On Stability of Electronic Circuits

Transcription

1 roceeding of the th WSAS International Conference on CIUITS On Stability of lectronic Circuit HASSAN FATHABADI lectrical ngineering Department Azad Univerity (South Tehran Branch) Tehran, IAN NIKOS MASTOAKIS Indutrial ngineering Department Technical Univerity of Sofia, Sofia, BULGAIA Abtract: -In thi paper, a neceary ufficient condition for robut tability of electronic circuit at high frequency which contain differential pair (emitter-coupled pair) i propoed In fact, when emitter-coupled pair are ued a one input ignal one output ignal, they have uncertainty in their tranfer function at high frequency ven a will be hown, thi uncertainty can be caued intability in cloed loop electronic circuit at high frequency The uncertainty will be modeled a multiplicative perturbation in the tranfer function of the differential pair at high frequency Baed on thi uncertainty model, a neceary ufficient condition for robut tability of above electronic circuit at high frequency will be preented Thi condition guarantee internal tability of electronic circuit at high frequency with repect to above uncertainty Key- Word: - Stability, robut, uncertainty, perturbation, electronic, circuit Introduction A group of electronic circuit that contain emittercoupled pair are very important They have many application in amplifier, communication circuit etc [] The differential pair or emitter-coupled pair i an eential building block in modern integrated circuit (IC) amplifier [] Thi circuit i hown in Fig Thi paper conider thi group of electronic circuit at high frequency A we know, emittercoupled pair ha two main propertie: A) It reject mot of common noie that arrive from two bae B) It tranfer characteritic ha linear region larger than a tage which conit of only one BJT A we know, the property (A) i decreaed by increaing frequency [] About property (B), in fact, for mall difference voltage ( V d 4 V = in Fig), the differential pair T V V V d behaved a a linear amplifier [] Conider differential pair which hown in Fig When V o = V O, V i = V V =, differential pair ha one input ignal(v i ) one output ignal ( V ) a hown in Fig Thu, we can conider differential pair a SISO (Single Input Single Output) block which hown in Fig Correponding author addre: Haan Fathabadi, No 7-Ghazanfar Khav Alley - Metry Jey Blover- Tehran-Iran OBox: mail: h4477@hotmailcom Fig mitter-coupled pair ISSN: ISBN:

2 roceeding of the th WSAS International Conference on CIUITS Uncertainty in ytem i one of the mot difficult problem [], [4], [5] Almot, all ytem, uch a linear, nonlinear, dicrete, neural etc have uncertainty in their tranfer function [6], [7], [8] In recent year, there are many reearche in analyi of circuit robut tability [9], [], [] In thi paper, we how, the block which hown in Fig ha uncertainty at high frequency Thi uncertainty i in the place of it pole We conider only domain pole it uncertainty will be modeled a multiplicative perturbation at high frequency in the tranfer function of the above block Then bae on thi uncertainty model, we will tudy the robut tability of the cloed loop electronic circuit which contain above block a a tion of their plant at high frequency Finally a neceary ufficient condition for robut tability tabilization of above electronic circuit at high frequency will be propoed Fig Differential pair with one input one output Multiplicative erturbation Suppoe that the nominal plant tranfer function i conider perturbed plant tranfer function of the form ( = [ + Δ( W ( ] () Here W ( i a fixed table tranfer function, the weigh, Δ ( i a variable table tranfer function atifying Δ( ) further more, it i aumed that no untable pole of are canceled on forming ( [8] Thu, ( ) have the ame untable pole Such a perturbation Δ ( i aid to be allowable The idea behind thi uncertainty model i that Δ ( W ( i normalized plant perturbation away from (): = Δ( W ( () hence if Δ( ), then W (, ω () So W ( provide the uncertainty profile The inequality () decribe a dik in the complex plane In fact, at each frequency the point lie in the dik with center radiu W ( Typically, W ( i a (roughly) increaing function of ω In other word, uncertainty increae with increaing frequency The main purpoe of Δ ( i to account for phae uncertainty to act a a caling factor on the magnitude of the perturbation Thu, thi uncertainty model i characterized by a nominal plant together with a weighting function W ( [] Fig SISO equivalence block erturbed Tranfer Function of Differential air Conider differential pair which hown in Fig We define: V V ( V ( ), V od id ( = Δ ( VO ( VO (, (4) = Δ ISSN: ISBN:

