14.3 The Decibel Scale

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14.3 The Decibel Scale

It is not always easy to get a quick plot of the magnitude and phase of the transfer function as we did above. A more systematic way of obtaining the frequency response is to use Bode plots. Before we begin to construct Bode plots, we should take care of two important issues: the use of logarithms and decibels in expressing gain.

For Example 14.2.

Figure 14.7 For Practice Prob. 14.2.

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Historical

Alexander Graham Bell (1847–1922) inventor of the telephone, was a Scottish-American scientist.

Bell was born in Edinburgh, Scotland, a son of Alexander Melville Bell, a well-known speech teacher. Alexander the younger also became a speech teacher after graduating from the University of Edinburgh and the University of London. In 1866 he became interested in transmitting speech electrically. After his older brother died of tuberculosis, his father decided to move to Canada. Alexander was asked to come to Boston to work at the School for the Deaf. There he met Thomas A. Watson, who became his assistant in his electromagnetic transmitter experiment. On March 10, 1876, Alexander sent the famous first telephone message: “Watson, come here I want you.” The bel, the logarithmic unit intro duced in this chapter, is named in his honor.

Since Bode plots are based on log arithms, it is important that we keep the following properties of logarithms in mind:

1. log P1P2 = log P1 + log P² 2. log P1∕P2 = log P¹ − log P² 3. log Pn = n log P 4. log 1 = 0

Historical note: The bel is named after Alexander Graham Bell, the inventor of the telephone.

In communications systems, g ain is measured in bels. Historically, the bel is used to measure the ratio of two levels of power or power gain G; that is,

$$G = \text{Number of bels} = \log_{10} \frac{P_2}{P_1} \tag{14.4}$$

The decibel (dB) provides us with a unit of less magnitude. It is 1∕10th of a bel and is given by

$$G_{\text{dB}} = 10 \log_{10} \frac{P_2}{P_1} \tag{14.5}$$

When P1 = P2, there is no change in po wer and the g ain is 0 dB. If P2 = 2P1, the gain is

$$G_{\rm dB} = 10 \log_{10} 2 \simeq 3 \text{ dB} \tag{14.6}$$

and when P2 = 0.5P1, the gain is

$$G_{\rm dB} = 10 \log_{10} 0.5 \simeq -3 \text{ dB} \tag{14.7}$$

Equations (14.6) and (14.7) sho w another reason wh y log arithms are greatly used: The logarithm of the reciprocal of a quantity is simply negative the logarithm of that quantity.

Alternatively, the g ain G can be e xpressed in terms of v oltage or current ratio. To do so, consider the network shown in Fig. 14.8. If P1 is the input power, P2 is the output (load) power, R1 is the input resistance,

Voltage-current relationships for a fourterminal network.

and R2 is the load resistance, then P1 = 0.5V^{2 1}∕R1 and P2 = 0.5V² 2 ∕R2, and Eq. (14.5) becomes

$$G_{\text{dB}} = 10 \log_{10} \frac{P_2}{P_1} = 10 \log_{10} \frac{V_2^2 / R_2}{V_1^2 / R_1}$$

= 10 \log_{10} \left(\frac{V_2}{V_1}\right)^2 + 10 \log_{10} \frac{R_1}{R_2} (14.8)

$$G_{\text{dB}} = 20 \log_{10} \frac{V_2}{V_1} - 10 \log_{10} \frac{R_2}{R_1}$$

(14.9)

For the case when R2 = R1, a condition that is often assumed when comparing voltage levels, Eq. (14.9) becomes

$$G_{\text{dB}} = 20 \log_{10} \frac{V_2}{V_1}$$

(14.10)

Instead, if P1 = I ¹ 2 R1 and P2 = I² 2 R2, for R1 = R2, we obtain

$$G_{\text{dB}} = 20 \log_{10} \frac{I_2}{I_1} \tag{14.11}$$

Three things are important to note from Eqs. (14.5), (14.10), and (14.11):

• 1. That 10 log 10 is used for po wer, while 20 log 10 is used for v oltage or current, because of the square relationship between them (P = V² ∕R = I 2 R).
• 2. That the dB value is a logarithmic measurement of the ratio of one variable to another of the same type. Therefore, it applies in expressing the transfer function H in Eqs. (14.2a) and (14.2b), which are dimensionless quantities, but not in expressing H in Eqs. (14.2c) and (14.2d).
• 3. It is important to note that we only use voltage and current magnitudes in Eqs. (14.10) and (14.11). Negative signs and angles will be handled independently as we will see in Section 14.4.

With this in mind, we now apply the concepts of logarithms and decibels to construct Bode plots.