Designing Operational Amplifier Oscillator Circuits for ... - Microchip

MAN866Designing Operational Amplifier Oscillator CircuitsFor Sensor ApplicationsAuthor:INTRODUCTIONJim LepkowskiMicrochip Technology Inc.Operational amplifier (op amp) oscillators can be usedto accurately measure resistive and capacitive sensors.Oscillator design can be simplified by using theprocedure discussed in this application note. The derivationof the design equations provides a method toselect the passive components and determine the influenceof each component on the frequency of oscillation.The procedure will be demonstrated by analyzingtwo state-variable RC op-amp oscillator circuits.SENSOR APPLICATIONSState-variable oscillators are often used in sensor conditioningapplications because they have a reliablestart-up and a low sensitivity to stray capacitance. Theabsolute and ratio state-variable oscillators can beused to accurately detect both resistive and capacitivesensors. However, this application note will only analyzecapacitive applications. The state-variable's threeop-amp topology provides for a more dependable oscillationstart-up than a single op amp oscillator. The virtualground voltage at the inverting terminal of theamplifiers provides for immunity from stray capacitance,which is important in sensor applications,because the sensor capacitance is often only 10 to100 pF. In addition, the state-variable oscillator doesnot require matched capacitors or capacitors that havea terminal connected to ground.The absolute oscillator provides an output frequencythat is proportional to the square root of the product oftwo capacitors (i.e., freq. ∝ (C 1 x C 2 ) 1/2 ). Absolutequartz pressure sensors and humidity sensors areexamples of capacitive sensors that can use the absoluteoscillator. Also, this circuit can be used with resistivesensors, such as RTDs, to provide a temperatureto-frequencyconversion.The ratio oscillator provides an output frequency that isproportional to the square root of the ratio of two capacitors(i.e., freq. ∝ (C 4 / C 3 ) 1/2 ). The ratio oscillator canbe used to cancel the effect of a fluid-level sensor witha varying dielectric constant, such as an oil level sensor.An oil level sensor consists of two capacitancesthat are formed by tubes where the fluid serves as thedielectric media. The measurement capacitor (C MEAS )is partially covered by fluid and detects the level of theoil in the tank. In contrast, the compensation sensor(C COMP ) is completely buried in the fluid. When theratio C MEAS / C COMP is calculated, the dielectric constantof the oil is canceled. Air pressure and accelerationsensors can also use the ratio sensor to minimizethe error that occurs from the variance of the dielectricconstant over temperature.TRANSDUCER SYSTEMA block diagram of a typical sensor system is shown inFigure 1. The oscillation frequency can be found bycounting the number of clock pulses (i.e., MHz) in atime window that is formed by the square wave output(i.e., kHz) of a comparator circuit. The counter andcomparator circuits can be implemented with aPICmicro ® microcontroller.The PICmicro microcontroller can be used to providecurve-fit temperature correction for precision sensingapplications. Temperature correction can be accomplishedby implementing a curve-fitting routine withdata obtained by calibrating the sensor over the operatingrange. The temperature correction data can bestored in the E 2 memory of the PICmicro microcontroller.A silicon IC sensor can provide the temperature ofthe sensor.© 2003 Microchip Technology Inc. DS00866A-page 1

