It goes about like this:

Now an alternating voltage or current is one which not only reverses its direction periodically, but also is one in which the value of the voltage or current is continually changing. Because of this continual change, energy will at times be stored in the magnetic field and shortly thereafter returned to the circuit, if the circuit contains inductance. Similarly with the electric field and capacitance. All of the energy stored during one part of a cycle is returned by the time the cycle is over.

Apparent Power

In other words, inductance and capacitance take energy (or power) from the power source only to hand all of it back again. A "pure" inductance or capacitance (i.e., without associated resistance) uses no power. Nevertheless, current does flow in the circuit when voltage is applied. If we multiply the voltage by the current, the same as we do to find power in dc circuits, we get a number which seems to represent power. It only seems to do so, because no real work is done unless there is resistance. This power is called apparent power or wattless power. To distinguish it from real power a different unit is used -- a volt-ampere. One volt-ampere is the same as one watt -- except that it doesn't do any work, while a real watt does.

You are undoubtedly curious as to how it is that there can be voltage and current but no power. A detailed examination of what goes on in the circuit is beyond the scope of this book. Briefly, however, it is a matter of timing (for which the technical term is phase). The voltage and current don't pull together, as they do in a simple resistance. When one is big the other is likely to be small; or, even, when the polarity of the voltage is positive the current may be negative -- that is, flowing in the "wrong" direction. It's something like a tug-of-war in which two teams expend a lot of effort in pulling each other back and forth without making any net progress one way or the other.

Inductive Reactance

We said earlier that the more rapidly the current changes, the larger the opposing voltage generated in an inductance. A high-frequency alternating current changes more rapidly than a low-frequency one, since there are more cycles per second. Thus the higher the frequency and the larger the inductance, the harder it is for current to flow through the inductance; it meets more opposition. The measure of this opposition is called inductive reactance. It is something like the opposition that resistance offers to current flow, and so the unit of reactance is also named the ohm.

Like the wattless watt, though, it is an ohm without resistance. It does act like a real ohm to this extent: Given a fixed frequency, the current through it will be directly proportional to the voltage applied. In other words, we can write for reactance the equivalent of Ohm's Law for resistance:

where X stands for reactance. But for a given value of inductance, reactance increases with the frequency, so it is not a constant like resistance is -- unless we specify that the frequency stays constant.

Capacitive Reactance

A capacitor acts in just the opposite way. The more rapidly the applied voltage changes in value, the faster the capacitor stores energy. This means that a high-frequency alternating voltage will put more current into a given capacitor than a low frequency voltage could. Thus the reactance of a capacitor goes down as the frequency increases. Nevertheless, the same formula applies if the frequency stays constant. All we have to remember is that X gets smaller as the capacitance is made larger, and that it also gets smaller as the frequency is made larger.

To distinguish inductive from capacitive reactance the former is usually designated X_L and the latter X_C. Just plain X can mean either one or a combination of both. In the form of equations, the ideas expressed above in words result in

and

In these formulas, f is the frequency, L the inductance, and C the capacitance. The proper units have to be used. * We won't attempt to explain the factor 2p here because that's a whole topic in itself, and is chiefly of mathematical interest.

* The most convenient units in amateur work are megaherts for frequency, microhenrys for inductance, and micromicro farads (also called picofarads) for capacitance. With these units the formulas are and in which 6.28 is the approximate value of 2p . The reactance as given by these formulas is in ohms in both cases.

Reactances Combined

The "oppositeness" of inductive and capacitive reactance has another important effect. When a coil and capacitor are connected in series in a circuit, one tends to undo what the other is trying to do. This is quite different from placing two resistances in series. The resistances both act the same way, and the total resistance is the sum of the two. But if we put inductive and capacitive reactance in series the total reactance is the difference. Conventionally, capacitive reactance is called "negative" and inductive reactance is called "positive." Thus a series circuit might have an inductive reactance of "plus" 15 ohms and a capacitive reactance of "minus" 10 ohms; the total reactance would be only 5 ohms (15 - 10) in that case.

