INF3410 Fall Book Chapter 6: Basic Opamp Design and Compensation

INF3410 Fall 2015 Book Chapter 6: Basic Opamp Design and Compensation

content Introduction Two Stage Opamps Compensation Slew Rate Systematic Offset Advanced Current Mirrors Operational Transconductance Amplifiers Current Mirror Opamps Folded Cascode Opamp Fully Differential Amplifiers Advanced Circuits Essentials Summary Operational Transconductance Amps Fully Differential OpAmps Advanced Current Mirrors Book Chapter 6: Basic Opamp Design and Compensation 2

Content Introduction Two Stage Opamps Compensation Slew Rate Systematic Offset Advanced Current Mirrors Operational Transconductance Amplifiers Current Mirror Opamps Folded Cascode Opamp Fully Differential Amplifiers Advanced Circuits Essentials Summary Operational Transconductance Amps Fully Differential OpAmps Advanced Current Mirrors Book Chapter 6: Basic Opamp Design and Compensation 3

Classic Uses of Opamps An Operational Amplifier (Opamp) is a high gain voltage amplifier with differential input. Classic applications are: Book Chapter 6: Basic Opamp Design and Compensation 4

Two Stage CMOS Opamps The classic way of getting high gain is a two stage solution, also providing high output swing (as opposed to e.g. cascode gain stages). General principle: The compensation capacitor C cmp in conjunction with the output resistance of the first stage limits the bandwidth, which can be handy to stabilize the circuit when employed in a feedback configuration. Book Chapter 6: Basic Opamp Design and Compensation 6

Two Stage CMOS Opamp Example A simple example: DC Gain (in a first, mostly valid approximation): A = A v1 A v2 Book Chapter 6: Basic Opamp Design and Compensation 7

First Order Approximation of Frequency Response In mid range (only C C matters) simplified to: A v1 = g m1 Z out1 (6.5) 1 g m1 r ds2 r ds4 sc C A v2 A v (s) A v2 g m1 sc C A v2 = g m1 sc C (6 ω ta < g m1 C C (6.9) = = I bias V eff 1 C C (6.10, strong inv.) Book Chapter 6: Basic Opamp Design and Compensation 8 qi bias 2nkTC C (weak inv.)

Second Order Approximation of Frequency Response (1/2) Second order becomes necessary for analysis close to ω ta. Without R C : g m1 g m7 R 1 R 2 1 sc C g m7 A v (s) = (6.15) 1 + sa + s 2 b Book Chapter 6: Basic Opamp Design and Compensation 9

Interrupt: Second Order Approximation Deduction 1 v 1( 1 R 1 + sc 1 + sc C) + v in g m1 = v outsc C 1 v out( 1 R 2 + sc 2 + sc C) + v 1g m7 = v 1sC C 2 v 1 = v out 1 R 2 + sc 2 + sc C sc C g m7 2 3 Book Chapter 6: Basic Opamp Design and Compensation 10

Interrupt: Second Order Approximation Deduction 2 v out 1 R 2 + sc 2 + sc C sc C g m7 ( 1 R 1 + sc 1 + sc C) + v in g m1 = v outsc C 3 1 = 4 1 R v out sc C 1 R + s( C 2 +C C 2 R + C 1 +C C 1 R ) + s 2 (C 1 + C 2 C)(C 2 + C C) = v in g m1 4 = 5 sc C g m7 1 scc(scc gm7) R v out 1 R s( C 2 +C C 2 R + C 1 +C C 1 R ) s 2 (C 1 + C 2 C)(C 2 + C C) = v in g m1 5 = 6 sc C g m7 Book Chapter 6: Basic Opamp Design and Compensation 11

Interrupt: Second Order Approximation Deduction 3 1 R v 1 R s(c 2 Cg m7 + C 2 +C C R 1 out + C 1 +C C R ) + s 2 C 2 2 C (C1 + CC)(C2 + CC) = v in g m1 6 = 7 sc C g m7 v out sc C g m7 = g m1 v in 1 R 1 R s(c 2 Cg m7 + C 2 +C C R + C 1 +C C 1 R ) + s 2 C 2 7 = 8 2 C (C1 + CC)(C2 + CC) Book Chapter 6: Basic Opamp Design and Compensation 12

