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Susceptance
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Sometimes, you ll come across the term susceptance in reference to an ac circuit containing a capacitive reactance or an inductive reactance. Susceptance is symbolized by the capital letter B. It is the reciprocal of reactance. That is, B 1/X. Susceptance can be either capacitive or inductive. These are symbolized as BC and BL respectively. Therefore, BC 1/XC, and BL 1/XL. There is a trick to determining susceptances in terms of reactances. Or, perhaps better stated, a trickiness. Susceptance is imaginary, just as is reactance. That is, all values of B require the use of the j operator, just as do all values of X. But 1/j j. This reverses the sign when you find susceptance in terms of reactance. If you have an inductive reactance of, say, 2 ohms, then this is expressed as j2 in the imaginary sense. What is 1/(j2) You can break this apart and say that 1/(j2) (1/j)(1/2) (1/j)0.5. But what is 1/j Without making this into a mathematical treatise, suffice it to say that 1/j j. Therefore, the reciprocal of j2 is j0.5. Inductive susceptance is negative imaginary. If you have a capacitive reactance XC 10 ohms, then this is expressed as XC j10. The reciprocal of this is BC 1/( j10) (1/ j)(1/10) (1/ j)0.1. What is 1/ j Again, without going into deep theoretical math, it is equal to j. Therefore, the reciprocal of j10 is j0.1. Capacitive susceptance is positive imaginary. This is exactly reversed from the situation with reactances.
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Suppose you have a capacitor of 100 pF at a frequency of 3.00 MHz. What is BC First, find the reactance XC by the formula XC 1/(6.28fC)
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276 Impedance and admittance Remembering that 100 pF 0. 000100 F, you can substitute in this formula for f 3.00 and C 0.000100, getting XC 1/(6.28 3.00 0.000100) 1/0.001884 531 j531
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The susceptance, BC, is equal to l/XC. Thus, BC 1/( j531) j0.00188. Remember that capacitive susceptance is positive. This can short-circuit any frustration you might have in manipulating the minus signs in these calculations. Note that above, you found a reciprocal of a reciprocal. You did something and then immediately turned around and undid it, slipping a minus sign in because of the idiosyncrasies of that little j operator. In the future, you can save work by remembering that the formula for capacitive susceptance simplified, is BC 6.28fC siemens j(6.28fC)
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This resembles the formula for inductive reactance.
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An inductor has L 163 H at a frequency of 887 kHz. What is BL First, calculate XL, the inductive reactance XL 6.28fL 908 6.28 j908 0.887 163
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1/j908 j0. 00110. RememThe susceptance, BL is equal to 1/XL Therefore, BL ber that inductive susceptance is negative. The formula for inductive susceptance is similar to that for capacitive reactance: BL 1/(6.28fl) siemens j(1/(6.28fL)
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Admittance
Conductance and susceptance combine to form admittance, symbolized by the capital letter Y. Admittance, in an ac circuit, is analogous to conductance in a dc circuit.
Complex admittance
Admittance is a complex quantity and represents the ease with which current can flow in an ac circuit. As the absolute value of impedance gets larger, the absolute value of 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 j0.5) mean that there is a net inductance in the circuit, and positive j factors (such as j3) mean there is net capacitance.
The GB plane 277 Admittance is the complex composite of conductance and susceptance. Thus, admittance takes the form Y G jB G jB.
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