Parasitic Pole Approximation Techniques for Active Filter Design

Transcription

1 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-27, NO. 9, SEPTEMBER Parasitic Poe Approximation Techniques for Active Fiter Design RANDALL L. GEIGER, MEMBER, IEEE Abstmei-A method of approximating the parasitic poes of active fiters empoying one, two, or three operationa ampifier(s) 8 presented. hese approximations are good for both a singe-poe and two-poe mode of the o&rationa ampifier and are usefu for determhhg fiter stabiity. The approximation expressions are sufficienty shpe that they may be I I I I I I incuded In the active fiter desiga process. Vi c RC NETWORK StabWty criteria for active fiters empoying two-poe operationa ampifiers are stated in terms of the parasitic poe approxmati0ds obtabed using the simper singe-poe mode of the operationa ampifier. %e necessity of acuding the more accwate two-poe mode of an operationa ampuier in some appications is discwsed. Detaied comparisons Of the actua and approximate parasitic poes in two exampes are made. I I. INTRODUCTION T IS A we-known fact that the frequency response of an operationa ampifier (op amp) affects the performance of active fiters empoying these devices. The frequency dependent gain of the op amp causes both a perturbation in the desired poes of the active fiter from their nomina position and the introduction of parasitic poes in the active transfer function. It has aso been observed by many fiter designers that the performance is affected to the extent that some active fiters that ook promising when designed assuming the op amp s are idea either perform poory in the aboratory when actua op amp s are used or are actuay unstabe. This degradation in performance and/or possibe instabiity can often be attributed primariy to the movement of the desired poes and zeros from their idea position []-[3]. Much attention has been devoted to the design of active fiters which are ess sensitive to the parameters of the op amp s [4]-[8]. This has often been achieved by incorporating an accurate mode of the op amp itsef [7], [8] into the design process. The desired poes in these designs exhibit a reduced dependence upon the parameters of the op amp s. Some designs in which the desired poes are very neary ocated at the idea position are, however, experimentay observed to be unstabe. This is often a resut of the fact that the parasitic poes ie in the right haf of the s-pane. Such instabiities are often characterized in the aboratory by a high frequency approximatey sinusoida osciation which may vary in ampitude from a few miivots to severa vots depending upon the circuit topoogy and particuar op amp used. Manuscript received November 9, 978; revised October The:author is with the Department of Eectrica Engiueermg, Texas A&M University, Coege Station, TX Fig. I /80/ $ IEEE : Genera three op amp active fiter. This paper is concerned with obtaining approximate expressions for the ocation of the parasitic transfer function poes of active fiters empoying one, two, and three operationa ampifiers. The expressions derived herein approximate the parasitic poes for both singe-poe and two-poe modes of the op amp s. By incorporating stabiity considerations obtainabe from these parasitic poe approximations into the design process itsef, some promising configurations that have in the past been observed to be unstabe may be modified sighty to obtain a practica usabe fiter. The magnitude of the parasitic poe approximation probem can be appreciated by reaizing that an active fiter empoying three op amp s, each modeed by a twopoe mode, whidh is designed to reaize a second-order transfer function actuay has an eighth-order denominator poynomia that must be factored to obtain the two desired poes and the six parasitic poes. Athough the frequency response of the op amp is usuay (and often quite successfuy) approximated by a singe-poe transfer function, it is shown by exampe that a two-poe mode may be more usefu for determining stabiity. In this exampe a fiter is given which has a poes in the eft haf-pane when a singe-poe mode of the op amp is used but which has a pair of parasitic poes in the right haf-pane when a two-poe mode is empoyed. The predicted instabiity is in agreement with observed aboratory resuts. II. POLE-DEFINING EQUATION FOR ACTIVE FILTERS The most genera active fiter empoying three op amp s is shown in Fig. The op amp s are a assumed to have infinite input impedance, infinite common-mode rejection ratio and zero-output impedance. The transfer function& J

