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* See note 15 in Appendix.
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195
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Fig. 129
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" where the notation !L I 908) is used to indicate that the voltage drop across the inductor leads the current by 90 degrees. We learned, in section 6.8, that COMPLEX NUMBERS obey the same parallelogram law as do vector quantities. Hence, as far as ALGEBRAIC OPERATIONS are " " concerned, we can regard V and I , in eq. (195), as being complex numbers of the forms " " V V 0 jV 00 and I I 0 jI 00 , where the primes indicate the real and imaginary " " parts of V and I . However, before applying this concept to eq. (195), it should be noted that multiplying a complex number by j has the SOLE EFFECT of rotating the complex " " number through an angle of +90 degrees; thus, if Z is a complex number, then j Z has the " same magnitude as Z but is rotated through 908.* " " Hence, in eq. (195), let I and V be represented as complex numbers in the complex " plane; thus, to indicate that the angular position of !LI is to be increased by 90 degrees, all " " we need do is write j!LI in place of !L I 908); thus " " " RI j!LI V " and thus, upon solving for I , we have " I " V R j!L 197 196
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" " in which I is the rms vector in the BASIC SERIES RL CIRCUIT of Fig. 127, where V " to be the is the applied sinusoidal voltage vector. In our work we ll generally take V " " reference vector, in which case V V =08 V, a real number (the magnitude of V ). In the above equation, the denominator is called the IMPEDANCE of the circuit, which, for the case of Fig. 127, is a measure of the combined e ect of the resistance R and the inductive reactance !L: " Impedance is denoted by Z ; thus, the impedance of the basic series RL circuit of Fig. 127 is " Z R j!L showing that impedance is a complex number; thus eq. (197) becomes " " V I " Z which is OHM s LAW, in complex form, for the sinusoidal steady-state condition.
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* See note 16 in Appendix.
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In regard to the above, we will sometimes wish to deal only with the magnitudes of the complex numbers. To do this, all we need remember is that* (a) (b) the MAGNITUDE of the PRODUCT of complex numbers is equal to the PRODUCTS OF THE MAGNITUDES, and the MAGNITUDE of the QUOTIENT of two complex numbers is equal to the QUOTIENT OF THE TWO MAGNITUDES.
Hence, applying these rules to eq. (199) we have that " " jV j " " " jI j " and jI j jZ j jV j jZ j
200
" Next, the vector diagram representation for Z R j!L (eq. (198)) is shown in Fig. 130.
Fig. 130
Comparison of Fig. 130 with Fig. 129 shows that  in Fig. 130, is the phase angle between the voltage and current vectors shown in Fig. 129. Figure 130 is spoken of as an impedance triangle and shows, for the series RL circuit of Fig. 127, that " " 201 Z jZ j= where, from Fig. 130, " jZ j and  arctan !L lagging R 203 p R2 !2 L2 202
Problem 116 In the series RL circuit of Fig. 127, let the sinusoidal reference voltage be 115 volts rms, the resistance be 28 ohms, and the inductance be 0.12 henry. If the frequency is 60 Hz, nd (a) magnitude of rms current, (b) phase angle of current, (c) reading of voltmeter placed across L. (Answer: 2.1615 amperes) (Answer: 58.2458 lagging)
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