3 roceeding of the th WSAS International Conference on CIUITS [ V ( + V ( ] V ic ( = Δ, [ VO ( + VO ( ] Voc ( = Δ (5) Δ V ( Δ od Voc ( ADM ( =, ACM ( = (6) Vid ( Vic ( where V id ( i differential-mode input V od ( i differential-mode output V ic ( i common-mode input V oc ( i common-mode output A CM ( i common-mode gain A DM ( i differential-mode gain A we aw, when V o = V O, V i = V differential pair which hown in Fig i obtained Thu, we have Vo ( = Vod ( + Voc ( o Vo ( = ADM ( Vid ( + ACM ( Vic ( (7) V ( = [ A ( A ( ] V ( (8) o DM + From (8), we can write the tranfer function of the differential pair which hown in Fig a Vo ( ADM ( = = [ ] ACM ( (9) Vi ( ACM ( Alo we know that [] A CM ( ( + ) () where C are output capacity reitance of current ource, repectively The differential-mode gain i obtained from following equation []: g m ADM ( = () ( + )( + ) p p where i tranfer conductance of BJT Alo p p are pole of above tranfer function In fact, p >> Thu, we can ignore of p conider only domain pole CM p i So we have p A DM gm ( () ( + ) p A we know, domain pole following equation []: p = { C + C μ where = ( + r S b ) r [( + g S m p C i computed from ) + ] + C () In above quation i total reitance of differential mode input ignal ource r, rb, C, Cμ Cc are hown in Fig4 which i equivalence circuit of BJT Fig 4 quivalence circuit of BJT It i reminding that, alo i upper (high) half-power frequency of differential pair Jut a we ee, p i a multivariable function of r, rb, C In a BJT, r i input dynamic C μ r b reitance, i bae pin reitance, i internal capacity between internal bae internal emitter Cμ i internal capacity between internal bae internal collector On the other h, we know r, r, C C are related to internal propertie of BJT, uch a ( width of bae ), ND (concentration of donor atom, N A (concentration of acceptor atom, or (contant ditribution of minority Dn Dp p carrier, β etc [], [] b C c μ W B } ISSN: ISBN:

4 roceeding of the th WSAS International Conference on CIUITS From () above note, we conclude that, domain pole p i changed when a BJT in emittercoupled pair i replaced by another BJT (for example, when we repair electronic circuit which contain thi differential pair) Thu, domain pole p ha uncertainty By replacing () () in equation (9), it i not difficult to verify that gm ( = [+ ( )( )]( )( + ) (+ ) + 4 p (4) So equation (4) i the tranfer function of the differential pair which hown in Fig In (4), we define Δ ( =, (5) ( + ) W g = p m ( (6) + A we ee in (4), lim jω) = lim ( + jω ) ω ω 4 So at high frequency ( ω p ), we have ( = ( + ), (7) 4 ( i the nominal tranfer function of ( ) Conider equation (8), in fact, at low frequency ( ω C ), we have ADM ( jω ACM ( jω, in the other word ADM ( jω CM( jω) =, ACM ( jω where CM ( jω) i "Common Mode eject atio" When ω i increaed, at ω C, the CM( jω) i decreaed with lope - db Dec for ω p, the CM( jω) i decreaed with lope -4 db Dec [] So for ω p, we have ADM ( jω ACM ( jω from (8), it follow that ( ACM ( = ( + ), 4 thi i the ame reult which ha been obtained in (7) ven, reader can note that, although at low frequency, the phae of ( jω ) i about, at high frequency, it phae i about 8 Thi caue to change negative feedback to poitive feedback thu, intability in cloed loop electronic circuit at high frequency A we ee, in (5), Δ( i variable (becaue of uncertainty of ) table tranfer function with p Δ( ) in (6), W ( (the weigh) i fixed table tranfer function Alo in (7), i the nominal tranfer function with no canceled untable pole in forming ( ) By replacing (5), (6), (7) in (4), ( ) i become a the form ( = [ + Δ( W ( ],which i the ame equation () a we aw, i multiplicative perturbed tranfer function Thu, the differential pair which hown in Fig or it equivalence SISO block diagram which hown in Fig at high frequency, ha multiplicative perturbed tranfer function ( ) which ha been preented in (4) with Δ (, W ( which have been obtained in (5), (6) (7), repectively 4 Internal Stability Conider a baic feedback loop a hown in Fig 5 The ignal hown in Fig5, have the following interpretation: r reference or comm input y output meaured ignal n enor noie d, d external diturbance x, x, x, x4 tate variable,, forward plant F feedback tranfer function or the tranfer function of enor Note that, all of above ignal tranfer function are in Laplace domain For example ( =, x ( = x etc In Fig5, we have ISSN: ISBN:

