55:041 Electronic Circuits
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1 55:04 Electronc Crcuts Feedback & Stablty Sectons of Chapter 2. Kruger Feedback & Stablty
2 Confguraton of Feedback mplfer S o S ε S o ( S β S ) o Negate feedback S S o + β β s the feedback transfer functon f + β + T Implct ssumptons Made T s the loop gan Input sgnal s transmtted through amplfer and not through β network Output sgnal s transmtted through β network only There are no loadng effects β network does not load amplfer mplfer (wth/wthout β network) does not load source. Kruger Feedback & Stablty 2
3 β s the feedback transfer functon T s the loop gan β β >> f + β β β Profound result: Closed-loop gan s ndependent from open-loop gan, and we can control the closedloop gan wth the amount of feedback. Kruger Feedback & Stablty 3
4 Example How good s the /β approxmaton? ssume open-loop gan s 0 5, and the closed-loop gan s f 50. Then f 50 + β β 0 5 β f / β ssume open-loop gan s 0 6, wth the same β f + β β Practcally the same closed-loop gan. Kruger Feedback & Stablty 4
5 Op-mp Example VV + Σ VV oo 2 β + 2 Ideal case: + 2 β >> β Kruger Feedback & Stablty 5
6 f d f d d f + β d ( + β) 2 Gan Senstty Ddng both sdes wth closed-loop gan yelds d f f β + β + d ( + β) + β 2 ( + β) 2 ( β) 2 + β d d f f d + β Ths shows that the % change n closed-loop gan s smaller, by a factor +β, than the % change n open-loop gan.. Kruger Feedback & Stablty 6
7 Gan Senstty n engneer desgned a feedback amplfer β , and 0 5. By how much does the closed-loop gan change when the same feedback network s used, but an amplfer wth open-loop gan 0 6 s used? d f f + β d + ( ) % In other words, the open-loop gan changed by a factor 0, whle the closed loop gan changed about 0.5%.. Kruger Feedback & Stablty 7
8 . Kruger Feedback & Stablty 8 Gan Versus Frequency ssume we can characterze the frequency response of an amplfer wth a sngle pole (ths s true for many op-amps) H o s s ω + ) ( ) ( ) ( ) ( s s s f β + ( ) o H o o f s s β ω β ) ( We assume the feedback network s ndependent from frequency The closed-loop gan s smaller than the open-loop gan by a factor (+β) The 3 db bandwdth s larger by a factor (+β)
9 Gan-Bandwdth Product Gan-bandwdth product of a feedback amplfer s constant We can ncrease bandwdth at the expense of gan. Kruger Feedback & Stablty 9
10 Nose Senstty o n S o N o S N o n n S N o o S 0 N Conclusons. Negate feedback can reduce nternallygenerated addte amplfer nose Same closed-loop gan as preous amplfers 2 2 o + n S (mproe S/N) + β 2 + β 2 N 2. Negate feedback per 00se + wll 0. not reduce S/N wth respect to external nose o β 2 ( + ) + 00 n n o S o N S N o o 000 S N. Kruger Feedback & Stablty 0
11 educton of Nonlnear Dstorton Open-Loop Gan Non-lnear because the gan depends on the sgnal f + β β Closed-loop gan, β Kruger Feedback & Stablty
12 dantages of Negate Feedback. Gan Senstty aratons n gan s reduced 2. Bandwdth Extenson larger than that of basc amplfer 3. Nose Senstty may ncrease S/N rato 4. educton of Nonlnear Dstorton 5. Control of Impedance Leels nput and output mpedances can be ncreased or decreased Dsadantages of Negate Feedback. Crcut Gan reduced compared to that of basc amplfer 2. Stablty possblty that feedback crcut wll become unstable and oscllate at hgh frequences. Kruger Feedback & Stablty 2
13 Ideal Basc Feedback Confguratons Voltage mplfer Very Common Current mplfer Transconductance mplfer (oltage n-current out) Conerts current to oltage Transresstance mplfer (current n-oltage out) Conerts oltage to current. Kruger Feedback & Stablty 3
14 Ideal Seres-Shunt Feedback f + β Sample output oltage and feed t back to nput Ideal: assume feedback network does not load output/nput Voltage mplfer KVL: V x + I x + β V o 0 V x I x + β V ε f V I x x + β I I + x I x x ( β ) ( + β ). Kruger Feedback & Stablty 4
