Analytic Model and Bilateral Approximation for Clocked Comparator

Transcription

1 Analyic Model and Bilaeral Approximaion for Clocked Comparaor M. Greians, E. Hermanis, G.Supols Insiue of, Riga, Lavia, Research is suppored by: 1) ESF projec Nr.1DP/ /09/APIA/VIAA/020, which is co-financed by EU, 2) Lavian Council of Science hrough he projec Nr ICSES2010, Gliwice, ICSES 2010, Sepember Gais 7-10, Supols Gliwice, Poland

2 Ouline Inroducion Scope of he work Clocked comparaor model Sysems differenial equaion Analyical soluion of sysems diff equaion Experimenal resuls Conclusion 2

3 Inroducion Clocked comparaors are used in: ADC Memory devices Mixed-signal sysems... Comparaors' main parameers are: Propagaion Delay, [μs,ns] Offse Volage, [mv] Criical hreshold curve... 3

4 Moivaion Knowledge abou comparaors' characerisics allows o build more flexible designs. Inpu Theoreically: Sep[( θ)] Inpu In praxis: Sep[( θ)] clock clock = τ = τ 4

5 Scope of he work To offer a model of a clocked comparaor To describe he simplified model wih he firs order differenial equaion and solve i analyically To offer bilaeral approximaion for models described by higher order differenial equaions To ge comparaor equivalen ransiion characerisics To confirm he resuls in an experimen 5

6 Simplified model wih parameric resisance Clocked comparaor Simplified equivalen circui R1, R2 inpu resisances Rn variable resisance wich changes he sign when he clock is acive C parasisic capaciance 6

7 One sep closer o he descripion of he model Inpu 1 Inpu 12 Sep[( θ)] = θ U REF = θ clk + Rn - = τ = τ 7

8 Differenial equaion of he model Firs order ordinary differenial equaion y ' [ ] a 1 y [ ]=a 2 U ref a 3 Sep[ ] If R1, R2 hen he erm 0 a 1 = 1 R n C R 1 R 2 R 1 R 2 C, a 2 = 1 R 2 C a 3 = 1 R 1 C a 1 1 R n C The only parameric coefficien in he equaion, ha changes is' sign, when he clock is acive 8

9 Equaion of he model y ' [ ] a 1 y [ ]=a 2 U ref a 3 Sep[ ] Consans: Variable coefficien: a 3 = 1 R 1 C A a 1 1 R n C a 2 = 1 R 2 C B K [ ] Equaion wih ime varying coefficien K(): y ' [ ] K [ ] y [ ]= A Sep[ ] B U REF K [ ] The coefficien changes he sign when = τ 9

10 Characerisic funcion To solve he differenial equaion we need o choose characerisic funcion of comparaor K[( τ )] y ' [ ] K [ ] y[ ]= A Sep[ ] B U ref? Possibiliies: Linear funcion: 0 Non-linear: arcan(), arcanh() + K[(-τ)] - = τ The chosen funcion: K [ ]=r anh[ r ] 10

11 Soluion o obain criical hreshold 1 Equaion is solved considering iniial condiions: y[ 0 ] = 0 2 Limi when 0 - is found 3 Reference level U REF is obained by solving y - = 0 4 Limi when gives he expression of he criical hreshold U C = A 4[ anh[ r ]] 2 2 B A = B = 1 Θ = 5 11

12 Impulse response If criical hreshold is considered as he ransiion characerisic, hen firs derivaive illusraes impulse response h = Ar sech[ r ] B A = B = 1 Θ = 5, r = 5 A response o any arbirary signal could be acquired by using composiion of inpu signal and impulse response funcion V = h f d 12

13 Example: Recangular pulse R = = 2 Sep[ 2] Sep[ 15 ] arcan[anh [ 15 2 sech[ ] d = ]] arcan[ anh[ 2 2 ]] A = B = 1 Θ = 5 13

14 Higher order model a N y N [ ] a N 1 y N 1 [ ]... a 0 y[ ]= f *The order depends on he number of he reacive elemens in he model 14

15 Bilaeral approximaion a N y N [ ] a N 1 y N 1 [ ]... a 0 y [ ]= f a N z N [ ] a N 1 z N 1 [ ]... a 0 z [ ]=0 a N p N a N 1 p N 1... a 0 =0 { G G = K + = k =0 K G - = k= K 1 A k exp p k if A k exp p k if U = G f d 15

16 Experimenal seup Comparaor's dynamic es seup auomaically compensaes he reference level and brings he comparaor ino a measable sae Comparaor's dynamic es seup Principal elecric scheme of he decision sage of he experimenal seup 16

17 Experimenal resuls Theoreical and experimenal curves b = a = /p 1 = -0.34ns 1/p 2 = 0.46ns θ = 4.75ns Bilaeral approximaion: b a U c ={ p 1 p 1 p 2 exp p 2 if p 2 p 2 p 1 exp p 1 1 if 17

18 Conclusions A simplified model can be consruced using firsorder ordinary differenial equaion, which can be solved analyically In cases where higher order models are needed, a mehod of bilaeral approximaion can be used o solve corresponding differenial equaion Major achievemen of he described approach is he possibiliy o obain he response of comparaor o arbirary inpu signal a arbirary clock poin. This response characerize he criical hreshold, which provides he measabiliy of comparaor. If his hreshold is crossed he logical answer of comparaor changes from 0 1 or from

19 Inroducion o equivalen ime Inpu 1 Sep[( θ)] = θ Inpu 12 U REF = θ clk = τ τ 19