Practical_Antenna_Handbook_0071639586

A p p e n d i x A : U s e f u l M a t h 725
serve that the current and voltage through either of these families of components are in
quadrature—that is, they are 90 degrees out of phase with each other. We say that capacitors
and inductors are reactive elements, and it turns out to be very helpful to represent
the reactive component of a signal or a circuit parameter such as impedance as
imaginary, and to plot it on the y, or imaginary, axis. The real part of a signal or circuit
parameter is plotted along the x axis.
Throughout this book we will find many antenna and transmission line impedances
expressed in the form
Z = IN
R + jX
(A.6.4)
Sometimes, however, we need the input admittance, Y IN , corresponding to that impedance.
Specifically,
Y
IN
1 1
= =
Z R + jX
IN
(A.6.5)
This often turns out to be very inconvenient to work with because we can no longer
easily see how to break the complex admittance into its real part (conductance) and a
separate imaginary part (susceptance). However, a little math sleight of hand will help
us. If we multiply an equation by 1, we don’t change the equation. Furthermore, 1 can
be any messy number form divided by itself; for instance:
A − jB
1 =
A − jB
So let’s multiply both the numerator and denominator of Eq. (A.6.5) by the term
(R – jX):
1 1 R jX
× 1 = × −
R + jX R + jX R − jX
R − jX
=
( R + jX)( R − jX)
R − jX
=
( R 2 + jXR − jXR − j 2 X
2 )
R − jX
=
2 2
R + X
In other words, for an impedance of the form Z IN = R + jX,
Y
R − jX
=
R + X
R
X
= (real part) − j (imaginary part)
2 2 2 2
R + X
R + X
= G (conductance) + jB (susceptance)
IN 2 2
(A.6.6)