5 roceeding of the th WSAS International Conference on CIUITS x x x x 4 = r Fx = d = d = n + x 4 + x (8) + x ( F in Fig 5) Then, at leat one internal ignal of thi electronic circuit i unbounded Thi mean that, at leat one ignal i clipped Conequently, thi electronic circuit can not operate a linear circuit or amplifier (about the operating point) In matrix form thee are Fig 5 Baic feedback loop F x r x d = (9) x d x4 n Thu, the ytem i well-poed if only if the above 4 4 matrix i noningular, that i, the determinant + F i not identically equal to zero [] From (9), it follow that x x = x + F x4 F F F F r F d F d n () Definition : If the ixteen tranfer function in () are table, then the feedback ytem which hown in Fig 5 i aid to be internally table [] A a conequence, if the exogenou input are bounded in magnitude, o too are x, x, x, hence u, y v So, for an electronic circuit, internal tability i very important, becaue it guarantee bounded internal ignal for all point of electronic circuit Suppoe that, a electronic circuit ha BIBO tability but don t have internal tability In other word, there i at leat one pole zero cancellation in e( when the cloed loop tranfer function of the electronic circuit i formed 5 obut Stability Suppoe that, in Fig 5 i replaced by ( ) which ha uncertainty belong to a et uch a M Thi i hown in Fig 6 Definition : The ytem hown in Fig 6 ha robut tability if it provide internal tability for every ( M [] Aume that, in Fig 6 ( ) ha uncertainty a multiplicative perturbation form In other word, ( ) i decribed with equation () Theorem : The cloed loop ytem which hown in Fig 6 ha robut tability iff ( W ( () + roof: ( ) Aume that ( W ( Contruct the + Nyquit plot of L ( =, indenting D to the left around pole on the imaginary axi Since the nominal feedback ytem i internally table, we know thi form the Nyquit criterion: The Nyquit plot of L doe not pa through - it number of counterclockwie encirclement equal the number of pole of in e( plu the number of pole of ( ) ( in e( Fix an allowable Δ ( Contruct the Nyquit plot of = [ + Δ( W ( ] No additional indentation are required ince Δ( W ( introduce no additional imaginary axi pole We have to how that the Nyquit plot of [ + Δ( W ( ] doe not pa through - it number of counterclockwie encirclement equal the number pole of [ + Δ( W ( ] in e( plu the number of pole of F ( in e( ; equivalently, the Nyquit plot of [ + Δ( W ( ] doe not pa through - encircle it exactly a many time a doe the ISSN: ISBN:

6 roceeding of the th WSAS International Conference on CIUITS Nyquit plot of We mut how, in other word, that the perturbation doe not change the number of encirclement The key equation i + [+Δ( W( ] = [+ ][ +Δ( W( ( ] + () Since ( ( Δ( ) W( W( + + () ( The point + Δ( W ( alway + lie in ome cloed dik with center, radiu, for all point on D Thu from (), a goe once around D, the net change in the angle of + [ + Δ( W ( ] equal the net change in the angle of + Thi give the deired reult ( ( ) Suppoe, W ( + We will contruct an allowable Δ( that detabilize ( the feedback ytem Since i + trictly proper, at ome frequency ω, we have ( jω) W ( jω) jω) = + jω) (4) Suppoe that ω =, then () W () ) i a real number, either + + ) or - () If Δ () = W () ), then Δ () + ) i allowable () + Δ() W () ) = + ) (5) From () with repect to (5), we conclude that, the Nyquit plot of [ + Δ( W ( ] pae through the critical point, o the perturbed feedback ytem i not internally table 6 obut Stability of the Circuit Conider an electronic circuit that it block diagram at high frequency hown in Fig 7 A we ee, thi circuit contain a differential pair, which ha one input ignal one output ignal A we obtained in (4), the tranfer function of thi differential pair at high frequency i ( ) which ha uncertainty a multiplicative perturbation with Δ ( W ( that were derived in (5) (6) Alo the nominal tranfer function of ( ), i which wa obtained in (7) Fig 6 Baic feedback loop with uncertainty In Fig 7, ( ) ( are the total tranfer function of all tage which are located before after differential pair, repectively Alo F ( i the tranfer function of negative feedback in circuit Theorem : The electronic circuit which hown in Fig 7, ha robut tability with repect to uncertainty of ( ) ( the tranfer function of thi differential pair) at high frequency, iff g m C 4 ( ( ( + C ) ( ( (6) roof: ( ) A we ee, the block diagram of electronic circuit which hown in Fig 7 the block diagram of the cloed loop ytem which hown in Fig 6, both are the ame ISSN: ISBN:

