OpAmp (OTA) Design The design process involves two distinct activities: Architecture Design Find an architecture already available and adapt it to present requirements Create a new architecture that can meet requirements Component Design Design transistor sizes Design compensation network

All op amps used as feedback amplifier: If not compensated well, closed-loop can be oscillatory or unstable. damping ratio z ≈ phase margin PM / 100 Value of z: 1 0.7 0.6 0.5 0.4 0.3 Overshoot: 0 5% 10% 16% 25% 37%

UGF: frequency at which gain = 1 or 0 dB PM: phase margin = how much the phase is above critical (-180o) at UGF Closed-loop is unstable if PM < 0 UGF PM

Two Stage Op Amp Architecture

z

UGF GM<0 p1 p2 z1 PM<0

UGF p1 p2

UGF GM p1 p2 z1 PM

Types of Compensation Miller - Use of a capacitor feeding back around a high-gain, inverting stage. Miller capacitor only Miller capacitor with an unity-gain buffer to block the forward path through the compensation capacitor. Can eliminate the RHP zero. Miller with a nulling resistor. Similar to Miller but with an added series resistance to gain control over the RHP zero. Self compensating - Load capacitor compensates the op amp (later). Feedforward - Bypassing a positive gain amplifier resulting in phase lead. Gain can be less than unity.

General Miller effect v2 v1 i v2= v1 i= AVv1 v1/Z1 i i = (v1-v2)/Zf =v1(1-AV)/Zf = - v2(1-1/AV)/Zf i= -v2/Z2

Miller compensator capacitor CC C1 and CM are parasitic capacitances

DC gain of first stage: AV1 = -gm1/(gds2+gds4)=-2 gm1/(I5(l2+ l4)) DC gain of second stage: AV2 = -gm6/(gds6+gds7)=- gm6/(I6(l6+ l7)) Total DC gain: gm1gm6 AV = (gds2+gds4)(gds6+gds7) 2gm1gm6 AV = I5I6 (l2+ l4)(l6+ l7) GBW = gm1/CC

Zf = 1/s(CC+Cgd6) ≈ 1/sCC When considering p1 (low freq), can ignore CL (including parasitics at vo): Therefore, AV6 = -gm6/(gds6+gds7) Z1eq = 1/sCC(1+ gm6/(gds6+gds7)) C1eq=CC(1+ gm6/(gds6+gds7))≈CCgm6/(gds6+gds7) -p1 ≈ w1 ≈ (gds2+gds4)/(C1+C1eq) ≈ (gds2+gds4)/(C1+CCgm6/(gds6+gds7)) ≈ (gds2+gds4)(gds6+gds7)/(CCgm6) Note: w1 decreases with increasing CC

At frequencies much higher than w1, gds2 and gds4 can be viewed as open. Total go at vo: M6 CC gds6+gds7+gm6 CC+C1 CC vo Total C at vo: C1CC C1 CL CL+ CC+C1 M7 -p2=w2= CCgm6+(C1+CC)(gds6+gds7) CL(C1+CC)+CCC1

As CC is increased, w2 increases also. gds6+gds7 Note that when CC=0, w2 = CL As CC is increased, w2 increases also. However, when CC is large, w2 does not increase as much with CC. w2 has a upper limit given by: gm6+gds6+gds7 gm6 ≈ CL+C1 CL+C1 When CC=C1, w2 ≈ (½gm6+gds6+gds7)/(CL+½C1) ≈ gm6/(2CL+C1) Hence, once CC is large, its main effect is to lower w1, and hence lower GBW.

Also note that, in contrast to single stage amplifiers for which increasing CL improves PM, for the two stage amplifier increasing CL actually reduces w2 and reduces PM. Hence, needs to design for max CL

There are two RHP zeros: z1 due to CC and M6 z1 = gm6/(CC+Cgd6) ≈ gm6/CC z2 due to Cgd2 and M2 z2 = gm2/Cgd2 >> z1 z1 significantly affects achievable GBW.

gm6/(CL+C1) f (I6) A0 z1 ≈ gm6/Cgd6 w1 w2 z2 ≈ gm2/Cgd2 -90 No PM -180

z1 ≈ gm6/Cgd6 z2 ≈ gm2/Cgd2 z1 ≈ gm6/Cc gm6/(CL+C1) f (I6) A0 w1 w2 -90 No PM -180

gm6/(CL+C1) f (I6) A0 w2 z1 ≈ gm6/CC w1 gm1/CC -90 PM -180

PM ≈ 90o – tan-1(UGF/w2) – tan-1(UGF/z1) It is easy to see: PM ≈ 90o – tan-1(UGF/w2) – tan-1(UGF/z1) To have sufficient PM, need UGF < w2 and UGF << z1 In such case, UGF ≈ GB ≈ gm1/CC = z1 * gm1/gm6. GB < w2 GB << z1 Hence, need: PM requirement decides how much lower: PM ≈ 90o – tan-1(GB/w2) – tan-1(GB/z1)

