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2 6 Oeratonal Amlers and Actve Flters: A Bond Grah Aroach Glberto González and oberto Taa Unversty o Mchoacan Mexco Oen Access Database Introducton The most mortant sngle lnear ntegrated crcut s the oeratonal amler. Oeratonal amlers (o-am) are avalable as nexensve crcut modules, and they are caable o erormng a wde varety o lnear and nonlnear sgnal rocessng unctons (Stanley, 994). In smle cases, where the nterest s the conguraton gan, the deal o-am n lnear crcuts, s used. However, the requency resonse and transent resonse o oeratonal amlers usng a dynamc model can be obtaned. The bond grah methodology s a way to get an o-am model wth mortant arameters to know the erormance. A bond grah s an abstract reresentaton o a system where a collecton o comonents nteract wth each other through energy orts and are lace n the system where energy s exchanged (Karno & osenberg, 975). Bond grah modellng s largely emloyed nowadays, and new technques or structural analyss, model reducton as well as a certan number o sotware ackages usng bond grah have been develoed. In (Gawthro & Lorcan, 996) an deal oeratonal amler model usng the bond grah technque has been gven. Ths model only consders the oen loo voltage gan and show an alcaton o actve bonds. In (Gawthro & Palmer, 003), the `vrtual earth' concet has a natural bcausal bond grah nterretaton, leadng to smled and ntutve models o systems contanng actve analogue electronc crcuts. However, ths aroach does not take account the tye o the o-am to consder ther nternal arameters. In ths work, a bond grah model o an o-am to obtan the tme and requency resonses s roosed. The nut and outut resstances, the oen loo voltage gan, the slew rate and the suly voltages o the oeratonal amler are the nternal arameters o the roosed bond grah model. In the develo o ths work, the Bond Grah model n an Integral causalty assgnment (BGI) to determne the roertes o the state varables o a system s used (Wellstead, 979; Sueur & Dauhn-Tanguy, 99). Also, the symbolc determnaton o the steady state o the varables o a system based on the Bond Grah model n a Dervatve causalty assgnment (BGD) s aled (Gonzalez et al., 005). Fnally, the smulatons o the systems reresented Source: New Aroaches n Automaton and obotcs, Book edted by: Harald Aschemann, ISBN ,. 39, May 008, I-Tech Educaton and Publshng, Venna, Austra

3 84 New Aroaches n Automaton and obotcs by bond grah models usng the sotware 0-Sm by Controllab Products are realzed (Controllab Products, 007). Thereore, the man result o ths work s to resent a bond grah model o an o-am consderng the nternal arameters o a tye o lnear ntegrated crcut and external elements connected to the o-am, or examle, the eedback crcut and the load. The outlne o the aer s as ollows. Secton and 3 summarzes the background o bond grah modellng wth an ntegral and dervatve causalty assgnment. Secton 4 the bond grah model o an oeratonal amler s roosed. Also, the requency resonses o the some lnear ntegrated crcuts that reresent oeratonal amler usng the roosed bond grah model are obtaned. Secton 5 gves a comarator crcut usng a bond grah model and obtanng the tme resonse. Secton 6 resents the roosed bond grah model o an eedback o-am; the nut and outut resstances, bandwdth, slew rate and suly voltages o a non-nvertng amler usng BGI and BGD are determned. Secton 7 gves the lters usng a bond grah model o an o-am. In ths secton, we aly the lters or a comlex sgnal n the hyscal doman. The bond grah model o an o-am to desgn a Proortonal and Integral (PI) controller and to control the velocty o a DC motor n a closed loo system s aled n secton 8. Fnally, the conclusons are gven n secton 9.. Bond grah model Consder the ollowng scheme o a Bond Grah model wth an Integral causalty assgnment (BGI) or a multort Lnear Tme Invarant (LTI) system whch ncludes the key vectors o Fg. (Wellstead, 979; Sueur & Dauhn-Tanguy, 99). Fg.. Key vectors o a bond grah. MS MS, ( CI, ) and ( ) In g., (, ) e energy dssaton elds, ( D ) the detector and ( 0,,, ) transormers, TF, and gyrators, GY. The state denote the source, the energy storage and the TF GY the juncton structure wth xt () n and xt () m are comosed o energy varables t () and qt () d assocated wth I andc elements n ntegral causalty and dervatve causalty, resectvely, ut () denotes the lant nut, yt () q the lant outut, zt () n the co-energy vector, zt () m the dervatve co-energy and Dt () r and Dt () r are a mxture d n out

4 Oeratonal Amlers and Actve Flters: A Bond Grah Aroach 85 o et () and t () showng the energy exchanges between the dssaton eld and the juncton structure (Wellstead, 979; Sueur & Dauhn-Tanguy, 99). The relatons o the storage and dssaton elds are, zt () = Fxt () () zt () = Fxt () () d d d The relatons o the juncton structure are, out ( ) = ( ) D t LD t xt ( ) xt & () S S S S 3 4 () () = Dt out Dt S S S 0 n 3 ut () yt () S S S x& () t d n (3) (4) d ( ) = T ( ) zt Szt The entres o S take values nsde the set { 0, ±, ±, ± } 4 (5) mn where m and n are transormer and gyrator modules; S and S are square skew-symmetrc matrces and S and S are matrces each other negatve transose. The state equaton s (Wellstead, 979; Sueur & Dauhn-Tanguy, 99), ( ) = ( ) + ( ) & x t Axt But (6) where ( ) = ( ) + ( ) yt Cxt Dut (7) ( ) = + A ESSMSF (8) ( ) = B E S SMS (9) ( ) C = S + SMS F (0) 3 3 beng DSSMS = + () E = ISFSF + () n T 4 d 4

