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Sometimes, you ll come across the term susceptance in reference to ac circuits. Susceptance is symbolized by the capital letter B. It is the reciprocal of reactance. Susceptance can be either capacitive or inductive. These quantities are symbolized as BC and BL, respectively. Therefore we have these two relations: BC = 1/XC BL = 1/XL All values of B theoretically contain the j operator, just as do all values of X. But when it comes to finding reciprocals of quantities containing j, things get tricky. The reciprocal of j is equal to its negative! Expressed mathematically, we have these two facts: 1/j = j 1/( j ) = j As a result of these properties of j, the sign reverses whenever you find a susceptance value in terms of a reactance value. When expressed in terms of j, inductive susceptance is negative imaginary, and capacitive susceptance is positive imaginary just the opposite situation from inductive reactance and capacitive reactance. Suppose you have an inductive reactance of 2 . This is expressed in imaginary terms as j2. To find the inductive susceptance, you must find 1/( j2). Mathematically, this expression can be converted to a real-number multiple of j in the following manner: 1/( j 2) = (1/j )(1 2) = (1/j )0.5 = j0.5 Now suppose you have a capacitive reactance of 10 . This is expressed in imaginary terms as j10. To find the capacitive susceptance, you must find 1/( j10). Here s how this can be converted to the straightforward product of j and a real number:
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1/( j10) = (1/ j )(1 10) = (1/ j)0.1 = j 0.1 When you want to find an imaginary value of susceptance in terms of an imaginary value of reactance, first take the reciprocal of the real-number part of the expression, and then multiply the result by 1.
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Problem 15-5 Suppose you have a capacitor of 100 pF at a frequency of 3.00 MHz. What is BC First, find XC by the formula for capacitive reactance:
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XC = 1/(6.28f C ) Note that 100 pF = 0.000100 F. Therefore: XC = 1/(6.28 3.00 0.000100) = 1/0.001884 = 531 The imaginary value of XC is equal to j531. The susceptance, BC, is equal to 1/X C. Thus, BC = 1/( j531) = j0.00188, rounded to three significant figures. The general formula for capacitive susceptance in siemens, in terms of frequency in hertz and capacitance in farads, is: BC = 6.28f C This formula also works for frequencies in megahertz and capacitances in microfarads.
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Problem 15-6 Suppose an inductor has L = 163 H at a frequency of 887 kHz. What is BL Note that 887 kHz = 0.887 MHz. You can calculate XL from the formula for inductive reactance:
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XL = 6.28fL = 6.28 0.887 163 = 908 The imaginary value of XL is equal to j908. The susceptance, BL = is equal to 1/XL. It follows that BL = 1/j 908 = j0.00110. The general formula for inductive susceptance in siemens, in terms of frequency in hertz and inductance in henrys, is: BL = 1/(6.28f L) This formula also works for frequencies in kilohertz and inductances in millihenrys, and for frequencies in megahertz and inductances in microhenrys.
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Real-number conductance and imaginary-number susceptance combine to form complex admittance, symbolized by the capital letter Y. This is a complete expression of the extent to which a circuit allows ac to flow. As the absolute value of complex impedance gets larger, the absolute value of complex admittance becomes smaller, in general. Huge impedances correspond to tiny admittances, and vice versa. Admittances are written in complex form just like impedances. But you need to keep track of which quantity you re talking about! This will be obvious if you use the symbol, such as Y = 3 j0.5 or Y = 7 + j3. When you see Y instead of Z, you know that negative j factors (such as in the quantity 3 j 0.5) mean there is a net inductance in the circuit, and positive j factors (such as in the quantity 7 + j3) mean there is net capacitance. Admittance is the complex composite of conductance and susceptance. Thus, complex admittance values always take the form Y = G + jB. When the j factor is negative, a complex admittance may appear in the form Y = G jB. Do you remember how resistances combine with reactances in series to form complex impedances In Chaps. 13 and 14, you saw series RL and RC circuits. Did you wonder why parallel circuits were ignored in those discussions The reason was the fact that admittance, not impedance, is best for working with parallel ac circuits. Resistance and reactance combine in a messy fashion in parallel circuits. But conductance (G ) and susceptance (B ) merely add together in parallel circuits, yielding admittance (Y ). Parallel circuit analysis is covered in detail in the next chapter.
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