Revision of intermediate electronics

Transcription

1 Revson of ntermedate electroncs ports, feedback and flters Imperal College London EEE

2 Generalsed Thevenn + Norton Theorems: port parameters Amplfers, flters etc have nput and output Input can be voltage or current Output can be voltage or current By conventon current postve nto postve termnal Negatve termnals usually consdered connected together General form of amplfer or flter: I I + Thevenn or Thevenn or + V V Norton Norton Amplfer Imperal College London EEE

3 The voltage amplfer G parameters Also called the reverse hybrd parameters Norton g v g v v =g v Thevenn =g Formal descrpton: g : Input admttance g : Reverse current gan g : Voltage gan g : Output mpedance = gv+ = gv+ g = v + g = gv+ g v g g v v v = = g g v G A voltage amplfer exhbts nonzero reverse current gan! The portreversed amplfer s a current amplfer Reverse path has less than unty power gan. If the reverse current gan s zero, the amplfer s called Unlateral Imperal College London EEE 3

4 The current amplfer H parameters Also called the hybrd parameters Thevenn v h h v Norton Formal descrpton h : Input mpedance h : Reverse voltage gan h : Current gan h : Output admttance v =h v v =h = h+ v = h+ h v = + hv = h+ hv v h h = h h v = H v A current amplfer exhbts a reverse voltage gan! The portreversed amplfer s a voltage amplfer Reverse path has less than unty power gan. If the reverse voltage gan s zero, the amplfer s called Unlateral Imperal College London EEE 4

5 The transconductance amplfer Y parameters Also called the short crcut parameters Norton v y y v Norton =y v Formal descrpton y : Input admttance y : Reverse admttance gan y : transadmttance (gan) y : Output admttance = yv+ = yv+ y v = + yv = yv+ yv =y v y y v v = y y v = Y v A transconductance amplfer exhbts a reverse transconductance gan! The portreversed amplfer s also a transconductance amplfer Reverse path has less than unty power gan. If the reverse gan s zero, the amplfer s called Unlateral Imperal College London EEE 5

6 The transresstance amplfer Z parameters Also called the open crcut parameters Thevenn v z z v Thevenn v =z v =z Formal descrpton z : Input mpedance z : Reverse mpedance gan z : transmpedance gan z : Output mpedance v v = z+ v = z+ z = v + z = z+ z v z z v = z z = Z A transresstance (also called a transmpedance) amplfer exhbts a reverse transmpedance gan! The portreversed amplfer s also a transresstance amplfer Reverse path has less than unty power gan. If the reverse gan s zero, the amplfer s called Unlateral Imperal College London EEE 6

7 Gan of a fully loaded voltage amplfer Z S VS G v G v G V Y L G We start wth the amplfer defnton, plus the sourceload boundary condtons: = gv+ g v = gv+ g = g( vs Z s) gvy L v = v Z v = g ( v Z ) g vy s s s s L = vy L After a lot of algebra we conclude that: v g =, g = g g g g v + g Z + g Y + Y Z s S L g L s Imperal College London EEE 7

8 Cascade connecton: Transmsson Parameters In a cascade connecton, V of network X = V of network X I of network X = I of network X We can defne a new set of parameters so that we have a smple way to calculate the response of cascades of amplfers. A sutable defnton s: v A B v = C D Wth ths defnton, the ABCD parameters of a cascade of two networks are found from the matrx product of the ndvdual ABCD matrces ports labelled for clarty): 3 3 X X X 3 =X X v A B v = C D v A B A B v3 v A3 B3 v3 v = A B v3 C D C D 3 = C3 D 3 3 = C D 3 Imperal College London EEE 8