5 86 New Aroaches n Automaton and obotcs ( ) 3. Bond grah n dervatve causalty assgnment M = ILS L (3) n We can use the Bond Grah n Dervatve causalty assgnment (BGD) to solve drectly the roblem to get A. Suose that A s nvertble and a dervatve causalty assgnment s erormed on the bond grah model (Gonzalez et al., 005). From (4) the juncton structure s gven by, ( ) ( ) ( ) zt ( ) ( ) ( ) xt & JJJ 3 Dt = JJJDt 3 nd outd y t JJJ d ut (4) where the entres o J have the same roertes that S and the storage elements n (4) have a dervatve causalty. Also, D and nd outd D are dened by outd ( ) = ( ) D t LD t (5) d nd and they deend o the causalty assgnment or the storage elements and that junctons have a correct causalty assgnment. From (6) to (3) and (4) we obtan, * * ( ) = ( ) + ( ) zt Axt But & (6) where * * ( ) = & ( ) + ( ) yt Cxt Dut d (7) beng AJJNJ = + (8) * * B = J + JNJ (9) * 3 3 CJJNJ = + (0) * 3 3 DJJNJ = + () ( ) NILJ = L () n d d The state outut equatons o ths system n ntegral causalty are gven by (6) and (7). It ollows, rom (), (6), (7), (6) and (7) that,

6 Oeratonal Amlers and Actve Flters: A Bond Grah Aroach 87 AFA = (3) * B FA B = (4) * CCA = (5) * DDCAB = (6) * Consderng xt & () = 0, the steady state o a LTI MIMO system dened by x ABu ss = (7) ss ( ) y = DCABu (8) ss ss where x ss and yss are the steady state o the state varables and the outut, resectvely. In an aroach o the BGD, the steady state s determned by x FBu ss = (9) ss * ss y Du = (30) * ss 4. A bond grah model o an oeratonal amler The standard oeratonal amler (o-am) symbol s shown n Fg.. It has two nut termnals, the nvertng (-) nut and the nonnvertng (+) nut, and one outut termnal. The tycal o-am oerates wth two Drect Current (DC) suly voltages, one ostve and the other negatve (Stanley, 994). Fg.. Oeratonal amler symbol. The comlex acton o the o-am results n the amlcaton o the derence between the voltages at the nonnvertng, V +, and the nvertng, V, nuts by a large gan actor, K, desgned oen loo gan. The outut voltage s, out ( ) V = KV V (3) +

7 88 New Aroaches n Automaton and obotcs The assumtons o the deal o-am are (Barna & Porat, 989): ) The nut medance s nnte. ) The outut medance s zero. 3) The oen loo gan s nnte. 4) Innte bandwdth so that any requency sgnal rom 0 to Hz can be amled wthout attenuaton. 5) Innte slew rate so that outut voltage charges smultaneously wth nut voltage charges. The mlcatons o the assumtons are: no current wll low ether nto or out o ether nut termnal o the o-am, also, the voltage at the outut termnal does not charge as the V loadng s vared and nally, rom H 3, VV = out, we take the lmt when K, note + that V V + K =, whch ndcates that the voltages at the two nut termnals are orced to be equal n the lmt. The assumtons o an o-am are not comletely true n ractce, and to be ully cometent n the analyss and desgn o o-ams crcuts, one must be aware o the lmtatons. Thereore, we roose a more realstc model alcable to DC and low requences based on Bond Grah wth Intregral causalty assgnment (BGI), snce an o-am s a multstage amler, t can be reresented by Fg. 3. Fg. 3. Bond grah model o an oeratonal amler. The ndvdual stages used n o-am are searately chosen to develo derent amler characterstcs. Those amler characterstcs whch are determned by a gven stage deend on whether t unctons as an nut stage, ntermedate stage or outut stage. So, the bond grah model o the o-am s comosed by 3 stages, whch are: Characterstcs o the derental nut stage o an oeratonal amler are the most crtcal actors whch aect the accuracy o an o-am n rovdng voltage gan. Errors eects o ollowng stages are reduced n sgncance by the gan solaton rovded by the rst stage. Ths nut stage consders the two nut termnals o o-am, the derental nut resstance, denoted as, whch s the resstance between the nvertng and non-nvertng nuts and K s the oen loo gan. The ntermedate stage ntroduces the requency comensaton o the o-am usng a lag network. Also, usng a MTF, the slew rate o the o-am s consdered. Followng the nut and ntermedate voltage gan stages o an o-am, t s desrable to rovde medance solaton rom loads. In ths way the characterstcs o the gan

