Oscillator Circuits -- INTRODUCTION

An oscillator is any non-rotating device which generates a signal at a frequency determined by the constants in the circuit.

Oscillators can be placed into almost as many categories as there are oscillators. One method of classification is by the type of wave form produced. A second is by the frequency of the waveform.

Oscillator waveforms can be broadly classified into one of two types-the sinusoidal, or harmonic; and the nonsinusoidal, or nonharmonic. A sinusoidal waveform, more familiarly known as a sine wave, is nothing more than a graphical representation of simple harmonic motion. Most oscillators in everyday life are sinusoidal, or harmonic. Examples are the pendulum of a clock, the spring of a watch, a child's swing, or the piston of a gasoline or steam engine. All the oscillations mentioned must be rein forced or replenished periodically, or they will die out because of natural losses. The pendulum of a clock, for instance, is reinforced once each cycle by a small amount of energy released at the appropriate time by the clock mechanism.

The harmonic oscillators discussed in this volume are further classified according to the frequency range within which they operate-either the audio- or radio-frequency range. These circuits are as follows:

• Radio-frequency
• Crystal oscillator
• Hartley oscillator
• Colpitts oscillator
• Tuned-plate-tuned-grid oscillator
• Electron-coupled oscillator
• Audio-frequency
• Phase-shift oscillator

Nonharmonic oscillators, also called relaxation oscillators, are discussed in this volume as follows:

• Thyratron sawtooth generator
• Blocking oscillator (pulse generator)
• Multi vibrators

The term relaxation oscillator is derived from a particular property of resistance-capacitance combinations: A capacitor is "charged" with a positive or negative voltage and then permitted to "discharge" through a resistance. This process of discharging a capacitor through a resistive path gave rise to the descriptive terms relaxing and relaxation.

There is a fundamental difference between relaxation oscillators and the harmonic oscillators mentioned previously: All harmonic oscillators must have energy fed back from output to input in order for oscillation to continue, whereas relaxation oscillators need no such energy feedback. Therefore, oscillators can also be classified as either feedback or relaxation type.

BASIC ELECTRICAL FUNDAMENTALS

All electronic circuit actions consist of free electrons moving about under controlled conditions in the three main types of components-inductors, capacitors, and resistors. Added to these three components are switching or regulating devices known as vacuum tubes (and in more recent years, solid-state devices such as diodes and transistors, which are taking over many of the functions previously performed only by tubes). The three circuit components are each closely related to certain fundamental electrical phenomena from which their names have been derived. Before these three phenomena --- inductance, capacitance, and resistance--can be understood, it will be necessary to understand the principle of electron drift. Electron drift is the foundation of all electric currents ( or more accurately, of all electron currents) --- except the electron flow within a battery and the electron flow across the evacuated space within a vacuum tube. The first is a chemical reaction, and the second is adequately described in practically every other Section of this guide and need not be belabored here.

Free Electrons and the Electron-Drift Process

In this day and age, almost every school child is aware of the atomic/molecular structure of matter. Each atom bears an astonishing resemblance to our solar system. At the center is the nucleus, where most of the atom's mass is concentrated. Orbiting around each nucleus are electrons, the number differing according to the type of material.

In our solar system the revolving planets are kept from flying off into space by the gravitational attraction between each planet and the sun, which is at the center, or nucleus, and contains most of the mass of the system. In each atom the planetary electrons are kept from flying off into space ( or into the surrounding material) by an electrical rather than a gravitational attraction.

This electrical attraction exists because each electron carries within itself one negative unit of charge. The atom as a whole is electrically neutral, but its nucleus will always contain the same number of positive charge (called protons) as there are negative charges (meaning electrons) orbiting around it. For example, an atom of material which has eight orbiting electrons will have a nucleus with a positive charge of eight units; so the two charges will cancel each other.

There are several methods of setting electrons free from their positions in orbit or, in other words, "stealing" these tiny planets from their solar systems. Heat is one method; it is used in most electronic tubes. Chemical reaction, another common method, is the one used in batteries.

There would be no point in setting electrons free from their orbits unless we had some use for them. The immensity of our requirements for free electrons in everyday life is suggested in part by the definitions of electric current and voltage.

Definitions

Electric current consists of free electrons in motion.

Free electrons in concentration constitute a negative voltage.

A concentration of positive ions, caused by a deficiency of free electrons from their planetary positions, constitutes a positive voltage.