9 Transmsson (or ABCD) parameters () Note the sgn of and also the reverse sense of sgnal flow. The sgn s chosen so the ABCD matrx of a cascade of two networks s just the matrx product of the ndvdual ABCD matrces (compare ths to the messy loadng calculaton before!) The reverse sense of sgnal flow s to keep the matrx fnte f an amplfer s unlateral. The converson from, say, Y to ABCD follows the same logc as the Y(H) calculaton: v A B v 0 v A B 0 v = = C D Y Y v C D Y Y v A B Y C D Y v A B v = C D =, y = YY YY Y Y Note that all ABCD parameters are nversely proportonal to the gan. Ths s the reason for formally choosng port as the nput port. The ntutve choce of nput at port would make all parameters nversely proportonal to the reverse gan, whch s small, and often not very accurately determned. Imperal College London EEE 9

10 port Connecton rules summary For the exact calculaton of crcut nterconnectons we can use port matrx algebra: Y Y Y +Y G G G +G shuntshunt: add Y matrces shuntseres: add G matrces Z Z +Z H H +H Z H seresseres: add Z matrces seresshunt: add H matrces X X X X cascade connecton: multply ABCD matrces Imperal College London EEE 0

11 more on port parameters We can determne port parameters from the defntons. For example, the Y parameter descrpton states that: = yv+ yv = y v + v v These relatons mply that the y parameters are partal dervatves: y y = y = v v v = 0 v = 0 = y = v v v = 0 v = 0 Note that these are small sgnal parameters, so, e.g. v =0 means that v s kept constant. The y parameter of a transstor s the famlar transconductance. Imperal College London EEE

12 Input and output mpedance of an amplfer We often use Impedance or Admttance to mply whch dervatve we have n mnd. The nput mpedance s the z or g whle the nput admttance the y or h parameter. For example, f the output s open crcuted: v Znput = Zn = = z 0 = / g Y L = = 0 On the other hand, f the output s shorted, then: Y = Y = = y = / h nput n Z L = 0 v v = 0 From our dscusson about parameter conversons we know that: Z ( ) y = Y = y y y y y Note that only f the amplfer s unlateral (y =0) we have z =/y Smlarly the output mpedance of an amplfer depends on whether the amplfer s drven by a voltage or a current source, and ndeed, on the value of the source mpedance n the general case. The argument can be reversed: If the nput or output mpedance of an amplfer does not depend on the load or the source mpedance respectvely, then the amplfer s necessarly unlateral Imperal College London EEE

13 Amplfers: modellng summary Name / Representaton Voltage Current Transconductance Transmpedance Parameters G H Y Z Input Norton Thevenn Norton Thevenn Output Thevenn Norton Norton Thevenn Forward gan Voltage Current Admttance Impedance Reverse gan Current Voltage Admttance Impedance Name / Representaton Voltage Current Transconductance Transmpedance Input V I V I Output V I I V Ideal form VCVS CCCS VCCS CCVS Termnal mpedance Ideal Real Input Output Input Output 0 Hgh Low 0 Low Hgh Hgh Hgh 0 0 Low Low Notes:. Choce of representaton s arbtrary. Representaton emphasses the ntended functon 3. Can convert one representaton nto any other by Thevenn Norton transforms Imperal College London EEE 3

14 Converson between amplfer representatons Some are obvous matrx nversons from the matrx equatons: Recall that the nverse of a x matrx A s: Z=Y a a a a a a a a a Other conversons, e.g. to express y n terms of h, start wth the defntons and express the varables of the y descrpton n terms of h parameters. The resultng equaton s an dentty vald for all values of the h descrpton ndependent varables. All ths can also be done by rearrangng the terms n the equatons. Ths may be easer at the begnnng. G = H A = = y y v 0 h h = = Y y y v h h v 0 v 0 h h h Y =, h hh hh h h 0 = = = h h h Imperal College London EEE 4

15 The nonnvertng amplfer v n G R R H=R /(R +R ) Feedback network Assume fnte opamp gan G. Treat the network connectng the output and the nput as an deal amplfer wth gan H=R /(R +R ) from output to nput R Gv G v R v = G v v v = = v lm = = + + G R + R n out out n out out n R+ R R + GH G v n H R Imperal College London EEE 5

