Susceptance

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 Ω. 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 Ω. 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:

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.

Problem 1

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:

XC = −1/(6.28fC)

Note that 100 pF = 0.000100 μF. Therefore:
XC = −1/(6.28 × 3.00 × 0.000100)
= −1/0.001884 = −531 Ω

The imaginary value of XC is equal to −j531. The susceptance, BC, is equal to 1/XC. 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 = 6.28fC

This formula also works for frequencies in megahertz and capacitances in microfarads.

Problem 2

Suppose an inductor has L = 163 μH 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:

XL = 6.28fL
= 6.28 × 0.887 × 163
= 908 Ω

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/(6.28fL)

This formula also works for frequencies in kilohertz and inductances in millihenrys, and for frequencies in megahertz and inductances in microhenrys.