An introduction to electronics
An introduction to electronics

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An introduction to electronics

Series and parallel networks (combining the laws)

The ordering of the components, and how they are connected, is important in a circuit. For example, two components (here resistors) can be arranged in two different ways, as shown in Figure 11. A circuit in which the current must take a single path, going through first one component and then the other in series, is shown in Figure 11(a); a circuit in which the current splits and takes two parallel paths at the same time is shown in Figure 11(b).

Described image
Figure 11  Circuit schematics showing two resistors arranged in (a) series and (b) parallel

By combining Ohm’s and Kirchhoff’s laws, it can be shown that:

  • the total resistance cap r of n resistors in series is given by

    cap r equals sum with, 4 , summands cap r sub one plus cap r sub two plus midline horizontal ellipsis plus cap r sub n
  • the total resistance cap r of n resistors in parallel is given by

    cap r equals one divided by sum with, 4 , summands one divided by cap r sub one plus one divided by cap r sub two plus midline horizontal ellipsis plus one divided by cap r sub n

SAQ 2

In Figure 11, let equation left hand side cap r sub one equals right hand side 470 postfix times normal cap omega and equation left hand side cap r sub two equals right hand side 1.3 postfix times k normal cap omega. To three significant figures:

  • a.What is their resistance in series?
  • b.What is their resistance in parallel?

Answer

  • a.In series, the combined resistance isequation left hand side cap r sub one plus cap r sub two equals right hand side open 470 plus 1300 close postfix times normal cap omega equals 1770 postfix times normal cap omega.
  • b.In parallel, the combined resistance is

    multiline equation line 1 equation left hand side one divided by one divided by cap r sub one plus one divided by cap r sub two equals right hand side one divided by one divided by 470 postfix times normal cap omega plus one divided by 1300 postfix times normal cap omega line 2 equation left hand side equals right hand side one divided by 0.00212 times horizontal ellipsis plus 0.000769 times horizontal ellipsis postfix times normal cap omega line 3 equation sequence equals one divided by 0.00289 times horizontal ellipsis postfix times normal cap omega equals 345 postfix times normal cap omega

    to three significant figures.

    Alternatively, note that for two resistors in parallel,

    equation sequence one divided by one divided by cap r sub one plus one divided by cap r sub two equals one divided by cap r sub two plus cap r sub one divided by cap r sub one times cap r sub two equals cap r sub one times cap r sub two divided by cap r sub one plus cap r sub two

    so the resistance in parallel can be written simply as

    470 multiplication 1300 divided by 470 plus 1300 postfix times normal cap omega equals 611 postfix times 000 divided by 1770 postfix times normal cap omega equals 345 postfix times normal cap omega left parenthesis to three s full stop f full stop right parenthesis

T212_1

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