The Decibel and its Usage

Transcription

1 The Decbel and ts Usage Consder a two-stage amlfer system, as shown n Fg.. Each amlfer rodes an ncrease of the sgnal ower. Ths effect s referred to as the ower gan,, of the amlfer. Ths means that the sgnal outut ower from an amlfer s greater than ts sgnal nut ower. Ths ower gan may be exressed as: 3 / ower o/ ower Fg. out n or o For the system shown, f the nut ower s mw, and the gans of the two amlfer stages are 0 tmes and 00 tmes resectely, then the fnal outut ower, 3, may be determned as follows. therefore, 3 ; so watt mw W ; so watt therefore, From these results t may be seen that the outut ower of the system s 000 tmes that of the nut.

2 The Decbel and ts Usage In other words, oerall sgnal ower gan, or, oerall sgnal ower gan, In general, when amlfers (or other deces) are cascaded n ths way, the oerall gan (or loss) s gen by the roduct of the nddual stage gans (or losses). Note: The effcency of any machne or dece s defned as the rato of ts outut ower to ts nut ower. Howeer, ths does NT mean that an amlfer s more than 00% effcent! The reason s that only the sgnal nut and outut owers are consdered when quotng the ower gan. No account s taken of the comarately large amount of ower njected from the d.c. ower suly, wthout whch the amlfer cannot functon. In ractce, small sgnal oltage amlfers wll hae an eff cency fgure of less than 5%. ower amlfers may hae an effcency n the order of 70%. It s often more conenent to exress ower gan ratos n a logarthmc form, known as the Bel (named after lexander Graham Bell). Thus a ower gan exressed n ths way s: o log Bel where nut ower; o outut ower; and logarthms to the base 0 are used. The Bel s an nconenently large unt for ractcal uroses, so the decbel (one tenth of a Bel) s used. Hence, o 0 log decbel () The unt symbol for the decbel s. For the two-stage amlfer system consdered, the ower gans would be exressed as follows: and 0 log0 0 0 log log Note that the oerall system gan,, when exressed n s smly the sum of the nddual stage gans, also exressed n.

3 The Decbel and ts Usage 3 Worked Examle communcatons system, nolng transmsson lnes and amlfers, s llustrated n Fg.. Each secton of transmsson lne attenuates (reduces) the sgnal ower by a factor of 35.5%, and each amlfer has a gan rato of 5 tmes. Calculate the oerall ower gan of the system as (a) a ower rato, and (b) n decbels. / o/ Fg. (a) o For each lne, so, total loss o For each amlfer, 5 so, total gan therefore, oerall gan tmes ns (b) For each lne, attenuaton s 0 log so total attenuaton For each amlfer, gan 0 log 5 7 so total gan 73 * Hence, oerall gan of the system 8 3 ns * Note that a loss or attenuaton exressed n has a negate alue; whereas a gan has a oste alue. further ont to note s that f the gans and attenuatons of the system had orgnally been exressed n, then the calculaton would smly hae been as follows: erall system gan (3 7) (4 4.5) 3 ns The aboe examle llustrates the conenence of usng decbel notaton, snce t noles only smle addton and subtracton to determne the oerall gan or attenuaton of a system. It has also been shown that a gan of tmes s equalent to a ower gan of 3 (more recsely, 3.0 ). It s left to the reader to confrm, by usng

4 4 The Decbel and ts Usage a calculator, that an attenuaton of tmes (.e. a gan of 0.5) s equal to 3. Ths fgure of 3 wll frequently be met when dealng wth the frequency resonse cures for amlfers and other frequencydeendent crcuts, such as seres and arallel tuned crcuts. In order to gan a feel for ower gans and losses exressed n, the followng lst shows the corresondng ower gan ratos tmes tmes tmes 0 gans 0 tmes 0 tmes 3 tmes tmes 3 0. tmes 0 losses 00. tmes tmes tmes 40 Worked Examle (a) Conert the followng gan ratos nto. () 50, ( ) 50, ( ) 0.4 (b) Conert the followng gans and losses nto ratos. () 5, () 8, () 5 (a) () 0 log 50 4 ns () 0 log 50 7 ns () 0 log ns (b) () 5 0 log (rato) so.5 log (rato) therefore, (rato) antlog.5 36 tmes ns () 8 0 log (rato) 08. log (rato) therefore, (rato) antlog tmes ns () 50 log (rato).5 og (rato) therefore, (rato) antlog (. 5) ns