2 794 IEEE TRANSACTIONS ON CIRCLJITS AND SYSTEbQVOL. as-27,n0. 9,SBpTeMa RR 980 TABLE POLE-DWINING EQUATIONS FOR Acrrw FILTE& EMPLOYING up TOTHR3EOPhP'S 3 2 (vi) D, 2(s,~) - D,(S) + rs(+qrs)piw + T*s*(~+W*P~LS) + r3s3(+ens)3~p(~) - 0 this fiter may be expressed as [7] v, -= v;: N"+A,+A,+A2A, N2 4 N23 D,(s) DI(s,A)=Do(s)+~=o. (4) The gain function of the singe-poe mode of the op amp is assumed to be of the form [] () where a N s and D s are poynomias in s dependent ony upon the passive RC network and A,, A,, and d, are gains of the op amp s and are ideay infinite. Do is the desired denominator poynomia and D, is the characteristic poynomia of the passive RC network. At this point, for the convenience of reduced notationa compexity, the assumption of identica op amp s wi be made. Thus for A, =A,=A,= A, the poe-defining equation (the denominator of ()) may be expressed as PI(S) P2(s) + DpW D&,A)=D,+-+- A2 - A3 =o (2) where P,(s)=D,+D,+D, and P,(s)=D,,+D,,+D,. The foowing deveopment can readiy be modified to hande the more genera nonidentica op amp situation except where otherwise noted by considering the actua denominator of () instead of the simpified expression of (2) as the poe-defining equation. The poe-defining equations in the two identica op amp and singe op amp cases may simiary be expressed, respectivey, as PI(S) + D,(s) =. D2(s, A)=Do(s)+ A - A2 (3) where 7 is the reciproca of the gain-bandwidth product of the op amp. The gain function of the two-poe mode of the op amp is assumed to be of the form A(s)= TS( +87s) where r is again the reciproca of the gain-bandwidth product and 8 is a parameter of the op amp that determines the ocation of the second poe. When 8= 0, the two-poe mode reduces to the singe-poe mode. The poe-defining equations (2)-(4) when the singe-poe mode and two-poe mode of the op amp is empoyed are isted in Tabe I. III. APPROXIMATE ROOT EXFWSSIONS If the op amp s are assumed to be idea, then r=o and the poe-defining equations isted in Tabe I a reduce to Do(s)=O. The roots of D,(s) are referred to as the desired roots. In practice, however, r is sma but not zero. As a consequence, there wii be a sma movement of the desired roots as we as the existence of additiona roots which are, in genera, strongy dependent upon r. These additiona roots are termed parasitic roots. In most good designs, the movement of the desired roots from their idea position due to the nonzero op amp time constants is sma. Approximations of the desired roots and the associ- (6)

3 GEIGER: PARASITIC POLE APPROXIMATION TJXHNIQUES 795 TABLE APPROXIMATE POLE-DEFINING EQUATIONS II ated stabiity probem can be found in the iterature []-[5], [7], [8], [o], [. The emphasis here is paced upon approximating the parasitic roots. Approximate OP Amp Root Equations with Singe-Poe Mode of the Let the degree of the passive RC network be np. Then the poynomias Q,(s), P,(s), P2(s), and D,(s) in the poe-defining equations of Tabe I are of degrees n,, n,, n2, and np, respectivey, where no, n,, and n2 are ess than or equa to np and where the highest order coefficient of D, is assumed to be equa to unity. The foowing assumption wi now be made. Assumption : Assume r is so sma that Do(s) is a factor of the poe-defining equations (i), (iii), and (v) of Tabe I. Athough D,-,(s) is in genera not a factor of the poedefining equations, the actua ocation of the desired poes are very near to the roots of De(s) for sma vaues of r since the roots of the poe defining equation are anaytic functions of r [ 2. With this assumption, the poe-defining equations (i), (iii), and (v) of Tabe I can be approximated by equations (i ), (iii ) and (v ) of Tabe II, where D,,(s)= 5 d,,s. i=o A reasonabe choice of the a coefficients in (i ) can be made by equating the coefficients of the highest np-n,, +2 powers of s on both sides of z in (i ). Equating these coefficients resuts in a set of inear equations in the a variabes which can readiy be soved. The a coefficients in (iii ) and (v ) are determined in a simiar manner. If r is sufficienty sma, it is often the case that each of these inear equations is essentiay dependent upon ony a singe a variabe making the soution very simpe. With r sma, the equations of Tabe II of the form fp-,,+j 2 a,(t.s) =O, j=,2,3 (7) i=o are the equations that can be soved to approximate the parasitic roots when the singe-poe mode of the op amp is used. For existing active fiters, it is most often the case that n,,=np so that the order of the equation that determines the parasitic roots is equa to the number of op amp s in the circuit. If one or two op amp s are used, the expressions for the parasitic roots are particuary simpe. Severa aternative methods for obtaining the parasitic roots are discussed by Heinein and Homes [3, p These methods give resuts simiar to those presented here. Approximate Op Amp s Root Expressions for Two-Poe Mode of the Two methods of obtaining the approximate parasitic roots when a two-poe mode of the op amp is empoyed wi be considered. The first directy paraes the singe-poe mode deveopment provided equations (ii), (iv), and (vi) of Tabe I are used in pace of (i), (iii), and (v) in the preceding derivation. If the number of op amp s is very arge, the order of the equation corresponding to (7) is rather high. The second method, which is generay more tractabe, extracts the two-poe mode parasitic roots directy from those obtained by appying the much simper singe-poe mode. This method is not, however, readiy extendabe to the nonidentica op amp situation.