15 Seres-Shunt Feedback Output esstance How do we determne output resstance? Fast V β V ε x. Turn off ndependent sources 2. dd test oltage V x 3. See what test current I x flows 4. Determne V x /I x I x V V of x x V x V o ε ( β V ) o ( + β ) V I x x o x o ( + β ) KCL. Kruger Feedback & Stablty 5
16 Equalent Crcut: Seres-Shunt Feedback Crcut f + ( β ) f + β of o + β ( ) Useful to thnk of ths as the mproement factor BW f ( + β )BW. Kruger Feedback & Stablty 6
17 Ideal Basc Feedback Confguratons Voltage mplfer Current mplfer Transconductance mplfer (oltage n-current out) Conerts current to oltage Transresstance mplfer (current n-oltage out) Conerts oltage to current. Kruger Feedback & Stablty 7
18 Ideal Shunt-Seres Feedback Sample output current and feed t back to nput Ideal: assume feedback network does not load output, so that I o s unaffected Current mplfer f + β Feedback Subtract, educe, Steal From Fast I I V ε f I I ε I ε ε + βi + β I + o ( I ) ( β ) V I e I ( + β ) ( + β ). Kruger Feedback & Stablty 8
19 Ideal Shunt-Seres Feedback How do we determne output resstance?. Turn off ndependent sources 2. dd test current I x output 3. See what test current V x results 4. Determne V x /I x Iε βi x V x ( I x Iε ) o ( I x ( β I x )) o I x ( β ) o + of V I x x ( + β ) o Fast. Kruger Feedback & Stablty 9
20 Equalent Crcut: Shunt-Seres Feedback Crcut Current mplfer f + β ( ) Useful to thnk of ths as the mproement factor f + β of V I x x ( + β ) o. Kruger Feedback & Stablty 20
21 ecap - Ideal Basc Feedback Confguratons Voltage mplfer Current mplfer Transconductance mplfer (oltage n-current out) Conerts current to oltage Transresstance mplfer (current n-oltage out) Conerts oltage to current. Kruger Feedback & Stablty 2
22 Ideal Seres-Shunt Feedback f + β Sample output oltage and feed t back to nput Ideal: assume feedback network does not load output/nput Voltage mplfer V x I I x x + + β V β V ε o Fast f V I x x + β I I + x I x x ( β ) ( + β ). Kruger Feedback & Stablty 22
23 Ideal Seres-Seres Feedback Crcut Sample output current and feed t back to nput as a oltage Ideal: assume feedback network does not load output, so that I o s unaffected Transconductance mplfer (oltage n-current out) gf g + β z g Unts of β Z? V/ Fast I + β I z o ( ) I + βz gv ε f? (neglect S ) f I + β z I g I ( + β ) z g Conerts output current to a oltage. Kruger Feedback & Stablty 23
24 Ideal Seres-Seres Feedback Crcut I x V x Fast I o I x I of V x x + I x o V x o V V β I z + ε gv ε g x o I x ( + β ) z x ( β I ) z x g of? (neglect S ). Kruger Feedback & Stablty 24
25 Equalent Crcut: Seres-Seres Feedback Crcut Transconductance mplfer (oltage n-current out) f ( + β z g ) gf of o ( + βz g ) Useful to thnk of ths as the mproement factor g + β z g. Kruger Feedback & Stablty 25
26 ecap-ideal Basc Feedback Confguratons Voltage mplfer Current mplfer Transconductance mplfer (oltage n-current out) Conerts current to oltage Transresstance mplfer (current n-oltage out) Conerts oltage to current. Kruger Feedback & Stablty 26
27 Ideal Shunt-Shunt Feedback Crcut Sample output oltage and feed t back to nput as a current Ideal: assume feedback network does not load output, so that V o s unaffected Transresstance mplfer (current n-oltage out) zf z + β g z f of + β ( ) g + β z o ( ) g z Conerts output oltage to a current. Kruger Feedback & Stablty 27
28 Equalent Crcut: Shunt-Shunt Feedback Crcut Transresstance mplfer (current n-oltage out) Useful to thnk of ths as the mproement factor f o of ( + β ) ( ) g z + β z + β g. Kruger Feedback & Stablty 28 g z zf z
29 Summary of Feedback mplfer Functons. Kruger Feedback & Stablty 29
30 Notes on Unts and Subscrpts Transresstance mplfer (current n-oltage out) z V I o Unts of resstance/mpedance (Z) I fb β g I V fb o Unts of conductance (g) Conerts output oltage to current f + β z zf of ( + β g z ) + β g z ( g z ) Product should be dmensonless. Kruger Feedback & Stablty 30
31 . Kruger Feedback & Stablty 3 Op-mp Seres-Shunt Feedback Crcut Seres-shunt feedback Input mpedance wll ncrease Output mpedance wll decrease Bandwdth wll ncrease Seres-shunt feedback Take some of the output oltage Feed t back n seres wth nput ssume ery large o o fb V V V β β ( ) f β f β 2 2 f + + ( ) o o of β
32 Op-mp Shunt-Shunt Feedback Current-n, oltage-out Transresstance Z Ideal op-amp, neglect nput current: Output oltage and 2 generate a feedback current that reduces current flowng nto op-amp β g 2 Feedback reduces nput mpedance Shunt-Shunt Feedback. Kruger Feedback & Stablty 32
33 Op-mp Shunt-Seres Feedback f + + F / + β. Kruger Feedback & Stablty 33
34 Op-mp Seres-Seres Feedback I L Transconductance amplfer (oltage n, current out) What s II LL f EE 00Ω and VV 2 V? nswer: m I L Voltage-controlled current source Feedback oltage s a functon of output current E conerts current to oltage Feedback: oltage n seres wth nput oltage Ideal op-amp, neglect base current: gf I V L E β Z E Seres-Seres Feedback What happens to and oo? Both ncrease. Kruger Feedback & Stablty 34
35 Dscrete Transstor Crcut Current-n, oltage-out Transresstance Z Output oltage and F generate a feedback current that reduces current flowng nto transstor base β g F Feedback reduces nput mpedance Shunt-Shunt Feedback. Kruger Feedback & Stablty 35
36 Dscrete Shunt-Seres Transstor Crcut. Kruger Feedback & Stablty 36
37 Dscrete Transstor Crcut Voltage out Current n Note that o C2 E 2 E 2 so that ths crcut s really samplng the output oltage Output oltage and F generate a feedback current that reduces current flowng nto amplfer Shunt-Shunt Feedback. Kruger Feedback & Stablty 37
38 Multstage Feedback Crcut I fb I Output oltage and F generate a feedback current that reduces current flowng nto transstor base Feedback reduces nput mpedance Shunt-Shunt Feedback. Kruger Feedback & Stablty 38
39 ecap-confguraton of Feedback mplfer Negate feedback β s the feedback transfer functon Implct ssumptons Made T s the loop gan f + β + T Input sgnal s transmtted through amplfer and not through β network Output sgnal s transmtted through β network only There are no loadng effects β network does not load amplfer mplfer (wth/wthout β network) does not load source. Kruger Feedback & Stablty 39
40 Stablty Secton 2.9 ecall defnton of loop gan: T β We assume β s not a functon of frequency Howeer, the amplfer gan,, s a functon of frequency (s), and we normally set s jω, so (jω). Thus T(jω) β(jω). Closed-loop gan: If T(jω) -, then ( jω) f ( jω) + T( jω) f ( jω) ( jω) Instablty We can wrte T ( jω) T ( jω) φ Equalent condtons for stablty T( jω) < or φ less than 80 Gan margn: when the amplfer phase shft s80 o, how much headroom/margn before the gan s and the amplfer becomes unstable? Phase margn: when the amplfer gan s, how much more headroom/margn before the phase shft s 80 o amplfer becomes unstable?. Kruger Feedback & Stablty 40
41 Complex Number eew T + ja 2 + ja a φ tan T + a T + ja T + ja + a 2 φ tan a tan ( a) T K + ja + jb T K + ja + jb K + a 2 + b 2 φ tan a tan b tan ( a) tan ( b). Kruger Feedback & Stablty 4
42 . Kruger Feedback & Stablty 42? ), ( tan 80 b b M 2 0 ) ( f j K f T ) ( f j K f T +?, 0 0 ) ( f f K f j K f T M b 80 tan f K
43 . Kruger Feedback & Stablty ) ( f j f j K f T ) ( + + f f K f T?, 0 0 ) ( f f f K f T Numercal soluton, tral-and-error
44 (jω). Kruger Feedback & Stablty 44
45 - -. Kruger Feedback & Stablty 45
46 - -. Kruger Feedback & Stablty 46
47 - -. Kruger Feedback & Stablty 47
48 - -. Kruger Feedback & Stablty 48
49 (jω). Kruger Feedback & Stablty 49
50 . Kruger Feedback & Stablty 50