8 Oeratonal Amlers and Actve Flters: A Bond Grah Aroach 89 stages are reserved under load, and adequate sgnal current s made avalable to the load. In addton, the outut stage rovdes solaton to the recedng stage and a low outut medance to the load. Ths stage s ormed by the outut termnal, the outut resstance o the o-am denoted as and the adjustment o suly voltages, ostve voltage V and negatve voltage c e o V, are aled usng a MTF element. The useulness o the bond grah model o an oeratonal amler can be shown, alyng ths model to μa 74 o-am by Farchld Semconductor Cororaton (Stanley, 994; Gayakward, 000) and, TL 084 and OP 7 by Texas Instruments (Stanley, 994; Gayakward, 000) whose data sheets are shown n Table. O-am μa 74 TL 084 OP 7 ( Ω) ( Ω) 6 0 o K S V s ( μ ) ( ), c e VVV ± 5 ± 8 ± Table. Data sheets o μa 74, TL 084 andop 7. Usng the data sheets o an o-am, the hgh cuto requency o the oen loo voltage gan,, s determned. Moreover, the comensaton arameters o the bond grah model are o dened by o πc = (3) From (3) and the data sheets o the μa 74, TL 084 and OP7 o-ams, the arameters to obtan the requency resonse are gven n Table. cc O-am μa 74 TL 084 OP 7 ( Ω) ( ) 3 h 0 6 CF h Table. Parameters o requency resonse Substtutng the arameters o Tables and to bond grah model o each o-am, the requency resonses o μa 74, TL 084 and OP7 o-ams are shown n Fg. 4, 5 and 6, resectvely. Note that, the requency resonses o the μa 74, TL 084 and OP 7 o-ams are very close resect to the data sheets o these o-ams (Stanley, 994; Gayakward, 000). Also, K = 0Log K. The next secton an alcaton o the bond grah model decbel gan s ( ) db o an o-am to rove the tme resonse s resented.

9 90 New Aroaches n Automaton and obotcs Fg. 4. Frequency resonse o μa 74 o-am. Fg. 5. Frequency resonse o TL 084 o-am. Fg. 6. Frequency resonse o OP 7 o-am. 5. Comarator crcut Comarator crcuts reresent to the rst class o crcuts we have consdered that are bascally nonlnear n oeraton. Seccally, comarator crcuts roduce two or more dscrete oututs, each o whch s deendent on the nut level (Floyd & Buchla, 999). In ths alcaton, the o-am s used n the oen loo conguraton wth the nut voltage on one nut and a reerence voltage on the other, whch s shown n Fg. 7.

10 Oeratonal Amlers and Actve Flters: A Bond Grah Aroach 9 Fg. 7. Bond grah model o a comarator o-am. Alyng a voltage ( ) = 3sn( π ) vt tv to the nonnvertng nut, where 0.5 Hz + + and v V to the nvertng nut o the bond grah model o Fg. 7. Also, the suly = voltages are V = V and V = V, the tme resonse o the comarator crcut usng c the μa 74 o-am s shown n Fg. 8. e + = Fg. 8. Tme resonse o the comarator o-am. In accordng wth the objectve o the comarator crcut, the tme resonse o the Fg. 8 s satsactory (Stanley, 994; Gayakward, 000; Floyd & Buchla, 999). However, the requency o the nut sgnal ncreases, we have to consder the resonse o hase sht versus requency, whch s obtaned usng the roosed bond grah model and s shown n Fg. 9 or μa 74 o-am hase sht between the nut and outut sgnals. I the nonnvertng nut voltage s ( ) = 0.sn ( π ) nvertng nut voltage s ( ) = 0 vt tv where 00Hz, the = vt V and, the suly voltages are V = V and c

11 9 New Aroaches n Automaton and obotcs V = V, then the tme resonse o the comarator crcut usng μa 74 o-am s a e 0 square waveorm o magntude ± V and a hase sht o θ = 86.8 resect the nut sgnal, whch s shown n Fg. 0. Fg. 9. Phase sht versus requency o μa 74 o-am. Fg. 0. Tme resonse o comarator wth ( ) 0 sn ( 00π ) Vt t. + = The hase sht o the outut sgnal o Fg. 0 can be vered rom the ollowng equaton, calculatng the lag tme o the outut sgnal, t = θ 360 (33) where θ s the hase sht o the bode lot or μa 74 o-am s the Fg. 8, and s the requency o the nut sgnal. In ths case t =.4ms. In next secton o-am conguratons wth negatve eedback n the hyscal doman are roosed.