CONDUCTORS AND RESISTORS

Some materials release electrons from their orbits with the greatest of ease; these materials are classed as conductors of electricity. Other materials hold their electrons in orbit very tightly and release them only with the utmost difficulty; these materials are classified as insulators. Examples of both types abound in everyday life. Gold, silver, copper, and various other metals and their alloys are examples of good conductors. Glass, wood, and rubber are three familiar insulating materials.

Under the influence of an external force such as an applied voltage, electrons will move through a conductor. This movement is known as electron drift. As an example, if each end of a conductive wire is connected to the two terminals of a battery, electrons will leave the negative terminal and move through the wire toward the positive terminal, pushing other electrons into the positive terminal. A continuous flow from the negative to the positive terminal will exist as long as the circuit is connected (and the battery lasts). No single electron completes the entire journey around even such a simple closed circuit as this one. Trillions upon trillions of electrons are in motion, even in a unit of current as small as one ampere. Also, the number of atoms which actually release free electrons is an infinitesimal fraction of the number of atoms within the material itself. Hence, in moving from the negative battery terminal and into the wire, each electron flows only the tiniest fraction of a millimeter before finding its path impeded by a planetary electron from another atom in the material.

Since each electron carries a like charge, the two repel each other. As a result, the planetary electron is dislodged from its orbit and starts down the wire in the same direction as-and probably ahead of-the approaching electron. The newly dislodge electron will in fact pick up some of the velocity of the other electron. The latter will be lowered down accordingly, and it then becomes an easy prey for being "recaptured" by the atom which has just lost an electron. The original electron "falls" into the orbit just vacated and again takes up planetary motion.

So we still have what we started with-an electron moving through the conductor and driven by an applied negative voltage.

The interchange just described is electron drift. It will occur countless quadrillions of times for even the tiniest measurable electron current to flow--even a micro-microampere ( one-millionth of one-millionth of an ampere!). The difference between a good conductor and a poor one is the relative ease or difficulty with which its electrons can be dislodged from their orbits and recaptured.

Resistors are aptly named, for they are inserted at certain points in the circuit to resist the flow of current. The value selected for each resistor is nothing more than a measure of the difficulty with which electron drift occurs within the resistor.

CAPACITANCE

Electron drift can also help us understand the important electrical characteristic known as capacitance, and from this to acquire a deeper appreciation of the role that capacitors play in electronic circuits. Consider again the situation as a moving electron approaches a planetary electron. There is a space between them, however small, when the planetary electron finally yields to the repulsive force of the approaching electron and jumps out of orbit. It does not really matter to the two electrons whether the space which separates them at that moment is the open space between copper atoms, air atoms . . . or the open space between atoms of any other material--either conductor or insulator.

What this suggests is that the movement of one electron can be transferred to another electron-irrespective of whether both are contained in the same conductive material, or in two different but adjacent conductive materials. As a clarifying example, suppose a conducting wire is severed between the points where the two electrons will be situated when the second one breaks away from its orbit. Suppose also that the two cut ends of the wire are separated by an infinitesimal distance, perhaps a millionth of an inch. Electrically this is an "open" circuit, through which current cannot flow in the normal sense. Nonetheless, drift action can be made to occur between the two electrons. How do we explain this paradox? We can do so by saying that the two ends of the wire have capacitance toward each other. It might be more descriptive, and also truer, to say that the two electrons have capacitance toward each other. Under these conditions, electron drift can be made to occur-provided the first electron can be made to -flow up to the break point in the wire.

This is a fairly large proviso, of course, but it is subject to a simple laboratory demonstration which is also one of the standard capacitor checks performed by all technicians. The capacitor in question is connected across a DC voltage source such as a battery, and an ammeter is placed in the line to indicate the current flow.

The instant the switch is closed to activate the circuit, the ammeter needle will deflect, indicating an initial flow of electrons (current) into the capacitor. (This current flow exists on both sides of the capacitor.) The ammeter needle quickly drops back to zero, indicating no further current flow. We say that the capacitor is "charged" to the value of the applied voltage. (If the ammeter needle remains in a deflected position-indicating some current flow-the capacitor is "leaking" and is defective. If no initial deflection occurs at all the capacitor is "open," also a defect. In either case, the capacitor should be discarded.) Fig. 1A shows the current condition for such a capacitor test the moment immediately after the switch is closed. An initial surge of charging current flows around the circuit and builds up the voltage across the capacitor, as shown in Fig. 1B, with negative electrons amassed on the upper plate. Since the electrons had to flow into the capacitor before they could become concentrated there, the current into a capacitor is said to lead ( or precede) the voltage across it.

