14 What You Should Know About Decibels

Transcription

1 14 What You Should Know About Decibels Every year dozens of students who should know much better lose a lot of exam marks because they haven t grasped the concept of the decibel. This is a great pity: a good understandg of the decibel is essential to just about any branch of electronic engeerg. To start with a defition: the decibel is ten times the logarithm (to the base ) of a power ratio 1. Remember that it s a power ratio, always a ratio of powers, and you won t go far wrong. For example: suppose you had an amplifier with an put signal of mw and an output power of 0 W. The power ga (usually just called the ga, or sometimes the lear ga to emphasise that the figure is not decibels) is the output power divided by the put power: output power 0 W power ga 000 put power 0.00 W (0.1) Sce this is a ratio of two powers, we can also express it db: 0 power ga db log log db 0.00 (0.) That s it. It s not difficult. However, there is a huge amount of confusion surroundg the use of the db, most of which arises due to some short-cuts made by engeers a particular field of electronics who do not always state all of their assumptions, and leave some thgs as ambiguous. It s very useful to know the correspondg values db and lear power ratios for some common cases: for example a factor of ten is a change db of, and a factor of is a change db of 3. log (0.3) log 3.01 (0.4) Engeers often talk about a signal beg 3 db down when they mean it s got half the power, and it s useful to know this without havg to reach for your calculator. Similary, 6 db up means four times the power, 9 db up is eight times the power, 1 db up is sixteen times the power, and so on. The relationship between the lear and decibel scales of power ga are shown the followg diagrams: 1 It derives from a unit called a Bel, which was just the logarithm (to the base ten) of the power ratio. However, this was found to be a rather large unit, and the deci-bel, which is one-tenth of a Bel, was used stead. 006 University of York Page 1 3/0/0

2 15 db db 50 db 0 db +5 db + db +15 db +0 db +5 db Figure 14-1 Relationship between db and Lear Power Ga db 7 db db 11.8 db 13.0 db 14.0 db 14.8 db 15.4 db 16.0 db Figure 14- Relationship between Lear Power Ga and db A few more examples: Lear Power Ratio Power Ratio db 1 0 db db db db 0 0 db db 3 db 4 6 db db 14.1 Usg Voltage Ratios to Calculate db In some textbooks, you will see the ga db beg calculated usg a formula like this: Vout ga (db) 0log V (0.5) Be careful if you see this. It is common practice, but it can be misleadg, as it is not always correct. It relies on the (often un-stated) assumption that the impedance the circuit at the put and the output is the same. Remember, a db is always a power ratio. If all you ve got is the voltage, then you need to calculate the power usg: V V out power power out Z Z (0.6) out 006 University of York Page 3/0/0

3 In the most general case, when the put voltage has an impedance Z and the output has an impedance Z out, then the power ga can be expressed as: Vout V Vout V Z out out power out power ga power Z Z Z (0.7) Now if, and only if, the output impedance is equal to the put impedance, then we can write: Vout Vout Vout V V V power ga db log log 0log (0.8) But, and it s worth emphasisg this pot yet aga, if the put impedance is not equal to the output impedance, then this simple formula ga (db) = 0log (V out /V ) is not valid. In the general case, we d have to write: V Z power ga db 0log log out out V Z (0.9) In many cases simple circuit analysis, the impedance at a certa pot the circuit is either not given, does not really exist (for example if the amplitude of a signal is stored as a digital number a computer or DSP chip) or is not easily calculable. In these cases, the impedance is often normalised to be one, and the 0log (V out /V ) formula is used. It usually works, but don t forget about the exceptions. 14. dbmw, dbw, dbw, etc A db is a unit of power ratio. If you want to express a real power terms of db, then you need some reference power to use, so the power can be expressed as a ratio with the reference power. For example, a real power of 30 W could be expressed as a power that is 30 times greater than a power of one watt. In these cases, where a reference unit of one watt is assumed, the result of the ratio is expressed terms of the units of dbw. P power dbw log log PW 1 W (0.) where P(W) is the power of the signal expressed Watts. For example, 0 Watts is 0 dbw, sce log (0) = 0, and one milliwatt (1 mw) is 30 dbw, sce log (0.001) = 30. To calculate the power dbw, just take times the logarithm (to the base ) of the ratio of the power to one Watt. So, for example, W is 3 dbw, W is dbw, and 500 mw is 3 dbw. The reference power doesn t have to be one Watt (sometimes one Watt is a rather large power to use, and smaller units of power are more convenient). For example, if the reference power used is one microwatt, then the units become dbw, and, for example, a power of 1 mw could be written as 30 dbw, sce: 3 P power dbw log log 6 30 dbw 6 (0.11) 006 University of York Page 3 3/0/0

