Communication Systems. Department of Electronics and Electrical Engineering

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1 COMM 704: Communicaion Lecure : Analog Mulipliers Dr Mohamed Abd El Ghany Dr. Mohamed Abd El Ghany,

2 nroducion Nonlinear operaions on coninuous-valued analog signals are ofen required in insrumenaion, communicaion, and conrol sysem design. he operaions include Modulaion Demodulaion Frequency ranslaion Muliplicaion li n analog-signal processing he need ofen arises for a circui ha akes wo analog inpus and produces an oupu proporional o heir produc Such circui are ermed analog mulipliers. n he following secions, we examine several analog mulipliers ha depend d on he exponenial ransfer funcion of bipolar ransisors. Winer 0

3 he Emier-coupled Pair as a Simple Muliplier From KL around he inpu loop Assume he collecor resisors are small enough ha he ransisor don operae in sauraion Assume he ransisors are idenical so ha s s And assuming high β Winer 0 3

4 he Emier-coupled Pair as a Simple Muliplier he emier-coupled pair, was shown in o produce oupu currens ha were relaed o he c c differenial inpu volage by: Q Q id Winer 0 4

5 he Emier-coupled Pair as a Simple Muliplier Assuming ( id / << id he emier-coupled pair by iself can be used as a primiive muliplier. he curren is acually he bias curren for he emier-coupled pair. Wih he addiion of more circuiry, we can make proporional o a second inpu signal. Winer 0 5

6 he Emier-coupled Pair as a Simple Muliplier K o ( i BE ( on R c cc R c he differenial oupu curren of he emier- coupled pair can be calculaed l o give o C K o id ( BE( on -A circui works as a muliplier under he assumpion ha id is small, and ha i is greaer han BE(on - his ype of circui is ermed a wo-quadran muliplier; he muliplier works in only wo quadrans of he id - i plane id i p id i R Q Q 3 Q Q 4 - Winer 0 6

7 Gilber Cell he Gilber cell muliplier is ou c3-5 - c4-6 a modificaion of he emier-coupled cell, which allows four-quadran c4 muliplicaion. is he basis for mos inegraed circui balanced muliplier sysems. c Q Q n he following analysis, we assume ha he ransisors are idenical. c3-5 c4-6 c3 c5 Q 3 Q 4 Q 5 Q 6 c c6 Winer 0 7

8 Gilber Cell he collecor curren of Q 3 and Q 4 are: C3 C + exp( / C 4 C + exp( Similarly, he collecor curren of Q 5 and Q 6 are: C C C5 C 6 exp( / + exp( / + / ou c3-5 - c4-6 c3-5 c4-6 c3 c4 c5 Q 3 Q 4 Q 5 Q 6 c c c6 he wo curren C and C can be relaed o C + exp( / C + exp( / Q Q Winer 0 8

9 Gilber Cell Subsiuing by C and C o obain expressions for collecor currens C3, C4, C5, and C6 in erms of inpu volages and. C3 C 4 C 5 C 6 [ + exp( / ][ + exp( / [ + exp( / ][ + exp( / [ + exp( / ][ + exp( / + exp( / ][ + exp( / [ he differenial oupu curren is hen given by ou c3-5 - c4-6 c3-5 c4-6 c3 c4 c5 Q 3 Q 4 Q 5 Q 6 c c Q Q c6 C 5 C 4 6 C 3 + C 5 ( C 4 + C 6 ( C 3 C 6 ( C 4 C 5 anh( / anh( 3 / Winer 0 9

10 Pracical Applicaion of Gilber Cell here are hree caegories of applicaion according o he magniude of he applied signals and :. f he magniude of and are kep small wih respec o anh(, /, / he circui behaves as a muliplier, developing he produc. f one of he inpus of a signal ha is large compared o Causing he ransisors o which ha signal is applied o behave like swiches raher han near-linear devices. his effecively muliplies he applied small signal by a square wave, and he circui acs as a modulaor 3. f boh inpu are large compared all six ransisors in he circui behave as nonsauraing swiches. his is useful for he deecion of phase difference beween wo ampliude-limied signals, as is required in phase-locked loops, and is someimes called he phase-deecor mode. Winer 0 0

11 Gilber Cell Muliplier ( / ( /, << -hus for small ampliude signals, he circui performs an analog muliplicaion. Unforunaely, he ampliudes of he inpu signals are ofen much larger han -An alernae approach is o inroduce a nonlineariy i ha predisors he inpu signals o compensae for he hyperbolic angen ransfer characerisic of he basic cell. -he required nonlineariy is an inverse hyperbolic angen characerisic anh - anh - ou c3-5 - c4-6 c3-5 c4-6 c3 c4 c5 Q 3 Q 4 Q 5 Q 6 c c Q Q c6 Winer 0

12 nverse Hyperbolic angen Circui We assume for he ime being ha he circuiry wihin he box develops a differenial oupu curren ha is linearly relaed o he inpu volage. o + K o K CC Here o is he dc curren ha flows in each oupu lead if is equal o zero, and K is he ransconducance of he volage-o-curren converer. he differenial volage developed across he wo diode-conneced ransisors is o + K o K o + K ln( ln( ( S S o K + x anh x ln( x K anh ( Using he ideniy: We ge: anh( / anh( / o K K ( ( o o Q 7 Q 8 olage-curren converer Winer 0

13 Complee Analog Muliplier he circui of differenial-o-single ended converer has a funcion given by: ou ou K 3 K K K 3 o o he oupu volage is hus proporional o produc over a wide range. he consans are usually chosen so ha ou 0. Winer 0 3

14 Gilber Cell as a Swiching Modulaor n communicaion sysems, he need frequency arises for he muliplicaion of a coninuously varying signal by a square wave. his is easily accomplished wih he muliplier circui by applying a sufficienly large signal direcly o he cross-coupled pair. m ( Small-signal inpu m ( m cos w m c c( ( o ( Large-signal g inpu oupu ou c ( ( K[ K n n C A n A n cos w c A n n sin nπ K is he magniude of he gain of he muliplier from he small-signal signal inpu o he oupu ( m m ( ] K [cos( nw c + n w m A n m cos( nw cosw cosw c n w m c Winer 0 4

15 Gilber Cell as a Phase Deecor f unmodulaed signals of idenical frequency wo are applied o he wo inpu, he circui behaves as a phase deecor and produces an oupu whose dc componen is proporial o he phase difference beween he wo inpus. in in & in >> cc R c R c in + in o - + in - - cc Winer 0 5

16 Gilber Cell as a Phase Deecor he average value of he oupu waveform is given by: average Π ( d ( w [ Π 0 o A A ] Π Where areas A and A are as indicaed. hus odc componen in phase deecor oupu average Π φ φ φ [ Rc Rc ] Rc ( Π Π Π Linearized PD his phase demodulaion echnique is widely used in phase-locked loops. Winer 0 6

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