12 Oeratonal Amlers and Actve Flters: A Bond Grah Aroach Feedback oeratonal amler An o-am that uses eedback s called a eedback s called a eedback amler. A eedback amler s sometmes reerred to as a closed loo amler because the eedback orms a closed loo between the nut and the outut. A eedback amler essentally conssts o two arts: an o-am and a eedback crcut. The eedback crcut can take any orm whatsoever, deendng on the ntended alcaton o the amler. Ths means that the eedback crcut may be made u o ether assve comonents, actve comonents, or combnatons o both (Gayakward, 000). A closed loo amler can be reresented by usng two blocks, one or an o-am and another or a eedback crcut. There are our ways to connect theses two blocks. These connectons are classed accordng to whether the voltage or current s ed back to the nut n seres or n arallel, as ollows: ) Voltage seres eedback, ) Voltage shunt eedback, 3) Current seres eedback and 4) Current shunt eedback (Gayakward, 000) and (Floyd & Buchla, 999). The voltage seres eedback conguraton s one o the most mortant because ths s commonly used. An n deth analyss o ths conguraton n the hyscal doman s resented n ths secton, comutng voltage gan, nut resstance, outut resstance and the bandwdth. 6. A bond grah model o a nonnvertng amler An o-am connected n a closed loo conguraton as a nonnvertng amler s shown n Fg.. The nut sgnal s aled to the nonnvertng nut. A orton o the outut s aled back to the nvertng nut through the eedback network n the hyscal doman. The BGI and the Bond Grah n a Dervatve causalty assgnment (BGD) are shown n Fg., n order to get the symbolc exressons o the closed loo system n steady state alyng the methodology gven n secton. Fg.. Bond grah models o a nonnvertng amler. The closed loo gan, A CL, can be determned usng the BGD aroach, rom (30) we have, A CL y ss * = = D (34) uss where y ss and u ss are the steady state values o the outut and nut, resectvely.

13 94 New Aroaches n Automaton and obotcs We now derve the closed loo gan o the nonnvertng amler usng the BGD. The key vectors o BGD are gven by, xt () = q (); t xt &() = (); t zt () = e () t ut () = et (); yt () = et () 9 [ ] [ ] D () tetetetttt = () () () () () () nd D () t = tttetetet () () () () () () outd the consttutve relatons are, (35) F = (36) C h and the juncton structure s, Ldag =,,,, d F h o (37) K K m m 0 m m m J = ; ; 0 0 J = = J3 0 0 m m m J J K = ; J = m m 3 K = ; J = J = J = 0 m [ ] (38) From (), (), (34), (36), (37) and (38), the closed loo gan o the nonnvertng amler s, A CL where ( )( ) ( ) mk + + = m L o mδ+ Km + mm ( + ) + o L L Δ= L (39)

14 Oeratonal Amlers and Actve Flters: A Bond Grah Aroach 95 Note that the closed loo gan (39) takes account the nternal arameters and the external elements connected to o-am. A normal oeraton o the o-am usng the bond grah model ndcates the modules o MTFs arem =, the slew rate s sucent o the o-am, andm =, the suly voltages allow to obtan the outut voltage deendng the nut voltage and the gan o the o-am. Consderng, m =, m = and = 0, we obtan o + ( A ) = CL K Alyng the Lm( A ) CL, the deal closed loo gan o ths amler, ( ) CL K ( A ) (40) A, s determned, CL = + (4) The tme resonse o the nonnvertng amler usng μa 74 o-am, = 0K Ω, = 90K Ω and the nut sgnal s vt ( ) = 0.5sn ( πtv ) where = KHz, s shown n Fg.. Fg.. Tme resonse o the nonnvertng amler. 6.. Inut resstance o the nonnvertng amler The nut resstance o the nonnvertng amler can be determned usng the BGD o Fg.. Consderng the outut ( t ), the submatrcesj, J andj rom (38) are changed by

15 96 New Aroaches n Automaton and obotcs [ ] J = ; J = J = 0 (4) From (), (), (37) and (38) the nut resstance s determned by, F where( ) ss ( )( ) ( ) = K o L L L e and( ) ss I = 0 we reduce, o ( )( ) o L o L (43) are the steady state values o the nute and outut, resectvely. K = ( ) F (44) nally, the term K + + amler ( F ) s dened by, hence the deal nut resstance o the nonnvertng = + ( ) F K + (45) The result o (45) ndcates that the deal nut resstance o the o-am wth eedback s ( + K+ ) tmes that wthout eedback. In addton, the equaton (43) allows to determne the nut resstance o the o-am consderng the nternal arameters and external elements or ths conguraton. Equaton (45) can be vered n (Stanley, 994; Gayakward, 000). 6.. Outut resstance o the nonnvertng amler Outut resstance s the resstance determned lookng back nto the eedback amler rom the outut termnal. A BGD that allows to obtan the outut resstance o a nonnvertng amler s shown n Fg. 3. The key vectors o the BGD o Fg. 3 are, xt () = q (); t xt &() = (); t zt () = e () t ut () = et (); yt () = t () [ ] [ ] D () tetetettet = () () () () () nd D () t = () t () t () t e () t () t outd (46)

16 Oeratonal Amlers and Actve Flters: A Bond Grah Aroach 97 Fg. 3. Dervaton o outut resstance o a bond grah model o a nonnvertng amler. the consttutve relatons are, Ldag =,,,, o h (47) and the juncton structure o the BGD s, F = C h (48) K T J = ; J = 0 ; J = J = [ ] J = K ; J = J = J = J = Substtutng (), (47), (49) nto () the outut resstance o a nonnvertng amler s obtaned, of ( e ) ( ) ss * ( D ) [ ( ) o ] ( + + ) + + ( + ) + + ss = = = K o (49) (50) Calculatng ( ) = lm of we have, of ( ) of = o ( + ) + + K + o (5)