However, reactances of the same kind add up just as resistors do. That is, an inductive reactance of 15 ohms placed in series with one of 8 ohms will result in a total of 23 ohms. The same would be true of two capacitive reactances of these same values, except that the sign would be negative.

Also, reactances of the same kind connected in parallel are combined by the same rules that we use for resistances. Not so with reactances of opposite kind in parallel! Things begin to get complicated in that case -- too much so to be considered in this book, except for one special case, the resonant circuit.

Resonant Circuits

Since the reactance of an inductance goes up when we increase the frequency, while the reactance of a capacitance goes down, it is reasonable to expect that at some frequency the reactances of a given inductance and capacitance will be equal. This is so. The frequency at which it happens is called the resonant frequency of the combination.

We're rarely able to ignore resonance in radio frequency circuits. It's important because at the resonant frequency the inductive reactance is balanced out by the capacitive reactance. This leaves us with only resistance operating in the circuit. There's more to it than just cancellation of reactive effects, though, as we shall see.

FIG. 2-7

Fig. 2-7 illustrates how reactance changes with frequency. In making up this graph we have chosen 5 pH for the inductance and 50 pF for the capacitance. The scale chosen lets us show a large range of values, of both frequency and reactance, with constant percentage accuracy at any point on curves drawn on it. The X_L curve shows that the reactance goes from about 30 ohms at 1 MHz to over 3000 ohms at 100 MHz. The capacitive reactance does just the opposite -- goes from 3000 ohms at 1 MHz to a little over 30 ohms at 100 MHz. Other values of inductance and capacitance would have different actual values, but would behave similarly; the ones picked just happen to be convenient for study.

The striking thing here is that these two curves cross each other at close to 10 MHz. At this frequency their reactances are equal numerically. In other words, the combination of 5 m H and 50 pF is resonant at 10 MHz. Remember that this is only one such combination, picked out simply to show graphically how resonance occurs. Theoretically there is an infinite number of combinations that will resonate at any given frequency. practically, however, we are confined to certain ranges of inductance and capacitance, because of constructional limitations of actual coils and capacitors.

Series Resonance

With respect to the circuit in which they are used, the coil and capacitor may be connected either in series or parallel. This is shown in Fig. 2-8. If we have a source of voltage, E, at the resonant frequency and the two are in series, the current is the same all around the circuit. By the Ohm's Law equations for reactance which we gave earlier, the current will cause a voltage to exist across each reactance. These voltages have been labeled E_L and E_C in the series circuit. Strange as it may seem, they can be many, many times larger than the source voltage, E. In fact, they are usually at least ten times as large, and may be as much as a few hundred times as large.

FIG. 2-8 -- Series and parallel circuits formed by inductive and Capacitive reactances, together with a source of voltage, E.

This can happen because the two reactances cancel each other's effects, since they are equal at resonance. Thus around the circuit there is zero reactance. There is nothing, then, to limit the flow of current except the resistance in the circuit. Although we haven't shown any resistance in Fig. 2-8, there is always some, because no components operate without at least a little power loss. Also, the source of voltage will have an internal resistance. But if the total resistance is small, the current will be large, by Ohm's Law. And a large current will result in large, but equal, voltage drops across each reactance. The time or phase of these voltages is such that the voltage across L has its positive maximum at the same instant that the voltage across C has its negative maximum. The two voltages always add up just to zero.

We can look at this another way, which may make it seem more reasonable: These unusually large voltages can develop because of energy stored in the reactances. The energy going into the magnetic field of the inductance is energy coming out of the electric field of the capacitor, during one part of a cycle. Then when it has all been stored in the magnetic field, it starts coming back into the circuit and goes into the electric field. This means that a lot of energy can be handed back and forth between L and C without making it necessary for the source to supply any. Of course, the energy "bank account" came from the source. But after an initial surge, the source has only to supply the actual power used up in resistance.