16 Negatve feedback: A control systems perspectve The sgnals on the network must be selfconsstent o ( ) v G v v H v = o 0 = + GH vg v + Forward Gan=G Feedback Gan=H v o V G + GH V o If GH s large, Taylor expanson gves: v o = + v ( ) H GH GH GH s called the LOOP GAIN G L Imperal College London EEE 6

17 The nvertng amplfer R R G KG vo = G( Kv voh) v0 = v + GH vk R lm vo = lm +... = v G G H GH R ( ) ( ) K = R / R + R, H = R / R + R Note: v out and v n are appled by superposton on the crcut. Ths way we obtan the voltage dvders K,H There are two negatve sgns on the summng juncton, snce both forward and feedback sgnals are appled on the nvertng nput Imperal College London EEE 7

18 v The nvertng amplfer: A smpler way to calculate Input vdvder Gan=K=R/(R+R) Opamp Gan=G Feedback vdvder Gan=H=R /(R +R ) v o K G/(+GH) lm G KG/(+GH) KG/(+GH) = K/H Imperal College London EEE 8

19 Feedback n electroncs There s both a voltage and a current at every termnal Precse defntons of measurements: Voltage s measured wth voltmeters. Voltmeters are connected n parallel to the crcut, and have nfnte nternal resstance (VM draw no current).. Current s measured wth ammeters. Ammeters are connected n seres to the crcut and have zero nternal resstance (AM develop no voltage). There are 4 ways to mplement electronc feedback: We may sample (measure) the output: Voltage, by connectng the nput (port!) of the FB network n shunt (parallel) Current, by connectng the nput (port!) of the FB network n seres We can then mx (feed back) the sgnal to the nput as: Voltage, by connectng the output (port!) of the FB network n seres Current by connectng the output (port!) of the FB network n shunt (parallel) Exact descrpton of electronc feedback nvolves port matrx addton. Ths s very tedous, we usually use approxmatons. Imperal College London EEE 9

20 The nonnvertng amplfer: seres shunt feedback v n G Amplfer R R H=R /(R +R ) Feedback Network The opamp acts lke a voltage amplfer The feedback network samples the output voltage, voltage dvdes t and feeds back a voltage nto the nput, so that v n s the sum of nput and fed back v. The feedback network shares nput current and output voltage wth the opamp Imperal College London EEE 0

21 Feedback on voltage amplfers SeresShunt connecton: Add H parameters V + I Port Port + Voltage Amp A + I + V Functon of feedback net: Measure output V Correct (mx) nput V.e. t mproves a Gamp Electrcal ports Functonal ports + Feedback Net B + Port Port Port Port Shared electrcal varables: Input: I, V Output: V,I The feedback network s functonally a voltage amplfer from Port Port Electrcally both networks must be treated as current amplfers P P to account for the shared (nput) electrcal varables. In the calculaton we consder the ELECTRICAL descrpton: V VA VB IA IB I I B ( ) I = I + A I = HA + = + = B V H A V HA HB B V HA+B V Add H parameter representatons of amplfer and feedback network Convert back to G parameter representaton for composte Vamp. Imperal College London EEE

22 The nvertng amplfer: Shunt Shunt feedback R R G Feedback Network Amplfer Amplfer and feedback network have dentcal nput and output voltages The feedback network samples the output voltage and contrbutes a current to correct the nput. The amplfer G functons as a CCVS (but ths should not confuse us, the representaton s arbtrary!) Snce the amplfer and the feedback network share voltages they must be treated as transconductors! Imperal College London EEE

