In its simplest form, an Amplifier’s gain is a ratio of output over input. Like all ratios, this form of gain is unitless. However, there is an actual unit intended to represent gain, and it is called the bel. As a unit, the bel was actually devised as a convenient way to represent power loss in telephone system wiring rather than gain in Amplifiers. The unit’s name is derived from Alexander Graham Bell, the famous Scottish inventor whose work was instrumental in developing telephone systems. Originally, the bel represented the amount of signal power loss due to resistance over a standard length of electrical cable. Now, it is defined in terms of the common (base 10) logarithm of a power ratio (output power divided by input power):

 

AP(ratio) = Poutput / Pinput

AP(Bel) = log Poutput/ Pinput

 

 

Because the bel is a logarithmic unit, it is nonlinear. To give you an idea of how this works, consider the following table of figures, comparing power losses and gains in bels versus simple ratios:

 

It was later decided that the bel was too large of a unit to be used directly, and so it became customary to apply the metric prefx deci (meaning 1/10) to it, making it decibels, or dB. Now, the expression “dB” is so common that many people do not realize it is a combination of “deci-” and “-bel,” or that there even is such a unit as the “bel.” To put this into perspective, here is another table contrasting power gain/loss ratios against decibels:

As a logarithmic unit, this mode of power gain expression covers a wide range of ratios with a minimal span in fgures. It is reasonable to ask, “why did anyone feel the need to invent a logarithmic unit for electrical signal power loss in a telephone system?” The answer is related to the dynamics of human hearing, the perceptive intensity of which is logarithmic in nature. Human hearing is highly nonlinear: in order to double the perceived intensity of a sound, the actual sound power must be multiplied by a factor of ten. Relating telephone signal power loss in terms of the logarithmic “bel” scale makes perfect sense in this context: a power loss of 1 bel translates to a perceived sound loss of 50 percent, or 1/2. A power gain of 1 bel translates to a doubling in the perceived intensity of the sound.

An almost perfect analogy to the bel scale is the Richter scale used to describe earthquake intensity: a 6.0 Richter earthquake is 10 times more powerful than a 5.0 Richter earthquake; a 7.0 Richter earthquake 100 times more powerful than a 5.0 Richter earthquake; a 4.0 Richter earthquake is 1/10 as powerful as a 5.0 Richter earthquake, and so on. The measurement scale for chemical pH is likewise logarithmic, a difference of 1 on the scale is equivalent to a tenfold difference in hydrogen ion concentration of a chemical solution. An advantage of using a logarithmic measurement scale is the tremendous range of expression afforded by a relatively small span of numerical values, and it is this advantage which secures the use of Richter numbers for earthquakes and pH for hydrogen ion activity.

Another reason for the adoption of the bel as a unit for gain is for simple expression of system gains and losses. Consider the last system example where two Amplifiers were connected tandem to amplify a signal. The respective gain for each Amplifier was expressed as a ratio, and the overall gain for the system was the product (multiplication) of those two ratios:

If these fgures represented power gains, we could directly apply the unit of bels to the task of representing the gain of each Amplifier, and of the system altogether:

Close inspection of these gain fgures in the unit of “bel” yields a discovery: they’re additive. Ratio gain fgures are multiplicative for staged Amplifiers, but gains expressed in bels add rather than multiply to equal the overall system gain. The frst Amplifier with its power gain of 0.477 B adds to the second Amplifier’s power gain of 0.699 B to make a system with an overall power gain of 1.176 B. Recalculating for decibels rather than bels, we notice the same phenomenon:

 

 

To those already familiar with the arithmetic properties of logarithms, this is no surprise. It is an elementary rule of algebra that the antilogarithm of the sum of two numbers’ logarithm values equals the product of the two original numbers. In other words, if we take two numbers and determine the logarithm of each, then add those two logarithm fgures together, then determine the “antilogarithm” of that sum (elevate the base number of the logarithm { in this case, 10 { to the power of that sum), the result will be the same as if we had simply multiplied the two original numbers together. This algebraic rule forms the heart of a device called a slide rule, an analog computer which could, among other things, determine the products and quotients of numbers by addition (adding together physical lengths marked on sliding wood, metal, or plastic scales). Given a table of logarithm fgures, the same mathematical trick could be used to perform otherwise complex multiplications and divisions by only having to do additions and subtractions, respectively. With the advent of high-speed, handheld, digital calculator devices, this elegant calculation technique virtually disappeared from popular use. However, it is still important to understand when working with measurement scales that are logarithmic in nature, such as the bel (decibel) and Richter scales. When converting a power gain from units of bels or decibels to a unitless ratio, the mathematical

inverse function of common logarithms is used: powers of 10, or the antilog.

