non-linear oscillaors The invering comparaor operaion can be summarized as When he inpu is low, he oupu is high. When he inpu is high, he oupu is low. R b V REF R a and are given by he expressions derived in he comparaor noes. Fiendish quesion: Wha would happen if we ied he oupu back o he inpu? The circui would become confused. When he oupu was high, hen inpu would be high. u wih he inpu high, he oupu would go low, causing he inpu o go low. The comparaor is faced wih a conundrum, and i would respond by oscillaing back forh beween high and low as fas as i possibly could. EE 230 non-linear oscillaors 1
This business of swiching back and forh migh be useful. This would be anoher form of oscillaor circui. u o be pracical, we need o pu a ime delay beween when he oupu swiches and when he inpu senses ha he oupu has swiched. We can do his by using an R circui o ie he oupu back o he inpu. The charging and discharging of he capacior will deermine he ime delays and hence he oscillaion frequency. R b 2. The oupu volage will charge he capacior. V REF R a 1. Suppose ha he inpu is low and he oupu is high. The capacior volage is also low. R 3. s he capacior charges, he inpu climbs higher. 4. Once he inpu volage becomes equal o, he comparaor will flip he oupu volage goes low. 5. Wih he inpu low, he capacior discharges, and decreases. 6. The inpu will decrease unil i his, causing he oupu o go high again. The cycle repeas. EE 230 non-linear oscillaors 2
Oupu high The comparaor will always be operaing a some poin on is ransfer characerisic. Le s sar by assuming he oupu has jus swiched high, puing us a poin on he ransfer characerisic. The oupu volage is a and he inpu (capacior) volage is a. R The capacior and resisor form a simple R circui wih a source volage of and iniial capacior volage of. The capacior will begin charging. V i = From EE 201: () =V f V f V i exp () = ( )exp R R () EE 230 non-linear oscillaors 3
s he capacior charges, he volage increases......unil =... which causes he comparaor o swich o he low-oupu condiion. V V EE 230 L L non-linear oscillaors 4
Oupu low Now he comparaor is a poin on he ransfer characerisic. The oupu volage is low ( ) and he capacior volage is high ( ). The difference in volages causes he capacior o discharge, moving he operaing poin along he lower branch of he curve. R () =V L (V L )exp R V ci = EE 230 non-linear oscillaors 5
Oupu low Now he comparaor is a poin on he ransfer characerisic. The oupu volage is low ( ) and he capacior volage is high ( ). The difference in volages will cause he capacior o discharge. () =V L (V L )exp <. R () R V i = EE 230 non-linear oscillaors 6
s he capacior discharges, he comparaor operaing poin moves along he lower branch of he ransfer characerisic fer some ime, he capacior volage drops o... D EE 230 non-linear oscillaors 7
which causes he comparaor o swich abruply o he highoupu condiion righ back o where we sared. D The cycle sars over and repeas indefiniely. The capacior charges and discharges beween and and he oupu swiches beween and. EE 230 non-linear oscillaors 8
Time high VL vo vi VTL VTH VL The ime required o move from o on he characerisic is found using he capacior charging equaion. This is he ime ha he oupu will be high. vc () = VL (VL VTL ) exp R EE 230 VTH = VL (VL TH = R ln VL VL VTL ) exp TH R VTL VTH non-linear oscillaors 9
Time low VL vo vi VTL VTH D VL gain, he ime required o move from o D on he characerisic is found using he capacior (dis)charging equaion. This is he ime ha he oupu will be low. vc () = VS EE 230 (VS VTL = VS (VS TL = R ln VS VS VTH ) exp VTH ) exp R TL R VTH VTL non-linear oscillaors 10
T H T L Period: T = T H T L = R ln (V S )(V S ) (V S )(V S ) EE 230 non-linear oscillaors 11
non-linear oscillaor - version 2 (funcion generaor) 2 variaion on he previous circui gives anoher oscillaor. In his case, a non-invering comparaor is used wih an invering inegraor ying he oupu back o he inpu. ssume symmeric supplies: = 2 () = R b R a vo1 V REF 1 R = V L R 2 (0) 0 R 1 (τ) dτ 2 (0) 1. Sar wih he inpu o he comparaor (2 ) high. Then he oupu is also high, 1 =. Since he comparaor oupu is also he inegraor inpu, he consan posiive volage causes he inegraor oupu o ramp down o lower volage. 2. When he oupu of he inegraor drops o, he comparaor will swich o he low-oupu sae, 1 =. 3. The inegraor oupu ramps up in volage unil i his. 4. The comparaor swiches o he high oupu sae, and he cycle repeas. EE 230 non-linear oscillaors 12
Oupu high The comparaor will always be operaing a some poin on is ransfer characerisic. Sar by assuming he oupu has jus swiched high he comparaor inpu (inegraor oupu) has jus reached. This is poin on he ransfer characerisic. The oupu volage is a and 2 =. The inpu o he inegraor is. This causes he inegraor oupu o ramp down from. R v o2 1 = 1 2 2 () = R EE 230 non-linear oscillaors 13
1 2 The inegraor oupu volage coninues o decrease... 1 2...unil i his (poin on he ransfer curve)... 1 2...causing he comparaor o swich o he low sae. EE 230 non-linear oscillaors 14
Oupu low Now he comparaor is a poin on he ransfer characerisic, wih 1 = (recall ha is negaive) and 2 =. The inegraor oupu volage will sar o ramp up. 2 () = V L R R v o2 1 = 1 1 2 EE 230 non-linear oscillaors 15
1 1 2 The inegraor oupu volage coninues o increase... 2 1 1 unil i reaches... 2 2 1 D vo1 2...and he comparaor swiches back o he high oupu sae. 2 nd he cycle sars over. EE 230 V V L D L non-linear oscillaors 16
The oal period of he oscillaion will be deermined by he ime needed for he inegraor o ramp up and down beween VTL and VTH. This is represened by he imes needed o move across he op and boom of he ransfer characerisic. v The ime ha he comparaor oupu is high is represened by he ransiion from o on he ransfer characerisic. Use he inegraor equaion wih vo1 = VL. vo2 () = VTH VTL = VTH VTL VTH vo2 VL VL R VL TH R VTH VTL TH = R VL EE 230 VL o1 ΔVT = R VL non-linear oscillaors 17
The ime ha he comparaor oupu is low is represened by he ransiion from o D on he ransfer characerisic. Use he inegraor equaion wih vo1 = VL. vo1 VL VTL VL vo2 () = VTL R VTH VTH VL vo2 D VL = VTL TL R VTH VTL TL = R = R VL ΔVT VL 1 1 T = TH TL = R (ΔVT ) VL VL If VL = VL (power supply volages are symmeric) ΔVT T = 2R VL EE 230 non-linear oscillaors 18
1 T H 2 T L The oupu of he comparaor will be a square wave, wih high and low volages given by he oupu limis and high and low imes deermined by he specifics of he hyseresis curve and he R ime consan of he inegraor. The oupu of he inegraor will be a riangle or ramp wave, ramping up and down beween and, wih he same period as he square wave. s an added bonus, i is relaively easy o conver he riangle o a sine wave using a filer ha aenuaes he higher order harmonics. EE 230 non-linear oscillaors 19