23 Feedback on transmpedance amplfers The ShuntShunt connecton: Add Y parameters V + I Port Port + Transmp + Amp A I V + Functon of feedback net: Measure output V Correct (mx) nput I.e. t mproves a Zamp Electrcal ports Functonal ports + Feedback Net B + Port Port Port Port Shared electrcal varables: Input: V,V Output: I, I The feedback network s functonally a transconductance amplfer from Port Port Electrcally both networks must be treated as transconductance amplfers P P to account for the shared (nput) electrcal varables. In the calculaton we consder the ELECTRICAL descrpton: I IA IB VA VB V V ( ) I = I + A I = YA + = = B V YB A V Y A +YB B V YA+B V Add Y parameter representatons of amplfer and feedback network Convert to Z parameter representaton for composte Zamp Imperal College London EEE 3

24 The Mller theorem: shuntshunt feedback Consder a shunt admttance connected between the nput and output of an nvertng voltage amplfer of gan G. Y In Out In Out G G (+G)Y (+/G)Y Lookng from the nput, the current gong through the feedback element s: ( ) ( ) ( ) = v v Y= + GYv Y = + GY n, F n out n M, n Lkewse, lookng from the output, the amplfer has a gan=/g, so the extra current gong nto the feedback element s: out, F = + Yvout YM, out = + Y G G These consderatons lead to the equvalence of the two dagrams above n terms of ther nput and output admttance. NOTE: Only f the amplfer s deal ts gan wll not change!!! Imperal College London EEE 4

25 Negatve Impedance converter From the Mller Theorem, Vn R R R Vout Z n R R = = = G R + R RR R Snce the opamp wth R and R form a voltage amplfer of gan G = + R R Ths method s used to synthesse negatve resstances, C s, L s Invert a gven mpedance (thnk of a capactor n the poston of R ) Multply or dvde mpedance magntudes (note the rato R /R ) Imperal College London EEE 5

26 The emtter degenerated CE amplfer: seresseres feedback V CC Z S V=V(Z E ) Z L I=I E V S I E Z E We can now apply the feedback equatons: (the lmt s for large transconductance) ( // // ) ( β ) β V gm ZL RCE CCE g Z Z = V + g R + / + g R R out m L L T m E m E E The closed loop amplfer behaves an amplfer wth a reduced transconductance gm gm g = m + g R β + / β + g R R ( ) m E m E E The nput mpedance can easly be calculated (note we have ncluded the shunt Mller effect) : β Z = Z + g R + Z = C + A + C The output mpedance s ( ( β )/ β) wth // ( ) ( ) n n0 m E n0 BE V BC gm ( ( β ) β) ( ) Z = Z / 0 + g R + R + g R out out m E CE m E The frequency response s agan calculated from the nput and output voltage dvders. Imperal College London EEE 6

27 Feedback on transconductance amplfers Seres Seres connecton: Add Z parameters V + I Port Port + Transcond + Amp A I + V Functon of feedback net: Measure output I Correct (mx) nput V.e. t mproves a Yamp Electrcal ports Functonal ports + Feedback Net B + Port Port Port Port Shared electrcal varables: Input: I, I Output: V,V The feedback network s functonally a transmpedance amplfer from Port Port Electrcally both networks must be treated as transmpedance amplfers P P to account for the shared (nput) electrcal varables. In the calculaton we consder the ELECTRICAL descrpton: V VA VB IA IB I I B ( ) V = V + A V = ZA + = = B I Z A I Z A +ZB B I ZA+B I We wll study ths type of connecton when we study transstor amplfers Imperal College London EEE 7

28 The nd form of the Mller theorem seres feedback consder an mpedance connected n seres wth the common termnal of a current amplfer of gan H. Lookng from the nput, the voltage developed on the feedback element s: ( ) ( ) ( ) V = + Z = + H Zv Y = + H Z Z n out n M, n Lkewse, lookng from the output, the amplfer has a gan=/h, so the voltage developed on the feedback element s: VZ = + Zout ZM, out = + Z H H These consderatons lead to the equvalence of the two dagrams above n terms of ther nput and output mpedance. NOTE: Only f the amplfer s deal ts current gan wll not change as a result of the seresseres feedback. Imperal College London EEE 8