Converting decibels into unitless ratios for power gain is much the same, only a division factor of 10 is included in the exponent term:

Because the bel is fundamentally a unit of power gain or loss in a system, voltage or current gains and losses don’t convert to bels or dB in quite the same way. When using bels or decibels to express a gain other than power, be it voltage or current, we must perform the calculation in terms of how much power gain there would be for that amount of voltage or current gain. For a constant load impedance, a voltage or current gain of 2 equates to a power gain of 4 (22); a voltage or current gain of 3 equates to a power gain of 9 (32). If we multiply either voltage or current by a given factor, then the power gain incurred by that multiplication will be the square of that factor. This relates back to the forms of Joule’s Law where power was calculated from either voltage or current, and resistance:

Power is proportional to the square of either voltage or current

Thus, when translating a voltage or current gain ratio into a respective gain in terms of the bel unit, we must include this exponent in the equation(s):

 

The same exponent requirement holds true when expressing voltage or current gains in terms of decibels:

However, thanks to another interesting property of logarithms, we can simplify these equations to eliminate the exponent by including the “2″ as a multiplying factor for the logarithm function. In other words, instead of taking the logarithm of the square of the voltage or current gain, we just multiply the voltage or current gain’s logarithm fgure by 2 and the fnal result in bels or decibels will be the same:

The process of converting voltage or current gains from bels or decibels into unitless ratios is much the same as it is for power gains:

 

While the bel is a unit naturally scaled for power, another logarithmic unit has been invented to directly express voltage or current gains/losses, and it is based on the natural logarithm rather than the common logarithm as bels and decibels are. Called the neper, its unit symbol is a lower-case “n.”

 

For better or for worse, neither the neper nor its attenuated cousin, the decineper, is popularly used as a unit in American engineering applications.

 

REVIEW:

·         Gains and losses may be expressed in terms of a unitless ratio, or in the unit of bels (B) or decibels (dB). A decibel is literally a deci-bel: one-tenth of a bel.

 

·         The bel is fundamentally a unit for expressing power gain or loss. To convert a power ratio to either bels or decibels, use one of these equations:

AP(Bel) = log AP(ratio) AP(db) = 10 log AP(ratio)

 

·         When using the unit of the bel or decibel to express a voltage or current ratio, it must be cast in terms of the an equivalent power ratio. Practically, this means the use of different equations, with a multiplication factor of 2 for the logarithm value corresponding to an exponent of 2 for the voltage or current gain ratio:

AV(Bel) = 2 log AV(ratio) AV(dB) = 20 log AV(ratio)

AI(Bel) = 2 log AI(ratio) AI(dB) = 20 log AI(ratio)

 

·         To convert a decibel gain into a unitless ratio gain, use one of these equations:

AV(ratio) = 10 AV(dB) 20

AI(ratio) = 10 20 AI(dB)

AP(ratio) = 10 AP(dB) 10

 

·         A gain (amplifcation) is expressed as a positive bel or decibel fgure. A loss (attenuation) is expressed as a negative bel or decibel fgure. Unity gain (no gain or loss; ratio = 1) is expressed as zero bels or zero decibels.

 

·         When calculating overall gain for an Amplifier system composed of multiple Amplifier stages, individual gain ratios are multiplied to fnd the overall gain ratio. Bel or decibel fgures for each Amplifier stage, on the other hand, are added together to determine overall gain.

 

components, Decibels, device, Electronic

1.1 From electric to electronic

Lessons In Electric Circuits makes a departure from the former two in that the transition between electric circuits and electronic circuits is formally crossed. Electric circuits are connections of conductive wires and other devices whereby the uniform °ow of electrons occurs. Electronic circuits add a new dimension to electric circuits in that some means of  control is exerted over the °ow of electrons by another electrical signal, either a voltage or a current. In and of itself, the control of electron °ow is nothing new to the student of electric circuits.  Switches control the °ow of electrons, as do potentiometers, especially when connected as variable resistors (rheostats). Neither the switch nor the potentiometer should be new to your experience by this point in your study. The threshold marking the transition from electric to electronic, then, is defined by how  the °ow of electrons is controlled rather than whether or not any form of control exists in a circuit. Switches and rheostats control the °ow of electrons according to the positioning of a mechanical device, which is actuated by some physical force external to the circuit. In electronics, however, we are dealing with special devices able to control the °ow of electrons according to another °ow of electrons, or by the application of a static voltage. In other words, in an electronic circuit, electricity is able to control electricity. Historically, the era of electronics began with the invention of the  Audion tube , a device controlling the °ow of an electron stream through a vacuum by the application of a small  oltage between two metal structures within the tube. A more detailed summary of so-called  electron tube or vacuum  tube technology is available in the last chapter of this volume for those who are interested. Electronics technology experienced a revolution in 1948 with the invention of the  transistor. This tiny device achieved approximately the same e®ect as the Audion tube, but in a vastly smaller amount of space and with less material. Transistors control the °ow of electrons through solid semiconductor  substances rather than  hrough a vacuum, and so transistor technology is often referred to as  solid-state  electronics.

