6.002 CIRCUITS AND ELECTRONICS Incremental Analysis 6.002 Fall 2000 Lecture 7 1 Review Nonlinear Analysis X Analytical method X Graphical method Today X Incremental analysis Reading: Section 4.5 6.002 Fall 2000 Lecture 7 2 Method 3: Incremental Analysis Motivation: music over a light beam Can we pull this off? iD + vD LED light intensity I D ∝ iD vI music signal vI (t ) + – iR t vI (t ) iD (t ) light AMP iR ∝ I R light intensity IR in photoreceiver LED: Light Emitting expoDweep ☺ iR (t ) sound nonlinear linear problem! will result in distortion 6.002 Fall 2000 Lecture 7 3 Problem: The LED is nonlinear distortion iD iD vD vD = vI t vD t iD 6.002 Fall 2000 vD Lecture 7 t 4 Insight: iD small region looks linear (about VD , ID) ID VD vD DC offset or DC bias Trick: vI vi (t ) + – VI + – iD = I D + id + vD LED vD = VD + vd VI 6.002 Fall 2000 Lecture 7 vi 5 Result iD id ID vD VD 6.002 Fall 2000 Lecture 7 vd very small 6 Result vD = vI vd vD VD t iD id iD ~linear! ID t Demo 6.002 Fall 2000 Lecture 7 7 The incremental method: (or small signal method) 1. Operate at some DC offset or bias point VD, ID . 2. Superimpose small signal vd (music) on top of VD . 3. Response id to small signal vd is approximately linear. Notation: iD = I D + id total DC small variable offset superimposed signal 6.002 Fall 2000 Lecture 7 8 What does this mean mathematically? Or, why is the small signal response linear? nonlinear iD = f (vD ) We replaced vD = VD + ∆vD large DC vd increment about VD using Taylor’s Expansion to expand f(vD) near vD=VD : iD = f (VD ) + + df (vD ) ⋅ ∆vD dvD vD =VD 1 d 2 f (v D ) 2! dvD 2 v 2 ⋅ ∆vD + " D =VD neglect higher order terms because ∆vD is small 6.002 Fall 2000 Lecture 7 9 iD ≈ f (VD ) + constant w.r.t. ∆vD d f (v D ) ⋅ ∆vD d vD vD =VD constant w.r.t. ∆vD slope at VD, ID We can write X : I D + ∆iD ≈ f (VD ) + d f (v D ) ⋅ ∆ vD d vD vD =VD equating DC and time-varying parts, I D = f (VD ) operating point d f (v D ) ∆iD = ⋅ ∆vD d vD vD =VD constant w.r.t. ∆vD so, ∆ iD ∝ ∆vD 6.002 Fall 2000 Lecture 7 By notation, ∆ iD = id ∆ v D = vd 10 In our example, iD = a e bv D From X : I D + id ≈ a e bVD + a e bVD ⋅ b ⋅ vd Equate DC and incremental terms, I D = a ebVD operating point aka bias pt. aka DC offset id = a ebVD ⋅ b ⋅ vd id = I D ⋅ b ⋅ vd constant 6.002 Fall 2000 Lecture 7 small signal behavior linear! 11 Graphical interpretation operating point I D = a ebVD id = I D ⋅ b ⋅ vd A slope at VD, ID iD ID id B VD operating point vd vD we are approximating A with B 6.002 Fall 2000 Lecture 7 12 graphically mathematically now, circuit We saw the small signal Large signal circuit: VI ID + LED VD - + – I D = a ebVD Small signal response: id = I D b vd + vd - behaves like: id R= small signal circuit: vi + – + vd - 1 ID b id 1 I Db Linear! 6.002 Fall 2000 Lecture 7 13