29 Feedback on current amplfers The Shunt Seres connecton: Add G parameters V + I Port Port + Current + Amp A I + V Functon of feedback net: Measure output I Correct (mx) nput I.e. t mproves an Hamp Electrcal ports Functonal ports + Feedback Net B + Port Port Port Port Shared electrcal varables: Input: V,I Output: I, V The feedback network s functonally a current amplfer from Port Port Electrcally both networks must be treated as voltage amplfers P P to account for the shared (nput) electrcal varables. In the calculaton we consder the ELECTRICAL descrpton: I IA IB VA VB V V ( ) V = V + A V = GA + = + = B I GB A I GA GB B I GA+B I We wll study ths type of connecton when we study transstor amplfers Imperal College London EEE 9

30 port network feedback connecton rules Shunt Shunt: add Y Seres Shunt: add H Y Y Y +Y H H H +H Seres Seres: add Z Shunt Seres: add G Z Z Z +Z G G G +G Imperal College London EEE 30

31 Seresshunt feedback: Effect on nput output mpedance A real opamp has: Fnte nput mpedance Fnte output mpedance Fnte Gan Treat the feedback network as f t draws no current. Ths s equvalent to: Zo [ R, R] Z The nput mpedance s derved from: ( ) ( ) ZG+ Z H= v Z n out o n n Z + GH = v ZH n n out o v Zn = = Z + GH n n = 0 out The output mpedance s: ( ) ( ) v Z v v H G = v Z Z = = + out n out out out o out out v n V I n R = 0 + Z o ( GH) + G R Z o I out V 0 Ideal amplfer Imperal College London EEE 3

32 Postve feedback Same analyss as negatve feedback, apart for H H Postve feedback an be used to do thngs negatve feedback cannot do: Introduce hysteress (e.g. Schmtt Trgger) Generate negatve mpedances Invert an mpedance Note that under postve feedback we can have F=GH=0 If F=0 we (n theory) can turn an amplfer nto an deal verson by a sutable feedback connecton and GH=. GH=, when t occurs at a fnte (.e. non zero) frequency, s the Barkhausen condton for oscllaton The opamp s called operatonal precsely because t can be used to perform mathematcal operatons on sgnals (addton, subtracton, ntegraton, dfferentaton, multplcaton by a scalar, ) on operators (nverson) on mpedances (negaton, nverson, multplcaton, dvson, ) Imperal College London EEE 3

33 The domnant pole approxmaton an opamp has never an nfnte gan at DC the opamp gan as a functon of frequency s adequately descrbed by: A v ( f ) ADC A ADC f DC p = = = + sτ + jf / f f + jf Where A DC s the DC gan of the amplfer, typcally f p s the domnant pole frequency, typcally 0 Hz. The product A DC f p s called the ganbandwdth product (GBW). The GBW s a characterstc constant of the opamp, typcally When we do AC analyss we must consder the fnte complex gan of The amplfer, especally when we try to get hgh gan at hgh frequences. p p Imperal College London EEE 33

34 Invarance of the gan bandwdth product Consder a nonnvertng amplfer, and a domnant pole opamp Applyng the feedback theory we get the closed loop gan: ADC fp fp + jf ADC fp GBW AV = = = ADC fp fp + jf + ADC f ph fp + jf + GBW H + H f + jf p The DC gan s: A ( f 0) V = = f p GBW + GBW H The pole of ths amplfer s at: f0 = fp + GBW H It follows that the product of the DC gan of a nonnvertng amp and ts pole poston equals the gan bandwdth product of the opamp! Ths s only true for a domnant pole opamp! (.e. most voltage mode amplfers) Imperal College London EEE 34

