Practical_Antenna_Handbook_0071639586

C h a p t e r 3 : A n t e n n a B a s i c s 85
Note Both R and X are functions of frequency, and their values depend in general on the
values of all components in the circuit. Thus, Z IN for an arbitrary circuit may be 50 + j47 W
(ohms) at one frequency and 35 – j17 W at another frequency!
• Resonance. If a simple series circuit contains a resistor, an inductor, and a
capacitor all connected end to end, a meter across the two end terminals would
“see” an input impedance at some frequency f of
Z = R + jX + jX
IN
L
( )
= R + j X + X
L
C
C
(3.9)
Since X C is always a negative number, this amounts to subtracting capacitive
reactance from inductive reactance. In other words, looking into this simple
network we can only tell what the difference between the inductive and capacitive
reactance is. Another way of saying this is that we can see only the net reactance.
It turns out that we can represent the two-terminal impedance of any
configuration of R, L, and C at a single frequency with Eq. (3.9) as long as we
understand that R, L, and C in the equation are functions of frequency and
generally bear absolutely no relationship to specific resistors, inductors, or
capacitors in the circuit.
A circuit is said to be resonant or in resonance at some frequency f when
X C = –X L so that the second (or “j”) term in Eq. (3.9) goes to zero. When that
occurs,
1
− = −2 πfL
(3.10)
2 π fC
Dividing both sides by –1 and multiplying both sides by 2πfC leads to
( f )
1 = 2π
= ( 2π)
2
2 2
LC
f LC
(3.11)
Swapping sides and rearranging a few factors produces
f
2
1
=
(3.12)
2
( 2π)
LC
or
⎛ 1 ⎞
f = ⎜ ⎟
⎝ 2π
LC ⎠
(3.13)
Equation (3.13) is one of the most important equations of all radio and
electronics design; we will have occasion to use this relationship at numerous
points in this book, such as when we discuss the use of parallel-resonant
lumped-component traps in shortened dipoles and Yagi beams.