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

52 part 1 dc circuits each resistor in the network is

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52PART 1DC CircuitsEach resistor in thenetwork is the sum of all possible products of Y resistorstaken two at a time, divided by the opposite Y resistor.The Y andnetworks are said to bebalancedwhenR1=R2=R3=RY,Ra=Rb=Rc=R(2.56)Under these conditions, conversion formulas becomeRY=R3orR=3RY(2.57)One may wonder whyRYis less thanR. Well, we notice that the Y-connection is like a “series” connection while the-connection is like a“parallel” connection.Note that in making the transformation, we do not take anything outof the circuit or put in anything new. We are merely substituting differentbut mathematically equivalent three-terminal network patterns to createa circuit in which resistors are either in series or in parallel, allowing usto calculateReqif necessary.EXAMPLE2.14Convert thenetwork in Fig. 2.50(a) to an equivalent Y network.cba10 15 (a)RbRaRc25 cba5 3 7.5 R2R1R3(b)Figure 2.50For Example 2.14: (a) originalnetwork, (b) Y equivalent network.Solution:Using Eqs. (2.49) to (2.51), we obtain
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CHAPTER 2Basic Laws53R1=RbRcRa+Rb+Rc=25×1025+10+15=25050=5R2=RcRaRa+Rb+Rc=25×1550=7.5R3=RaRbRa+Rb+Rc=15×1050=3The equivalent Y network is shown in Fig. 2.50(b).P R A C T I C EP R O B L E M2 . 1 4Transform the wye network in Fig. 2.51 to a delta network.20 R2bac10 R1R340 Figure 2.51For Practice Prob. 2.14.Answer:Ra=140, Rb=70, Rc=35.EXAMPLE2.15Obtain the equivalent resistanceRabfor the circuit in Fig. 2.52 and use itto find currenti.aaibbcn120 V5 30 12.5 15 10 20 +Figure 2.52For Example 2.15.Solution:Inthiscircuit, therearetwoYnetworksandonenetwork. Transformingjust one of these will simplify the circuit. If we convert the Y networkcomprising the 5-, 10-, and 20-resistors, we may selectR1=10,R2=20,R3=5Thus from Eqs. (2.53) to (2.55) we haveRa=R1R2+R2R3+R3R1R1=10×20+20×5+5×1010=35010=35Rb=R1R2+R2R3+R3R1R2=35020=17.5Rc=R1R2+R2R3+R3R1R3=3505=70With the Y converted to, the equivalent circuit (with the voltagesource removed for now) is shown in Fig. 2.53(a). Combining the threepairs of resistors in parallel, we obtain
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54PART 1DC Circuits7030=70×3070+30=2112.517.5=12.5×17.512.5+17.5=7.29171535=15×3515+35=10.5so that the equivalent circuit is shown in Fig. 2.53(b). Hence, we findRab=(7.292+10.5)21=17.792×2117.792+21=9.632Theni=vsRab=1209.632=12.458 Aab30 70 17.5 35 12.5 15 (a)ab21 (b)7.292 10.5 Figure 2.53Equivalent circuits to Fig. 2.52, with the voltage removed.P R A C T I C EP R O B L E M2 . 1 5For the bridge network in Fig. 2.54, findRabandi.24 100 Vi30 10 50 13 20 +baFigure 2.54For Practice Prob. 2.15.Answer:40,2.5 A.2.8APPLICATIONSResistors are often used to model devices that convert electrical energyinto heat or other forms of energy.Such devices include conductingwire, lightbulbs, electric heaters, stoves, ovens, and loudspeakers. In this
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