Possible design steps for max GB For a given CL and Itot Assume a current share ratio q, i.e. I6+I5 = Itot, I5 = qI6 , I1 = I2 = I5/2 Size W6, L6 to achieve max gm6/(CL+Cgs6) which is > w2 C1  W6*L6, gm6  (W6/L6)0.5 Size W1, L1 so that gm1 ≈ 0.1gm6 this make z1 ≈ 10*GBW Select CC to achieve required PM by making gm1/CC < 0.5 w2 Check slew rate: SR = I5/CC Size M5, M7, M3/4 for current ratio, IMCR, etc

Comment If we run the same total current Itot through a single stage common source amplifier made of M6 and M7 Single pole go/CL Gain gm6/go Single stage amp GB = gm6/CL >gm6/(CL+C1) > w2 > gm1/CC = GB of two stage amp Two stage amp achieves higher gain but speed is much slower! Can the single stage speed be recovered?

Other considerations Output slew rate: SR = I5/CC Output swing range: VSS+Vdssat7 to VDD – Vdssat6 Min ICM: VSS + Vdssat5 + VTN + Von1 Max ICM: VDD - |VTP| - Von3 + VTN Mirror node approx. pole/zero cancellation Closed-loop pole stuck near by Can cause slow settling

When vin is short, the D1 node sees a capacitance CM and a conductance of gm3 through the diode con. So: p3 = -gm3/CM When vin is float and vo=0. gm4 generate a current in id4=id2=id1. So the total conductance at D1 is gm3 + gm4. So: z3 = -(gm3+gm4)/CM =2*p3 If |p3| << GB, one closed-loop pole stuck nearby, causing slow settling!

Eliminating RHP Zero at gm6/CC icc = vg gm6 = CCdvCC/dt vg= RZCCdvCC/dt +vcc CCdvCC/dt (gm6RZ-1)CCdvCC/dt + gm6vcc=0

For the zero at M6 and CC, it becomes z1 = gm6/[CC(1-gm6Rz)] So, if Rz = 1/gm6, z1 →  For such Rz, its effect on the p1 node can be ignored so p1 remains as before. Similarly, p2 does not change very much. similar design approach.

Realization of Rz vb

VDD M8 M9

VDD M8 M9

Another choice of Rz is to make z1 cancel w2: z1=gm6/CC(1-gm6Rz) ≈ - gm6/(CL+C1) CC+CL+C1  Rz = gm6CC CL+C1 1 = (1+ ) gm6 CC

Let ID8 = aID6, size M6 and M8 so that VSG6 = VSG8 Then VSGz=VSG9 Assume Mz in triode Rz = bz(VSGz – |VT| - VSDz) ≈ bz(VSGz – |VT|) = bz(2ID8/b9)0.5 = bz(2aID6/b6)0.5(b6/b9)0.5 = bz/b6 *b6VON6 *(ab6/b9)0.5 = bz/b6 *1/gm6*(ab6/b9)0.5 Hence need: bz/b6 *(ab6/b9)0.5 =(CC+CL+C1)/CC

gm6/(CL+C1) f (I6) A0 -z1 ≈ w2 w1 gm1/CC -90 PM -180

With the same CC as before Z1 cancels p2 P3, z3, z2, not affected P1 not affected much Phase margin drop due to p2 and z1 nearly removed Overall phase margin greatly improved Effects of other poles and zero become more important Can we reduce CC and improve GB?

z1 ≈ p2 z2 ≈ gm2/Cgd2 z4 ≈ gm6/Cgd6 A0 gm6/CL Operate not on this but on this or this z1 ≈ p2 z2 ≈ gm2/Cgd2 z4 ≈ gm6/Cgd6 w1 w2 pz=-1/RZCC -90 -180

Increasing GB by using smaller CC It is possible to reduce CC to increase GB if z1/p2 pole zero cancellation is achieved Can extend to gm6/CL Or even a little bit higher But cannot push up too much higher Other poles, zeros Imprecise mirror pole/zero cancellation P2/z1 cancellation GB cannot be too high relative to these p/z cancellation Z2, z4, and pz=-1/RZCC must be much higher than GB