AN866Step 2: Solve N(s) = 0The second step in the procedure determines thezeroes of N(s). Routh’s stability criterion, shown inAppendix B, provides a method that determines thezeroes of the characteristic equation without thenecessity of factoring the equation.First, the Routh test consists of forming a coefficientarray from N(s). Next, the procedure substitutes s = jω οfor s, with the summation of the row set to zero. If therow equation produces a non-trivial solution for ω ο , theprocedure is complete and the frequency of oscillationis equal to ω ο . If the row equation does not yield anequation that can be solved for ω ο , the procedure continueswith the next row in the Routh array. Usually, it isnecessary only to complete the first two or three rowsof the Routh array to produce an equation that can besolved for ω ο .Step 3: Sub-Circuit Design EquationsThe third step in the design procedure analyzes thesub-circuits formed at each amplifier. The sub-circuitequations are formed by obtaining the gain equationand pole/zero locations for each amplifier.Step 4: Verify⏐LG⏐ ≥ 1The final step in the procedure verifies that the loopgainis equal to, or greater than, one after the R and Ccomponent values have been chosen. This step is alsorequired to verify that the amplifiers do not saturate,which will result in an error in the oscillation frequency.AMPLIFIER SELECTION CRITERIAThe appropriate op amp to use in a sensor oscillator isdetermined by the required accuracy and acceptabledistortion of the oscillation frequency. The design equationsassume that the amplifiers are ideal. However, opamps have a finite gain bandwidth product (GBW), alimited slew rate (SR) and full power bandwidth (f P ).The non-ideal characteristics of the amplifier will lowerthe oscillation frequency at high frequencies and mayalso result in a design with poor start-up characteristics.Note that the total harmonic distortion specificationof the amplifiers is critical for oscillators that are usedas sine wave references. However, the shape of thewaveform is not critical in most sensor applicationsbecause only the frequency of the output is measured.Several general design rules can be used to select anop amp for an oscillator circuit. First, the GBW shouldbe a factor of 10 to 100 higher than the maximum oscillationfrequency. Next, the full-power bandwidth,defined as f P = SR / (2πV P ), where V P is the voltageswing (V O(max) - V O(min) ) of the output signal, should beat least 2 times greater than the maximum oscillationfrequency. For example, the MCP6024 quad amplifierhas a GBW = 10 MHz (typ.), SR = 7 V/µs (typ.) and a f Pof 400 kHz, with V DD = 5V. An oscillator with amaximum frequency of 100 kHz can be implementedwith the MCP6024 with enough design margin that thenon-ideal characteristics of the amplifier can beneglected.ABSOLUTE STATE-VARIABLEOSCILLATORCircuit DescriptionThe schematic of the absolute circuit is shown inFigure 3. The state-variable oscillator consists of twointegrators and an inverter circuit. Each integrator providesa phase shift of 90°, while the inverter adds anadditional 180° phase shift. The total phase shift of360° of the feedback loop produced by the three amplifiersresults in the oscillation. The first integrator stageconsists of amplifier A 1 , resistor R 1 and sensor capacitanceC 1 . The second integrator consists of amplifierA 2 , resistor R 2 and sensor capacitance C 2 . The inverterstage consists of amplifier A 3 , resistors R 3 and R 4 andcapacitor C 4 . The addition of capacitor C 4 helps ensureoscillation start-up by providing an additional phaseshift.The absolute oscillator does not require a limit circuit ifrail-to-rail input/output (RRIO) amplifiers are used andthe gain of the inverter stage (A 3 ) is equal to one(i.e., R 3 = R 4 ). The sinewave output of the signal willswing within approximately 50 mV of the V DD and V DDpower rails as shown in Figure 5.A complementary output voltage comparator (A 4 ) isused to convert the oscillator’s sinewave output to asquare wave digital signal. The comparator functionsas a zero-crossing detector and the switching point isequal to the virtual ground voltage (i.e., V DD /2). ResistorR 9 is used to provide additional hysteresis (V HYS ) tothe comparator. Listed below is the hysteresisequation.EQUATION:RV = -------------------- 8× ( V – V )HYS R 8+ R Omax ( ) Omin ( )9R 8V HYS≅ -------------------- × VR 8+ R DD 9© 2003 Microchip Technology Inc. DS00866A-page 3

AN866V DDR 6V DD /2R 7C 4V 0C 1C 2R 1A 1A 2R 4R 9V DD /2R 2R 3A 3R 8VA 41V DD /2 V 2V DD /2 V 3V DD /2FIGURE 3:Absolute Oscillator Schematic.⎛R A 4 ⎞------1A = 3⎜ ⎟⎛-------------------------⎞2= ----------------– 1sR C ⎝R 3 ⎠⎝sR 4C 4+ 1⎠2 2V 1 V 3V 2A– 11= ----------------sR 1C 1FIGURE 4:Absolute Oscillator Signal Flow Diagram.© 2003 Microchip Technology Inc. DS00866A-page 4

