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.
86 P a r t I I : F u n d a m e n t a l s V s + _ I s • Sources. Electrical and electronic circuits (including antennas and transmission lines) are made operational by being supplied with electrical charge. Devices designed for this specific purpose are called power supplies, batteries, or Battery sources. A battery, for instance, is nothing but a supplier, or source, of readily accessible electric charge resulting from a carefully designed internal chemical reaction. Depending on the exact circuit or antenna we are examining, we may prefer to power it with a voltage source or a current source. These are idealized representations of practical batteries and power supplies. An ideal voltage source delivers a specified voltage to a pair of terminals on a circuit or antenna regardless of the current through the source. Similarly, an ideal current source Voltage Source passes a specified current between two terminals regardless of the resulting voltage across the source. In real life, no source is ideal but most are good enough that these approximations are valid for the range of practical currents and voltages we will deal with in this book. Unlike batteries and most power supplies, sources can supply alternating (ac) voltages and currents. This is especially important Current for Source radio frequency (RF) systems. Antennas, in particular, have value only when attached to a source of RF energy. For virtually all the real-world antennas and RF circuits we will be considering in this book, the fundamental unit values (farads and henries, respectively) for capacitors and inductors are much too large. Thus, you will see reference to millihenries, microhenries, and nanohenries, and to microfarads, nanofarads, and picofarads, where these prefixes have the following meanings: milli m 10 –3 or 0.001 times micro m 10 –6 or 0.000 001 times nano n 10 –9 or 0.000 000 001 times pico p 10 –12 or 0.000 000 000 001 times Similarly, often the fundamental unit of resistance (the ohm) is too small and we will find it more convenient to speak in terms of kilohms or megohms: kilo k 10 3 or 1 000 times mega M 10 6 or 1 000 000 times Example 3.1 In commonly accepted terminology, a 0.001-mF capacitor is spoken of as a “double ought one”. The “microfarad” portion is implied. A 0.001-mF capacitor is the same as a 1-nF (nanofarad) or 1000-pF (picofarad) capacitor (obtained by moving the decimal point either three or six places to the right, respectively).
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86 P a r t I I : F u n d a m e n t a l s<br />
V s<br />
+<br />
_<br />
I s<br />
• Sources. Electrical and electronic circuits (including antennas and transmission<br />
lines) are made operational by being supplied with electrical charge. Devices<br />
designed for this specific purpose are called power supplies, batteries, or<br />
Battery<br />
sources. A battery, for instance, is nothing but a supplier, or source, of readily<br />
accessible electric charge resulting from a carefully designed internal chemical<br />
reaction. Depending on the exact circuit or antenna we are examining, we may<br />
prefer to power it with a voltage source or a current source. These are idealized<br />
representations of practical batteries and power supplies. An ideal voltage<br />
source delivers a specified voltage to a pair of terminals on a circuit or antenna<br />
regardless of the current through the source. Similarly, an ideal current source<br />
Voltage Source<br />
passes a specified current between two terminals regardless of the resulting<br />
voltage across the source. In real life, no source is ideal but most are good<br />
enough that these approximations are valid for the range of practical currents<br />
and voltages we will deal with in this book. Unlike batteries and most power<br />
supplies, sources can supply alternating (ac) voltages and currents. This is<br />
especially important Current for Source radio frequency (RF) systems. <strong>Antenna</strong>s, in particular,<br />
have value only when attached to a source of RF energy.<br />
For virtually all the real-world antennas and RF circuits we will be considering in<br />
this book, the fundamental unit values (farads and henries, respectively) for capacitors<br />
and inductors are much too large. Thus, you will see reference to millihenries, microhenries,<br />
and nanohenries, and to microfarads, nanofarads, and picofarads, where these<br />
prefixes have the following meanings:<br />
milli m 10 –3 or 0.001 times<br />
micro m 10 –6 or 0.000 001 times<br />
nano n 10 –9 or 0.000 000 001 times<br />
pico p 10 –12 or 0.000 000 000 001 times<br />
Similarly, often the fundamental unit of resistance (the ohm) is too small and we<br />
will find it more convenient to speak in terms of kilohms or megohms:<br />
kilo k 10 3 or 1 000 times<br />
mega M 10 6 or 1 000 000 times<br />
Example 3.1 In commonly accepted terminology, a 0.001-mF capacitor is spoken of as a<br />
“double ought one”. The “microfarad” portion is implied. A 0.001-mF capacitor is the<br />
same as a 1-nF (nanofarad) or 1000-pF (picofarad) capacitor (obtained by moving the<br />
decimal point either three or six places to the right, respectively).
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