Designing Operational Amplifier Oscillator Circuits for ... - Microchip

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AN866APPENDIX A: MASON’S REDUCTIONTHEOREMThe oscillation frequency is determined by finding thepoles of the denominator of the transfer equation T(s)or equivalently the zeroes of the numerator N(s) of thecharacteristic equation ∆(s). Mason’s theory is especiallyuseful for analyzing oscillators that have multiplefeedback loops.Mason’s theorem [5] states that the transfer functionfrom input X to output Y is:EQUATION:Ts ( )YΣ iP i∆s iΣ iP i∆s i= -- = ------------------ = ------------------X ∆s Ns ( )-----------Ds ( )Where:P i = the direct transmittance or path form input X tooutput Y∆s i = the system determinant. (∆s i = 1 if P i touchesall of the loops)∆s = 1 - ΣL j + ΣL k L l - ΣL m L n L o +....ΣL j = the sum of all loops (i.e. loop gains)ΣL k L l = the sum of products of pairs of nontouchingloopsΣL m L n L o = the sum of products of gains of non–touching loops taken three at a time.APPENDIX B: ROUTH STABILITYTESTThe Routh Stability Test [5] can be used to test thecharacteristic equation to determine whether any of theroots lie on the imaginary axis. Routh’s test consists offorming a coefficient array from N(s). Next, the proceduresubstitutes s = jω ο for s, and the summation of therow is set to zero. If the row equation produces a nontrivialsolution for ω ο , the procedure is complete and thefrequency of oscillation is equal to ω ο . If the row equationdoes not yield an equation that can be solved forω o , the procedure continues with the next row in theRouth array. This technique arranges the numerator ofthe characteristic equation (i.e., denominator of thetransfer equation) into the array listed below.EQUATION:N(s) = a o s n + a 1 s n-1 + a 2 s n-2 + a 3 s n-3 +...+ a n-1 s + a nNote for simplicity, only the first three rows of the Routhcoefficient array are shown below.s n a 0 a 2 a 4 .... a ns n–1 a 1 a 3 a 5 .... a n–1s n–2 b 1 b 2 b 3 .... b. n–2where the coefficients b 1 , b 2 , b 3 , etc., are defined as:b 1 = (a 1 a 2 - a 0 a 3 ) / a 1b 2 = (a 1 a 4 - a 0 a 5 ) / a 1b 3 = (a 1 a 6 - a 0 a 7 ) / a 1The Routh stability criterion states:1. A necessary and sufficient condition for stabilityis that the first column of the array does not containsign changes.2. The number of sign changes in the entries of thefirst column of the array is equal to the numberof roots in the right half s–plane.3. If the first element in a row is zero, it is replacedby ε, and the sign changes when ε → 0 arecounted after completing the array.4. The poles are located in the right half plane oron the imaginary axis if all the elements in a roware zero.© 2003 Microchip Technology Inc. DS00866A-page 12
AN866CONCLUSIONOperational amplifier oscillators can be used to producea frequency that is proportional to resistive andcapacitive sensors. Design equations defining theoscillation frequency are readily available for severalcommon oscillators, such as Wein bridge and phaseshift oscillators. However, detailed design equationsthat show the relationship of the resistors and capacitorsare generally not available. Thus, there is a needfor a design procedure that derives the equations inorder to select the resistor and capacitor componentsthat maximize the accuracy of the oscillation frequency.The design procedure was demonstrated by analyzingtwo state-variable oscillators for capacitive sensingapplications.BIBLIOGRAPHY1. Celma, C., Martinez, P., and Carlosens, A.,“Approach to the Synthesis of Canonic RC–Active Oscillators Using CCII”, IEE Proc. Circuits,Devices and Systems, Vol. 141, No. 6,December 1994, pp. 493–497.2. Lepkowski, J. and Young, C, “AND8054 -Designing RC Oscillator Circuits with Low VoltageOperational Amplifiers and Comparatorsfor Precision Sensor Applications”, ONSemiconductor, Phoenix, Arizona, 2002.3. Martinez, P., Aldea C. and Celma, S., “Approachto the Realization of State-Variable Based Oscillators”,IEEE International Conference on ElectronicsCircuits and Systems, Vol. 3, 1998,p.139–142.4. Sidorowicz, R., “An Abundance of SinusoidalRC–Oscillators”, Proc. IEE, Vol. 119, No. 3,March 1972, pp. 283–293.5. Truxal, J., “Introductory System Engineering”,McGraw–Hill, N.Y., 1972.6. Van Valkenburg, M., “Analog Filter Design”,Saunders College Publishing, Fort Worth, 1992.© 2003 Microchip Technology Inc. DS00866A-page 13
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