EECS240 Spring Lecture 13: Settling. Lingkai Kong Dept. of EECS

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1 EES240 Spring 203 Lecture 3: Settling Lingkai Kong Dept. of EES Settling Why intereted in ettling? Ocillocope: track input waveform without ringing AD (witchedcap amplifier): gain a ignal up by a precie amount within T ample Φ 2 Φ Φ f Φ2 Φ2 o EES240 Lecture 3 2

2 Step Repone Two type of ettling error : Static Finite gain, capacitor mimatch Dynamic Take time to reach final value o /A v time EES240 Lecture 3 3 Static Error i x f A vo o KL à " o i = + c FA vo T o o /A v tatic error with" f F = + + c = f i f time EES240 Lecture 3 4 2

3 Static Error (cont.) o i c = c + FA vo FA { vo relative error Example: loed loop gain: c = 4, f = pf, = 4pF, i = pf F = /6 ( i hurt!) Error pecification: <0.% FA vo > 000 A vo > 6000 over output range EES240 Lecture 3 5 Dynamic Error Many poible dynamic effect that impact ettling error: Finite bandwidth Feedforward zero Nondominant pole Doublet Slewing Approximate analyi approach: Decompoe each error ource, iolate interaction Add all error together EES240 Lecture 3 6 3

4 Single Time ontant Linear Settling For dynamic ettling (and for T 0 >> ), can generally ignore r o i x f G m o L o i f Gm = c L + + FG ( F ) m f EES240 Lecture 3 7 Time Domain Step Repone Frequency domain: Time domain: o, tep + z = c + p tep Note: "For p=z the error i zero and the circuit ha infinite bandwidth." "Application?" EES240 Lecture 3 8 4

5 Time Domain Step Repone Frequency domain: input tep: i, tep = output tep: tep + z = c + p o, tep i, tep + z = c + p tep v o,tep Time domain: (invere Laplace tranform) ( * " % * $ ( t) = tep c p ' * $ ' * $ z e pt ' ideal repone* $ initial error ' * #(feedforward) & ) * exponentially decaying error +, EES240 Lecture 3 9 ae : p/z << ( t) v o, tep tepc e ideal repone t τ Relative ettling error: ε ( ) ( = ) v ( t ) v t v t t o o = = o t = ln ε τ e t τ Eaiet number to remember: 2.3τ per decade Example: % ettling, 4.6n clock cycle: τ = n L,eff uually et by noie ue ettling to determine required g m EES240 Lecture 3 0 5

6 ae 2: p/z not negligible ( t) c e t τ vo, tep tep ideal repone p z Relative ettling error: vo( t ) vo( t = t) p ε = = e v ( t ) z t ε = ln τ Ff + Leff Example: c = 0.25, f = pf, = 250fF, i = 250fF, L = pf F = 0.67, L,eff =.33pF ε = 0.%: t (no feedforward) = 6.9τ t (with feedforward) = ln[e3/(+0.67*0.75)]=7.3τ EES240 Lecture 3 o t τ NonDominant Pole Ignore feedforward zero for implicity (Jut increae final wing by +F f / L,eff ) H ( ) = Model for nondominant pole: G p ω m 2 u ( ) = Kω o in = c + FG Gm0 = + p u 2 Leff m ( ) i unity gain bandwidth of T EES240 Lecture 3 2 6

7 Step Repone EES240 Lecture 3 3 NonDominant Pole (cont.) Relative error : ε = v ( t, K) EES240 Lecture 3 4 7

8 Settling Time Settling time : t 3 ( K) for ε = 0, τ = Optimum at K=3.3 EES240 Lecture 3 5 NonDominant Pole v. ε Optimum K actually depend on required accuracy Still, alway want to avoid K<~2 EES240 Lecture 3 6 8

9 Doublet Amplifier model: G () = G m mo + ωz + ω p with ω p = βω 3dB, ω 3dB ω p ωz = α α = + ε with ε << ibandwidth of T ( ) loedloop gain (ignore feedforward zero): o c + ω z = c + ω + ω + FG in Leff 3dB pp with" m ( ) ω 3dB ω pp FG ω p mo Leff EES240 Lecture 3 7 Doublet Analyi Step repone tω t 3dB ( ) tep ( = ) v t c Ae Be ω pp o, tep with" B ε β ( β ) 2 A B EES240 Lecture 3 8 9

10 Doublet Example α=.5 β=0.3 EES240 Lecture 3 9 Doublet oncluion ae A: τ2 τ i.e. β Doublet ettle fater than amplifier Ha no impact on overall ettling time τ > ae B: 2 τ Doublet ettle more lowly than amplifier Determine overall ettling time (unle ε within ettling accuracy requirement) Avoid low doublet! EES240 Lecture

11 Final Note on Doublet EES240 Lecture 3 2 Slewing Tranconductor ΔI v. Δ: Model for (nonlinear) lewing amplifier Piecewie linear approximation: Slewing with contant current, followed by Linear ettling exponential t = t lew + t,lin EES240 Lecture 3 22

12 Slewing Analyi ircuit model during lewing: f p x I SS L o EES240 Lecture 3 23 Slewing Analyi (cont.) i,tep x x,tep * o t lew t lin o,tep time EES240 Lecture

13 Slewing Analyi (cont.) Slewing period: f L x, tep = i, tep with 2 = i f L Δ Δ x = x, tep * Δ o = F Δ Δ o x Leff tlew = = SR FI SS Linear ettling during final * of wing at x : Step during linear ettling: * F t, lin ci, tep F Linear ettling time: = ln ε τ * EES240 Lecture 3 25 x 3