17 98 New Aroaches n Automaton and obotcs nally, K K, the deal outut resstance ( ) o nonnvertng amler s gven by, o of the ( ) of = + o K + Ths result shows that the deal outut resstance o the nonnvertng amler s ( K ) tmes the outut resstance (5) o o the o-am. That s, the outut + + resstance o the o-am wth eedback s much smaller than the outut resstance wthout eedback. In addton (5) can be vered n (Stanley, 994; Gayakward, 000) Bandwdth o the nonnvertng amler The bandwdth o an amler s dened as the band (range) o requency or whch the gan remans constant. Manuacturers generally secy ether the gan-bandwdth roduct or suly oen loo gan versus requency curve or the o-am (Gayakward, 000). Fg. 4 shows the oen loo gan versus requency curve o the μa 74 o-am. From ths curve or a gan o 00,000, the bandwdth s aroxmately 5Hz ; or the gan-bandwdth roduct s MHz. On the other extreme, the bandwdth s aroxmately MHz when the gan s. Hence, the requency at whch the gan equals s known as the unty gan bandwdth (UGB). Snce or an o-am wth a sngle break requency o, the gan-bandwdth roduct s constant, and equal to UGB, we can wrte, where bandwdth wth eedback. F ( )( ) ( )( ) Thereore, the bandwdth o an eedback o-am s, UGB = K = A (53) o CL F F ( K )( o) A = (54) CL Fg. 4. Frequency resonse o the nonnvertng amler.

18 Oeratonal Amlers and Actve Flters: A Bond Grah Aroach 99 The requency resonse o the nonnvertng amler based on BGI o Fg., usng μa 74 o-am, K = Ω, = 5 Ω = 30KHz, s shown n Fg. 4. K and the nut sgnal s vt ( ) =.0 sn ( π ) tv where Note that the requency resonse o ths amler ndcates that the closed loo gan s A = 6 = 5.56dB untl aroxmately 66KHz, whch s vered rom (54). CL 6..4 Slew rate Another mortant requency related arameter o an o-am s the slew rate. The slew rate s the maxmum rate o change o outut voltage wth resect to tme, usually seced n Vs. μ Ideally, we would lke an nnty slew rate so that the o-am's outut voltage would change smultaneously wth the nut. Practcal o-ams are avalable wth slew rates rom 0.Vs μ to well above 000Vs μ. The slew rate ( S ) can be obtaned by, πv S = V / μs (55) 6 0 where s the nut requency and V s the eak value o the outut sne wave. In order to show the eect o the slew rate o an o-am, the tme resonses o the nonnvertng amler based on BGI o Fg., usng μa 74 and OP 7 o-ams, K = Ω, = 5K Ω and the nut sgnal s vt () =.0sn( πtv ) where = 30KHz, are shown n Fg. 5. The deal closed gan s then ( A ) = 6. CL Fg.5. Eect o the slew rate o the nonnvertng amler usng μa 74 and OP 7 oams.

19 300 New Aroaches n Automaton and obotcs From (55), the mnmum slew rate o an o-am wth the revous condtons s S =.3V μs. Thereore, the slew rate o the OP 7 o-am, whch s.8vs μ s enough or the nut sgnal wth the gven condtons. Fg. 5, shows the outut sgnal o the μa 74 o-am has dstorton because o the slew rate s 0.5Vs. μ 6..5 Suly volages The most lnear ntegrated crcuts, artcularly o-ams, use one or more derental amler stages, and derental amlers requre both a ostve ( ) c V and negatve ( V ) ower suly or roer oeraton o the crcut. The suly voltages o the roosed bond grah model o an o-am can be tested alyng an nut sgnal vt () = 3sn( πt ), = 30KHz, K = Ω, = 5K Ω and V = V and V = V to μa74 o-am, the tme resonse s shown n Fg. 6. c e The deal closed loo gan s ( A ) = 6, o ths orm the deal outut sgnal would be CL Vt () = 8sn( πt ). However, the suly voltages o the o-am are ± V, lmtng the out outut sgnal to ths rank o voltage. Next secton, the o-am ntegrator that s one the most nterestng conguratons on a bond grah model s desgned. e Fg.6. Tme resonse wth slew rate and suly voltages usng μa 74 o-am. 6. The oeratonal amler ntegrator An o-am ntegrator smulates mathematcal ntegraton, whch s bascally a summng rocess that determnes the total area under the curve o a uncton (Floyd & Buchla, 999). A basc ntegrator on a BGD model s shown n Fg. 7. Notce that the eedback element s a caactor that orms an C crcut wth the nut.

20 Oeratonal Amlers and Actve Flters: A Bond Grah Aroach 30 Fg. 7. Bond grahs o an oeratonal amler ntegrator. In order to determne the transer uncton o the o-am ntegrator based on bond grah model o Fg. 7, the methodology gven n Secton s aled. The key vectors are, qt () t () et () ut () = et () xt () = ; () = ; () = ; () xt & () zt qt t et () yt () = et () [ ] Dt () = et () et () et () t () et () n [ ] Dt () = t () t () t () et () t () out the consttutve relatons are, T T (56) Fdag =, (57) CC h Ldag = and the juncton structure matrx s,,,,, o L h m 0 m K 0 K 0 m [ ] [ ] S S 0 0 S = ; S = ; S = 0 S = m m ; m S = 0 ; = = S m 0 0 = ; S = (58) (59)

21 30 New Aroaches n Automaton and obotcs From (8), (3), (57), (58) and (59) the A matrx s, ( m + )( + ) ( + ) C Ch A = m C C hh where Δ= ( + ) + + m [ ] m o L L ( mmk + ) ( Δ+ mk ) o L L o L L Substtutng (3), (58) (59) nto (9), From (0), (3), (57), (58) and (59) the C B B s determned by, ( + ) ( mk ) ol m = Δ h o L C matrx s, = ( + ) m Δ C C o L L Fnally, substtutng (3), (58) and (59) nto (), h D s gven by, (60) (6) (6) D = (63) ol Δ The transer uncton o a system reresented n sace state can be calculated by, ( ) Gs () = C si A B + D (64) n Substtutng (60), (6), (6) and (63) nto (64), the transer uncton o the o-am ntegrator o Fg. 7 s, ( scc + scm Kmm ) hho o L Gs () = scc sc hh ( CKmm Cm hh L ) where Λ= ( m + )( + o L). Δ + Δ+ + Δ + mλ Consderng a normal oeraton o the o-am, mm = = and = 0, we have, o (65) Gs () = CC C C K K K K hh hh s s C (66)

22 Oeratonal Amlers and Actve Flters: A Bond Grah Aroach 303 In addton, alyng lm Gs ( ), the deal transer uncton, Gs (), s determned by, K Gs () sc = (67) Note that, (67) s the tycal transer uncton o a ntegral controller. However, the equaton (65) s the transer uncton o a ntegral controller based on o-am and consderng ther nternal arameters and external elements. The tme resonse o the ntegrator crcut usng μa74 o-am and = 0Ω, C = 00μF and vt () = sn( πtv ) where = KHz shows n Fg. 8. The outut sgnal roves that the bond grah model o Fg. 8, s an ntegrator o the nut sgnal. Fg. 8. Tme resonse o the ntegrator. The requency resonse o the ntegrator crcut o Fg. 7 s shown n Fg. 9. Note that (67) s reresented by Fg. 9. Fg. 9. Frequency resonse o the ntegrator crcut. Next secton an mortant alcaton o the o-am that s an actve lter n the hyscal doman s roosed.

23 304 New Aroaches n Automaton and obotcs 7. Actve lters usng bond grah models An electrc lter s oten a requency selectve crcut that asses a seced band o requences and blocks or attenuates sgnals o requences outsde ths band. Flters may be classed n a number o ways (Gayakward, 000):. Analog or dgtal. Analog lters are desgned to rocess analog sgnals, whle dgtal lters rocess analog sgnals usng dgtal technques.. Passve or actve. Passve lters use resstors, caactors and nductors n ther constructon and the actve lters emloy transstor or o-ams n addton to resstors and caactors. An actve lter oers the ollowng advantages over a assve lter: a) Gan and requency adjustment lexblty. b) No loadng roblem. c) Cost. A lter wll usually conorm to one o our basc resonse tyes: low-ass, hgh-ass, bandass and band-reject. 7. Low-ass lter A low-ass lter allows only low requency sgnals to ass through, whle suressng hgh-requency comonents. The bond grah model o a low-ass lter s shown n Fg. 0. Fg. 0. Bond grah model o a low-ass lter. The key vectors o the bond grah o Fg. 0 are, qt () t () et () ut () = et () xt () = ; () = ; () = ; () xt & () zt qt t et () yt () = et () [ ] Dt () = et () et () et () t () t () et () et () n [ ] Dt () = t () t () t () et () et () t () t () out T T (68) The consttutve relatons o the elements are, Fdag =, (69) CC h e

24 Oeratonal Amlers and Actve Flters: A Bond Grah Aroach 305 Ldag = and the juncton structure s 3 3 [ ],,,,,, o L h c P 3 m K 0 m P 0 T S = P 0 0 ; S = 0 S S = 0 ; S = 0 0 = 0 0 ; S = S = S = S = (70) (7) where 0 0 = K P m m ; P = (7) 0 0 From (8), (3), (69), (70) and (7), the A matrx o the low-ass lter s, A = m mmk mk mk L ( + ) + o L m L C ΔC C ΔC hh h hh h m ΔC L h c c h ( + )( m + ) + C ΔC o L o L (73) [ ] [ ( ) ] Δ= m where ( )( ) o L L L Substtutng (70) and (7) nto (3), the B matrx s obtaned B = 0 c T (74) Fnally, rom (0), (3), (69), (70) and (7), the C matrx s

25 306 New Aroaches n Automaton and obotcs C m ΔCh ( ) L = + + Lo ΔC h (75) and D = 0 From (73) to (75), and (64), and consderng deal characterstcs o the o-am, = 0, =, K = and L Gs () s obtaned Gs () =, the deal transer uncton o the low-ass lter, denoted by A C CL C C = s + C where = + A CL. cc The deal transer uncton o the low-ass lter ndcates that on the ass band the gan s almost A and the ole o the system s located at the hgh cuto requency dened by h CL = πc. CC The requency resonse o the low-ass lter based on bond grah model usng the μa74 o-am and consderng the numercal values o the Table 3 s shown n Fg. 0K Ω 0K Ω L Table 3. Numercal values o the low-ass lter. c K Ω 5.9 Ω C C K 8 0 F h o Fg.. Frequency resonse o the low-ass lter. The Fg shows that = KHz. Also, we have a rst order lter, because the lter has a h ole and the rollo rate o the lter s 0 db er decade. 7. Hgh-ass lter A hgh ass lter allows only requences above a certan break ont to ass through. In other words, t attenuates low requency comonents. A rst order hgh-ass lter on bond grah model s shown n Fg..

