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) d 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) d o = (6) To sole for the output oltage o (t), multply ths equaton by, sole for the dfferental d o, and ntegrate to get d o = 1 (t) (7) Assumng the output oltage s known at tme t =, the ntegraton lmts are o(t) o() d 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 condton 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 d o(t) = (15) Solng ths equaton for o (t) produces the crcut nput-output relatonshp o (t) = d (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), d 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 d 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 d = 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 d = 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. 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 output shown n Fgure? 3. Usng the cursors, measure the response alues of the output waeform at tme alues of τ, τ, 3τ, and 4τ. Frst, place the Y1 cursor at V and the Y cursor at 63.% of the fnal steady-state alue of 1 V or.63 V. Then, place the X cursor so that t ntersects the Y cursor at ths alue of.63 V. ecord the tme constant τ = X as the tme dfference between the two cursors X1 and X. Usng the measured alue of τ, fnd and record the output oltage at τ, 3τ and 4τ. 5
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. τ (ms) Theoretcal Measured Percent Error (%). 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. (τ) (V) (3τ) (V) (4τ) (V) Theoretcal Measured Percent Error (%) 6
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.