Fig. 1. Current and voltage conditions for capacitor test. ( A) Initial flow of current from DC source into capacitor. ( B) Current has stopped, but volt-age exists across capacitor.

Fig. 2. Enlarged view of conductor with a small break or opening. ( A) Initial flow of current. (B) Condition after initial current surge. Fig. 2 shows the hypothetical example discussed previously, in which the two ends of a severed wire have capacitance toward each other, so that an initial charging current flows around the circuit but a continuous current does not.

When a capacitor plate has charged to the value of the applied voltage as in Figs. 1B and 2B, no more electrons can be driven onto the negative plate and the flow of current around the circuit stops.

Although capacitors will not "pass" a pure direct current, the initial surge of electrons into the capacitor ( or up to the break point in the wire, which resembles a capacitor) is not a smooth, steady flow and hence not a pure direct current. Capacitors will "pass" an alternating current because electrons are continuously being driven onto one plate of the capacitor (charging it), and then pulled back off (discharging it) as the applied voltage reverses. Thus, electron current can be made to flow back and forth regularly on each plate of a capacitor, and in this sense the capacitor is said to be passing an alternating current.

The ability to visualize the action of a capacitor is absolutely essential to a quick and easy understanding of electronic· circuit action. For this reason, wherever capacitors are mentioned in this guide, their action has almost always been re-described to fit the conditions for the circuit under consideration.

Capacitor Construction

A capacitor (sometimes referred to as a condenser) is a manufactured device for making use of two important electrical properties associated with capacitance. These are the ability of a capacitor to:

1. Store an electric charge, either negative electrons or positive ions.

2. Pass an alternating current from one plate to the other.

This property is frequently stated as the ability of a capacitor to oppose any change in applied voltage.

The amount of capacitance exhibited by any capacitor depends on several interrelated factors which can be tied together with the following formula: where, C = 22.35 KA (n -1) 10xd C is the amount of capacitance in microfarads, K is the dielectric constant (the dielectric is explained in a later paragraph), A is the area of one plate in square inches (it is assumed all plates are the same size and shape) , n is the number of plates, d is the distance between plates ( thickness of the dielectric, as explained a little later) . It can be seen from the formula that the amount of capacitance varies directly with the area of the capacitor plates, and inversely with the distance between them. These considerations become highly significant at radio frequencies, where unwanted capacitive coupling may occur between components and cause interference. For example, two wires passing close to each other will be capacitively coupled, the amount being determined by their diameters and by the distance between them. Also within vacuum tubes each electrode will exhibit some capacitance to the others, al though the resulting interelectrode capacitance may have advantages as well as disadvantages.

The material between the plates of a capacitor is called the dielectric. It may be air or any other insulator, and its insulating ability is known as the dielectric constant, or K. The dielectric constant of air is unity, or 1. Mica has a dielectric constant of 5.5, and ordinary glass, slightly over 4.

Of equal if not greater importance is the dielectric strength of a material, or how high a voltage it can withstand before breaking down. The dielectric strength is expressed in volts per unit of distance. For air it is about 80 volts per .001 inch. Mica is 25 times stronger, or 2,000 volts per .001 inch.

Assuming two capacitors have equal plate area and separation, this means the breakdown voltage of one with a mica dielectric will be 25 times greater than one with an air dielectric.

The dielectric strength of the material (and its thickness) determine the working-voltage rating of a capacitor. This is the rating used by the manufacturer to indicate the maximum voltage a capacitor can safely withstand without breaking down.

Capacitive Reactance

Capacitors offer a certain opposition to the passage of alternating currents. The amount varies inversely with the frequency of the current and the value of the capacitor, in accordance with this formula:

1 Xe= 2 pi fC

where, Xe is the capacitive reactance (opposition to current flow) in ohms, f is the frequency of the applied current in cycles per second, C is the capacitance in farads.

INDUCTANCE

The third important circuit characteristic is inductance, and the manufactured components most directly associated with it are called inductors, which include all coils and transformers.

We will find an endless variety of such devices, each manufactured for a specific application. As a group they cover the entire frequency spectrum, are called on to carry tiny currents or huge ones, and often must be able to withstand enormous voltages without rupturing or otherwise failing. Regardless of the conditions under which they are used or the circuit functions they are expected to fulfill, all inductors--coil, choke, transformer, etc.--take advantage of the electrical property known as inductance.

Inductance is the electrical equivalent of mechanical inertia.