7 roceeding of the th WSAS International Conference on CIUITS β g m = Ω, r = r K = = gm = 76 K = 7K By replacing above value in (), we obtain that, p 66 M rad For BJT ( ), we have T Fig 7 Block diagram of electronic circuit at high frequency In fact, the block with multiplicative perturbed tranfer function ( ) in Fig 6 ha been replaced by differential pair which ha multiplicative perturbed tranfer function ( ), a wa obtained in (4) with Δ (, W ( that were derived in (5), (6) (7) So, from theorem, by replacing (6) (7) in (), we have gm ( + C ( + ) ( ( 4 ) ( + ) ( ( 4 (7) It i not difficult to verify (6) Thi complete the proof xample : Conider the imulated circuit which hown in Fig 8 with following parameter For BJT ( T ): C C = Cc = 5 pf, μ = β =, r Ω V T 5 mv b T For BJT ( T ): C C = Cc = pf, μ = β =, r Ω V T 5 mv b We have ( i the ymbol of the parallel reitance S = 4 7K K 8K = 76K, C = K K = 5 45K T For biaing of the BJT ( T ), we have IC = I ma T C = 5 T So we have I C = ma, g mt = Ω, 5 r = β = 5 K, gmt LT = 4 7K K 8K = 76K, = [(5K 68K 68K) + rb ] r = K g mt LT AV = = 7 + g mt (68K 5K) By replacing above value in following equation []: p LT LT { C + Cμ [( + gmtlt ) + ] + Cc} (8) the domain pole of T, i computed a p 7 M rad Comparing imulated circuit in Fig 8 the block diagram in Fig 7, it follow that Fig8 Simulated electronic circuit ISSN: ISBN:

8 roceeding of the th WSAS International Conference on CIUITS 7 68K ( =, F ( = 57 68K + 5K ( = Taking thee other value in (6), we have 7 57 ( + ) = [ ( ] ( + ) 6 7 up (4 6 ω) So, for ω > 4888M rad >> p g m C 4 ( ( ( + C ) ( ( thu, imulated circuit ha robut tability alo it i untable for ω < 4888M rad The input of the imulated circuit i mall ignal inuoidal form A we ee, the reult of imulation which hown in Fig 9, validate the preented propoal 7 Concluion In thi paper, a group of electronic circuit which contain an emitter-coupled pair with one input one output ignal, ha been conidered We howed that, the tranfer function of thi differential pair at high frequency ha uncertainty Thi uncertainty ha been modeled a multiplicative perturbation Bae on thi multiplicative uncertainty model, the neceary ufficient condition for robut tability at high frequency ha been preented in (6) Thi condition guarantee internal tability of electronic circuit with repect to the uncertainty of the tranfer function of differential pair at high frequency A direct benefit of the reult in thi paper i a condition which mut be atified, when we deign integrated circuit at high frequency It i an intereting reearch topic to obtain imilar condition for other differential pair which contain MOSFT JFT Acknowledgment: Implementation of the project ha been done in Azad Univerity (South Tehran Branch) Fig9 The reult of imulated circuit a) b) c) ω = 6667 M rad ω = 49 M rad ω = 667 M rad eference: [] Gray, J Hurt, S H Lewi G Meyer, Analyi Deign of Analog Integrated Circuit, New York: Wiley, [] J Millman A Grabel, Microelectronic, nd ed New York: Mc Graw-Hill, 988 [] C Yuhua, M J Deen C Chih-Hung, MOSFT modeling for F IC deign, I Tran lectron Device, vol 5, 5, pp 86 - [4] MNA arlakci, obut tability of uncertain time-varying tate-delayed ytem, ISSN: ISBN:

9 roceeding of the th WSAS International Conference on CIUITS I roc Control Theory Application, vol 5, [9] A Buonomo A Lo Schiavo, Analyi of 6, pp emitter (ource)-coupled multivibrator, I [5] Sanqing Hu Jun Wang, Global robut tability Tran Circuit Sytem I, vol 5, 6, pp of a cla of dicrete-time interval neural network, 9- I Tran Circuit Sytem I, vol 5, 6, pp [] A Ghulchak A antzer, obut control 9-8 under parametric uncertainty via primal-dual [6] A Schmid Y Leblebici, obut circuit convex analyi, I Tran Automatic Cont, ytem deign methodologie for nanometer-cale vol 47,, pp 6-66 device ingle-electron tranitor, I Tran [] S Dagupta, J arker, B D O Very Large Scale Integration (VLSI) Sytem, vol, Anderon, F J Krau M Manour, 4, pp Frequency domain condition for the robut [7] N Ozcan S Arik, Global robut tability tability of linear nonlinear dynamical analyi of neural network with multiple time delay, ytem, I Tran Circuit Sytem, vol I Tran Circuit Sytem I, vol 5, 6, pp 8, 99, pp [] J C Doyle, B A Franci A [8] L Liu J Huang, Adaptive robut tabilization Tannenbaum, Feedback Control Theory, New of output feedback ytem with application to Chua' York:Macmillan,99 circuit, I Tran Circuit Sytem II, vol 5, 6, pp 96-9 ISSN: ISBN: