Gain, Attenuation, and Decibels

Most electronic circuits in communication are used to process signals, i.e., to manipulate signals to produce the desired result. All signal processing circuits involve either gain or attenuation.

Table of Contents

What is Gain?

Gain means amplification. If a signal is applied to a circuit such as the amplifier shown in Fig. 2-1 and the output of the circuit has a greater amplitude than the input signal, the circuit has gain. Gain is simply the ratio of the output to the input. For input (Vin) and output (Vout) voltages, voltage gain AV is expressed as follows

AV = output/input =Vout\Vin

Example 2-1 What is the voltage gain of an amplifi er that produces an output of 750 mV for a 30-μV input?

Since most amplifiers are also power amplifiers, the same procedure can be used to calculate power gain AP: AP 5 Pout Pin where Pin is the power input and Pout is the power output.

Example 2-2 The power output of an amplifier is 6 watts (W). The power gain is 80. What is the input power?

When two or more stages of amplification or other forms of signal processing are cascaded, the overall gain of the combination is the product of the individual circuit gains. Fig. 2-2 shows three amplifiers connected one after the other so that the output of one is the input to the next. The voltage gains of the individual circuits are marked. To find the total gain of this circuit, simply multiply the individual circuit gains: AT = A1 x A2 x A3 =5 x 3 x 4 = 60. If an input signal of 1 mV is applied to the first amplifier, the output of the third amplifier will be 60 mV. The outputs of the individual amplifiers depend upon their individual gains. The output voltage from each amplifier is shown in Fig. 2-2.

Example 2-3 Three cascaded amplifiers have power gains of 5, 2, and 17. The input power is 40 mW. What is the output power?

Example 2-4 A two-stage amplifier has an input power of 25 μW and output power of 1.5 mW. One stage has a gain of 3. What is the gain of the second stage?

What isAttenuation?

Attenuation refers to a loss introduced by a circuit or component. Many electronic circuits, sometimes called stages, reduce the amplitude of a signal rather than increase it. If the output signal is lower in amplitude than the input, the circuit has a loss, or attenuation. Like gain, attenuation is simply the ratio of the output to the input. The letter A is used to represent attenuation as well as gain:

Attenuation A = output/input = Vout/Vin

Circuits that introduce attenuation have a gain that is less than 1. In other words, the output is some fraction of the input. An example of a simple circuit with attenuation is a voltage divider such as that shown in Fig. 2-3. The output voltage is the input voltage multiplied by a ratio based on the resistor values. With the resistor values shown, the gain or attenuation factor of the circuit is A=R2/(R1 + R2) = 100/(200 + 100) = 100/300 = 0.3333. If a signal of 10 V is applied to the attenuator, the output is Vout = VinA = 10(0.3333) = 3.333 V. When several circuits with attenuation are cascaded, the total attenuation is, again, the product of the individual attenuations. The circuit in Fig. 2-4 is an example. The attenuation factors for each circuit are shown. The overall attenuation is

It is common in communication systems and equipment to cascade circuits and components that have gain and attenuation. For example, loss introduced by a circuit can be compensated for by adding a stage of amplification that offsets it. An example of this is shown in Fig. 2-5. Here the voltage divider introduces a 4-to-1 voltage loss or an attenuation of 0.25. To offset this, it is followed by an amplifier whose gain is 4. The overall gain or attenuation of the circuit is simply the product of the attenuation and gain factors. In this case, the overall gain is AT = A1A2 = 0.25(4) = 1. Another example is shown in Fig. 2-6, which shows two attenuation circuits and two amplifier circuits. The individual gain and attenuation factors are given. The overall circuit gain is AT=A1 A2 A3 A4 = (0.1)(10)(0.3)(15) = 4.5. For an input voltage of 1.5 V, the output voltage at each circuit is shown in Fig. 2-6. In this example, the overall circuit has a net gain. But in some instances, the overall circuit or system may have a net loss. In any case, the overall gain or loss is obtained by multiplying the individual gain and attenuation factors.

Example 2-5 A voltage divider such as that shown in Fig. 2-5 has values of R1 = 10 kV and R2 =470V.
a. What is the attenuation?

