12 resistor b c d r eq r 6 r 11 and e r eq given

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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. | e-Text Main Menu | Textbook Table of Contents | Problem Solving Workbook Contents Continue the process. Find the equivalent resistance of R 1||2 = i VS i1||2 + − R1 R 2 in parallel with R 3 . R1 + R 2 i i3 R1||2 + − 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 i1||2 = VS R 1||2 and VS = i eq R eq i3 = VS R3 i eq = VS R eq Using KCL, i = i1||2 + i 3 i = i eq Use the equations for i1||2 and i 3 and i eq to find i in terms of the source voltage and resistors. i= VS VS + R 1||2 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, | e-Text Main Menu | Textboo...
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