HO13_315aSP09_single_stage_OTA_1

HO13_315aSP09_single_stage_OTA_1 - Single Stage OTAs Part 1...

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Unformatted text preview: Single Stage OTAs Part 1 Boris Murmann Stanford University murmann@stanford.edu Copyright © 2009 by Boris Murmann EE315A ― HO #13 B. Murmann 1 Basic Differential Pair OTA • Common mode feedback • Half circuit model • Return ratio analysis – Loop gain – Closed-loop gain • Noise N i analysis l i • Step response – Linear settling g – Slewing B. Murmann EE315A ― HO #13 2 Basic Differential Pair OTA VDD MP1a MP1b MPB Vom Vop - Vod + Vip Vim MN1a IT/2 MN1b IT • Suppose th t in the operating point Vip=Vim, i Vid=0 S that i th ti i t V i.e. 0 • What is the output common mode voltage Voc = (Vom+Vop)/2 ? EE315A ― HO #13 B. Murmann 3 Operating Point Sensitivity T D Operating Point 1 Operating Point 2 OC DD • The operating point is very sensitive to small changes in the device characteristics • Solution: Common mode feedback (CMFB) B. Murmann EE315A ― HO #13 4 CMFB • Common mode feedback loop adjusts ΔI such that VOC is very close to the desired voltage EE315A ― HO #13 B. Murmann 5 Idealized CMFB Implementation ΔI Voc = Voc,des + B. Murmann ΔI GCMFB = Voc,des EE315A ― HO #13 for GCMFB → ∞ 6 Ideal Balun xfmr xfmr • Useful for separating common mode and differential mode signal components p • Bi-directional, preserves port impedances • Uses ideal, inductorless transformers that work down to DC B. Murmann EE315A ― HO #13 7 CMFB Implementation • In practice, we won’t be able to let GCMFB p , ∞ for loop stability p y – Nonetheless, the loop will get us to within a few mV of where we need to be – And most importantly help absorb variations in the device characteristics • In the first few lectures on OTA design, we will use the idealized common mode f db k circuit ( shown previously) t avoid d feedback i it (as h i l ) to id distraction from the main design task • Practical CMFB implementation examples (using transistors) will follow later in this course B. Murmann EE315A ― HO #13 8 Differential Mode Small Signal Half Circuit • With the circuit at the proper operating point, we can analyze its small-signal behavior using a differential mode half circuit model • Note that (to first order) the CMFB loop does not influence the behavior of the differential mode signals B. Murmann EE315A ― HO #13 9 OTA with Capacitive Feedback • Let’s get started by placing our simple OTA into a capacitive feedback loop ( encountered e.g. i an SC circuit) f db k l (as t d in i it) • Questions – What is the phase margin? p g – What is the closed loop transfer function? – What is the total integrated noise? – How fast does this circuit settle (in response to a step)? B. Murmann EE315A ― HO #13 10 Half Circuit Model Gm = g mn Cftot Ro = rop rop Co = Cdbp + Cdbn Cx = Cgsn + Cgbn EE315A ― HO #13 B. Murmann 11 Return Ratio Analysis Cftot v x = β ⋅ vo CLtot ir β= Cftot Cftot + Cs + Cx vo vx Cs Cx Gmvx it Ro CL+Co C “Feedback factor” ⎛ 1 v 0 = −it ⋅ ⎜ R0 ⎜ sCLtot ⎝ T (s ) = − B. Murmann ⎞ ⎟ ⎟ ⎠ CLtot = CL + Co + (1 − β ) Cftot ⎛ ir 1 = β ⋅ Gm ⋅ ⎜ R0 ⎜ it sCLtot ⎝ ⎞ β ⋅ Gm R0 β ⋅ a0 = ⎟= ⎟ ⎠ 1 + sRoCLtot 1 + sRoCLtot EE315A ― HO #13 12 Frequency Response of T(s) T ( jω) T 0 = βGmR0 ωpo = Ro→∞ T0 1 R0CLtot ωpo T (s ) = βGmR0 βGm βGm = ≅ 1 1 + sRoCLtot + sCLtot sCLtot Ro βGm =1 j ωcCLtot ⇒ ωc ≅ β Gm CLtot for ωc Ro >> ω ω 1 ⇔ >> 1 sCLtot ωpo Ro is irrelevant for understanding high frequency behavior around ωc EE315A ― HO #13 B. Murmann 13 Phase Margin T ( jω) Ro→∞ T ( j ω) = T0 ωpo ωc ω ⎡T ( j ω) ⎤ ⎣ ⎦ 0° ωpo ω −45° −90° B. Murmann βGmR0 ω 1+ j ωpo T ( j ωc ) = βGm R0 ω 1+ j c ωpo ⎛ ω ⎡T ( j ω) ⎦ ⎤ = − tan −1 ⎜ c ⎣ ⎜ ωpo ω=ωc ⎝ ⎞ ⎟ ≅ −90° ⎟ ⎠ PM ≅ 180° − 90° ≅ 90° EE315A ― HO #13 14 Closed Loop Transfer Function A (s ) = • vo T(s ) d = A∞ + vi 1+ T ( s ) 1+ T ( s ) Need to find A∞ and d – Let’s start with A∞ Let s Gm → ∞ Cftot Cs vi vx vo ix Cx Ro CL+Co Gmvx ⇒ v xm → 0 ⇒ i x → 0 0 = v i sCs + v o sCftot A∞ = vo vi =− Gm →∞ EE315A ― HO #13 B. Murmann Cs Cftot 15 Finding d at Low Frequencies • Capacitors are open circuits d0 = B. Murmann vo vi =0 Gm = 0 EE315A ― HO #13 16 Low-Frequency Closed-Loop Gain A0 = A∞ T0 d + 0 1 + T0 1 + T0 A∞ = − ⇒ A0 = − • Cs Cftot T0 = βGmRo d0 = 0 Cs 1 1 Cftot 1 + βGm Ro Error in low-frequency closed-loop g q y p gain ε0 = A0 − A∞ A∞ ε0 = ⎛ A0 T 1 1⎞ 1 −1= 0 −1= − 1 ≅ ⎜1− ⎟ − 1 = − 1 A∞ 1 + T0 T0 ⎝ T0 ⎠ 1+ T0 ε0 ≅ 1 T0 EE315A ― HO #13 B. Murmann 17 d at High Frequencies ieq = v i Cs ⋅ sCftot = v i β ⋅ sCs Cs + Cx + Cftot d= B. Murmann vo vi = Gm = 0 Ceq = (1 − β ) Cftot ieq C 1 =β s v i s Ceq + CL + Co CLtot ( EE315A ― HO #13 ) 18 High-Frequency Closed-Loop Gain (1) A( s ) ≅ A∞ ( ≅− • T(s ) d + 1+ T ( s ) 1+ T ( s ) Cs Cftot βCs βGm C s 1 − s ftot sCLtot CLtot Cs Cs 1 − z Gm + =− =− βGm βGm Cftot 1 + s CLtot Cftot 1 − s 1+ 1+ p sCLtot sCLtot βGm Pole frequency: ωp ≅ βGmRo βGm ≅ ωc ≅ ≅ T0 ⋅ ωpo CLtot RoCLtot As expected. EE315A ― HO #13 B. Murmann 19 High-Frequency Closed-Loop Gain (2) Cftot s Cs Cs 1 − z Gm =− A( s ) = − Cf 1 + s CLtot Cf 1 − s p βGm 1− s • Zero frequency: ωz = Gm Cftot ωz CLtot = ωp βCftot usually >> 1 • Therefore, the closed-loop -3dB frequency is approximately ω−3dB ≅ ωp ≅ B. Murmann βGm CLtot EE315A ― HO #13 20 Putting it All Together s Cs 1 1 − z A( s ) ≅ − Cf 1 + 1 1 − s T0 p T0 = βGmR0 p≅− βGm CLtot z≅+ Gm Cftot β= Cftot EE315A ― HO #13 B. Murmann Cftot + Cs + Cx 21 Noise Analysis (1) R≅ 1 βg mn Neglecting finite output resistance of the MOSFETs 2 vo 1 = 4kT γ n g mn + γ p g mp ⋅ R Δf j ωCLtot ( ) γ p g mp ⎛ = 4kT γ n g mn ⎜ 1 + γ n g mn ⎝ B. Murmann EE315A ― HO #13 2 ⎞ R ⎟⋅ ⎠ 1 + j ωRCLtot 2 22 Noise Analysis (2) 2 vo ∞ γ p g mp ⎛ p = ∫ 4kT γ n g mn ⎜ 1 + γ n g mn ⎝ 0 γ p g mp ⎛ = 4kT γ n g mn ⎜ 1 + γ n g mn ⎝ ⎞ R ⎟⋅ ⎠ 1 + j ωRCLtot 2 df ⎞ 2 1 ⎟⋅R ⋅ 4RCLtot ⎠ γ p g mp ⎞ ⎛ 1 1 = 4kT γ n g mn ⎜ 1 + ⋅ ⎟⋅ γ n g mn ⎠ βg mn 4CLtot ⎝ ⎛ ⎜ γ p g mp 1 kT = γn ⎜1+ β CLtot γ n g mn ⎜ ⎜ ⎝ B. Murmann ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ EE315A ― HO #13 23 Noise Analysis (3) • For low noise – Make gm2 as small as possible, i.e. use small gm/ID for current source device • Issue: Smaller gm/ID means larger “Vdsat” i.e. less available voltage swing – Maximize feedback factor β β= Cftot Cftot 1 1 1 = ≅ = Cggn g 1 + Cs + Cx 1 + Cs + Cx 1 + A∞ + mn 1 + A∞ + Cftot ωT Cftot Cftot Cftot Want ωt ∞ (short channel) B. Murmann EE315A ― HO #13 24 Noise in Differential Circuits • In differential circuits, the noise power is doubled (because there are two half circuits contributing to the noise) • But, the signal power increases by 4x – Looks like a 3dB win? DRsingle ˆ V2 ∝ o kT C DRdiff ( 2Vˆ ) ∝ o 2 kT C 2 =2 ˆ2 Vo kT C • Yes, there’s a 3dB win in DR, but it comes at twice the power dissipation (due to two half circuits) • Can get the same DR/power in a single ended circuit by doubling all cap sizes and gm EE315A ― HO #13 B. Murmann 25 OTA1 Behavioral Model Parameters: Gm, a0, fT, γ OTA1 in + vx - Gm 2π fT Rnoise = Cin Rnoise 1 in Gmvx Ro noiseless 1 γ ⋅ Gm 2 i n = 4kT 1 Rnoise Δf = 4kT γ ⋅ Gm Δf CCCS B. Murmann a0 Gm Cin = - Ro = EE315A ― HO #13 26 ...
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