The RX plane

Chapter The RX plane

Teach Yourself Electricity and Electronics Third Edition Book
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Teach Yourself Electricity and Electronics Third Edition Book

  • Problem 15-1Find the absolute value of the complex number 22 – j0.Note that this is a pure real. Actually, it is the same as 22j0, because j0 = 0.Therefore, the absolute value of this complex number is –(–22)22.Problem 15-2Find the absolute value of 0 j34.This is a pure imaginary number. The value of b in this case is 34, because 0 j340j(34). Therefore, the absolute value is ( 34)34.Problem 15-3Find the absolute value of 3 j4.In this number, a3 and b4, because 3j4 can be rewritten as 3 j( 4).Squaring both of these, and adding the results, gives 32( 4)291625. Thesquare root of 25 is 5; therefore, the absolute value of this complex number is 5.You might notice this “3, 4, 5” relationship and recall the Pythagorean theorem forfinding the length of the hypotenuse of a right triangle. The formula for finding thelength of a vector in the complex-number plane comes directly from this theorem.If you don’t remember the Pythagorean theorem, don’t worry; just remember theformula for the length of a vector.The RX planeRecall the planes for resistance (R) and inductive reactance (XL) from chapter 13. Thisis the same as the upper-right quadrant of the complex-number plane shown in Fig.15-2.Similarly, the plane for resistance and capacitive reactance (XC) is the same as thelower-right quadrant of the complex number plane.Resistances are represented by nonnegative real numbers. Reactances, whetherthey are inductive (positive) or capacitive (negative), correspond to imaginary num-bers.No negative resistanceThere is no such thing, strictly speaking, as negative resistance. That is to say, one can-not have anything better than a perfect conductor. In some cases, a supply of direct cur-rent, such as a battery, can be treated as a negative resistance; in other cases, you canhave a device that acts as if its resistance were negative under certain changing condi-tions. But generally, in the RX (resistance-reactance) plane, the resistance value is al-ways positive. This means that you can remove the negative axis, along with theupper-left and lower-left quadrants, of the complex-number plane, obtaining a halfplane as shown in Fig. 15-5.Reactance in generalNow you should get a better idea of why capacitive reactance, XC, is considered negative.In a sense, it is an extension of inductive reactance, XL, into the realm of negatives, in aThe RX plane269