Parallel Resonance

We have a different, but comparable, state of affairs when L and C are connected in parallel-. Here there are two current paths, with the same voltage, E, applied to both. The two branch currents, I_C and I_L, each depend only on the same Ohm's Law formula for a reactive circuit. If the reactance is small and the voltage E is large, each branch current will be large. But the same voltage is applied to both reactances (in the series circuit both had the same current). So in this case it is the currents that add up to zero around the circuit. Their phase is such that they cancel each other, in the part of the circuit outside the coil and capacitor. In this case, then, there is no current flowing around the circuit as a whole.

A parallel-resonant circuit "looks like" an open circuit to the source of voltage. Compare this with the series circuit, which "looked like" a short circuit. The reason for this behavior of the parallel circuit is the same as in the series case -- stored energy is tossed back and forth between the inductance and capacitance.

These ideas of "short circuit" and "open circuit" must be taken with caution. They would be literally true if we could have coils and capacitors without any losses. But these components always do have losses. If the losses are very small the series circuit is approximately a short circuit, and the parallel circuit is approximately an open circuit. Losses mean that the voltages in the series circuit don't quite balance each other, and the currents in the parallel circuit don't quite cancel each other. Some of the energy is lost each time it is handed back and forth. This lost energy has to be supplied continuously by the source, in order to keep things going at an even rate. The source "sees" a resistance, therefore -- a very small one in the case of the series circuit, and a very large one in the case of the parallel circuit.

Impedance

If you've digested what has gone before, you're ready to tackle impedance, a word that gets a lot of bandying around in amateur conversations. Its basic definition is simple, but the details are far from being so. In fact, we can't hope to do more than give you a speaking acquaintance with some of them in this book. The inner workings of actual circuits really belong in the field of engineering rather than hobbying. Fortunately, you don't need to know them in order to build and operate amateur equipment.

FIG. 2-9 -- Simple measurements of voltage and current don't give a clue to what the unknown impedance may actually be.

In broad terms, impedance is a number you get by dividing the voltage applied to a circuit by the current flowing into it. Resorting to the black box again (Fig. 2-9), suppose we measure the current I flowing into the box and find it to be 1/2 ampere when the applied ac voltage E is 250 volts. Dividing 250 by 1/2 gives 500 as the answer. Although this is not a dc circuit we say that we have 500 "ohms," since the ohm got established as a unit representing the ratio of voltage to current (that is, E/I) in dc work. But we don't know what's in the box, so we can't say that these "ohms" are either resistance or reactance. It would take more than a simple measurement of current or voltage to determine that, because -- as, we have seen -- there is an element of timing or phase that has to be taken into account. The ammeter and voltmeter don't give any information about phase.

The fact is that we could get the same answer whether we had 500 ohms of "pure" resistance or "pure" reactance. But we could also get the same answer if the box contained 500 ohms of something that was a combination of both. Such a combination not only can exist but actually is likely to be more common than either alone. The ohms, then, in this case could be either resistance or reactance, but more probably would be impedance, the name for the combination.

You can see why things start to get complicated at this point. In pure reactance they are out of step by exactly one-quarter of a cycle. These are both very special cases. In a complex impedance one made up of both resistance and reactance the current and voltage may be out of step in any degree between zero and one quarter cycle. The number of possible cases is infinite.

Resistive Impedance

It happens that in rf work we are concerned mostly with resonant circuits. These, as we have explained, "look like" pure resistances when they are exactly tuned to the frequency. Off the exact resonant frequency the resistance is no longer pure, but in many cases the adulteration of resistance by reactance isn't too great. It is customary, therefore, to use the term impedance rather loosely to mean a resistive impedance. A resistive impedance is one in which the resistive effects far override the reactive effects -- enough so that the latter can be neglected for practical purposes.

You will see references to impedance in amateur publications, with occasional rules and formulas for one or another special case. Don't fall into the trap of thinking that these rules and formulas are wholly accurate. In most cases they won't be. They are nearly always approximations based on assuming that something can be neglected. This is done to simplify them. Property used, such rules and formulas can be highly useful in practical work. Occasionally, though, they are misused through ignorance of their hidden limitations. Be cautious, therefore, about applying them to cases other than the one they actually are intended to satisfy.