Stated more simply, inductance is electrical inertia. Let us com pare it briefly with mechanical inertia.

A given mass possesses inertia, and for this reason requires a certain amount of force to set the mass in motion, or to speed it up or slow it down. Thus, a rolling ball-in the absence of friction-will continue to roll in the same direction and at the same speed. The same ball at rest will remain at rest.

The concept of electrical inertia is easy to visualize if the foregoing analogy is kept in mind. Remember that although the mass of one lone electron seems insignificant, its contribution be comes significant when we consider that there are trillions upon trillions of electrons in a single conductor! A given quantity of electrons, &wing through any conductive material, will tend to keep flowing in the same quantity. The reason is that the electrical property known as inductance will always operate to support this natural law. Some common descriptive statements which apply to inductance are:

1. An inductance always tries to keep the total current constant.

2. An inductance always opposes any change in the current.

3. In an inductance, the voltage "leads" the current; or conversely, the current "lags" the voltage.

4. Inductors generate a back electromotive force which opposes the applied voltage.

These statements can be better understood from Fig. 3. In Fig. 3A the two adjacent closed circuits are "connected" only by the mutual coupling between the two wires placed side by side in the center. Even though shown as straight wires, they can be considered as two windings of a transformer, since all conductors exhibit some inductance toward all other conductors.

The instant the switch in the left circuit is closed, a current will flow from the upper (negative) terminal of the battery, through the switch, downward through the wire we have designated as one winding of a transformer, and on around the circuit to the lower (positive) terminal. This current is shown in blue.

Fig. 3. Current flow in an inductive circuit. (A) Initial current flow when switch is closed. (B) Steady-state (DC) conditions. (C) Decrease in current flow when switch is opened.

Simultaneously, in the right circuit another current begins to flow upward through the other transformer winding, which we will call the secondary. With a sensitive ammeter or galvanometer in the circuit as shown, the needle will be deflected to the right slightly.

This secondary current has been shown in red. Where did it come from? Once we answer this question we can understand much about inductance.

This secondary current is nature's way of keeping the total current constant. Before the switch was closed, there was zero current flowing in both windings. The instant after the switch is closed, this zero. total-current condition must be maintained, even though a substantial current begins flowing downward through the primary winding. As the electrons which make up the· primary current begin to accelerate in the downward direction, the negative charge carried by them causes other electrons in the secondary winding to be accelerated upward. When the current flowing upward is subtracted from the one flowing downward, the remainder is zero for the tiniest fraction of a second.

This secondary current cannot flow forever-it is sustained only by changes in the amount of primary current, and the rate of change becomes less and less as the primary current approaches its steady-state value. The secondary current dies out gradually, and when the primary current finally reaches its full steady-state value, the secondary current will drop to zero as shown in Fig. 3B. Fig. 3C depicts the conditions immediately after the switch in the primary circuit has been opened. When the primary current flow is cut off, it must rapidly decrease (decelerate) until it reaches zero. Just as an increasing current flow in the primary winding will set up a current flow in the secondary winding, so too will a decreasing current. But this time, the current in the secondary winding will flow in the opposite direction ( downward) to keep the total current through the two windings constant.

A sensitive galvanometer in the secondary circuit will actually give a momentary deflection in the opposite direction from that shown in Fig. 3A, indicating a current reversal has occurred.

This secondary current lasts only while the primary current is changing. As soon as the primary current reaches zero, the secondary current will drop to zero also.

For every electron flow, or current, there must be a companion voltage to supply inertia. In the primary circuit of Fig. 3A, for example, the applied voltage is the companion and motivating force for the primary circuit. Because of the way the battery is connected, this applied voltage is negative at the top of the primary winding and positive at the bottom, causing the current to flow downward through the primary winding.

Suppose you were asked this question: "What polarity of voltage must exist across the secondary winding in order for the secondary current to flow upward through the winding?" There can be only one answer, since electrons always flow from a point of negative to a point of positive voltage-never in the opposite direction.

In Fig. 3A we can see that the polarity of the secondary voltage is positive at the top and negative at the bottom, and that the applied voltage in the primary is negative at the top and positive at the bottom. As a result, the voltage generated in the secondary "opposes" the applied voltage in the primary.

This voltage associated with secondary currents in transformers and other inductors has been given the name of counter electro motive force, or more simply, counter emf.

We can also see that the applied voltage actually "leads" the resulting current. The current referred to here is the total transformer or inductor current, which is not achieved until sometime after the primary voltage is applied, as shown in Fig. 3B. Thus, attainment of the final current value has "lagged," or fallen behind, the voltage producing it.

