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Note 27.
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Arrow and Double-Subscript Notation
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algebraic sum of voltage drops algebraic sum of generator voltages 0
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Network equations are written in accordance with Kirchho s voltage and current laws. The voltage law, for example, says that around any closed loop in a network the algebraic sum of the voltage drops and generator voltages must be equal to zero; thus
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so that a plus or a minus sign must be attached to each quantity in the equation for each loop of a network. Actually, however, in writing network equations we ll adhere to the THREE RULES stated in section 4.4, and write that algebraic sum of voltage drops algebraic sum of generator voltages
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in which we must remember that the signs are written in accordance with the rules given in " " " that section (except that now, in using vector algebra, we ll write Z; V , and I instead of R, V, and I see discussion given with Fig. 135 in Chap. 8). In regard to the above-mentioned signs, it should be noted that if a network contains just one generator, then no serious problem exists. If, however, a network contains TWO OR MORE GENERATORS then it is absolutely necessary that some method be used to indicate the DIRECTIONS of the voltage vectors. One such method uses the doublesubscript notation, in which the positive sense of the vector quantity is denoted by the order in which the subscripts are written. This is illustrated in the series circuit below (in which, for comparison, the familiar arrow notation is also shown).
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" Thus the notation Vab says there is an increase in vector voltage in going through that " particular generator in the direction from a to b, while Vdc says there is an increase or rise in voltage in going through that generator from d to c. (Note that the voltage arrows can " be used to give us the same information.) In regard to the current notation, Ide says that the vector current is positive in the direction of d to e. Hence, using double-subscript notation, and with the previously-mentioned rules in mind, the equation for the above gure can be written as (going around in the cw sense) " " " " " Ide Z Vab Vbc Vdc
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Square Root of 3 in Three-Phase Work
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p The factor 3 is prominent in three-phase work; our object here is to show how this comes about. To do this, let us note that the factor rst appears in eq. (424), because p sin 1208 sin 608 3=2 0:8660 a fact we can establish with the aid of the equilateral triangle shown below.
Appendix
For convenience, we ve let the sides of the triangle be 2 units in length, as shown. Hence, by the Pythagorean theorem, x2 1 4 so that x and therefore, by de nition, we have sin 1208 sin 608 x=2 p 3 p 3=2 0:8660
thus the factor
p 3 fundamentally appears in three-phase work.
Note 29.
Let and let
Proof of Eq. (467) (True Power)
" V V =a V ja generator voltage " I I =b I jb generator current
" " hence (a b angle between V and I , and hence, by eq. (117) in Chap. 5, P VI cos a b true power produced by the generator Now note that " I I = b I jb and therefore "" V I VI ja  jb VI j a b hence "" V I VI cos a b j sin a b "" showing that the REAL PART of the product V I is equal to the TRUE POWER produced by the generator, as eq. (467) states.
Note 30.
The Transistor as Ampli er
In general, in electronics, an ampli er is a circuit having an input signal and an output signal, in which the output signal is a reasonably good replica of the input signal and in which the POWER of the output signal is greater than the power of the input signal. An ampli er basically consists of a SOURCE OF DC POWER, an OUTPUT LOAD IMPEDANCE ZL , and a CONTROL DEVICE capable of controlling the instantaneous output current of the dc power source. This is illustrated in block-diagram form in Fig. 32-A.
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