Practical_Antenna_Handbook_0071639586

84 P a r t I I : F u n d a m e n t a l s
= × ∆ I
V L
∆t
(3.5)
where V (in volts) is the voltage across the inductor of value L when the current
through the inductor changes by some amount DI (in amperes) over a short
time interval Dt (in seconds). A simple inductor you can make at home is formed
from a length of insulated wire (such as the wire left over from the installation
of your new garage door opener) wrapped around your hand a few times.
Note The only time a voltage can exist across the two terminals of an inductor is when the
current through the inductor is changing: V = L DI/Dt.
As discussed for capacitors, an ideal inductor does not dissipate power, nor
are the current through, and the voltage across, the inductor in phase. The
magnitude of inductive reactance is
XL = 2πfL
(3.6)
where f is in hertz and L is in henries (or megahertz and microhenries). Note
that inductive reactance X L is defined as positive, in contrast to capacitive
reactance in Eq. (3.4).
• Complex impedance. Suppose we apply a sinusoidal wave of frequency f across
two terminals or nodes in a circuit consisting of interconnected resistors, inductors,
and capacitors. If we measure, with an oscilloscope or other appropriate
piece of test equipment, the voltage and current relationship at those terminals,
we will generally find that while the current also varies sinusoidally at frequency
f, it is not exactly in phase with the voltage as it would be if the circuit consisted
of only resistors. The capacitors and inductors in the circuit cause phase shifts of
varying degrees between currents and voltages in different parts of the circuit.
At any one frequency we can describe the relationship between the voltage
and the current at the two measurement terminals in terms of a single value of
resistance in series with a single value of reactance. In equation form, we say
ZIN = RIN + jX
(3.7)
IN
or, more simply,
Z = R + jX (3.8)
This is known as a complex impedance because of the use of the complex plane
mathematical operator j ( j = − 1 ) to represent reactances. The magnitude of X
may be either positive or negative, depending on whether the net reactance
seen appears to be inductive or capacitive, respectively. For correctness, we
speak of R as the resistive part and X as the reactive part of the impedance Z. Both
R and X are in ohms or multiples thereof.