AN866ABSOLUTE STATE-VARIABLEDesign EquationsSTEP 1: FIND LG AND ∆STs ( )A A= ---------------- = ----- =1 – LG ∆sA-----------Ns ( )-----------Ds ( )The loop-gain is found by breaking the loop in thesignal flow diagram of Figure 4, as shown below.A 1 A 2A 3V 1 V 2V 3The procedure continues by analyzing row s 2 to determinewhen the equation is equal to zero.Let s = jω οa 1 s 2 + a 3 = ( R 1 R 2 R 3 C 1 C 2 )( jω o ) 2 + R 4 = 020 = R 4 – R 1 R 2 R 3 C 1 C 2 ω oP2ω o = [ R 4 ⁄ ( R 1 R 2 R 3 C 1 C 2 )]ω o = [ R 4 ⁄ ( R 1 R 2 R 3 C 1 C 2 )] 1/2= 2π ⁄ ω o = 2π ⁄ [ R 4 ⁄ ( R 1 R 2 R 3 C 1 C 2 )] 1/2Note that C 4 does not appear in the oscillation equation.C 4 and R 4 form a low-pass filter. The gain of amplifierA 3 will not be a function of C 4 if the oscillationfrequency is less than the cut-off frequency of the filter.A 1 = – 1 ⁄ ( sR 1 C 1 )A 2 = – 1 ⁄ ( sR 2 C 2 )A 3 = – Z 4 ⁄ Z 3= –( R 4|| C 4 ) ⁄ R 3= –[( R 4 ⁄ R 3 )( 1 ⁄ ( sR 4 C 4+ 1))]If:1. R 1 = R 2 = R2. C 1 = C 2 = C3. R 3 = R 4Then:P=2πRCLG = A × A × A 1 2 3= [–1 ⁄ ( sR C )][– 1 ⁄ ( sR C )][(–R ⁄ R )( 1 ⁄ ( sR C + 1))]1 1 2 2 4 3 4 4= – R ⁄ ⎛s 3 R R R R C C C C + s 2 R R R C C ⎞4 ⎝ 1 2 3 4 1 2 3 4 1 2 3 1 2 ⎠∆s= Ns ( ) ⁄ Ds ( ) = 1 – LG= 1 – [– R 4 ⁄ ( s 3 R 1 R 2 R 3 R 4 C 1 C 2 C 4 + s 2 R 1 R 2 R 3 C 1 C 2 )][ s 3 R 1 R 2 R 3 R 4 C 1 C 2 C 4 + s 2 R 1 R 2 R 3 C 1 C 2 + R 4 ]= -----------------------------------------------------------------------------------------------------------------[ s 3 R 1 R 2 R 3 R 4 C 1 C 2 C 4 + s 2 R 1 R 2 R 3 C 1 C 2 ]Ns ( ) = s 3 R 1 R 2 R 3 R 4 C 1 C 2 C 4 + s 2 R 1 R 2 R 3 C 1 C 2 + R 4STEP 3: SUB-CIRCUIT DESIGN EQUATIONSIntegrator A 1Gain A 1 = – 1 ⁄ ( 2πfR 1 C 1 )Pole f p1 = 1 ⁄ ( 2πR 1 C 1 )Integrator A 2Gain A 2 = – 1 ⁄ ( 2πfR 2 C 2 )Pole f p2 = 1 ⁄ ( 2πR 2 C 2 )Gain = –[( R 4 ⁄ R 3 )( 1 ⁄ ( sR 4 C 4 + 1))]Gain ≅ – R 4 ⁄ R 3Integrator A 3A 1 A 2 A 3STEP 2: SOLVE N(s) = 0The zeros of the characteristic equation are determinedby using the Routh stability test.Ns ( ) = a 0 s 3 + a 1 s 2 + a 2 s + a 3a 0 = R 1 R 2 R 3 R 4 C 1 C 2 C 4a 1 = R 1 R 2 R 3 C 1 C 2a 2 = 0a 3 = R 4Routh Stability Test Coefficient Arrayrow s 3 a 0 + a 2 = 0row s 2 a 1 + a 3 = 0Row s 3 produces a trivial solution (ω ο = 0):a 0 ( jω o ) 3 + 0 = 0STEP 4: VERIFY ⏐LG⏐ ≥ 1Assume:1. R 1 = R 2 = R2. C 1 = C 2 = C3. R 3 = R 4= = = 1LG = A 1 × A 2 × A 3 = 1V 1 = A 1 × V 3V 2 = A 2 × V 1V 3 = A 3 × V 2Note that a voltage limit circuit should be added if rail-torailinput/output operational amplifiers are not used, or ifthe gain of the inverter is not equal to one (i.e., R 3 ≠ R 4, ).A limit circuit is required to prevent the frequency errorthat will result from the saturation delay time of theamplifiers.© 2003 Microchip Technology Inc. DS00866A-page 5