4 Similarly, usg a reference power of 1 mw leads to the unit dbmw, and therefore: 1 mw 30dBW =0dBmW =30 dbw (0.1) The dbm, dbu and db At first sight these common units don t make any sense. A db is a power ratio. A isn t a unit of power, it just means a factor of -6, and m means -3, and what is u about? While dbm is usually shorthand for dbmw, if you thought that db was just shorthand for dbw, then most cases you d be wrong. There are several different uses of the dbu or db. Sce occasionally the is written as a u (due to difficulties writg Greek letters some situations), often the only way you can tell what is meant is by context. For example: 1. Audio Engeerg: dbu means use as a reference power V (rms) an impedance of 600 ohms (a standard impedance used for audio work). So this case, the reference power is one milliwatt (sce 0.775^ / 600 = -3 ).. Radio propagation/emc: Here, db is most often used as a shorthand for dbv/m. It s impossible to defe a reference power this case although you can defe a reference power density ( Watts per square metre). In the case of free space, it corresponds to a power density of W/m. 3. Other fields: db is a shorthand for dbw. In this case the reference power is one microwatt. It is confusg, but if you remember that a db is always calculated usg a power ratio, you should be fe you just have to know / remember / work out what the reference power is each case Usg db If the decibel is so confusg, why does anyone bother usg them? The answer is that they make calculations much easier by allowg multiplications and divisions to be replaced by additional and subtractions. Before the days of pocket calculators that made a lot of difference, and it s still easier to add and subtract your head than to multiply and divide. It s also convenient to be able to express a very wide range of possible values without havg to use exponents, or a large number of zeros after the decimal pots. (It s very easy to miscount the number of zeros a large number, and that could be disastrous.) Many formulas can be expressed conveniently terms of db by takg the logarithm of the formula and multiplyg by ten. For example, consider the formula for the received power to a matched receiver for a radio lk free space: P r Pt Gt Gr (0.13) 4 d / where P r is the received power, P t the transmitted power, G t and G r the gas of the transmit and receive antennas, d the distance between the antennas and the wavelength. Take ten times the logarithm (to the base ten) of this equation, and we get: 006 University of York Page 4 3/0/0

5 P P G G d log log log log 0log 4 / (0.14) r t t r If the transmit power P t and receive power P r are expressed Watts, then we can write this as: dbw dbw db db 0log 4 / P P G G d (0.15) r t t r Specifyg the gas of the antennas decibels, and the transmitted and received powers dbw saves a lot of multiplication and division, replacg these operations with additions and subtractions, and speedg up the calculations significantly. Note that the power transmitted and power received are expressed terms of dbw, but the gas of the antennas are expressed terms of db. Sce antenna gas are defed terms of the ratio of two powers (the power a given direction divided by the power that would have been transmitted this direction by an omnidirectional antenna), this is fe we don t need a reference power for a simple ga. Note that we could equally well have written: dbμw 60 dbw db db 0log 4 / P P G G d (0.16) r t t r sce 60 dbw = 0 dbw, and 60 + P t dbw = P t dbw. Similarly: dbμw 30 dbmw db db 0log 4 / P P G G d (0.17) r t t r sce 30 + P t dbw = P t dbmw. A useful rule: if you fd you are dividg or multiplyg two quantities expressed terms of db or dbw, then you re almost certaly dog somethg wrong. Equally, if you fd you are addg two quantities both expressed dbw, you are probably dog somethg wrong: this is the equivalent of multiplyg two powers together Some Examples 1) An amplifier has a power put of Watts, and an output power of 8 Watts. What is the ga db? 8 log log (4) 6.0 db In practice, it s usually just quoted as 6 db. This is a very useful thg to know a factor of two power is 3 db, a factor of four power is 6 db, a factor of eight power is 9 db, etc. This is accurate enough for most purposes. ) An amplifier has a voltage put of Volts, and an output voltage of 8 Volts. What is the ga db? One exception is if you re usg the log-normal distribution to represent a distribution of receive signal powers, and usg the Q or erf functions to calculate the probability of a fade. In these cases, the standard deviation is expressed db, and to work out the argument of the Q-function requires a division by the standard deviation. 006 University of York Page 5 3/0/0

6 The simple answer is: it s impossible to tell. You are not told what the put and output impedances are, so you can t tell what the put power or output power are. 3) An amplifier has a voltage put of 3 Volts with an 50-ohm impedance, and supplies an output voltage of 4 Volts to a 0-ohm impedance. What is the ga db? Pout Vout R 4 50 log log log 0.51 db P 0 Rout V 3 Note that even though the voltage at the output is greater than the voltage at the put, the power the output is less than the power the put, so the ga db is negative. 4) Convert the formula E = P / to decibel form, where P is the power Watts and E is the electric field strength V/m. Take times the log of both sides, and we get: E P log log log E P 0log dbw ) Express a power of 70 dbw dbw, dbmw, and dbv assumg an impedance of 0 ohms. Therefore, P( W ) 70 (dbμw) log 6 ( W ) 7 P μw W P( W ) P (dbw) log log dbw 1( W ) ( W ) 4 P (dbmw) log log 3 40 dbmw ( W ) ( W ) 15 P (db V 0ohms) log log 150 dbμv 6 / Problems 1) Express the followg lear power ratios db: 0.1; -7 ; 00; 16; ) Express the followg quantities db lear power ratios: 13 db; 9 db; 30 db. 3) A high power amplifier is rated as havg a ga of 36 db. It receives an put power of mw. What is the output power? 006 University of York Page 6 3/0/0

7 4) A radio transmitter transmits a total of 0 W. A radio receiver, listeng to the transmitter receives pw. What is the loss of the radio channel, expressed db? 5) An impedance-matchg transformer accepts a se-wave put from a 0-ohm transmission le at 0. V peak-to-peak, and outputs a 0.1 Volt peak-to-peak signal to a 50- ohm matched transmission le. Express the loss of this transformer db. (Note that the mean power a se wave of peak-to-peak amplitude A Volts on a transmission le of amplitude Z ohms is A /8Z Watts.) 6) The ga of antennas is often expressed terms of dbi, the power ga relative to an isotropic antenna. The ga of the antenna used is dbi, and the radio signal strength received at a receiver is 50 pw. The antenna is then replaced by a new antenna, and the received signal strength creases to 00 pw. What is the ga of the new antenna dbi? 7) A se wave of amplitude Volts is sent to a long length of cable. At the other end, 50 meters away, the se wave emerges with an amplitude of 0. mv. What is the attenuation of the cable, expressed db/m (decibels per metre)? 006 University of York Page 7 3/0/0