Possible design steps for max GB For a given CL and Itot Assume a current share ratio q, i.e. I6+I5 = Itot, I5 = qI6 , I1 = I2 = I5/2 Size W6, L6 to achieve max single stage GB1 = gm6/(CL+Coutpara) A good trade off is to size W6 so that Cgs6 ≈ CL If L_overlap ≈ 5% L6, this makes z4=gm6/Cgd6 ≈ 20*GB1 Choose GB = aGB1, e.g. 0.5gm6/(CL+C1) Choose CC to make p2 < GB, e.g. Cc=CL/4, p2 ≈ GB/1.5 Size W1, L1 and adjust q so that gm1/CC ≈ GB Make z2=gm2/Cgd2 > (10~20)GB, i.e. Cgd2 < 0.1Cc Size Mz so that z1 cancels p2 Make sure PM at f=GB is sufficient Size other transistors so that para |p| > GB/(10~20) Check slew rate, and size other transistors for ICMR, OSR, etc

If CL=C1=4Cc, -p2=gm6/(C1+CL+C1CL/Cc) =1/3 * gm6/(C1+CL) -pz=1/RzCc, Rz=1/gm6 *(1+CL/Cc+C1/Cc); -pz=gm6/(Cc+C1+CL) ≈ 3*(-p2) Pole/zero cancellation cancelled p2, but introduced a new pole pz at just a few times the p2 frequency, if done right;

For input common mode range Vi+ = Vi- = Vicm should be allowed to vary over a large range without causing transistors to go triode Vicm_max = (VDD – Vdssat_tail) – VT – Vdssat1 Vicm_min = Vs of M1c – VT = VG of M1c/2c + Vdssat VG of M1c must be low But must be higher than Vo1 – VT1c Room for Vo1 variation: +- VEB of 2nd stage

Hence, Vicm_min depends on differential signal  bias M1c adaptively, based on actual input signal

For Balanced Slew Rate During output slewing On the otherhand All of 1st stage current goes to Cc network I-Rz drop ≈ constant 2nd stage Vg variation << Vd or Vo |Cc d(Vo-Vg)/dt| ≈ |Cc dVo/dt| <= |I1st st | Slew rate = max |dVo/dt| = I1st st /Cc On the otherhand I2nd st bias - I1st st is to charge CL+Cdbs max |dVo/dt| = (I2nd st bias - I1st st )/(CL+Cdbs) Want (I2nd st bias - I1st st )/(CL+Cdbs) = I1st st /Cc I2nd st drive max - I1st st is to discharge CL+Cdbs

VDD Av=gm6/go -p=go/(CL+Cdb) GB=gm6 /(CL+Cdb) To maximize GB, size M1 so that Cdb ≈ CL  W1 ≈CL/(CjLd) GBmax ≈rt(I*uCox/(2L*CL*Cj*Ld)) =rt(SR*uCox/(2Cj*L*Ld)) This is greater than: gm6’/(CL+C1) ≈gm6’/(CL+Cgs)

Common Mode feedback All fully differential amplifier needs CMFB Common mode output, if uncontrolled, moves to either high or low end, causing triode operation Ways of common mode stabilization: external CMFB internal CMFB

Cause of common mode problem Unmatched quiescent currents Vbb=VbbQ+Δ Vbb I2 Vin=VinQ Vbb=VbbQ Vo1 Vo2 Vin I1 Vo1Q Vo1 Vin=VinQ+ΔVin actual Q point M2 is in triode

Vxx Ix Vo Ix(Vo) VOCM Vin Iy(Vo) Vyy Iy Vo

Basic concept of CMFB: Dvb e desired common mode voltage CM measurement Vo+ +Vo- 2 Vo+ Vo- Voc - CMFB Dvb e VoCM + desired common mode voltage

Basic concept of CMFB: Dvb e e measurement Vo+ +Vo- 2 Vo+ Vo- Voc - CMFB Dvb e e VoCM + Find transfer function from e to Voc, ACMF(s) Find transfer function from an error source to Voc Aerr(s) Voc error due to error source: err*Aerr(0)/ACMF(0)

example Vb2 CC CC Vi+ Vi- Vo+ Vo- VCMFB Vb1 Vo+ VCMFB Voc - Vo- +

Need to make sure to have negative feedback Example Voc ? ? VoCM Need to make sure to have negative feedback