Interrupt: Second Order Approximation Deduction 4 v out v in = R 1R 2g m1g m7( sc C g m7 1) 1 s(c Cg m7r 1R 2 + R 2(C 2 + C C) + R 1(C 1 + C C)) + s 2 R 1R 2(C 2 C (C1 + CC)(C2 + CC)) rearrangin And that s the same as (6.15) with minimal rearranging. Thus: z 1 = gm7 C C Book Chapter 6: Basic Opamp Design and Compensation 13

Second Order Approximation of Frequency Response (2/2) ω 1 1 g m7 R 1 R 2 C C (6.19) ω 2 g m7 C 1 + C 2 (6.20) z 1 = g m7 C C The problem with positive zeros is negative phase shift, here dependent on C C : Increasing C C will reduce ω ta but also the frequency at which the phase shift becomes -180 o Book Chapter 6: Basic, making Opamp Design a feedback and Compensationsystem no more stable. 14

Compensation Tools Dominant pole compensation: Moving (only) the dominant pole of the open loop gain to a lower frequency. (Shifting ω t to a frequency smaller than the second most dominant pole) Lead compensation: Introducing a right hand side zero that shifts the -180 o phase shift to higher frequencies. Book Chapter 6: Basic Opamp Design and Compensation 15

Interrupt: Second Order Approximation Deduction 5 (With R C ) with R C in series the admittance sc C becomes sc 1+sC C R C and thus the term sc C g m7 1 in the nominator of (6.15) becomes (neglecting influences of R C elswhere): sc C g m7 1 sc C (1 + sc CR C)g m7 1 scc gm7 sgm7ccrc (1 + sc CR C)g m7 The nominator of that term becomes: /g m7 1 s(c C g m7c CR C) g m7 sc C( R C) 1 (Philipp 9) g m7 Book Chapter 6: Basic Opamp Design and Compensation 16

Lead compensation (1/2) With R C the zero becomes (without much influencing the poles!): 1 z 1 = (6.43) C 1 C R g m7 C R C can now be chosen to eliminate the zero (see equation(philipp 9)!): R C = 1 g m7 (6.44) or to negate the non-dominant pole ω 2 (using (6.20)): R C = 1 1 + C 1 + C 2 (6.45) Book Chapter 6: Basic Opamp Design and Compensation 17 g m7 C C

Lead compensation (2/2) Or to choose R C even higher to not cancel phase shift due to ω 2/eq to -180 o entirely but to delay it (create a phase lead), e.g. (dependent on a β in a closed loop application): 1 R C 1.7βg m1 IN all of the above R C may conveniently be implemented as transistor Q 9 in triode region (this β is the EKV notation β = μc ox W L ): R C = 1 βv eff 9 Book Chapter 6: Basic Opamp Design and Compensation 18

Slew Rate Concept The speed of an OpAmp output is not only limited by bandwidth but also by the bias current, as the output current cannot be bigger than the bias current. Thus, a big input step will get the transconductance out of its linear range and the output current saturates. Thus the maximum output gradient of an OpAmp is called slew rate (SR) in units [V/s]. Book Chapter 6: Basic Opamp Design and Compensation 19

Slew Rate Illustration V step,max < SR τ Book Chapter 6: Basic Opamp Design and Compensation 20

Increasing the Slew Rate The slew rate is dictated by the bias current and the compensation capacitor: SR = I D5 C C However, simply increasing the bias current or decreasing C C will raise ω ta, potentially making the circuit unstable. Thus, one needs also to increase ω 2 and/or V eff 1 (i.e. reduce (W/L) 1 ) to maintain proper compensation, which the book says are the only ways to design for higher slew rate. Book Chapter 6: Basic Opamp Design and Compensation 21

Systematic Offset Basically the zero output of stage one has to closely match the zero input of stage two. (What happens oterwise?) Zero input of stage two means the currents in Q 6 and Q 7 need to be equal. Zero output from stage one means that Q 4 is sinking half the bias current (while Q 5 is sourcing the whole bias current). Thus, if for instance Q 5 and Q 6 have the same W/L, then Q 7 needs to have twice the W/L of Q 4. More generally: W/L 7 W/L 4! = 2 W/L 6 W/L 5 (6.38) Book Chapter 6: Basic Opamp Design and Compensation 22

Cascode Current Mirror The cascode current mirror in chapter 3 reduces the output headroom by V out > 2V eff + V t0 (3.42). The problem is that the sources of the transistor closest to the output is at V eff + V t0. There are alternatives that provide equally high output resistance with less loss of headroom/output-swing. Book Chapter 6: Basic Opamp Design and Compensation 24

Wide-Swing Current Mirrors Think this circuit through for the case where I bias = I in. Then the current through all transistors is the same. For constant current in strong inversion (!) if you scale W/L by 1/a 2, V eff scales with a. V s1 = V g5 V gs1 = (V eff (n + 1) + V t0 ) (V eff n + V t0 ) = V eff and thus V out > (n + 1)V eff (6.78) for all transistors to be saturated. For instance for n=1 the optimum V out > 2V eff (6.79) is obtained. For I in < I bias, the minimum V out will shrink in absolute terms, but will no longer be optimal in terms of V eff. For I in > I bias the output resistance drops dramatically as the transistors Book Chapter 6: Basic Opamp Design and Compensation 25 enter the triode region.

