Decibel

Abstract

This chapter introduces the topic decibel in a compact form. Decibel is defined as the ratio of two quantities, typically power, voltage, or current. In the field of EMC, the decibel must be understood. One advantage of decibels is that a gain of 10 dB means a gain of 10 dB for current, voltage, and power likewise. This fact helps to prevent misinterpretations and helps to simplify things. This is the main reason why EMC and high-frequency system engineers like to work with decibels.

This chapter assumes that values of voltages V [V], currents I [A], power P [W], electric field strengths E [V/m], and magnetic field strength H [A/m] are given as root-mean-square (RMS) values.

If you cannot measure it, you cannot improve it.

—Lord Kelvin

3.1 Gain and Loss [dB]

Let’s have a look at the amplifier or attenuation network in Fig. 3.1. The power, voltage, and current gain of this network can be expressed in [dB] as [2]:

$$\displaystyle \begin{aligned} \text{Power gain in [dB]} = 10\cdot\log_{10}\left(\frac{P_{out}}{P_{in}}\right) = 10\cdot\log_{10}\left[\left(\frac{V_{out}}{V_{in}}\right)^2\cdot\left(\frac{R_{in}}{R_{load}}\right)\right] \end{aligned} $$

(3.1)

$$\displaystyle \begin{aligned} \text{Voltage gain in [dB]} = 20\cdot\log_{10}\left(\frac{V_{out}}{V_{in}}\right) + 10\cdot\log_{10}\left(\frac{R_{in}}{R_{load}}\right) \end{aligned} $$

(3.2)

$$\displaystyle \begin{aligned} \text{Current gain in [dB]} = 20\cdot\log_{10}\left(\frac{I_{out}}{I_{in}}\right) + 10\cdot\log_{10}\left(\frac{R_{load}}{R_{in}}\right) \end{aligned} $$

(3.3)

In case R _in and R _load are equal (typically 50 Ω), then the following term is equal to zero:

$$\displaystyle \begin{aligned} 10\cdot\log_{10}\left(\frac{R_{in}}{R_{load}}\right) = 10\cdot\log_{10}\left(\frac{R_{load}}{R_{in}}\right) = 10\cdot\log_{10}\left(1\right) = 0 \end{aligned}$$

Fig. 3.1

An arbitrary amplifier or attenuation network

Now we can write the following for power/voltage/current gain:

$$\displaystyle \begin{aligned} \mathbf{Power~gain~in~[dB]} = 10\cdot\log_{10}\left(\frac{P_{out}}{P_{in}}\right) \end{aligned} $$

(3.4)

$$\displaystyle \begin{aligned} \mathbf{Voltage~gain~in~[dB]} = 20\cdot\log_{10}\left(\frac{V_{out}}{V_{in}}\right) \end{aligned} $$

(3.5)

$$\displaystyle \begin{aligned} \mathbf{Current~gain~in~[dB]} = 20\cdot\log_{10}\left(\frac{I_{out}}{I_{in}}\right) \end{aligned} $$

(3.6)

Points to remember when it comes to gain and loss calculations in decibel :

• Absolute vs. relative. Decibels are always ratios of numbers, never an absolute quantity, even if they are named absolute levels.

• Amplification. If P _out is bigger than P _in, the gain value in [dB] is positive. This means that in case of an amplification, the power gain in [dB] is positive.

• Attenuation. If P _out is smaller than P _in, the gain value in [dB] is negative. This means that in case of an attenuation (loss), the power gain in [dB] is negative.

• Gain = −Loss. Power loss is indicated by a negative decibel power gain. For example, if an interconnection shows a loss of 1 dB, the power gain of that interconnection is −1 dB.

