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Equivalent circuit referred to transformer primary
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Figure 743 Maximum power transfer in an AC circuit with a transformer
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illustrates how the re ected load impedance, as seen by the source, is equal to ZL /N 2 , so that maximum power transfer occurs when ZL = ZS N2 RL = N 2 RS XL = N 2 XS
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EXAMPLE 714 Maximum Power Transfer Through a Transformer
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Problem
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Find the transformer turns ratio and load reactance that results in maximum power transfer in the circuit of Figure 744
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S LS RS ~ VS
1: N RL
Solution
Known Quantities: Source voltage, frequency and impedance; load resistance Find: Transformer turns ratio and load reactance
LS = 01 H; RL = 400
Schematics, Diagrams, Circuits, and Given Data: VS = 240 (0) V; RS = 10 ; = 377 rad/s
Assumptions: Use rms values for all phasor quantities in the problem
Analysis: For maximum power transfer, we require that RL = N 2 RS (equation 748)
Thus, N2 = RL 400 = 40 = RS 10 N= 40 = 6325
Further, to cancel the reactive power we require that XL = N 2 XS , ie, XS = 01 = 377 and XL = 40 377 = 1,508 Thus, the load reactance should be a capacitor with value C= 1 1 = = 176 F XL ( 1,508) 377
Check Your Understanding
712 If the transformer shown in Figure 745 is ideal, nd the turns ratio, N, that will ensure maximum power transfer to the load Assume that ZS = 1,800 and ZL = 8 713 If the circuit of Exercise 712 has ZL = (2 + j 10) and the turns ratio of the transformer is N = 54, what should ZS be in order to have maximum power transfer
Part I
Circuits
+ v (t) ~ S _
+ ZL vout (t) _ 1: N
THREE-PHASE POWER
The material presented so far in this chapter has dealt exclusively with singlephase AC power, that is, with single sinusoidal sources In fact, most of the AC power used today is generated and distributed as three-phase power, by means of an arrangement in which three sinusoidal voltages are generated out of phase with each other The primary reason is ef ciency: The weight of the conductors and other components in a three-phase system is much lower than in a single-phase system delivering the same amount of power Further, while the power produced by a single-phase system has a pulsating nature (recall the results of Section 71), a three-phase system can deliver a steady, constant supply of power For example, later in this section it will be shown that a three-phase generator producing three balanced voltages that is, voltages of equal amplitude and frequency displaced in phase by 120 has the property of delivering constant instantaneous power Another important advantage of three-phase power is that, as will be explained in 17, three-phase motors have a nonzero starting torque, unlike their single-phase counterpart The change to three-phase AC power systems from the early DC system proposed by Edison was therefore due to a number of reasons: the ef ciency resulting from transforming voltages up and down to minimize transmission losses over long distances; the ability to deliver constant power (an ability not shared by single- and two-phase AC systems); a more ef cient use of conductors; and the ability to provide starting torque for industrial motors To begin the discussion of three-phase power, consider a three-phase source connected in the wye (or Y) con guration, as shown in Figure 746 Each of the three voltages is 120 out of phase with the others, so that, using phasor notation,
+ ~ ~ Van _
~ Ia
Za n
~ Vcn
_ +~
_ ~ ~ Vbn +
b b
~ Ic
~ Ib
Figure 746 Balanced three-phase AC circuit
7
AC Power
we may write: Van = Van 0 Vbn = Vbn 120 Vcn = Vcn 240 = Vcn 120 where the quantities Van , Vbn , and Vcn are rms values and are equal to each other To simplify the notation, it will be assumed from here on that Van = Vbn = Vcn = V (751) (750)
17 will discuss how three-phase AC electric generators may be constructed to provide such balanced voltages In the circuit of Figure 746, the resistive loads are also wye-connected and balanced (ie, equal) The three AC sources are all connected together at a node called the neutral node, denoted by n The voltages Van , Vbn , and Vcn are called the phase voltages and form a balanced set in the sense that Van + Vbn + Vcn = 0
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