In Fig. 3C we see that the polarity of the counter emf has reversed itself-now the negative voltage is at the top and the positive voltage is at the bottom. This is the only possible polarity consistent with the downward flow of secondary current at this moment. The fact that this counter emf is opposing the applied voltage may be a little difficult to visualize, since the two are of the same polarity. However, by opening the switch we have removed the negative voltage previously applied at the top of the primary winding. This is actually equivalent to applying a positive voltage at that point. In this sense, the applied voltage and counter emf are opposite in polarity.

The current in the primary winding flows in only one direction, and yet we have seen that it can cause a two-way current better known as an alternating current-to flow in an adjacent winding. The reason is that it is a unique form of direct current known as pulsating DC. There will be several examples of its use in later Sections, such as in the Hartley and blocking oscillators and in the grid circuit of an electron-coupled oscillator.

Self-Inductance Fig. 4 shows how induced currents can be made to flow and a counter emf generated within a single conductor (shown enlarged in Fig. 4 for clarity). As before, the sudden application or removal of an applied voltage is required-or more specifically, a change in the applied voltage across the circuit is necessary for a self-induced current to flow.

As soon as the switch in Fig. 4 is closed and battery voltage is applied across the circuit, the primary current will attempt to rise to its full value. However, because all circuit components have self-inductance, the total current does not do so immediately, but remains at zero for an instant. The reason is simple as the primary current tries to increase, the "electrical inertia" of the electrons in it will cause other electrons in the circuit to be accelerated in the opposite direction. Hence, the two currents cancel each other at first.

As the primary current approaches its full value, its rate of increase is steadily dropping, permitting the induced current to die out. The counter emf of Fig. 4A no longer exists when the steady-state condition shown in Fig. 4B has been replaced.

Fig. 4C shows the conditions an instant after the switch is opened. The primary current will now try to drop to zero.

However, all inductances act to keep the total current from changing. Thus, as the electrons of the primary current start to decelerate, they induce other electrons in the same conductor to be accelerated in the same downward direction. Therefore, instead of the zero current we would expect to find upon first opening the switch, we find the full current flowing! Once again, a counter emf has come into existence, only this time its polarity is opposing the polarity we tried to apply by opening the switch to remove the negative applied voltage.

Inductor Construction

The entire art of inductors is built upon this property of electrical inertia. In theory, a coil of wire with many thousands of closely spaced turns is still the same straight wire shown earlier in Fig. 4. The object of the additional turns is to provide more inductance (by multiplying the effects of electron inertia). The opposition offered by inductors to the flow of alternating current varies directly with the frequency of the applied current and with the size of the inductor, in accordance with this formula: Xr, = 2 pi fL where, Xr, is the inductive reactance ( opposition to current flow) , in ohms, f is the frequency of the applied current in cycles per second, L is the inductance of the coil in henrys. Fig. 5 shows cross-sectional views of the two common types of coils. Their approximate inductances can be calculated from the following formulas: For a single-layer solenoid (Fig. 5A): r2 x N2 L=9r+l0l where, L is the inductance in microhenrys, N is the number of turns, l is the length of the coil in inches, r is the radius of the coil in inches.

Fig. 4. Self-inductance in a single wire (enlarged view). (A) Current flow when switch is (B) Steady-state (DC) current closed. (C) Current flow when switch is opened.

For a multilayer coil (Fig. 5B): r2 X N2 L = 6 r + 9Z + 10t where t, the only new term, is the thickness of the coil winding in inches. It will not be necessary for you to refer to these formulas in order to understand the circuit actions discussed in the text.

But like the capacitance formula offered previously, this one shows the correlation between physical size and configuration of a coil and its resultant inductance.

Fig. 5. Cross-sectional views of single-layer and multilayer coils. ( A) Single layer. (B) Multilayer.

THE INDUCTANCE-CAPACITANCE COMBINATION

Inductance and capacitance in combination possess the unique characteristic of resonating at a particular frequency. This important and basic circuit action is employed in all types of electronic equipment. Tuned inductance-capacitance (L-C) circuits are used in practically every piece of electronic equipment-and oscillators are no exception, as we shall learn in later Sections.

Fig. 6 shows the conditions at the ends of the four quarter cycles of an oscillation between an inductor L and a capacitor C. At the end of the first quarter-cycle (Fig. 6A), electrons are amassed on the upper plate of the capacitor (thereby constituting a negative voltage) , and there is a deficiency of electrons (positive voltage) on the lower plate. In other words, no current is flowing through the inductor yet.