51 Bode Plot: Phase and Gan Margns Loop Gan T(jω) β ( jω) Secton db Gan Margn 0 T ( f ) 80 db f f 80 Phase Margn 80 +φ( f). Kruger Feedback & Stablty 5
52 Bode Plot: Phase and Gan Margns The loop gan TT jjjj for a feedback amplfer s shown. Is the amplfer stable? Gan Margn 0 TT ff 80 db db dddd 0.8 db Phase Margn 80 + φφ(ff ) 80 + ( 75 ) 5 Yes, amplfer s stable.. Kruger Feedback & Stablty 52
53 Plottng Loop Gan If β s ndependent of frequency, then the loop gan T(jω) β(jω) s smply a scaled erson of the open-loop gan.. Kruger Feedback & Stablty 53
54 Plottng Loop Gan If β s ndependent of frequency, then the loop gan T(jω) β (jω) s smply a scaled erson of the open-loop gan. T( jω) 20log β 20log + 20log β db 20log 20log β We can determne the loop gan by graphcally subtractng a plot of 20log(/β) from the open loop Bode plot. powerful araton of the graphcal subtracton technque s the followng. Often 20log s aalable n graphcal form. It s ery conenent to plot 20log(/β) on ths graph, and consder ths the new frequency axs.. Kruger Feedback & Stablty 54
55 Graphcal Subtracton n amplfer has open-loop gan shown below. The amplfer s used n a feedback confguraton and the closed-loop gan s 4,000. What are the phase- and gan margns? Ths s where loop gan s (0 db) 0 db New axs closed loop o + β o β 32 db closed loop 4,000 β 20 log β 72 db Gan margn s 32 db 87 o Phase margn s 87 o mplfer s stable. Kruger Feedback & Stablty 55
56 Graphcal Subtracton n amplfer has open-loop gan shown below. The amplfer s used n a feedback confguraton and the closed-loop gan s,000. What s the phase margn? Open Loop Voltage Gan and Phase s Frequency closed loop closed loop o + β o,000 β β 20 log β 60 db Phase margn s 90 o. Kruger Feedback & Stablty 56
57 Draw ββ n amplfer has open-loop gan shown left. The amplfer s used n a feedback confguraton and the closed-loop gan s 00. What are the phase- and gan margns? What s the closed-loop bandwdth? 45 db closed loop o + β o β closed loop 00 β 20 log β 40 db 90 o Phase margn s 90 o Gan margn s 45 db Bandwdth ~ MHz. Kruger Feedback & Stablty 57
58 Wll ths amplfer be stable f used as a 30 amplfer? log db Draw ββ ββ 30 db Frequency where loop gan ββ oo, s 2 MHz. Kruger Feedback & Stablty 58
59 Wll ths amplfer be stable f used as a 30 amplfer? log db Phase 90 Phase margn 90 Draw ββ ββ 30 db. Kruger Feedback & Stablty 59
60 What s the phase margn f the amplfer s used n a feedback confguraton wth so the feedback amplfer has a gan of 5-dB? Open Loop Voltage Gan and Phase s Frequency Phase 80 Phase margn 00 Draw ββ ββ 5 db. Kruger Feedback & Stablty 60
61 What s the phase margn and bandwdth f closed loop gan 2 db?. What wll the rse tme for a step nput be? 2 db Method : Draw loop gan ββ oo ββ s ndependent of ff so the loop gan s just a scaled erson of the open loop gan TT ββ oo Open Loop Voltage Gan (db) Closed-loop response Loop Gan ff 2 db 3.98~ 4 shch mples that ββ Further, oooo 65 dddd , so that ββ oooo db. Consequently, the loop gan magntude plot s smply the open-loop magntude plot, shfted down 2 db. The phase plot s the same. The phase s 95 where the loop gan s 0 db, so the phase margn s 85. The 3-dB bandwdth of the feedback amplfer s 00 MHz. The rse tme s 0.35 BB 3.5 ns Open Loop Voltage Gan (db). Kruger Feedback & Stablty 6
62 What s the phase margn and bandwdth f closed loop gan 2 db?. What wll the rse tme for a step nput be? Method 2: Graphcal subtracton Draw ββ, whch s practcally the same as the closed-loop gan. Open Loop Voltage Gan (db) The phase s 95 where the loop gan s 0 db, so the phase margn s 85. The 3-dB bandwdth of the feedback amplfer s 00 MHz. The rse tme s 0.35 BB 3.5 ns Open Loop Voltage Gan (db). Kruger Feedback & Stablty 62
