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Unformatted text preview: n N resistors in parallel, determine if they can be replaced by a single Req
R1 R2 RN Figure 2.1
To find the equivalent resistance of the parallel resistors in Figure 2.1 combine the resistors two
at a time to discover a pattern. First, consider the circuits below, i
VS i1 +
− i i2 R1 +
− VS R2 ieq
Req Note that two resistors in parallel have the same voltage across them. Ohm's law gives us VS = i1 R 1 = i 2 R 2 VS = i eq R eq or i1 = VS
R1 and i2 = VS
R2 i eq = VS
R eq Using KCL, i = i1 + i 2 i = i eq Use the equations for i1 and i 2 and i eq to find i in terms of the source voltage and resistors. i=
Thus, 1
VS VS
1
+
= VS R + R R1 R 2
2
1
1
1
1
=
+
or
R eq R 1 R 2 i=
R eq = VS
R eq R1 R 2
R1 + R 2  v v This implies that the equivalent resistance of two resistors in parallel is the product of their
resistances divided by the sum of their resistances.  eText Main Menu  Textbook Table of Contents  Problem Solving Workbook Contents Continue the process. Find the equivalent resistance of R 12 = i
VS i12 +
− R1 R 2
in parallel with R 3 .
R1 + R 2
i i3 R12 +
− VS R3 ieq
Req Note that two resistors in parallel have the same voltage across them. Ohm's law gives us VS = i 1 2 R 1 2 = i 3 R 3
i12 = VS
R 12 and VS = i eq R eq
i3 = VS
R3 i eq = VS
R eq Using KCL, i = i12 + i 3 i = i eq Use the equations for i12 and i 3 and i eq to find i in terms of the source voltage and resistors. i= VS
VS
+
R 12 R 3 i= VS
R eq 1 1 = V R1 + R 2 + 1 = V 1 + 1 + 1 i = VS +
R S
S
R3 R1 R 2 R1 R 2 R 3 1 2 R 3 1
1
1
1
=
+
+
R eq R 1 R 2 R 3 Thus, Continuing this process for N resistors would show that the reciprocal of the equivalent resistance
of N resistors in parallel is the sum of the reciprocals of each resistance. In general, 1
1
1
=
+
+
R eq R 1 R 2 Problem 2.13 + N
1
1
=∑
R N n=1 R n If you know the current through two resistors in parallel ( R 1 and R 2 ), is there a simple way to determine the current through either R 1 or R 2 ? i
VS +
− i1
R1 i2
R2  v v Clearly, the two resistors in parallel have the same voltage across them. Using Ohm's law,  eText Main Menu  Textboo...
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 Spring '12
 bkav

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