AN866ABSOLUTE STATE-VARIABLETest ResultsThe components used in the evaluation design arelisted below. Note that the capacitive sensor (i.e., C 1and C 2 ) was simulated with discrete capacitors.The measured and calculated oscillation frequency isshown in Table 1. Figure 5 shows the oscillation waveformwhen C 1 = C 2 = 220 pF. The error in the measuredoscillation is attributed to the accuracy of the testequipment.R 1 = R 2 = R 6 = R 7 = 32.7 kΩR 3 = R 4 = 10 kΩR 8 = 1 kΩR 9 = 1 MΩC 1 = C 2 = see Table 1C 4 = 18 pFV DD = 5.0VA 1 , A 2 , A 3 ≡ MCP6024 (quad RRIO,GBW = 10 MHZ)A 4 ≡ MCP6541Push-Pull OutputComparatorTABLE 1:ABSOLUTE OSCILLATOR TEST RESULTSCapacitorValuesCalculated OscillationPeriod (Frequency)Measured OscillationPeriod (Frequency)C 1 = C 2 = 47 pF 9.66 µs (103.6 kHz) 10.0 µs (100.0 kHz)C 1 = C 2 = 56 pF 11.5 µs (86.9 kHz) 12.0 µs (83.3 kHz)C 1 = C 2 = 82 pF 16.9 µs (59.4 kHz) 16.6 µs (60.2 kHz)C 1 = C 2 = 100 pF 20.6 µs (48.7 kHz) 21.0 µs (47.6 kHz)C 1 = C 2 = 150 pF 30.8 µs (32.6 kHz) 30.0 µs (33.3 kHz)C 1 = C 2 = 220 pF 45.2 µs (22.1 kHz) 46.0 µs (21.7 kHz)Output of A 3 Amplifier (V 3 )Comparator Output (V O )FIGURE 5:Absolute Oscillator Test Results (C 1 = C 2 = 220 pF).© 2003 Microchip Technology Inc. DS00866A-page 6

AN866RATIO STATE-VARIABLEOSCILLATORCircuit DescriptionThe schematic of the ratio circuit is shown in Figure 6.This circuit consists of two integrators and a differentiatorcircuit. The integrators formed by amplifiers A 1 andA 2 are identical to the integrators used in the absolutecircuit. The differentiator stage is formed by amplifierA 3 , resistors R 3 , R 4 and R 5 , and the sensor capacitorsC 3 and C 4 to provide a 180° phase shift. The comparator(A 4 ) used to convert the sinewave output to asquare wave digital signal is identical to the absoluteoscillator circuit.A Bode plot of the differentiator stage is provided inFigure 8. The values of resistors R 3 , R 4 and R 5 areselected to set the break frequencies of the differentiatorstage so that the gain of the stage is equal to -C 3 /C 4 at the oscillation frequency. Resistor R 5 is alsoused to provide a DC current path around capacitor C 3in order to initiate oscillation at power-up.V DD /2V DDQ1C 4C 1CC 32R 1 R 4A 1 R 2A 2R 5A 3V DD /2V 1V DD /2 V 2 V DD /2 V 3R 6V DD /2R 7R 9R 8R 3A 4V DD /2FIGURE 6:Ratio Oscillator Schematic.- R ( sR C + 1)– 1A 4 5 3A = ---------------------------------------------------------------------------------------2= ----------------sR C 3 ( sR 3C 5C 3+ R 3+ R 5)( sR 4C 4+ 1)2 2V 1 V 3V 2A 11=----------------–sR 1C 1FIGURE 7:Ratio Oscillator Signal Flow Diagram.© 2003 Microchip Technology Inc. DS00866A-page 7