The voltage across the capacitor represents stored electrical energy. Energy in storage is potential energy. So, during the second quarter-cycle the electrons from the top plate will start to flow downward through the inductor in an effort to redistribute themselves equally between the two plates and thus neutralize the electric field. The self-inductance of the coil prevents an instantaneous build-up of current by generating a counter emf of the opposite polarity. This action is not shown in Fig. 6B, but because of it, the current is delayed exactly one quarter-cycle in reaching its peak. So, at the end of the second quarter-cycle the current is finally flowing at its maximum rate downward through the inductor. At this moment, the charge has been completely redistributed-no voltage remains across the capacitor or tuned tank.

Magnetic Lines of Force

As the current through the coil increases from zero to maxi mum, lines of magnetic force come into existence and expand outward from the coil. In essence, the coil becomes an electro magnet, and since the lines of force in Fig. 6B are entering at the top and exiting from the bottom, the coil has its north pole at the top and its south pole at the bottom.

These, then, are the conditions at the start of the third quarter cycle-zero voltage across the tank, maximum electron current flowing downward through the inductor (coil), and maximum lines of force surrounding it.

The potential energy represented by the charged capacitor has now become kinetic energy, in the form of moving electrons.

The lines of force are also a means of energy storage. So when the current tries to collapse during the third quarter-cycle, these lines of force will collapse and thereby drive additional current downward through the coil. The lines of force are trying to accomplish the effect we already know occurs with inductors-it is trying to keep the current from changing value.

As a result of this sequence, almost all the electrons which were on the upper plate of the capacitor will be delivered to the lower plate during the third quarter-cycle, and the charge distribution will look like Fig. 6C. As in Fig. 6A, this charge distribution again represents potential energy in storage, except now it has the opposite polarity.

At the start of the fourth quarter-cycle, the electrons massed on the lower plate must again redistribute themselves in order to neutralize the voltage across the capacitor. So they begin to flow upward through the inductor, and another familiar sequence begins: A counter emf with a polarity opposite that of the capacitor comes into existence. An induced current then flows downward in the coil and bucks the upward-flowing current.

A new set of magnetic lines of force ( also known as flux lines) develops and expands outward as long as the total current in the coil is expanding. These lines of force have a different direction from the one they had in Fig. 6B, because in Fig. 6D the main current through the coil is flowing in the opposite direction.

Now the lines of force are entering at the bottom and exiting at the top, again making the coil a small electromagnet but with its north pole at the bottom and its south pole at the top.

These, then, are the conditions at the start of the first quarter cycle--zero voltage across the capacitor and tank, maximum upward flow of electrons through the inductor, and maximum number of magnetic lines of force around it.

The collapse of these lines of force during the first quarter cycle tries to keep the current from dying out by drawing additional electrons upward through the inductor. The end result is that almost as many electrons are delivered to the upper plate as were amassed on the lower plate a half-cycle earlier.

Thus, we see that one cycle of an oscillation is characterized by (1) a periodic changing of the voltage across the tank from plus to minus and back to plus; (2) periodic up-and-down alter nations of the current through the inductor; and (3) periodic expansion and contraction of the magnetic lines of force around the inductor, the lines changing direction every half-cycle.

In brief, an electric oscillation consists of a cyclic interchange of energy between an electric and a magnetic field. The electric field is represented by the voltage across the capacitor in Figs. 6A and 6C; and the magnetic field, which is associated with a high current flow, is represented by the expanded lines of force in Figs. 6B and 6D.

Fig. 6. Oscillation in an L-C circuit. (A) At end of first quarter-cycle. (B) At end of second quarter-cycle. (C) At end of third quarter-cycle. (D) At end of fourth quarter-cycle.

The number of cycles occurring each second is called the frequency. Tuned circuits of the type shown in Fig. 6 are used in the generation of radio frequencies ranging from a few thou- sand cycles per second, up to hundreds of kilocycles and even megacycles (a megacycle is one million cycles per second). The values of the capacitor and inductor determine the basic operating frequency of any tuned tank, in accordance with the standard formula which states that: where,

f= 1/ 2pi _/LC

f is the frequency in cycles per second, L is the inductance in henrys, C is the capacitance in farads. A later Section shows how this formula is derived from the two simple reactance formulas for inductance and capacitance.

The strength of an oscillation is measured by the amount of voltage (amplitude of the voltage peaks) across the tuned tank.

