Bose State Unersty Department of Electrcal and omputer Engneerng EE 1L rcut Analyss and Desgn Lab Experment #8: The Integratng and Dfferentatng Op-Amp rcuts 1 Objectes The objectes of ths laboratory experment are: To demonstrate the nput-output relatonshps of the ntegratng and dfferentatng amplfer confguratons usng operatonal amplfers. To smulate a frst-order lnear dfferental equaton usng an ntegratng op-amp crcut. Theory The dynamc characterstcs of capactors and nductors produce sgnal processng functons that cannot be obtaned usng resstors. The op-amp crcut of Fgure 1(a) s smlar to the nertng amplfer except for the capactor n the feedback path. To determne the nput-output relatonshp of ths crcut, we use the deal op-amp equatons as well as Krchhoff s oltage and current laws. Begn by wrtng Krchhoff s current law (KL) at the nertng node, (t) (t) = (t) (1) and substtute the - characterstcs of the resstor and capactor, (t) (t) Then use the deal op-amp equatons = (t) () (t) = (3) (t) = P (t) = (4) together wth the fact that (t) = o (t) (t) = o (t) P (t) = o (t) (5) and rewrte KL as (t) o = (6) To sole for the output oltage o (t), multply ths equaton by, sole for the dfferental o, and ntegrate to get o = 1 (t) (7) Assumng the output oltage s known at tme t =, the ntegraton lmts are o(t) o() o = 1 t (t) (8) 1
P P o o (a) (b) whch yelds o (t) = 1 Fgure 1: Op-Amp (a) Integrator and (b) Dfferentator rcuts t (t) o () (9) From Equaton (5), the ntal conon o () s actually the oltage on the capactor at tme t =. When ths oltage s ntally zero at tme t =, the crcut nput-output relatonshp reduces to o (t) = 1 t (t) (1) The output oltage s proportonal to the ntegral of the nput oltage when the ntal capactor oltage s dscharged. Ths crcut s an nertng ntegrator snce the proportonalty constant (1/) s negate. Ths constant has the unts of sec 1 so that both sdes of Equaton (1) hae unts of olts. Interchangng the resstor and the capactor n Fgure 1(a) produces the op-amp dfferentator n Fgure 1(b). To dere the nput-output characterstc of ths crcut, wrte KL as before at the nertng node : (t) (t) = (t) (11) Then, use the deal op-amp equatons (t) = (1) (t) = P (t) = (13) together wth the fact that (t) = (t) (t) = (t) P (t) = (t) (14) and rewrte KL as o(t) = (15) Solng ths equaton for o (t) produces the crcut nput-output relatonshp o (t) = (16)
The output oltage s proportonal to the derate of the nput oltage. Ths crcut s called an nertng dfferentator snce the proportonalty constant () s negate. Ths constant has unts of sec so that both sdes of the equaton hae unts of olts. 1% 63.% (t)/ ss τ τ 3τ 4τ t Fgure : Tme esponse of a Frst-Order Lnear Dfferental Equaton The soluton of the followng lnear frst-order equaton n a arable (t), s gen by = a(t) b, () = (17) (t) = b a ( 1 e at) = ss (1 e t/τ ) (18) where ss = b/a s the fnal steady state of the quantty (t) and τ = 1/a s the tme constant. The soluton x(t) gen by Equaton (18) can be erfed by drect substtuton nto Equaton (17). As shown n Fgure, the tme constant τ s the tme that t takes the arable x(t) to reach approxmately 63.% of ts fnal steady state alue ss. ote also that the ntal slope at tme t = s equal to t= = b so that a new arable followng ths tangent lne would reach the fnal steady state ss n a tme ss = bt 1 = b a = t 1 = 1 a = τ () It can easly be erfed that the crcut n Fgure 3 yelds the followng nput-output relatonshp: (t) = 1 t [ ( V dc 1 ) ( ) ] 4 (t), () = (1) 5 1 1 3 4 Dfferentatng ths ntegral equaton wth respect to tme yelds = 1 [ ( V dc 1 ) ( ) ] 4 (t), 5 1 1 3 4 () = () (19) 3
1 5 V dc 3 4 (t) Fgure 3: Integratng Op-Amp rcut = 1 ( 1 ) ( 4 5 1 3 4 ) (t) 1 5 ( 1 ) V dc, () = (3) Identfyng ths last equaton wth Equaton (17), we get ( 1 a = 1 ) ( ) 4 5 1 3 ( ) 4 1 b = V dc 5 1 (4) (5) 3 Equpment Aglent DSO514A Dgtal Storage Osclloscope Aglent 33A Functon/Arbtrary Waeform Generator Fluke 115 True MS Multmeter Assorted resstors and capactors. 4
4 Procedure 1k 1n 1k 1k 1 Vdc 1k 1k (t) Fgure 4: Integratng Op-Amp Test rcut Smulaton of a Frst-Order Lnear Dfferental Equaton 1. Buld the crcut of Fgure 4 on your protoboard. (Assume that the large resstor 6 s nfnte.) Use precson resstors and try to match the capactor as close to 1 nf as possble by measurement.. Apply a 1-Hz, 1-V peak-to-peak square wae wth.5-v offset and 5% duty cycle to the nput of the ntegratng op-amp. Make sure the functon generator s set on HIGH Z. Obsere the nput square waeform 1 (t) on hannel 1 and the output waeform (t) on hannel of the scope. Does the output resemble the theoretcal soluton gen by Equaton (18)? 3. Usng the cursors, measure the response alues of the output waeform at tme alues of τ, τ, 3τ, and 4τ. 5 Data Analyss and Interpretaton 1. ompare the measured tme constant τ wth the predcted alue of τ = 5 and calculate the relate error between the two alues.. Gen that (t) = 1 e (t/τ), calculate the theoretcal alues of (t) at t = τ, τ, 3τ, 4τ. ompare these alues to the ones measured n the lab. 3. Draw a tangent lne to the output waeform at tme t = and erfy that ts crosses the steady state output at tme t = τ. 5
Bose State Unersty Department of Electrcal and omputer Engneerng EE 1L rcut Analyss and Desgn Lab Experment #8: The Integratng and Dfferentatng Op-Amp rcuts Date: Data Sheet ecorded by: Equpment Lst Equpment Descrpton Aglent DSO514A Dgtal Storage Osclloscope Aglent 33A Functon/Arbtrary Waeform Generator HP/Aglent E3631A Trple Output Power Supply BSU Tag umber or Seral umber Smulaton of a Frst-Order Lnear Dfferental Equaton: τ τ 3τ 4τ t (ms) (t) (V).63 apactance Measurement: = nf.