4 796 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. ~~-27, NO. 9, SEPTEMB ER 980 The foowing deveopment of the second method is dependent upon the assumption: Assumptiqn 2: im,, a,.#o for a a coefficients of (7). This assumption roughy says that the a coefficients are independent of r for sma r. This assumption is satisfied in most stabe existing fiters. With this assumption and for sma vaues of r it foows that (7) may be written in factored form as np-no+j f (TS-pi)=O, j=,2,3 (8) i=, where for a &pi is not a function of 7. The parasitic poes obtained with the singe-poe mode are thus ocated at s=pjr, i= ; -., np-no+j. At this point, observe that the use of the two-poe mode of the op amp of (6) actuay invoves making the transformation TS+m( + 87s) (9) in the poe-defining equations (i), (iii), and (v) of Tabe I. Since the 7s term appears ony in the second part of the product in the poe-defining equations of Tabe II, it foows that a factored form of the poe-defining equation for active fiters using the two-poe approximation of the op amp s can readiy be obtained by repacing 7s by rs( +&rs) in (8). The parasitic poes can thus be determined from a knowedge of the singe-poe parasitic poes from the expression np-no+j if, (TS[ +&s] -pi)=o, j=,2,3. (0) The parasitic poes obtained ocated at from this expression are thus () This equation can be used to approximate a parasitic poes with a two-poe mode of the op amp once the parasitic poes with the singe-poe mode have been either determined or approximated. In the case that pi is a compex number, pi= -a+jj3 (2) p,? is aso a root of (0). In this case it can readiy be shown that the four second-order mode roots obtainabe from the first-order-mode roots pi and pf are given by the expression by the parasitic poe ocations. In the singe-poe mode case the parasitic roots themseves if soved for or the Routh-Hurwitz criterion can be used to determine stabiity. If a two-poe mode of an op amp is necessary the stabiity question can be answered based soey upon the ocation of the more readiy determined singe-poe mode parasitic roots provided the op amp s are identica. This is formaized in the foowing caim. Caim: If-r is sufficienty sma so that (7) is a good approximation of the poe-defining equation of the parasitic roots for the singe-poe mode of the op amp, then the fiter wi be stabe if a two-poe mode of the op amp is empoyed when the foowing equation is satisfied for each singe-poe mode parasitic root pi = - (Y + jp. epqa. (4) This caim foows from (3) with some routine agebraic manipuations by requiring that the rea part of a compex-conjugate poes be negative. The approximations presented thus far and utimatey the stabiity criterion have reied upon the fact that 7 is sufficienty sma. At this point the statement shoud be made that if a circuit is stabe with sufficienty sma vaues of r, then it wi often be stabe for vaues of r in which the approximations of Tabe II can not be justified provided the op amp s are identica. This can be estabished by showing that the parasitic poe movement as a function of increasing r for parasitic poes with a nonzero imaginary part is approximatey in a constant Q manner towards the origin of the s-pane. This fact wi now be derived for the singe-poe mode case directy from Assumptions and 2. Assume r is sufficienty sma that the approximate parasitic poe-defining equation (7) is justified and assume that the a coefficients in (7) are not dependent upon 7. Let pk be any parasitic root. Thus (7) may be written in fhe form I.--no+ i=o Since the a coefficients are independent of 7, it foows that 3Pk -=- -Pk ar 7 * (6) Thus the change in pk from its vaue at r=r, where 7, is sufficienty sma that (7) is justifiabe to its position d - 8&x+ 68% + 2 6e2p2 d - 8f?o + 602az + 6e2p2 2 (3) IV. STABILITYCONSIDERATIONS r2=r, +k may be approximated by Since it is assumed in these approximations that r is sufficienty sma that the desired roots are essentiay at Apkc+.&=-Pk&. (7) the idea ocations, the fiter stabiity wi be determined 7-7, 7