63 mplfer has an open-loop gan f j f j 0 5 Is amplfer stable f used so that closed-loop gan - 00? closed loop + β β 00, β β 0.0 Loop gan T s T ( f ) β (0.0) T( f ) + f j 0 3 ( 0 ) 3 + f j 0 5 Stable? examne phase- and gan margns T ( f ) f ( 0 ) φ tan tan f Kruger Feedback & Stablty Sole for f How? f 30 khz 8 o phase margn > amplfer s stable
64 . Kruger Feedback & Stablty 64
65 . Kruger Feedback & Stablty 65
66 . Kruger Feedback & Stablty 66
67 . Kruger Feedback & Stablty 67
68 . Kruger Feedback & Stablty 68
69 Next: Same Problem, but use Graphcal Subtracton. Kruger Feedback & Stablty 69
70 . Kruger Feedback & Stablty 70
71 Draw ββ ββ 00 Phase 90 Phase margn 90. Kruger Feedback & Stablty 7
72 . Kruger Feedback & Stablty 72
73 . Kruger Feedback & Stablty 73
74 . Kruger Feedback & Stablty 74
75 Draw ββ ββ Phase 90 Frequency where ββ oo s 0 6 Phase margn 90. Kruger Feedback & Stablty 75
76 . Kruger Feedback & Stablty 76
77 . Kruger Feedback & Stablty 77
78 . Kruger Feedback & Stablty 78
79 . Kruger Feedback & Stablty 79
80 . Kruger Feedback & Stablty 80
81 . Kruger Feedback & Stablty 8
82 . Kruger Feedback & Stablty 82
83 Nyqust Stablty Crteron (Secton 2.9.3) Not Coered. Kruger Feedback & Stablty 83
84 Determnng Loop Gan The term (+β) s used frequently n feedback amplfer analyss Loop gan T β s mportant n determnng stablty Step : Break the feedback loop Step 2: Termnate so that ports see same mpedances Step 3: Insert a test current/oltage and measure response: T -V r /V t Can be used as an analyss technque, n SPICE, as well as n some actual crcuts. Kruger Feedback & Stablty 84
85 T β V V r t. Kruger Feedback & Stablty 85
86 p n Loop gan? t t 0. t β n t > n 20 p n t > β p o 000 ( ) , p n o 3,000 t T o t 00. Kruger Feedback & Stablty 86
87 Frequency Compensaton Secton 2.0 Consder a feedback amplfer wth the loop gan TT as shown. How many poles? Stable or Unstable?. Kruger Feedback & Stablty 87
88 Frequency Compensaton Where loop gan TT, the phase s nearly 270, and the amplfer s unstable. How can we make the amplfer stable?. Kruger Feedback & Stablty 88
89 Frequency Compensaton Introduce a new, lowfrequency pole (ff DD ) at ery low frequency. In ths example, at ff DD 0 Hz. ff DD. Kruger Feedback & Stablty 89
90 Frequency Compensaton Introduce a new, lowfrequency pole (ff DD ) at ery low frequency. In ths example, at ff DD 0 Hz. ff DD Where TT, the phase not yet 80, so the amplfer s now stable. Where the phase s 80, TT <, stable. Kruger Feedback & Stablty 90
91 Frequency Compensaton How do we ntroduce low-frequency (domnant) pole nto a crcut? Nae approach Mller Compensaton Perhaps n f 3 db 2π C C M (+)C For compensaton to work f 3dB must be low > large compensaton capactors can be ery large > small (pf) capactors can be used. Kruger Feedback & Stablty 9
92 Frequency Compensaton Many op-amps hae the followng structure Dfferental Input Hgh-gan, nertng amplfer Power stage (gan ) The purpose of CC FF s to create a domnant pole at a low frequency, usng the Mller effect. Ths s called Mller compensaton.. Kruger Feedback & Stablty 92
93 Frequency Compensaton Smplfed Schematc of a 3-stage Op mp Dff Input Compensaton Capactor Mrrors Hgh-gan stage (nertng) Buffer. Kruger Feedback & Stablty 93
94 Beware Internally-compensated amplfers can also become unstable n way that are not obous to the unntated. Ths s often the case when drng capacte loads. Example of a capacte load? C L L C L nteracts wth L o and forms another pole wth addtonal phase shft If C L and/or L o s small, the pole s at ery hgh frequency and may not be mportant. Closed-loop gan 5 db Howeer, remember that phase shft starts at /0 of pole frequency Een what seems small capactances, the addtonal phase shft may remoe phase margn and the system s unstable. Phase margn 70 o Smlar problems wth capactances at nput Soluton? External compensaton. Kruger Feedback & Stablty 94
95 . Kruger Feedback & Stablty 95
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