26 Oeratonal Amlers and Actve Flters: A Bond Grah Aroach 307 Fg.. Bond grah model o a hgh-ass lter. The mathematcal model o the lter can be determned n the same manner that the lowass lter. However, or all the remanng lters o ths aer, we use the sotware 0-sm to know the erormance o the lter on the hyscal doman, consderng the nternal characterstcs o the o-am and the external comonents connected to the o-am n order to get the lter tye. Usng the same numercal values that the low-ass lter, the requency resonse o the lter shows n Fg. 3. Fg. 3. Frequency resonse o the hgh-ass lter. The Fg. 3 ndcates that we have a rst order lter and the low cuto requency KHz. L s 7.3 Band-ass lter The band-ass lter can be thought o as a combnaton o hgh and low ass lters. It allows only requences wthn a seced rang to ass through. In Fg. 4 a band-ass lter on bond grah model s roosed. The characterstc o ths lter s a narrow band-ass usng multle eedback. In order to get the erormance o ths lter, we use the nternal arameters o the μa 74 o-am gven n Tables and, and the external comonents are shown n Table 4.

27 308 New Aroaches n Automaton and obotcs Fg. 4. Bond grah model o a band-ass lter. c L c C C c c 4.77K Ω 00K Ω 0K Ω 0.K Ω 0.0μF 0.0μF Table 4. Numercal values o the external comonents The requency resonse o the band-ass lter s shown n Fg. 5. Fg. 5. Frequency resonse o the band-ass lter. Note that, the resonse o the lter allows to ass the requency comonent o KHz wth a closed loo gan o 0 and the low and hgh requences resect to KHz have a small magntude. 7.4 Band-reject lter The band-reject lter allows everythng to ass through wth the exceton o a secc range o requences. The band-reject lter s oten called notch lter, because t s commonly used or the rejecton o a sngle requency. The Fg. 6 shows ths lter on a bond grah model. Usng the arameters o the μa 74 o-am gven on Tables and, and the external elements o the o-am o Fg. 6 are gotten n Table 5, the requency resonse o the lter s shown n Fg. 7.

28 Oeratonal Amlers and Actve Flters: A Bond Grah Aroach 309 Fg. 6. Bond grah model o a band-reject lter. 3 39K Ω 9.5K Ω 39K Ω Table 5. External elements o the band-reject lter. CC = C L 3 K Ω 0.068μF 0.36μF Fg. 7. Frequency resonse o the band-reject lter. In Fg. 7, we show that 60Hz ower lne requency can be attenuated usng ths lter n the hyscal doman. 7.5 Alyng actve lters A lter s a crcut whch nhbts the transer o a secc range o requences. So, we can aly actve lters to select a requency comonent o a comlex sgnal. An nterestng alcaton o lters on bond grah models s shown n Fg. 8 where we have an nut sgnal gven by, [ π( ) ] [ π( ) ] [ π( ) ] V = 0.sn 0Hz t + 0.sn 500Hz t + 0.sn 0KHz t (76) In Fg. 8, the nut s aled to low-ass lter denoted by oam, to hgh-ass lter denoted by oam and to band-ass lter denoted by oam3.

29 30 New Aroaches n Automaton and obotcs Also, the external elements o Fg. 8 are obtaned on Table 6 and usng the arameters o the 74 μa o-am gven on Tables and, the requency resonses are shown n Fg. 9. Fg. 8. Comose sgnal aled to bond grah lters. = = C n ot M Ω Ω μf 5.9K Ω K Ω CCC = = 4 n 9K Ω 0.0μF 5.6K Ω Table 6. External elements o the comose system. In Fg. 9, the hgh cuto requency s = 0Hz or the low-ass lter, the low cuto H requency s = 0KHz or the hgh-ass lter and the center requency s = 500Hz. L Fnally, the tme resonse o each lter s shown n Fg. 30 and we note that the objectve o the lters s successul. c

30 Oeratonal Amlers and Actve Flters: A Bond Grah Aroach 3 Fg. 9. Frequency resonse o a comlete system. Fg. 30. Tme resonse o the comlete system. Note that the bond grah model o the o-am to desgn controllers, analog comuters and general lnear alcatons can be aled. 8. Controller based on bond grah model o an oeratonal amler Another mortant alcaton o oeratonal amlers s to mlement controllers and can be used to mrove the erormance o a closed loo system. A roortonal and ntegral (PI) controller desgned n the hyscal doman to control the velocty o a DC motor s aled (Barna & Porat, 989).

31 3 New Aroaches n Automaton and obotcs The nonnvertng and ntegrator conguratons o an o-am are used to derve the PI controller. Also, the nut reerence o the system s voltage V n and on the eedback a element GY to connect the outut wth the summng juncton s aled. The BGI o the closed loo system s shown n Fg. 3. Fg. 3. PI Controller o DC motor on a bond grah model. The BGI o Fg. 3 can be used to determne the state sace reresentaton or the transer uncton o the closed loo system. However, n ths case we only show the smulaton wth the objectve to test the PI controller connected to DC motor n a closed loo system. Fg. 3. Tme resonse o the outut o the DC motor usng a PI controller.