Once started, an oscillation could continue indefinitely at the same amplitude, if it were not for the inevitable losses due to wire resistance and other effects. Consequently, unless an oscillation is replenished from an outside source of energy, each succeeding cycle will be a little weaker than the preceding one until eventually the oscillation will die out entirely. (This "dying out" process is called damping of the oscillation).

Fig. 7. Effect of circuit "Q" on oscillatory waveform. (A) Light damping from high-"Q" (B) Heavy damping from low-"Q" circuit. circuit. The quality, or Q, of a tuned circuit is a comparative measure of its freedom from the losses which damp out an oscillation.

Once started, a high-Q circuit will oscillate for many thousands of cycles, whereas an oscillation in a low-Q circuit will die out in a relatively few cycles. A convenient means of visualizing the meaning of Q is to consider it a ratio between the number of electrons in oscillation and the number which drop out each cycle due to losses. Mathematically, circuit Q is stated by several different formulas, two of which are: Q = 2 pi X Energy in Storage Energy Lost Each Cycle or, where, L is the coil inductance in henrys, R is the circuit resistance in ohms. Fig. 7A shows a lightly damped waveform for an oscillation in a high-Q tuned circuit, where losses are low. The heavily damped voltage waveform in Fig. 7B is for an oscillation in a low-Q tuned circuit, where losses are heavy.

THE RESISTOR-CAPACITOR COMBINATION

The action between resistors and capacitors in combination is the last in our discussion of the five basic actions taking place in electronic circuitry. These combinations are most easily classified into one of two broad groups, depending on (1) the values of the resistor and capacitor, and (2) the frequency of the current/ voltage combination to which they must respond in the particular circuit. These two groups are the long time-constant and the short time-constant combinations. The same R-C combination will pro vide a long time-constant at one applied frequency and a short time-constant at a lower frequency.

Almost all circuits discussed in later Sections have at least one long time-constant R-C combination. For this reason, it is certainly worth your while to achieve an early mastery of the basic action in a resistor-capacitor combination. Of the five, it is probably the easiest to understand.

Fig. 8. A water tank with a fluctuating input and steady output is com parable to a "long time-constant" resistor-capacitor combination. (A) Charge. ( B) Discharge.

The drawings in Figs. 8 through 12 are based on an interesting set of analogies. Notice that the capacitance of a capacitor is likened to the capacity of a water tank, the resistance of a resistor is likened to the resistance of a water pipe, water level or pressure is compared to "electron level" or pressure (meaning voltage), and water flow or current is compared to electron flow or current.

Each of the five combinations in Figs. 8 through 12 shows two periods (or half-cycles) of operation-the charging half cycle, and the "other" half-cycle. Just to keep things simple, we will assume the amount of water added during each one of the five charging half-cycles is always the same-say, one bucketful.

Moreover, let us keep the frequency the same throughout by adding water to each tank at the rate of one bucketful per second.

Hence, we can say that each one of the five combinations operates at a frequency of one cycle per second.

Fig. 9. If tank is too narrow and thus has too little "capacity," the output current will surge. This is comparable to a "short time-constant" resistor-capacitor combination.

Fig. 10. With a low-resistance outlet (wide, short nozzle), pressure cannot build up and the water runs out as fast as it is added. This compares to a "short time-constant" R-C combination. In Fig. 8 the tank is large enough, and the output nozzle small enough, that the addition of one bucketful of water each second does not significantly raise the water level. Consequently, the water pressure does not change during the charging half-cycle depicted in Fig. 5A, because the same amount of water flows out through the nozzle during the discharging half-cycle of Fig. 5B. In electronic circuitry, this is analogous to a long time constant R-C combination.

Fig. 11. With a high-resistance outlet (narrow, long nozzle), a high water level and pressure will build up and current will be steady. This is also a long time-constant combination, but the same amount of input will produce a higher pressure than in Fig. 8.

Fig. 9 depicts the same conditions as before, only with a smaller water tank. Now a single bucketful of water does make a difference in the water level and consequently the pressure.

Because the pressure rises as each bucketful is poured in, more water flows out through the nozzle during the charging than the discharging half-cycle, since rate of flow (current) depends on water pressure. The same is true of electric current, which depends on electrical pressure, or voltage. Whenever the pressure level and consequently the amount of current in a circuit changes significantly during one complete cycle, the combination has a short time-constant at the particular frequency.