5 GEIGER: PAiUSTIC POLE APPROXBfhON TECHNIQUeS 797 Fig. 2. Biquadratic bandpass fiters. This shows that the movement as a function of r of V. EXAMPLE parasitic poes with a nonzero imaginary part is towards The ocations of the poes of the two second-order the origin in a constant Q manner as caimed above. bandpass active fiters of Fig. 2 wi now be approximated The above derivation extends readiy in the case that and a comparison made of the approximate poe ocations the two-poe mode of the op amp is empoyed. with the actua computer generated poes. The first circuit It can be concuded that fiter stabiity reated to the is a form of the popuar state-variabe fiter [I. The position of the parasitic poes with either a singe-poe or second is of interest because it is an exampe of a promistwo-poe mode of the op amp can be approximated by ing fiter that appears stabe when a singe-poe mode of investigating the position of the parasitic poes obtained the op amp is empoyed but which is actuay unstabe by considering the op amp s to have a singe-poe mode because of a right haf-pane parasitic poe that appears with r sufficienty sma so that the approximations of when the typica second poe of an op amp such as the Tabe II are justifiabe. 74 is incuded in the anaysis. The assumption of identica op amp s is necessary to An anaysis wi now be made of the approximate poe obtain reasonaby simpe expressions for the approximate ocations for the state-variabe fiter of Fig. 2(a). The parasitic poes in the genera singe-poe mode case and is anaysis of the remaining configuration is straightforward, crucia for determining the parasitic poes with a two-poe h ence ony the resuts wi be stated. mode of the op amp directy from those obtained with When a singe-poe mode of the op amp is empoyed, the singe-poe mode via (). It is, however, reasonabe the poe-defining equation for the circuit of Fig. 2(a) is to expect that a fiter which is predicted to be unstabe when identica op amp s are used wi not be usefu if D3,(sn, A)=s,+~+,+~~s~(4s~+s~(3+;)+~) unmatched op amp s are empoyed. It is aso reasonabe to expect that the approximations isted above wi be quite good as ong as the parameters of the op amp s do not differ greaty. +T;s;(~s;+s,[ 9+ ;] +2+ ;) No mention has been made to this point about how sma r shoud be to make the parasitic-poe approxima- +2+;( s;+sj 3+ ;] +2+ ;) tions previousy derived. The answer to this question depends in part upon the required accuracy of the ap- --(sjf+ 2 + )[ a3(7,s,)3+a2(7,s,)2 proximations. Based upon a few exampes it appears that the approximations derived above wi probaby be quite +ahn)+ao]. (8) good if for a particuar vaue of r the rea and imaginary parts of the approximate desired-poe ocations obtainabe The normaizations s,=s/w,, and rn =rwo have been made from the references previousy cited differ by at most a to eiminate the w. dependence from these equations thus few percent from their idea r=o vaue. making the anaysis simper. By equating the coefficients

6 798 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CA.+27, NO. 9, SEPTEMRER 980 TABLE III TABLEIV C~M~ARI~~N OF APPROXIMATE AND Ex.kcr PARAsrnc Pow LOCATIONSFORA SINGLE-P LEMODELOFTHEOPAMP ROOT LOCATIONS SINGLE-POLE MOOEL OF OP AMP -- ~MPARISONOF~PROXIMATF~ANDEXACT~RASITICPOLE LOCATIONS FOR A TWO-POLE MODEL OF THE OP h ROOT LOCATIONS TWO-POLE MODEL OF OP AMP EXACT STATE VARIABLE FILTER OF FIGURE 2a APPROXIMATE EXACT STATE VARIABLE FILTER OF FIGURE 2$ I,00-4,7907E-02 + j etoo E-02 t j etoo -.OOOGE+03 -.OOOOOE+03 -.O029E E EtOZ -5.OOOOOE+OZ _-_-_-- -_-_-.O E-02 + j e+oo * E-02 + j etoo -.o2562e+02 -.OOOOOE Et02 -.OOOOOE E+O -5.OOOOOE+O -. ZERO SENSITIVITY CIRCUIT OF FIGURE 2b,00-5.OOOOOE-02 + j etoo -5.OOOOOOE-02 *j,999687etoo E-0 + j et j.oet03 -.OOOE+03 -.OE _ O -5,0009E-02 +j,998760e+oo -5.OOOOOOE-02 + j etoo EtOO + j e j.oe+oz -.054EtOZ -.OE+OZ *Indicates desired root t,00 -.O,00.O E-02 + j,997348e+oo * -.OOOZOE+03 + j.99885e Et03 + j.o0268e E+03 + j,9657ze+oo E-02 + j etoo -.OOGE+02 + j.o07e+oz E-02 + j,998788e+oo -.OOOOOE+03 + j.oooooe+03 -.OOOOOE+03 t j.oooooe+03 -.OOOOOE+03 -.OOOOOE E-02 + j e+oo -.OOOOOE+02 + j.oooooe+02 -.O300E+OZ + j.o32e+oz -.OOOOOE+02 + j.oooooe+02 -.O0040E+OZ + j.93346e+oo -.OOOOOE+02 -.OOOOOE+02 ZERO SENSITIVITY CIRCUIT OF FIGURE 2b -5.OOOOOE-02 + j.99875oe+oo * -5.OOOOOE-02 + j e.00 -.O0505E+03 + j.o0498e+03 -.OOOOOE+03 + j.oooooe e+03 + j.7993e e+03 + j.7866et E+03 + j et E+OZ + j.7865e _- -5.OOOZE-02 + j.99876oetoo * -5.OOOOOE-02 + j e+oo -.O5488E+OZ + j.o470e+02 -.OOOOOE+03 + j etoo of the four highest powers of s the a coefficients of (7), subject to the assumption r,<, are readiy found to be a3-2) up-5 a,-4 (9) a,- i The singe-poe mode parasitic poe-defining equation (7) is thus 2(r,~,)~+5(r,s,)~+4(r,s,)+=O. The roots of this cubic poynomia s, = -.0/r, are (20) s r! =-.0/r,. (2) sn= -0.5/r, These approximate parasitic roots as we as the singe-poe mode approximate parasitic roots for the circuit of Fig. 2(b) are isted in Tabe III for r,, = 0.00 and 0.0 and for Q= 0. If the gain-bandwidth product of the op amp s are 277~ o6 rad/s, a typica vaue for the 74, then the r,,=o.oo and 0.0 cases correspond to bandpass fiters centered at khz and 0 khz, respectivey. The computer generated poes are aso isted in the tabe. A comparison of the parasitic poes of the two circuits of Fig. 2 with the computer generated poes for r,=o.oo shows that the approximations are exceent. The approximations are quite good even when r,, = 0.0. The approximate desired roots [2, p. 390 are aso isted in the tabe E+OZ tz.z90e+o * Indicates desired root + j.g06809e+oz + j,754664e+oz E+02 + j.7865e+oz E+O + j.7865etoz _.._ ~._.. Using ( ), the parasitic roots for the circuit of Fig. 2(a) when a two-poe mode of the op amp with 8= 0.5 (a typica vaue for the 74) is used can be readiy obtained from (2) to obtain the six approximate parasitic roots s, = ( -.O * j.0)/r, sn=(- s = -.0/7,.O+j.O)/r, s, = -.0/r, These approximate parasitic roots as we as those for the circuit of Fig. 2(b) are isted in Tabe IV for r,,=o.oo and 0.0 and Q= 0 aong with the computer generated roots. The approximate desired roots [2, p. 68 are aso incuded. The exact and approximate parasitic roots for the circuits of Fig. 2 are ikewise seen to be in cose agreement when the two-poe mode of the op amp is empoyed. The necessity of incuding the second poe of the op amp in the anaysis of some active fiters is borne out by the circuit of Fig. 2(b). It can be seen from Tabe III that if the simper but ess accurate singe-poe mode of the op amp is used an exact computer anaysis indicates the fiter is stabe since a poes have a negative rea part. By incuding the typica second-poe of the op amp in the anaysis, the approximate and exact root ocations isted * (22)

7 GEIGER: PARASITIC POLE APPROXIMATION 799 in Tabe IV indicate that the fiter is actu-ay unstabe because of a right haf-pane parasitic poe. This instabiity is in agreement with experimenta resuts. The instabiity coud, of course, have been predicted by considering the singe-poe mode of the op amp and using (4). It is interesting to note that the desired poe ocations for this circuit are essentiay unchanged from the vaues obtained by assuming the op amp s to be idea. This circuit is thus an exampe of an active fiter in which it appears that the performance in the region of interest is essentiay independent of the nonidea parameters of the op amp but which is actuay unstabe due to the presence of a