32 Oeratonal Amlers and Actve Flters: A Bond Grah Aroach 33 The tme resonse o the outut usng a μa 74 o-am, = 0K Ω, = 90K Ω, = Ω, L = 0.0mH, a a J = 0.00Kgm and / / bnmrads =, s shown n Fg. 3. Note that the PI Controller yelds a steady state outut equal to nut reerence o the system and the transent resonse deends o the arameters o the system and controller. 9. Conclusons In ths work an oeratonal amler s shown n a bond grah model. The roosed bond grah takes account the nut and outut resstances, oen loo gan, suly voltages, slew rate and requency comensaton o an oeratonal amler. Thereore, an advantage o ths model s to determne the erormance o alcatons based on oeratonal amlers consderng the tye o lnear ntegrated crcut to obtan the nternal arameters rom the data sheets. Fnally, oen loo and closed loo conguratons o the oeratonal amler n the hyscal doman have been shown. 0. eerences Arad Barna and Dan I. Porat, (989). Oeratonal Amlers, John Wley & Sons,ISBN: , Canada C. Sueur, G. Dauhn-Tanguy, (99), Bond grah aroach or structural analyss o MIMO lnear systems, Journal o the Frankln Insttute, vol. 38, no., (55-70). Dean C. Karno, onald C. osenberg, (975). System Dynamcs: A Uned Aroach, Wley, John & Sons, Forbes T. Brown (00)., Engneerng System Dynamcs, Marcel Dekker Inc, ISBN: , Unted States o Amerca. Glberto Gonzalez, Dauhn- Tanguy G. Galndo and De Leon J. (005), Steady State Error or a Closed Loo Physcal System wth a Bond Grah Aroach, Proceedngs o Internatonal Conerence on Bond Grah Modelng and Smulaton,. 07-, ISBN: , New Orleans, January 005, SCS. Norman S. Nse, (000). Control Systems Engneerng, John Wley & Sons, ISBN: , Unted States o Amerca. Peter Gawthro, Lorcan Smth (996)., Metamodellng, Prentce-Hall,ISBN: , Great Brtan. P.E. Wellstead, (979). Physcal System Modellng, Academc Press, ISBN: London. P. J. Gawthro and D. Palmer, (003). A bcausal bond grah reresentaton o oeratonal amlers", Proceedngs o the Insttuton o Mechancal Engneers Part I: Journal Systems and Control Engneerng, Vol. 7 amakant A. Gayakward, (000), O-ams and Lnear Integrated Crcut, Prentce Hall, ISBN: , Unted States o Amerca.

33 34 New Aroaches n Automaton and obotcs Thomas L. Floyd and Davd Buchla, (999), Basc Oeratonal Amlers and Lnear Integrated Crcuts, Prentce-Hall, ISBN: , Unted States o Amerca. Wllam D. Stanley, (994). Oeratonal Amlers wth Lnear Integrated Crcuts, Maxwell Macmllan Internatonal 0-Sm, Controllab Products (007) B. V., 0-Sm, htt:// roduct/0sm.html

34 New Aroaches n Automaton and obotcs Edted by Harald Aschemann ISBN Hard cover, 39 ages Publsher I-Tech Educaton and Publshng Publshed onlne 0, May, 008 Publshed n rnt edton May, 008 The book New Aroaches n Automaton and obotcs oers n chaters a collecton o recent develoments n automaton, robotcs as well as control theory. It s dedcated to researchers n scence and ndustry, students, and ractcng engneers, who wsh to udate and enhance ther knowledge on modern methods and nnovatve alcatons. The authors and edtor o ths book wsh to motvate eole, esecally under-graduate students, to get nvolved wth the nterestng eld o robotcs and mechatroncs. We hoe that the deas and concets resented n ths book are useul or your own work and could contrbute to roblem solvng n smlar alcatons as well. It s clear, however, that the wde area o automaton and robotcs can only be hghlghted at several sots but not comletely covered by a sngle book. How to reerence In order to correctly reerence ths scholarly work, eel ree to coy and aste the ollowng: Glberto Gonzalez and oberto Taa (008). Oeratonal Amlers and Actve Flters: A Bond Grah Aroach, New Aroaches n Automaton and obotcs, Harald Aschemann (Ed.), ISBN: , InTech, Avalable rom: htt:///books/new_aroaches_n_automaton_and_robotcs/oeratonal_amlers_and_ actve_lters a_bond_grah_aroach InTech Euroe Unversty Camus STeP Slavka Krautzeka 83/A 5000 jeka, Croata Phone: +385 (5) Fax: +385 (5) InTech Chna Unt 405, Oce Block, Hotel Equatoral Shangha No.65, Yan An oad (West), Shangha, 00040, Chna Phone: Fax:

35 008 The Author(s). Lcensee IntechOen. Ths chater s dstrbuted under the terms o the Creatve Commons Attrbuton-NonCommercal- ShareAlke-3.0 Lcense, whch ermts use, dstrbuton and reroducton or non-commercal uroses, rovded the orgnal s roerly cted and dervatve works buldng on ths content are dstrbuted under the same lcense.