In Fig. 10 we have restored the tank to its original size but enlarged the outlet nozzle (thus lowering its resistance). Now it is difficult to build up water pressure, since the water flows out almost as fast as it flows in. Because the water level fluctuates, the current through the nozzle is pulsating rather than steady.

Thus, as in Fig. 8, we have a short time-constant combination here.

In Fig. 11 we have narrowed the outlet nozzle down and thereby substantially increased its resistance. Thus, it will impede the flow, and more water will be added each cycle than can flow out, until the water level builds up to the point where there is sufficient pressure to force out the same amount that is added each cycle. However, since the water level does not decrease during the discharging half-cycle, there is no surging-the flow will gradually build up and then remain constant. Hence, this combination can be classified as a long time-constant.

Fig. 12. If tank has excessive capacity, it will require more water and more time to build up to the same pressure as in Fig. 8. This level will be achieved eventually, however; therefore this is also a long time-constant combination. The current output is steady from half-cycle to half-cycle, but will increase as the pressure does. Fig. 12 shows the effect of greatly increasing the tank capacity. One bucketful of water each cycle will not change the water level substantially; many cycles will be required to build up a pressure equal to that of Fig. 8. The moral is that the water tank is probably bigger than necessary, if its function is merely to maintain a constant water pressure and thus a constant water flow through the outlet nozzle. Anyway, this combination is a long time-constant one.

Time-Constant Formula

From the previous analogies it is possible to demonstrate the underlying meanings of several important formulas you will en counter in later Sections-in fact, in all electronic-circuit applications. The first of these is the time-constant formula.

The time constant of a resistor-capacitor combination is the time it requires to complete 63.2% of its charging or discharging action.

The formula states that this time is equal to the product of the component values as follows: T=RXC where, T is the time in seconds, R is the resistance of the charging path in ohms, C is the capacitance in farads.

When we consider the water tanks and their outlets, it is evident that the emptying, or discharge, time from any given water pressure or level (voltage) will vary directly with the size, or capacity, of the tank and with the amount of resistance the nozzle offers to the water flow. (A wide, short nozzle offers low resistance; a narrow, long nozzle offers high resistance.)

Coulomb's Law

Coulomb's law states another important principle referred to repeatedly in the following Sections, and it can also be demonstrated with the water-tank analogies. This law says that the attraction or repulsion between two electric charges is proportional to the product of their magnitudes, and is inversely proportional to the square of the distance between them. Hence, from this we can deduce that the amount of charge (negative electrons or positive ions) stored in any capacitor is proportional to the voltage across the capacitor. These three quantities are related arithmetically by the following formula:

Q=CxE

where, Q is the quantity of charge in coulombs (1 coulomb = 6.25 X 10 electrons), C is the capacitance in farads, E is the voltage, or electrical pressure, in volts across the capacitor as a result of the stored charge.

From the water-tank analogies it is evident that an increase in the amount of water stored in the tank will raise the water level and consequently the pressure. It is also apparent that a relatively small amount of water will fill up a narrow tank with a small capacity, thereby creating a high water level and pressure. This same quantity of water, when transferred to a wide tank, how ever, will barely cover the bottom of the tank, and the water level and pressure will be very low.

Ohm's Law

Another important formula whose meaning can be demonstrated with the tank analogies is Ohm's law, which states that the voltage developed across a resistor is proportional to the cur rent flowing through it. These three quantities are related arithmetically as follows: E=IXR where, E is the voltage across the resistive path in volts, I is the current through the resistor in amperes, R is the resistance of the path in ohms.

If the water level in one of the tanks is raised, it should be clear that the higher pressure will force more water (current) through the nozzle. Also, if the water level is not changed but a larger nozzle is substituted, its resistance will be lower and more water will flow out than before.

Conclusion

Each of these three formulas is elaborated on whenever an R-C combination appears in one of the later Sections and it seems necessary to do so in order to clarify circuit operation. There is a truly enormous body of literature making up all the mathematics of electronics. However, it is not necessary to master it all to understand how electronic circuits operate. A working understanding of the preceding three formulas, plus the two for capacitive and inductive reactance and the frequency formula for tuned circuits, is adequate in helping you visualize the action of any circuit. The reason is that all circuit actions can be placed into one of seven major categories. These might be called basic actions, of the types described briefly in this Section. They are:

Resistor.

Capacitor.

Inductor.

Inductor-capacitor combination (both at resonance and off resonance.)

Resistor-capacitor combination.

Resistor-inductor combination (which enjoys only limited usage).

Vacuum tubes (which provide the necessary regulation).