5 The Decbel and ts Usage 5 lthough the decbel s defned n terms of a ower rato, t may also be used to exress both oltage and current ratos, roded that certan condtons are met. These condtons are that the resstance of the load s the same as that of the source,.e. the condtons for maxmum ower transfer. Consder such a system, whereby the two resstance alues are ohm, the nut oltage s olt, and the outut oltage s olt. Let the corresondng currents be I and I amere. watt and gan 0 log 0 log watt hence, oltage gan, 0 log lso, usng the fact that I watt and I watt t s left to the reader to erfy that: I current gan, 0 log () I s shown n Further Electrcal and Electronc rncles, Chater, a tuned crcut used as a ass-band or sto-band flter has a bandwdth. The same concet also ales to a.c. amlfers. In the latter case, the frequency resonse cure for a oltage amlfer would be smlar to that shown n Fg. 3. The bandwdth s defned as that range of frequences oer whch the oltage gan s greater than, or equal to, m /, where m s the md-frequency gan. smlar resonse cure for the amlfer current gan could also be lotted. Now, ower gan oltage gan current gan or, gan m m Bandwdth f f f(hz) Fg. 3

6 6 The Decbel and ts Usage The cut-off frequences, f and f, defne the bandwdth, and at these frequences, the current and oltage gans wll be: thus m m and m m m resectely These onts on the resonse cure are therefore referred to as ether the cut-off onts, the half-ower onts, or the 3 onts. Worked Examle 3 n amlfer s fed wth a 50 m, 00 µ sgnal. The amlfer has a oltage gan of 75 tmes and a current gan of 50 tmes. Determne (a) the oltage, current and ower gans, exressed n, and (b) the outut oltage, current and ower. You may assume that nut and outut resstances are the same alue ; I ; 75; 50 (a) 0 log ns 0 log ns tmes so, 0 log ns (b) olt ns I I am m ns I watt mw ns Worked Examle 4 n amlfer has a oltage gan of 0 tmes at a frequency of 50 Hz; 60 tmes between khz and khz; and 5 tmes at 35 khz; Determne (a) the oltage gan, exressed n decbel, for each case, and (b) the oltage gan at the lmts of ts bandwdth. (a) at 50 Hz: 0 log 0 0 ns at md-frequences: 0 l o g6 0d B ns at 35 khz: 0 log ns (b) The lmts of the bandwdth occur when s 3 down on the mdfrequency alue, or /. m Therefore, ns or, m m tmes hence, 0 log ns

7 The Decbel and ts Usage 7 The decbel s defned n terms of a ower rato. Thus, the ower outut of a dece cannot be quoted drectly n. For examle, to say that an amlfer has an outut ower of 0 s comletely meanngless. In ths case t only makes sense to say that the amlfer has a ower gan of 0. Howeer, actual alues of outut ower may be exressed n decbel form roded a known reference ower nut s used. In electronc work, ths reference ower s one mllwatt ( mw), and owers exressed n ths way are desgnated as m. Consderng such an amlfer, t could be sad to hae a ower outut of 0 m. Ths would mean that ts actual outut ower was 0. W. Ths fgure s obtaned thus: 0 log 0 m log antlog 00 and snce mw, then W or 00 mw. Sulementary Worked Examle n attenuator crcut has nut and outut oltages of 8.5 and.4 resectely. Determne the attenuaton, exressed n decbels. 8.5 ;.4 attenuaton 0 log decbel 4. 0 log 0 log so, attenuaton ns o Sulementary Worked Examle n amlfer has a ower gan of 40. Exress ths gan as a ower rato log (rato) decbel 4 log (rato) antlog 4 rato so, rato 0 4 ns

8 8 The Decbel and ts Usage Sulementary Worked Examle 3 For the attenuator ad shown between termnals B and CD n Fg. 4, calculate (a) the outut ( ) and nut ( ) oltages, (b) the oltage attenuaton n decbels, and (c) the outut ower. 70 Ω C r 40 Ω E 5 3 L 00 Ω B D Fg. 4 (a) Usng Théenn s theorem and lookng n at termnals CD the crcut wll aear as shown n Fg. 5. E C 70 Ω r 40 Ω 3 o F D EF EF r r EF3 EF Fg Ω Ω ohm ohm To obtan the Théenn generator emf E, consder Fg. 6

9 The Decbel and ts Usage 9 I C 70 Ω r 40 Ω B 3 E o E 5 B D Fg. 6 Total resstance, ( 3) r ( ) ohm Ω E 5 I am E Ir olt 5( ) B B. 05 esstors and 3 act as a otental dder wth B aled across them, so E B olt E.575, and the Théenn equalent crcut s as Fg. 7 C o 58.7 Ω o L 00 Ω E o.575 D Fg. 7 L 00 E olt L ns

10 0 The Decbel and ts Usage Lookng n at termnals B of the orgnal crcut, the resstance seen s llustrated n Fg. 8, and may be obtaned as follows: 3 L GH ohm 3 L GH Ω GH ohm GH Ω G 70 Ω 3 L 00 Ω B H Fg. 8 So the crcut s equalent to that shown n Fg. 9, from whch: r 40 Ω E Ω E olt 5 r so, 0. 7 ns (b) attenuaton 0 log decbel log 0. 7 attenuaton 3 ns B Fg. 9 (c) L watt W ns