Folded cascode amplifier VDD M7A 150/3 150/3 M2A M2B 300/3 300/3 75/3 M13A M13B BIAS4 averager 1.5pF 1.5pF M7B 75/3 M3B BIAS3 OUT+ OUT- 20K 20K M3A 300/2.25 300/2.25 300/2.25 300/2.25 M6C 75/2.25 IN- IN+ Source follower M1A M1B M12B M6AB M12A 1000/2.25 75/2.25 1000/2.25 200/2.25 BIAS2 M11 M10 M9A M9B CL=4pF 4pF 150/2.25 50/2.25 50/2.25 BIAS1 M8 M5 200/2.25 M4A M4B 150/2.25 50/2.25 50/2.25 VSS Folded cascode amplifier

VDD M2A M2B BIAS4 M13A M13B BIAS5 M3B OUT+ OUT- M3A BIAS3 IN- IN+ M1A M1B M12B M12A BIAS2 M10 M9A M9B M5 BIAS1 M4A M4B VSS

This corresponding part for vo1- to vo+ not shown VDD M5 M7 M5c Vin+ Vin- M1 M2 M1c M2c Vo- Vo+ Mz Vo1- Vo1+ CC M3c M4c M6 M3f M3 M4 M4f This corresponding part for vo1- to vo+ not shown Mz bias from the same circuit

VDD

Small signal analysis of CMFB Example: IB IB VCM M4 M3 Vo+ Vo- M1 M2 -Δi +Δi +Δi M5 +Δi -Δi -Δi VCMFB Δi=0 2Δi Differential signal Common mode signal

Differential Vo: Vo+↓ by ΔVo, Vo-↑ by ΔVo Common mode Vo: Vo+↑ by ΔVo, Vo-↑ by ΔVo

IB IB VCM M4 M3 Vo+ Vo- M1 M2 +Δi +Δi M5 -Δi -Δi VCMFB Δi=0 2Δi M7 Δi7 + - 1 gm6 -2Δi -2Δi M6

CMFB loop gain: example Vb2 CC CC Vi+ Vi- Vo+ Vo- VCMFB Vb1 Vo+ VCMFB Voc - Vo- +

All very similar, except go1 is now half Vo v gm5v -gm5vro4 -gm5vro4gm6 Compare Poles: p1 at Vo1 node: p2 at Vo node: z1 due to compensation All very similar, except go1 is now half

A = gm1/gm6 IB IB VCM M4 M3 Vo+ Vo- M1 M2 M5 M3 VCMFB Vo M1 VCM VCMFB Low impedance node M6

Simple transistor circuits Can use any # of ideal current or voltage sources, resisters, and switches Use one or two transistors Examine various ways to place the input and output nodes Find optimal connections for high gain high bandwidth high or low output impedance low input referred noise

Single transistor configurations It’s a four terminal device Three choices of input node For each input choice, there are two choices for the output node The other two terminals can be at VDD, GND, virtual short (V source), virtual open (I source), input, or output node Most connections are non-operative or duplicates D and S symmetric;

3 valid input choice and 1 output choice Connection of other terminals: or Resister

Capacitor Gnd or virtual Common source

This is D To VDD Source follower

N-channel common gate p-channel common gate

Diode connections

Building realistic circuits from simple connections flip vertical  Combine  N common source

flip left-right  N common source Combine to form differential pair 

Vbb flip upside down to get current source load  Vbb Combine to form differential amp

Can also use self biasing and convert to single ended output  Replace virtual gnd by current source

two transistor connections Start with one T connections, and add a second T Many possibilities many useless some obtainable by flip and combine from one T connections some new two T connections Search for ones with special properties in terms of AV, BW, ro, ri, etc

First MOST is CS D1 connects to D2: (with appropriate n-p pairing) -kvo vo vin CS with negative gm at output node CS Push pull CS

VDD VDD M3 M4 k k vo -vo M1 M2 vin -vin gm1 M5 AV= gm1vin+gds1vo+ gds3vo-kvogm3=0 gds1+gds3-kgm3 gds1+gds3 AV=  when k = gm3 GBW=gm1/Co = GBW of simple CS

When Vx = gnd T2 is not useful VDD When Vx = Vin, T2 and T1 are just one T Vo When Vx = kVo what do we get? Vx

KCL: gm1*Vin + Vo * (gds1+gds2) - Vo*gm2gm3/gm4=0 When Vx = kVo what do we get? VDD M3 Vo M1 M2 M4 Vx KCL: gm1*Vin + Vo * (gds1+gds2) - Vo*gm2gm3/gm4=0 -gm1 Av = gds1+gds2 - gm2gm3/gm4

VDD Vx=gnd, M2 is I source Vx = vin, ? Vo Vx = ─ vin, ? M1 M2 Vx = vo, capacitor Vx Vx = kvo, negative gds feedback

Vx = kvo, negative gds feedback VDD Vo kVo M1 M2 Vx -gm1 Av = gds1+gds2 – k*gm2

D1 connects to S2 VDD VDD VDD Cascode just a single NMOST any benefits?