Enhanced Output Impedance Current Mirrors (1/2) Similarly to the cascode current mirror V d2 (and thus the current through Q 2 ) is attempted to be kept as constant as possible. While V g1 is constant in the cascode current mirror, here it is actively moved to compensate the influence of V out on V d2 So while the a circuit with constant V g1 would have R out g m1 r ds1 r ds2 (like a cascode current mirror), this circuit has: R out (A + 1)g m1 r ds1 r ds2 (6.82) Book Chapter 6: Basic Opamp Design and Compensation 26

Enhanced Output Impedance Current Mirrors (2/2) Note: V bias needs to be big enough to keep Q 2 in saturation! Stability of feedback loop needs to be veryfied! Parasitic resistance from drain to bulk may become the actual limiting factor! Book Chapter 6: Basic Opamp Design and Compensation 27

Enhanced Gain Cascode Gain Stage A V (s) = g m2 R out 1 The same technique can be used to enhance the output resistance, and thus the gain of a cascode gain stage. Note: The current source needs a similarly enhanced output resistance! sc L (6.83) R out (s) = g m1 r ds1 r ds2 (1 + A(s)) (6.84) Book Chapter 6: Basic Opamp Design and Compensation 28

Enhanced Output Impedance Current Mirrors Implementation The amplifier is a common source gain stage. Note: Again the output swing is quite limited by V out > V eff 3 + V tn + V eff 1 (one way of looking at this is that the amp s V bias = V eff 3 + V tn ) r out (s) g m1 r ds1 r ds2 (g m3 r ds3 2 ) (6.93) Book Chapter 6: Basic Opamp Design and Compensation 29

Wide Swing AND enhanced impedance Book Chapter 6: Basic Opamp Design and Compensation 30

Space and Power Conserving Variant Quite equivalent with worse current matching but less power and layout space consumption. More modular with splitting Q 2 and presumably better stability. Book Chapter 6: Basic Opamp Design and Compensation 31

Operational Transconductance Amplifiers These are operational amplifiers with high output impedance, limited in bandwith by the output load (and not in internal nodes that are low impedance nodes). Thus, mainly suited for capacitive output loads only! Book Chapter 6: Basic Opamp Design and Compensation 33

Current Mirror Opamp A simple concept boosting the output current resulting in good bandwidth and good slew rate assuming C L is dominant. A V (s) = ω ta SR = Kg m1 C L Kg m1 r out 1+sr out C L (6.119) = 2KI D1 C L V eff 1 (6.121) KI b C L (6.124) Book Chapter 6: Basic Opamp Design and Compensation 34

Folded Cascode Opamp Ignore Q 12 and Q 13 for an initial analysis. Think of it as an extension of a differential pair: the cascodes simply increase the output resistance of the differential output current higher voltage gain given the same transconductance. Book Chapter 6: Basic Opamp Design and Compensation 35

Interrupt: Common Mode Rejection Ratio On the white board... Book Chapter 6: Basic Opamp Design and Compensation 37

Basic TransAmp with Diff Output Book Chapter 6: Basic Opamp Design and Compensation 38

Small Signal Considerations Book Chapter 6: Basic Opamp Design and Compensation 39

Fully Differential Current Mirror Opamp Book Chapter 6: Basic Opamp Design and Compensation 40

Dual Single Ended Structure Actively pulling the output up and down. (Class AB amplifier as opposed to class A). Better symmetrical slew rate. CMFB needed! Book Chapter 6: Basic Opamp Design and Compensation 41

Partially Dual Single Ended Structure Actively pulling the output up and down. Also better (symmetrical) slew rate, but maybe worse bandwith (due to more capacitance in current mirrors). CMFB needed! Book Chapter 6: Basic Opamp Design and Compensation 42