right haf-pane poe caused by the second-poes of the op amp s. VI. CONCLUSION A method of approximating the parasitic roots of active fiters empoying mutipe op amp s has been presented. These approximations are good for either a singe-poe or two-poe mode of the op amp when the op amp time constant r is sufficienty sma. These approximation techniques invove soving ow-order poynomias rather than the higher order poynomias necessary to obtain the exact roots. Of particuar importance is the fact that for two op amp s or ess a cosed-form expression for the parasitic poes in terms of the components of the circuit is usuay possibe for either a singe-poe or two-poe mode of the op amp. Such a cosed-form expression shoud be usefu to the fiter designer in the design process itsef. It has aso been shown that since the parasitic poe movement is in a constant Q manner as a function of increasing T, stabiity criterion can be approximated for vaues of r at which the poe approximations can not be justified based upon the approximations when r is sufficienty sma. An exampe has been given of a circuit which appears to perform very we when a singe-poe mode of the op amp is used but which is actuay unstabe when a typica second poe is incuded in the mode of the op amp. The importance of considering the second poe of the op amp during the design process is borne out by this exampe. Finay, comparisons of the readiy obtained parasitic poe approximations are made with the actua computer generated poes for two circuits. These comparisons show that the approximations are quite good. PI VI [3 [4 VI WI REFERENCES A. Budak and D. M. Petrea, Frequency imitations of active fiters using operationa ampifiers, IEEE Trans. Circuit Theory, vo. CT-9, pp , Juy 972. A. Budak, Passiw and Active Network Anaysis and Synthesis. Boston, MA: Houghton Miffin, 974. W. E. Heinein and W. H. Homes, Act& Fiters for Integrated Circuits. New York: Springer-Verag, 974, ch. 4. R. Tarmy and M. S. Ghausi, Very high-q insensitive active RC networks, IEEE Aue Tram. Circuit Theory, vo. (X-7, pp , W. B. Mikhae and B. B. Bhattacharyya, A practica design for VI insensitive RC-active fiters, IEEE Trans. Circuits Syst., vo. CAS- 22, pp , May 975. M. A. Reddv. Guerationa-amoifier circuits with variabe ohase shift and their appication to high-q active RC-fiters and RCosciators, IEEE Trans. Circuits Syst., vo. CAS-23, pp , June 976. [7 R. L. Geiger and A. Budak, Active fiters with zero ampifiersensitivity, IEEE Trans. Circuits Syst., vo. CAS-26, pp , Am RI L. Geiger, Low-sensitivity active fiter design, Ph.D. disserta- tion, Coorado State University, Fort Coins, CO, D. Akerberg and K. Mossberg, A versatie active RC buiding bock with inherent compensation for the finite bandwidth of the ~np~~7>. IEEE Trans. Circuits Syst., vo. CAS-2, pp ,.. WI R. Sninivasagopaan and G. 0. Martens, A comparison of a cass of active fiters with respect to the operationa-ampifier gainbandwidth product, IEEE Tram. Circuits Syst., vo. CAS-2, pp , May 974. L. C. Thomas, The biquad: Part I-Some practica design consgc~~,tm,~ IEEE Trans. Circuit Theory, vo. CT-8, pp , -----I -_ -. E. Hie, Anaytic Function Theov. vo. II, 962, ch. 2, Sec Watham, MA: Ginn, Randa L.. Geiger (S 75-M 77) was born in Lexington, NB, on May 7, 949. He received the B.S. degree in eectrica engineering and the M.S. degree in mathematics from the University of Nebraska, Lincon, in 972 and 973, respectivey. He received the Ph.D. degree in eectrica engineering from Coorado State University, Ft. Coins, in August 977. He joined the facuty of the Department of Eectrica Engineering at Texas A&M University in September 977 and is currenty an Assistant Professor in the department. His present area of research is in the active circuits fied. Dr. Geiger is a member of Em Kappa Nu, Sigma Xi, Pi Mu Epsion,.-. - ana atgma tau.