VDD VDD Vo -kVx -kVo Vx Cascode with positive Vx feedback Cascode with positive Vo feedback

VDD VDD VDD Vin Vo Vo Vo Effects on GBW? Folded cascode

VDD VDD Vx -kVo -Vx Vo Vo folded cascode with positive feedback

VDD M2 M1 Vbb Vin CL Rb flip up-down for source M2 M1 Vbb Vin CL M4 M3 Vyy Vxx VDD connecting D1 to S2 cascoding

VDD M4 M2 Vbb Vin- CL M8 M6 M3 M1 Vin+ M7 M5 Vyy Vxx M9 flip left-right to get this differential telescopic cascoded amplifier add M9 to change gnd to virtual gnd GBW=gm1/Co

M4 M2 Vin- CL M8 M6 M3 M1 Vin+ M7 M5 Vyy M9 VDD Vx Vo How to connect G3 to –Vx, –kVx, or – kVo Same GBW Gain can be very high

M4 M2 Vin- CL M8 M6 M3 M1 Vin+ M7 M5 Vyy M9 VDD Vx Vo How to connect G3 to –Vx, –kVx, or – kVo Same GBW Gain can be very high

VDD Vin CL Vbb flip up-down for I sources VDD M1 Vin CL M2 Vbb connecting n-D to p-S

VDD VDD folded cascode amp Same GBW Vbb Vin+ Vin- CL

VDD VDD How to connect for positive feedback? Vbb Vin+ Vin- CL

D1 connects to G2, two stages VDD VDD VDD VDD two stage CS amplifier CS amplifier with a source follower buffer

VDD VDD VDD VDD VDD VDD Needs compensation and CM feedback Can gain be higher than single stage? Can GBW be improved vs single stage?

VDD VDD VDD VDD Vx -Vx -vin Can you connect without loading effect?

VDD VDD VDD VDD Vomin = Vin-min + Vdssat or = VT + 3Vdssat Biasing?

VDD VDD VDD VDD VDD VDD But is the gain improved? Is GBW improved? Vomin = 2Vdssat

VDD VDD VDD V? Vx Vx Same as above, only T2 is pMOS Connecting S1 to D2 makes ro really small buffer or output stage

VDD VDD or

VDD VDD VDD VDD M1 

connecting S1 to G2 VDD Vx Vx

VDD VDD

VDD VDD Vx  Vx?

connecting S1 to S2 Vo Vo -Vin

connecting S1 to D2 V? V?

? ? e.g. 

M1 is common gate: D1 connects to G2 VDD Vin

D1 connects to S2 Vin

PSRR

Vout = AddVdd + Av(V1-V2) = AddVdd - AvVout  Vout(1+Av) = AddVdd Good as long as Av >> 1, or f < GB

For zeros, set vdd = 0, vout float. DC gain: ignore all caps and find relationship between vdd and vout at vout Dgm1 at Id1same at Id2Dgm1/(gds2+gds4) at G6vg6gm6/gds6 across DS6 vdd= Dgm1/(gds2+gds4) *gm6/gds6 Vdd/vout = gm6gm1/gds6(gds2+gds4) For zeros, set vdd = 0, vout float. This is the unity gain buffer configuration of the amp. Hence, char roots are: -GB and p2

For poles, make vout = 0, vdd float. Three nodes: S3/S4/S6, G3/G4/D1: ignore Write KCL at D2/D4/G6 node: v(gds2+gds4+sCI+sCC)=vdd(gds4+gds1*1) Current balance in M6: gm6(v-vdd)=gds6vdd v=(1+gds6/gm6)vdd gds6/gm6*(gds2+gds4)+(1+gds6/gm6)s(CI+sCC)=0 gds6/gm6*(gds2+gds4)= -s(CI+sCC) Pole at - gds6(gds2+gds4) /(gm6(CC+CI))

Similar computation for PSRR- Get DC gain Get zeros: they are the same as in PSRR+, and the same as poles of unity feedback Avd Get dominant pole: Practice this, and see if you get similar results as in book

Two-Stage Cascode Architecture Why Cascode Op Amps? Control the frequency behavior Increase PSRR Simplifies design Where is the Cascode Technique Applied? First stage - Good noise performance Requires level translation to second stage Requires Miller compensation Second stage - Self compensating Reduces the efficiency of the Miller compensation Increases PSRR