49 252 superposition of y and networks as an aid in

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Unformatted text preview: . (2.49) to (2.51) that R1 R 2 + R 2 R 3 + R 3 R 1 = Ra Rb Rc (Ra + Rb + Rc ) (Ra + Rb + Rc )2 Ra R b Rc = Ra + R b + R c c Figure 2.49 (2.52) Superposition of Y and networks as an aid in transforming one to the other. Dividing Eq. (2.52) by each of Eqs. (2.49) to (2.51) leads to the following equations: Ra = R1 R2 + R2 R3 + R3 R1 R1 (2.53) Rb = R1 R 2 + R 2 R 3 + R 3 R 1 R2 (2.54) Rc = R 1 R 2 + R 2 R 3 + R 3 R1 R3 (2.55) | v v From Eqs. (2.53) to (2.55) and Fig. 2.49, the conversion rule for Y to is as follows: | e-Text Main Menu | Textbook Table of Contents | Problem Solving Workbook Contents 52 PART 1 DC Circuits Each resistor in the network is the sum of all possible products of Y resistors taken two at a time, divided by the opposite Y resistor. The Y and networks are said to be balanced when R1 = R2 = R3 = RY , Ra = Rb = Rc = R (2.56) Under these conditions, conversion formulas become RY = R 3 R = 3RY or (2.57) One may wonder why RY is less than R . Well, we notice that the Yconnection is like a “series” connection while the -connection is like a “parallel” connection. Note that in making the transformation, we do not take anything out of the circuit or put in anything new. We are merely substituting different but mathematically equivalent three-terminal network patterns to create a circuit in which resistors are either in series or in parallel, allowing us to calculate Req if necessary. EXAMPLE 2.14 Convert the network in Fig. 2.50(a) to an equivalent Y network. a b 5Ω Rc a b 7.5 Ω R2 R1 25 Ω Rb 10 Ω 15 Ω Ra R3 c c (b) (a) Figure 2.50 3Ω For Example 2.14: (a) original network, (b) Y equivalent network. | v v Solution: Using Eqs. (2.49) to (2.51), we obtain | e-Text Main Menu | Textbook Table of Contents | Problem Solving Workbook Contents CHAPTER 2 R1 = Basic Laws 53 250 Rb Rc 25 × 10 = =5 = Ra + R b + R c 25 + 10 + 15 50 R2 = R c Ra 25 × 15 = 7.5 = Ra + R b + R c 50 R3 = R a Rb 15 × 10 = =3 Ra + R b + R c 50 The equivalent Y network is shown...
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