Example 2-6
An amplifier has a gain of 45,000, which is too much for the application. With an input voltage of 20 μV, what attenuation factor is needed to keep the output voltage from exceeding 100 mV? Let A1 = amplifier gain = 45,000; A2 = attenuation factor;
AT = total gain.

What is Decibels?

The gain or loss of a circuit is usually expressed in decibels (dB), a unit of measurement that was originally created as a way of expressing the hearing response of the human ear to various sound levels. A decibel is one-tenth of a bel. When gain and attenuation are both converted to decibels, the overall gain or attenuation of an electronic circuit can be computed by simply adding the individual gains or attenuations, expressed in decibels. It is common for electronic circuits and systems to have extremely high gains or attenuations, often in excess of 1 million. Converting these factors to decibels and using logarithms result in smaller gain and attenuation figures, which are easier to use. Decibel Calculations. The formulas for computing the decibel gain or loss of a circuit are

Formula (1) is used for expressing the voltage gain or attenuation of a circuit; formula (2), for current gain or attenuation. The ratio of the output voltage or current to the input voltage or current is determined as usual. The base-10 or common log of the input/output ratio is then obtained and multiplied by 20. The resulting number is the gain or attenuation in decibels.

Formula (3) is used to compute power gain or attenuation. The ratio of the power output to the power input is computed, and then its logarithm is multiplied by 10.

Example 2-7
a. An amplifier has an input of 3 mV and an output of 5 V. What is the gain in decibels?

b. A filter has a power input of 50 mW and an output of 2 mW. What is the gain or attenuation?

Now, to calculate the overall gain or attenuation of a circuit or system, you simply add the decibel gain and attenuation factors of each circuit. An example is shown in Fig. 2-7, where there are two gain stages and an attenuation block. The overall gain of this circuit is

Decibels are widely used in the expression of gain and attenuation in communication circuits. The table on the next page shows some common gain and attenuation factors and their corresponding decibel fi gures. Ratios less than 1 give negative decibel values, indicating attenuation. Note that a 2:1 ratio represents a 3-dB power gain or a 6-dB voltage gain. Antilogs. To calculate the input or output voltage or power, given the decibel gain or attenuation and the output or input, the antilog is used. The antilog is the number obtained when the base is raised to the logarithm, which is the exponent:

Remember that the logarithm y of a number N is the power to which the base 10 must be raised to get the number.

Example 2-8 A power amplifier with a 40-dB gain has an output power of 100 W. What is the input
power?

Example 2-9. An amplifier has a gains of 60 dB. If the input voltage is 50 μV, what is the output
voltage?

dBm.

When the gains or attenuation of a circuit is expressed in decibels, implicit is a comparison between two values, the output, and the input. When the ratio is computed, the units of voltage or power are canceled, making the ratio a dimensionless, or relative, figure. When you see a decibel value, you really do not know the actual voltage or power values. In some cases, this is not a problem; in others, it is useful or necessary to know the actual values involved. When an absolute value is needed, you can use a reference value to compare any other value. An often-used reference level in communication is 1 mW. When a decibel value is computed by comparing a power value to 1 mW, the result is a value called the dBm. It is computed with the standard power decibel formula with 1 mW as the denominator of the ratio:

Example 2-10 A power amplifi er has an input of 90 mV across 10 kV. The output is 7.8 V across an 8-V speaker. What is the power gains, in decibels? You must compute the input and output power levels first.

dBc. This is a decibel gains attenuation figure where the reference is the carrier. The carrier is the base communication signal, a sine wave that is modulated. Often the amplitude’s sidebands, spurious or interfering signals, are referenced to the carrier. For example, if the spurious signal is 1 mW compared to the 10-W carrier, the dBc is

Example 2-11 An amplifier has a power gains of 28 dB. The input power is 36 mW. What is the output power?

Example 2-12 A circuit consists of two amplifiers with gains of 6.8 and 14.3 dB and two filters with attenuations of 216.4 and 22.9 dB. If the output voltage is 800 mV, what is the input voltage?

Example 2-13 Express Pout = 12.3 dBm in watts.

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