Atoms

All matter is made up of countless tiny particles whizzing around. These particles are extremely dense; matter is mostly empty space. Matter seems continuous because the particles are so small, and they move incredibly fast.

Each chemical element has its own unique type of particle, known as its atom. Atoms of differ- ent elements are always different. The slightest change in an atom can make a tremendous differ- ence in its behavior. You can live by breathing pure oxygen, but you can’t live off of pure nitrogen. Oxygen will cause metal to corrode, but nitrogen will not. Wood will burn furiously in an atmos- phere of pure oxygen, but will not even ignite in pure nitrogen. Yet both are gases at room temper- ature and pressure; both are colorless, both are odorless, and both are just about of equal weight. These substances are so different because oxygen has eight protons, while nitrogen has only seven. There are many other examples in nature where a tiny change in atomic structure makes a major dif- ference in the way a substance behaves.

The part of an atom that gives an element its identity is the nucleus. It is made up of two kinds of particles, the proton and the neutron.These are extremely dense. A teaspoonful of either of these par- ticles, packed tightly together, would weigh tons. Protons and neutrons have just about the same mass, but the proton has an electric charge while the neutron does not.

Isotopes and Atomic Weights

For a given element, such as oxygen, the number of neutrons can vary. But no matter what the num- ber of neutrons, the element keeps its identity, based on the atomic number. Differing numbers of neutrons result in various isotopes for a given element.

Each element has one particular isotope that is most often found in nature. But all elements have numerous isotopes. Changing the number of neutrons in an element’s nucleus results in a dif- ference in the weight, and also a difference in the density, of the element. Thus, hydrogen contain- ing a neutron or two in the nucleus, along with the proton, is called heavy hydrogen. The atomic weight of an element is approximately equal to the sum of the number of protons and the number of neutrons in the nucleus. Common carbon has an atomic weight of about 12, and is called carbon 12 or C12. But sometimes it has an atomic weight of about 14, and is known as car- bon 14 or C14.

Imaginary Numbers

Have you been wondering what j actually means in expressions of impedance? Well, j is nothing but a number: the positive square root of −1. There’s a negative square root of −1, too, and it is equal to −j. When either j or −j is multiplied by itself, the result is −1. (Pure mathematicians often denote these same numbers as i or −i.) The positive square root of −1 is known as the unit imaginary number. The set of imaginary numbers is composed of real-number multiples of j or −j. Some examples are j4, j35.79, −j25.76, and −j25,000. The square of an imaginary number is always negative. Some people have trouble grasping this, but when you think long and hard about it, all numbers are abstractions. Imaginary numbers are no more imaginary (and no less real) than so-called real numbers such as 4, 35.79, −25.76, or −25,000. The unit imaginary number j can be multiplied by any real number on a conventional real number line. If you do this for all the real numbers on the real number line, you get an imag- inary number line (Fig. 15-1). The imaginary number line should be oriented at a right angle to the real number line when you want to graphically portray real and imaginary numbers at the same time. In electronics, real numbers represent resistances. Imaginary numbers represent reactances.

IAn Example of Zo in Practice

In rigorous terms, the ideal characteristic impedance for a transmission line is determined accord- ing to the nature of the load with which the line works. For a system having a purely resistive impedance of a certain number of ohms, the best line Zo value is that same number of ohms. If the load impedance is much different from the characteristic impedance of the transmission line, excessive power is wasted in heating up the transmission line. Imagine that you have a so-called 300-Ω frequency-modulation (FM) receiving antenna, such as the folded-dipole type that you can mount indoors. Suppose that you want the best possible re- ception. Of course, you should choose a good location for the antenna. You should make sure that the transmission line between your radio and the antenna is as short as possible. But you should also be sure that you purchase 300-Ω TV ribbon. It has a value of Zo that has been optimized for use with antennas whose impedances are close to 300 + j0. Characteristic Impedance 237 15-8 Edge-on views of coaxial transmission line (A) and parallel-wire line (B). In either type of line, Zo depends on the conductor diameters and spacing, and on the nature of the dielectric material between the conductors. See text for discussion. Impedance matching is the process of making sure that the impedance of a load (such as an an- tenna) is purely resistive, with an ohmic value equal to the characteristic impedance of the transmis- sion line connected to it. This concept will be discussed in more detail in the next chapter.

Admittance

Real-number conductance and imaginary-number susceptance combine to form complex admit- tance, symbolized by the capital letter Y. This is a complete expression of the extent to which a cir- cuit allows ac to flow. As the absolute value of complex impedance gets larger, the absolute value of complex admit- tance becomes smaller, in general. Huge impedances correspond to tiny admittances, and vice versa. Admittances are written in complex form just like impedances. But you need to keep track of which quantity you’re talking about! This will be obvious if you use the symbol, such as Y = 3 − j0.5 or Y = 7 + j3. When you see Y instead of Z, you know that negative j factors (such as in the quan- tity 3 − j0.5) mean there is a net inductance in the circuit, and positive j factors (such as in the quantity 7 + j3) mean there is net capacitance. Admittance is the complex composite of conductance and susceptance. Thus, complex admit- tance values always take the form Y = G + jB. When the j factor is negative, a complex admittance may appear in the form Y = G − jB. Do you remember how resistances combine with reactances in series to form complex imped- ances? In Chaps. 13 and 14, you saw series RL and RC circuits. Did you wonder why parallel cir- cuits were ignored in those discussions? The reason was the fact that admittance, not impedance, is best for working with parallel ac circuits. Resistance and reactance combine in a messy fashion in parallel circuits. But conductance (G ) and susceptance (B ) merely add together in parallel circuits, yielding admittance (Y ). Parallel circuit analysis is covered in detail in the next chapter.

Admittance can be depicted on a plane similar to the complex impedance (RX ) plane. Actually, it’s a half plane, because there is ordinarily no such thing as negative conductance. (You can’t have a component that conducts worse than not at all.) Conductance is plotted along the horizontal, or G, axis on this coordinate half plane, and susceptance is plotted along the B axis. The GB plane is shown in Fig. 15-9, with several points plotted. It’s Inside Out The GB plane looks superficially identical to the RX plane. But mathematically, the two could not be more different! The GB plane is mathematically inside out with respect to the RX plane. The center, or origin, of the GB plane represents the point at which there is no conduction for dc or for ac. It is the zero-admittance point, rather than the zero-impedance point. In the RX plane, the ori- gin represents a perfect short circuit, but in the GB plane, the origin corresponds to a perfect open circuit. As you move out toward the right (east) along the G, or conductance, axis of the GB plane, the conductance improves, and the current gets greater. When you move upward (north) along the jB axis from the origin, you have ever-increasing positive (capacitive) susceptance. When you go down (south) along the jB axis from the origin, you encounter increasingly negative (inductive) susceptance.

Pure Reactances

Pure inductive reactances (XL ) and capacitive reactances (XC) simply add together when coils and capacitors are in series. Thus, X = XL + X C. In the RX plane, their vectors add, but because these vec- tors point in exactly opposite directions—inductive reactance upward and capacitive reactance downward (Fig. 16-1)—the resultant sum vector inevitably points either straight up or straight down, unless the reactances are equal and opposite, in which case they cancel and the result is the zero vector.

Aurther

Stan Gibilisco

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