35 Conclusons Feedback reduces gan Feedback reduces component and envronmental senstvty Feedback ncreases lnearty There are 4 ways to apply electronc feedback Feedback can be used to modfy nput and output mpedances: A seres negatve feedback ncreases mpedance A seres postve feedback decreases mpedance A shunt negatve feedback ncreases admttance A shunt postve feedback decreases or zeroes admttance Postve feedback can lead to dynamc nstablty Opamps are modelled as domnant pole systems Opamps have fnte DC gan Opamps have a low frequency pole. Amplfers bult wth opamp are subject to a constant GBW Imperal College London EEE 35

36 nd order flter transfer functons: Revew Second order flter transfer functons are all of the followng form: ( ) ( s ) ω0 ζs ω0 C s/ ω + Bζs/ ω + A H( s) = H, Q= / + / + ζ H 0 s the overall ampltude, ω 0 the break (or peak) frequency, and ζ the dampng factor ζ srelated to the qualty factor Q by: Q=/ζ The 3dB bandwdth of an underdamped nd order flter s approx /Q tmes the peak frequency. Functon Low Pass Hgh Pass Band Pass A 0 0 B 0 0 C 0 0 The coeffcents A, B, C determne the functon of the flter: Band Stop All Pass 0 Imperal College London EEE 36

37 Tee P transformatons When analysng actve band pass or band stop actve flters we often encounter the twntee passve notch flter topology Ths requres qute a bt of algebra to compute, so we prove a, useful for smplfyng networks, lemma: Y3 Z Z s equvalent to Y Y Z3 Proof: wrte the z matrx of the Tee< and the y matrx of the P and requre that the two crcuts are representatons of same network: Z+ Z3 Z3 Y+ Y3 Y3 ZTee =, P, P Tee Z3 Z Z Y = 3 Y3 Y Y Y = Z Y Y Y3 Z =, Z =, Z3 = YY + YY 3+ YY 3 YY + YY 3+ YY 3 YY + YY 3+ YY 3 Z Z Z3 Y =, Y =, Y3 = ZZ + ZZ3+ ZZ3 ZZ + ZZ3+ ZZ3 ZZ + ZZ3+ ZZ3 Imperal College London EEE 37

38 Hgher order flter synthess usng nd order sectons A general flter transfer functon s of the form: ( ) H s n k ax k Pn( s) s z s z s z = 0 m m( ) k 0 n bx k = 0 ( 0)( ) ( n) ( )( ) ( ) = = = Q s s p s p s p P(s) and Q(s) have real coeffcents. To make a hgher order flter: factor P(s) and Q(s) nto quadratc and lnear factors Implement factors as bquads Cascade bquad sectons to obtan the orgnal transfer functon Note that the roots of P, Q are real or come n conjugate pars. The centre frequences and dampng factors of the sectons requred to mplement standard forms (Butterworth, Chebyshev, Ellptc etc) are tabulated. Tables are ncluded n CAD software for automated synthess Imperal College London EEE 38

39 A useful network transformaton: Impedance nverson and the gyrator A gyrator can perform mpedance nverson (L C) Impedance scalng seres parallel connecton converson! Proper symbol of gyrator Alternate symbol Smple actve mplementaton (very popular by analogue CMOS desgners. Each gm s made of a MOSFET or two!) Imperal College London EEE 39

40 ¼ wavelength transmsson lne P and Tee networks wth negatve elements negatve values of components wll be added to precedng and subsequent stage mpedances resultng n overall postve mpedances! Note that for narrowband sgnals, eg, L s a capactor! Ladder LC flters can be synthessed only wth capactors and gyrators *Passve Gyrators Z, Z s completely arbtrary, can be a flter transfer functon and more Imperal College London EEE 40

41 Gyrator functon bascs A seres (floatng) component between two gyrators appears nverted and grounded Two dentcal gyrators n seres are the dentty operator A grounded component between two gyrators appears nverted and n seres Two dfferent gyrators n seres perform drecton senstve mpedance multplcaton by a constant: Imperal College London EEE 4

42 Some more gyrator denttes or, how to make e.g. a seres resonance crcut when you only have parallel resonators n your component box and vce versa Imperal College London EEE 4