• Cutoff frequency f _c . At the cutoff frequency, the output power (P _out) is half the input power (P _in), and the power/voltage/current gains are all −3 dB.

  $$\displaystyle \begin{aligned} \text{Power gain in [dB] at }f_c = 10\cdot\log_{10}\left(\frac{1}{2}\right) = -3 \text{dB} \end{aligned}$$
  $$\displaystyle \begin{aligned} \text{Voltage gain in [dB] at }f_c = 20\cdot\log_{10}\left(\frac{1}{\sqrt{2}}\right) = -3 \text{dB} \end{aligned}$$
  $$\displaystyle \begin{aligned} \text{Current gain in [dB] at }f_c = 20\cdot\log_{10}\left(\frac{1}{\sqrt{2}}\right) = -3 \text{dB} \end{aligned}$$

• Ratio to [dB]. Table 3.2 presents some common ratio to [dB] value conversions. For example, if power increases by factor 2, the power/voltage/current gain increases by 3 dB.

3.2 Absolute Levels [dBm, dBμV, dBμA]

The most common absolute power, voltage, and current levels in EMC are [dBm], [dBμV], and [dBμA] (Fig. 3.2). They are calculated like this [2]:

(3.7)

(3.8)

(3.9)

Fig. 3.2

Commonly used absolute decibel levels in EMC

More information about physical quantities and their units (also in decibel) can be found in the Appendix D. The same concept of absolute levels can also be applied to electrical fields E [V/m], magnetic fields H [A/m], or radiated power density S [W/m²]:

(3.10)

(3.11)

(3.12)

Points to remember when it comes to calculations with absolute power levels in decibel:

• Zero values.

  • 0 dBm 1 mW

  • 0 dBµV 1 µV

  • 0 dBµA 1 µA

  • 0 dBµV/m 1 µV/m

  • 0 dBµA/m 1 µA/m

• Negative values. A negative [dBm]-value means that the power is smaller than 1 mW. A negative [dBμV]-value means that the voltage is smaller than 1 µV. A negative [dBμA]-value means that the current is smaller than 1 µA.

• Positive values. A [dBm]-value bigger than 0 means that the power is higher than 1 mW. A [dBμV]-value bigger than 0 means that the voltage is higher than 1 µV. A [dBμA]-value bigger than 0 means that the current is higher than 1 µA.

• Gain and loss. Gain values in [dB] G _dB can just be added to the absolute power levels in order to get the output power. The linear calculation with input power P _in, linear gain G, and output power P _out can be written as:

  $$\displaystyle \begin{aligned} P_{out} = P_{in} \cdot G \end{aligned}$$

  In decibel, the output power P _out,dB is the sum of the input power P _in,dB and the gain G _dB:

  $$\displaystyle \begin{aligned} P_{out,dB} = 10\cdot\log_{10}\left(P_{out}\right) = 10\cdot\log_{10}\left(P_{in}\right) + 10\cdot\log_{10}\left(G\right) = P_{in,dB} + G_{dB} \end{aligned}$$

  Let’s assume a signal with P _in,dBm = 0 dBm at the input of an amplifier with gain G _dB = 20 dB. The output power is:

  

• Never sum up absolute levels. Do never sum up absolute decibel levels, because adding decibels means multiplying the linear values and therefore:

  

  What does power squared mean? Thus, never add up absolute decibel levels.

• Sum of absolute levels and decibel. It is allowed and useful to sum up [dBm]-, [dBμV]-, or [dBμA]-values with gain G values in [dB]:

  

• Subtraction of two absolute levels. Subtracting two absolute [dBm]-, [dBμV]-, or [dBμA]-values is equivalent to computing the ratio of their linear values:

  

3.3 Summary

The Tables 3.1, 3.2, 3.3 and 3.4 present conversions between different absolute decibel levels, linear rations to relative decibel values and vice versa.

Table 3.1 Conversion between [dBμV], [dBμA], and [dBm] for systems with system impedance Z ₀ = 50 Ω[1]

Table 3.2 Conversion from (linear) ratios to [dB] values and vice versa

Table 3.3 Conversion between voltages in [V] and [dBμV], between currents in [A] and [dBμA], and between power in [dBm] and [mW] for different system impedances Z ₀ [1]

Table 3.4 Conversion between field strengths [V/m], [A/m], [dBμV/m], [dBμA/m], [dBpT], and [μT] for free-space (far-field) where Z ₀ = 377 Ω [1]