By G. I. Atabekov and P. K. M'Pherson (Auth.)
ISBN-10: 0080134831
ISBN-13: 9780080134833
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Example text
Thus, when complex numbers are being added or subtracted the algebraic (or trigonometrical) form of writing a complex number is used: A + Β = (Α! +B1)+j(A2 A-B**^ + B2), ) -BJ+Mz-Bz). 39) In a geometrical interpretation for obtaining a vector representing the sum or the difference of complex numbers, vectors representing these numbers should be added or subtracted according to the rule of operations governing the vectors (Fig. 13). 40) A- = ± B B J(a ß) e - = ΑΔα-β. 41) B As can be seen from (40) and (41), the modulus of the product is equal to the product of the moduli of the factors, and the argument of the product is equal to the sum of their arguments; the modulus of the quotient APPLICATIONS OF COMPLEX QUANTITIES 43 is equal to the quotient of the moduli of the dividend and divisor, and the argument of the quotient is equal to the difference of their arguments.
When x > 0; the current then lags behind the voltage (Fig. 7). e. when x < 0; the current then leads the voltage (Fig. 8). Thus, angle φ is positive for a lagging and negative for a leading current. e. when the inductive and capacitive reactances are equal. Such operating conditions in an electrical circuit are known as series resonance (Chapter VIII, Section 1). By dividing the sides of a right-angled voltage triangle by Im we obtain a right-angled impedance triangle (Figs, lb and 86) which satisfies con dition (21).
Other complex numbers not associated with sinusoids are referred to as phasors. 6. Ohm's and KirchhofFs Laws in Complex Notation Let us examine the application of the complex method to a case of r, L and C elements connected in series and in parallel. 57) OHM'S A N D KIRCHHOFF'S LAWS 47 the parameters of r, L and C and the sinusoidal voltage u = Um sin(co/ + ψ) on the terminals of the circuit are given, and current i is the value to be sought. In view of the fact that steady conditions in a circuit with a sinus oidal current are being considered here, the solution of the given differ ential equation will be found in the form of a sinusoidal function i = Im sin (cot + ψ — φ).
Linear Network Theory by G. I. Atabekov and P. K. M'Pherson (Auth.)
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