Wide Input Fully Differential Cascode OpAmp A problem with low supply voltage is the minimum requirement for the common mode voltage. Complementary input pairs help. Book Chapter 6: Basic Opamp Design and Compensation 43

Two Stage Differential OpAmp Another challenge with low supply voltage is the output swing. Common source output stages do comparatively well: just one V eff away from the rails. Book Chapter 6: Basic Opamp Design and Compensation 44

Common Mode Feedback Principle Carefull: A feedback loop that needs to be stable! Book Chapter 6: Basic Opamp Design and Compensation 45

Continuous Common Mode Feedback Variant 1 Book Chapter 6: Basic Opamp Design and Compensation 46

Continuous Common Mode Feedback Variant 2 Saturation of the diff-pairs is a problem as the outputs swing much wider as the input reduce gain. Book Chapter 6: Basic Opamp Design and Compensation 47

Continuous Common Mode Feedback Variant 3 Book Chapter 6: Basic Opamp Design and Compensation 48

Switched Cap Common Mode Feedback Book Chapter 6: Basic Opamp Design and Compensation 49

DiffPair At the core of almost all differential CMOS amplifiers is the diff-pair. The diff-pair invariably translates a differential input voltage into a differential output current around a small signal operating point. (Interestingly this is true for any large signal monotonic function of I D (V GS )...) From here on it is best to think in currents for a while rather than voltages. i + i - g v m +x r ds g v m -x r ds v x Book Chapter 6: Basic Opamp Design and Compensation 51

Differential Current Source With complementary inputs (v +x = v x ), v x will be clamped to 0V, simplifying things considerably: two small signal current sources with a parallel output-resistance (common source gain stages, in fact). The simplification holds even for the large signal model where the output current is limited by [0, I B ] and the DC V x± is the average of both inputs. The large signal model is a sinking(!) current source for an nfet pair: so practically you can only connect anything at the top terminal. i + i - g v m +x r ds g v m -x r ds Book Chapter 6: Basic Opamp Design and Compensation 52

Differential Transconductance Amplifier Using a current mirror you can turn one of your large signal sinking current sources into a sourcing current source. Thus, you can connect the two output currents in a single node thereby subtracting them: you get a single ended current output if you connect it to a low (input) impedance node. g m v+x g mv-x rds rds i+ i- i out Book Chapter 6: Basic Opamp Design and Compensation 53

Differential Operational Transconductance Amplifier (1/2) g v m +x r ds Or you get a single ended voltage output if you connect it to a high (input) impedance node. v out C L g v m -x r ds Book Chapter 6: Basic Opamp Design and Compensation 54

Differential Operational Transconductance Amplifier (2/2) Rearranging the circuit yields a very simple small signal (DC) model. If the output is connected to a significant capacitive load, this model is even good enough for AC. g (v - v ) m +x v out r ds -x r ds C L Book Chapter 6: Basic Opamp Design and Compensation 55

Folded Casode OpAmp The only difference from a small signal perspective of the folded cascode opamp is an increased output resistance. Simply see the cascode gain stage in chapter 3 if you want to understand how this is achieved starting from two Differential Current Sources, or more precisely from two differntial common source gain stages. g (v - v ) m +x g r r m ds ds -x v out g r r m ds ds C L Book Chapter 6: Basic Opamp Design and Compensation 56

Current Mirror OpAmp And the current mirror opamp r simply increases the ds transconductance Kg m (v +x- v -x) v out r ds C L Book Chapter 6: Basic Opamp Design and Compensation 57

Fully Differential OpAmps Fully Differential Opamps are in a way simpler, as one can go back to only considering the diff-pair small signal model with complementary inputs. The tricky bit is the large signal point of operation, as one needs to provide exactly matched current sources of I b for each 2 branch of the diff pair to ensure zero output for zero input. Thus, the common mode feedback circuits. i + i - g v m +x r ds g v m -x r ds Book Chapter 6: Basic Opamp Design and Compensation 58

Cascode Principle i out This whole section deals (again) with the marvels of a cascode transistor that hugely enhances a common source stage output resistance, bringing it closer to a ideal current source. It basically adds a series resistance of g m r ds : g (-v ) m r ds x v x r ds g mvin Book Chapter 6: Basic Opamp Design and Compensation 59

Advanced Current Mirrors The rest of this section in the book introduces various ways to a) deduce an optimal V bias to maximize the output swing and b) to make V bias dynamic to increase the output resistance even more. The basic principle of b) is illustrated in: Cadence demonstrations live... Book Chapter 6: Basic Opamp Design and Compensation 60