]R_[[:Template:7]]R_[[:Template:8]]+R_[[:Template:1]]R_[[:Template:3]]R_[[:Template:5]]R_{ {7}}R_[[:Template:11]]+R_[[:Template:1]]R_[[:Template:3]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:8]]+R_[[:Template:1]]R_[[:Template:3]]R_[[:Template:6]] R_[[:Template:7]]R_[[:Template:10]]+R_[[:Template:1]]R_[[:Template:3]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:11]]+R_[[:Template:1]]R_[[:Template:3]]R_{ {6}}R_[[:Template:8]]R_[[:Template:10]]+R_[[:Template:1]]R_[[:Template:3]]R_[[:Template:6]]R_[[:Template:10]]R_[[:Template:11]]+R_[[:Template:1]]R_[[:Template:3]]R_[[:Template:7]]R_[[:Template:8]]R_[[:Template:10]]+R_[[:Template:1]]R_[[:Template:3]]R_[[:Template:7]]R_[[:Template:10]]R_[[:Template:11]]+R_[[:Template:1]] R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:7]]+R_[[:Template:1]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:8]]+R_[[:Template:1]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:11]]+R_[[:Template:1]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:7]]R_[[:Template:8]]+R_ [[:Template:1]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:7]]R_[[:Template:11]]+R_[[:Template:1]]R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:8]] +R_[[:Template:1]]R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:10]]+R_[[:Template:1]]R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:11]]+R_[[:Template:1]]R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:8]]R_[[:Template:10]]+R_[[:Template:1]]R_[[:Template:4]]R_[[:Template:6]]R_{{10} }R_[[:Template:11]]+R_[[:Template:1]]R_[[:Template:4]]R_[[:Template:7]]R_[[:Template:8]]R_[[:Template:10]]+R_[[:Template:1]]R_[[:Template:4]]R_[[:Template:7]]R_ [[:Template:10]]R_[[:Template:11]]+R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:7]]+R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:8]]+R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:11]]+R_[[:Template:2]]R_[[:Template:3]]R_ [[:Template:4]]R_[[:Template:7]]R_[[:Template:9]]+R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:8]]R_[[:Template:9]]+R_[[:Template:2]]R_[[:Template:3]] R_[[:Template:4]]R_[[:Template:9]]R_[[:Template:11]]+R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:7]]+R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:8]]+R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:11]]+R_[[:Template:2]]R _[[:Template:3]]R_[[:Template:5]]R_[[:Template:7]]R_[[:Template:8]]+R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:5]]R_[[:Template:7]]R_[[:Template:11]]+R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:9]]+R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:6]]R_[[:Template:8]]R_[[:Template:9]]+R_{ {2}}R_[[:Template:3]]R_[[:Template:6]]R_[[:Template:9]]R_[[:Template:11]]+R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:7]]R_[[:Template:8]]R_[[:Template:9]]+ R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:7]]R_[[:Template:9]]R_[[:Template:11]]+R_[[:Template:2]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:7]]+R_[[:Template:2]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:8]]+R_[[:Template:2]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:6]]R_{ {11}}+R_[[:Template:2]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:7]]R_[[:Template:8]]+R_[[:Template:2]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:7]] R_[[:Template:11]]+R_[[:Template:2]]R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:9]]+R_[[:Template:2]]R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:8]]R_[[:Template:9]]+R_[[:Template:2]]R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:9]]R_[[:Template:11]]+R_[[:Template:2]]R_[[:Template:4]]R_[[:Template:7]]R _[[:Template:8]]R_[[:Template:9]]+R_[[:Template:2]]R_[[:Template:4]]R_[[:Template:7]]R_[[:Template:9]]R_[[:Template:11]]+R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:7]]+R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:8]]+R_[[:Template:3]]R_[[:Template:4]]R_{ {5}}R_[[:Template:6]]R_[[:Template:11]]+R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:7]]R_[[:Template:9]]+R_[[:Template:3]]R_[[:Template:4]] R_[[:Template:5]]R_[[:Template:7]]R_[[:Template:10]]+R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:8]]R_[[:Template:9]]+R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:8]]R_[[:Template:10]]+R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:9]]R_[[:Template:11]]+R_[[:Template:3]] R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:10]]R_[[:Template:11]]+R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:9]]+R_{ {3}}R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:8]]R_[[:Template:9]]+R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:9]]R_[[:Template:11]]+ R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:7]]R_[[:Template:9]]R_[[:Template:10]]+R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:8]]R_[[:Template:9]]R_[[:Template:10]]+R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:9]]R_[[:Template:10]]R_[[:Template:11]]+R_[[:Template:3]]R_[[:Template:5]]R_[[:Template:6]]R_{{7} }R_[[:Template:8]]+R_[[:Template:3]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:9]]+R_[[:Template:3]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:10]]+R_[[:Template:3]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:11]]+R_[[:Template:3]]R_[[:Template:5]]R_[[:Template:6]] R_[[:Template:8]]R_[[:Template:9]]+R_[[:Template:3]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:8]]R_[[:Template:10]]+R_[[:Template:3]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:9]]R_[[:Template:11]]+R_[[:Template:3]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:10]]R_[[:Template:11]]+R_[[:Template:3]]R_{{5} }R_[[:Template:7]]R_[[:Template:8]]R_[[:Template:9]]+R_[[:Template:3]]R_[[:Template:5]]R_[[:Template:7]]R_[[:Template:8]]R_[[:Template:10]]+R_[[:Template:3]]R_{ {5}}R_[[:Template:7]]R_[[:Template:9]]R_[[:Template:11]]+R_[[:Template:3]]R_[[:Template:5]]R_[[:Template:7]]R_[[:Template:10]]R_[[:Template:11]]+R_[[:Template:3]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:8]]R_[[:Template:9]]+R_[[:Template:3]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:9]]R_[[:Template:10]]+R_ [[:Template:3]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:9]]R_[[:Template:11]]+R_[[:Template:3]]R_[[:Template:6]]R_[[:Template:8]]R_[[:Template:9]]R_{{10} }+R_[[:Template:3]]R_[[:Template:6]]R_[[:Template:9]]R_[[:Template:10]]R_[[:Template:11]]+R_[[:Template:3]]R_[[:Template:7]]R_[[:Template:8]]R_[[:Template:9]]R_ [[:Template:10]]+R_[[:Template:3]]R_[[:Template:7]]R_[[:Template:9]]R_[[:Template:10]]R_[[:Template:11]]+R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:8]]+R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:9]]+R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:6]]R_ [[:Template:7]]R_[[:Template:10]]+R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:11]]+R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:8]]R_[[:Template:9]]+R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:8]]R_[[:Template:10]]+R_[[:Template:4]]R_[[:Template:5]]R_ [[:Template:6]]R_[[:Template:9]]R_[[:Template:11]]+R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:10]]R_[[:Template:11]]+R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:7]]R_[[:Template:8]]R_[[:Template:9]]+R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:7]]R_[[:Template:8]]R_[[:Template:10]]+R_[[:Template:4]]R _[[:Template:5]]R_[[:Template:7]]R_[[:Template:9]]R_[[:Template:11]]+R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:7]]R_[[:Template:10]]R_[[:Template:11]]+R_{ {4}}R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:8]]R_[[:Template:9]]+R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:9]]R_[[:Template:10]]+ R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:9]]R_[[:Template:11]]+R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:8]]R_[[:Template:9]]R_[[:Template:10]]+R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:9]]R_[[:Template:10]]R_[[:Template:11]]+R_[[:Template:4]]R_[[:Template:7]]R_[[:Template:8]]R_{{9} }R_[[:Template:10]]+R_[[:Template:4]]R_[[:Template:7]]R_[[:Template:9]]R_[[:Template:10]]R_[[:Template:11]]}{R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:4]] R_[[:Template:7]]+R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:8]]+R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:11]]+R_{ {2}}R_[[:Template:3]]R_[[:Template:6]]R_[[:Template:7]]+R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:6]]R_[[:Template:8]]+R_[[:Template:2]]R_[[:Template:3]]R _[[:Template:6]]R_[[:Template:11]]+R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:7]]R_[[:Template:8]]+R_[[:Template:2]]R_[[:Template:3]]R_[[:Template:7]]R_[[:Template:11]]+R_[[:Template:2]]R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:7]]+R_[[:Template:2]]R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:8]]+R_[[:Template:2]] R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:11]]+R_[[:Template:2]]R_[[:Template:4]]R_[[:Template:7]]R_[[:Template:8]]+R_[[:Template:2]]R_[[:Template:4]]R_[[:Template:7]]R_[[:Template:11]]+R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:7]]+R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:8]]+ R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:11]]+R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:7]]+R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:8]]+R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:11]]+R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:7]]R _[[:Template:10]]+R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:8]]R_[[:Template:10]]+R_[[:Template:3]]R_[[:Template:4]]R_[[:Template:10]]R_[[:Template:11]]+R _[[:Template:3]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:7]]+R_[[:Template:3]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:8]]+R_[[:Template:3]]R_{{5} }R_[[:Template:6]]R_[[:Template:11]]+R_[[:Template:3]]R_[[:Template:5]]R_[[:Template:7]]R_[[:Template:8]]+R_[[:Template:3]]R_[[:Template:5]]R_[[:Template:7]]R_{ {11}}+R_[[:Template:3]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:8]]+R_[[:Template:3]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:10]]+R_[[:Template:3]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:11]]+R_[[:Template:3]]R_[[:Template:6]]R_[[:Template:8]]R_[[:Template:10]]+R_[[:Template:3]]R_[[:Template:6]]R _[[:Template:10]]R_[[:Template:11]]+R_[[:Template:3]]R_[[:Template:7]]R_[[:Template:8]]R_[[:Template:10]]+R_[[:Template:3]]R_[[:Template:7]]R_[[:Template:10]]R_ [[:Template:11]]+R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:7]]+R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:8]]+R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:6]]R_[[:Template:11]]+R_[[:Template:4]]R_[[:Template:5]]R_[[:Template:7]]R_[[:Template:8]]+R_[[:Template:4]]R_[[:Template:5]]R_ [[:Template:7]]R_[[:Template:11]]+R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:8]]+R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:10]]+R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:7]]R_[[:Template:11]]+R_[[:Template:4]]R_[[:Template:6]]R_[[:Template:8]]R_[[:Template:10]]+R_[[:Template:4]] R_[[:Template:6]]R_[[:Template:10]]R_[[:Template:11]]+R_[[:Template:4]]R_[[:Template:7]]R_[[:Template:8]]R_[[:Template:10]]+R_[[:Template:4]]R_[[:Template:7]]R_ [[:Template:10]]R_[[:Template:11]]}} \)