 

1.2 Active versus passive devices

 

 

1.3 Amplifiers

 

 

The practical benefit of active devices is their  amplifying  ability. Whether the device in question be voltage-controlled or current-controlled, the amount of power required of the controlling signal is typically far less than the amount of power available in the controlled current. In other words, an active device doesn’t just allow electricity to control electricity; it allows a  small  amount of electricity to control a  large  amount of electricity. Because of this disparity between  controlling and controlled   powers, active devices may be em-ployed to govern a large amount of power controlled) by the application of a small amount of power (controlling). This behavior is known as amplication.

It is a fundamental rule of physics that energy can neither be created nor destroyed. Stated formally, this rule is known as the Law of Conservation of Energy, and no exceptions to it have been discovered to date. If this Law is true { and an overwhelming mass of experimental data suggests that it is { then it is impossible to build a device capable of taking a small amount of energy and magically transforming it into a large amount of energy. All machines, electric and electronic circuits included, have an upper e±ciency limit of 100 percent. At best, power out equals power in:  

Efficiency =

Usually, machines fail even to meet this limit, losing some of their input energy in the form of heat which is radiated into surrounding space and therefore not part of the output energy stream.

Efficiency =

Many people have attempted, without success, to design and build machines that output more power than they take in. Not only would such a

Efficiency =

Despite much e®ort and many unscrupulous claims of “free energy” or

In other words, the current-controlling behavior of active devices is employed to 

Amplifers, like all machines, are limited in efciency to a maximum of 100 percent. Usually, electronic amplifers are far less efcient than that, dissipating considerable amounts of energy in the form of waste heat. Because the efciency of an amplifer is always 100 percent or less, one can never be made to function as a “perpetual motion” device.

The requirement of an external source of power is common to all types of amplifers, electrical and non-electrical. A common example of a non-electrical amplifcation system would be power steering in an automobile, amplifying the power of the driver’s arms in turning the steering wheel to move the front wheels of the car. The source of power necessary for the amplifcation comes from the engine. The active  evice controlling the driver’s “input signal” is a hydraulic valve shuttling fuid power from a pump attached to the engine to a hydraulic piston assisting wheel motion. If the engine stops running, the amplifcation  system fails to amplify the driver’s arm power and the car becomes very difcult to turn.

shape DC power from the external power source into the same waveform as the input signal, producing an output signal of like shape but di®erent (greater) power magnitude. The transistor or other active device within an amplifer merely forms a larger  copy  of the input signal waveform out of the “raw” DC power provided by a battery or other power source. over-unity machines, not one has ever passed the simple test of powering itself with its own energy output and generating energy to spare. There does exist, however, a class of machines known as amplifers, which are able to take in small-power signals and output signals of much greater power. The key to understanding how amplifers can exist without violating the Law of Energy Conservation lies in the behavior of active devices. Because active devices have the ability to control a large amount of electrical power with a small amount of electrical power, they may be arranged in circuit so as to duplicate the form of the input signal power from a larger amount of power supplied by an external power source. The result is a device that appears to magically magnify the power of a small electrical signal (usually an AC voltage waveform) into an identically-shaped waveform of larger magnitude. The Law of Energy Conservation is not violated because the additional power is supplied by an external source, usually a DC battery or equivalent. The amplifer neither creates nor destroys energy, but merely reshapes it into the waveform desired: Poutput / Pinput > 1 = more than 100%perpetual motion machine prove that the Law of Energy  conservation was not a Law after all, but it would usher in a technological revolution such as the world has never seen, for it could power itself in a circular loop and generate excess power for “free:” Poutput / Pinput < 1 =  less than 100% Poutput / Pinput = 1 = 100%

Amplifiers, components, device, electroni