AN866Gain (dB)⎛C 3 ⎞20log 10⎜------⎟⎝C 4 ⎠F1z1= --------------------2πR 5C 3OscillationRangeAssumptions1. C 3 > C 42. R 3 < R 4 < R 53. R 3 < < R 5 and R 4 ≈ R 5-----100 Frequency (Hz)⎛ R20log 4 ⎞10⎜--------------------⎟⎝R 3+ R 5 ⎠1F p1= -------------------- F = --------------------1p22πR C 2πR 4 43C 3FIGURE 8:Bode Plot of Differentiator Amplifier.Voltage Limit CircuitThe ratio oscillator uses a limit circuit to accommodatethe varying gain requirement of the circuit. It may benecessary to add a voltage limit or clamp circuit to theoscillator to prevent the amplifiers from saturating andavoid slew rate limitations. The voltage limit circuitformed by PNP transistor Q 1 is used to create the maximumvoltage limit. The clamping voltage of the limitcircuit is provided below:EQUATION:V Max_Limit= V Q1_base+ V Q1_base-to-emitterV Max_Limit≅ V Q1_base+ 0.7VIn single-supply applications, it is not necessary to useboth maximum and minimum limit circuits. Only one ofthe limit circuits is required due to the symmetry of thesinewave that is centered around the virtual groundvoltage at the non-inverting terminal of the amplifiers(V DD /2). In the reference design of Figure 6, V DD isequal to 5V and V Q1-base is equal to 2.5V. Thus, theoscillation waveform at V 2 will swing from 1.8V to 3.2Vor 2.5V ±0.7V.Note that the transistor adds a small capacitance (C Q1 )to the integrator capacitor of the circuit (C 2 ). If C 2 is relativelysmall, the effective capacitance of the limit circuit(C Limit ) can be reduced by connecting a diode inseries between the emitter junction and the output ofthe amplifier (i.e. 1/C Limit = 1/C Q1 + 1/C Diode ).© 2003 Microchip Technology Inc. DS00866A-page 8

AN866The procedure continues by analyzing row s 3 to determinewhen the row equation is equal to zero.STEP 3: SUB-CIRCUIT DESIGN EQUATIONSIntegrator A 1Gain A 2 1a s 3 + a s = s ⎛ a s 2 + a ⎞=01 3 ⎝ 1 3 ⎠Let s = jω οjω oa 1ω o2Gain A 1 = – 1 ⁄ ( 2πfR 1 C 1 )Pole f = 1 ⁄ ( 2πR 1 C 1 )⎛– ⎝+ a ⎞3 ⎠= 02a 3ω o= -----a 1= – ⁄ ( 2πfR 2 C 2 )Pole f = 1 ⁄ ( 2πR 2 C 2 )Integrator A 2DC Gain R 4 R 3 R 5R R C 4 5 3= ---------------------------------------------------------------------------------------------------------------------( R 1R 2C 1C 2)( R 3R 5C 3+ R 3R 4C 4+ R 4R 5C 4)( Rω 4R 5C 3)o = ---------------------------------------------------------------------------------------------------( R 1R 2C 1C 2)( R 3R 5C 3+ R 3R 4C 4+ R 4R 5C 4)If:1. R 1 = R 2 = R2. C 1 = C 2 = CThen:If:1. R 5 >> R 32. R 4 >> R 3Then:P=2π-----ω oP 2πRC R 3 C 1/24C----------- 4R----- 3=+ + -----R 5C C 3 3R 4P 2πRC C 4≅ ----- 1/2C 31/2Differentiator A 3= – ⁄ ( + )Gain at Oscillation =C 3–-----C 4Pole f p1 = 1 ⁄ ( 2πR 4 C 4 )Pole f p2 = 1 ⁄ ( 2πR 3 C 3 )Zero f z = 1 ⁄ ( 2πR 5 C 3 )STEP 4: VERIFY ⏐LG⏐ ≥ 1Assume:1. R 5 >> R 3 and R 4 >> R 3 , then A 3 = - C 3 / C 42. V 2 = V Max_Limit (i.e. place limit circuit at A 2 )Next, calculate the voltages at the output of eachamplifier starting at V 2 .V 2=V Max_LimitV 3 = A 3 × V 2 =V 1 = A 1 × V 3 =Oscillation will be sustained if:A 2 × V 1 ≥ V Max_LimitC----- 3× VC Max_Limit4---------------------1× V2πfR 1 C 31⎛ 1--------------------- ⎞ × V⎝2πfR 2 C ⎠ 1 ≥ V Max_Limit2© 2003 Microchip Technology Inc. DS00866A-page 10

AN866RATIO STATE-VARIABLETest ResultsR 1 = R 2 = R 6 = R 7 = 32.7 kΩC 1 = C 2 = 220 pFC 3 = C 4 = see Table 2R 3 = 5 kΩR 4 = 3.3 MΩR 5 = 10 MΩR 8 = 1 kΩR 9 = 1 MΩV DD = 5.0VA 1 , A 2 , A 3 ≡ MCP6024 (quad RRIO,GBW = 10 MHZ)A 4 ≡ MCP6541 Push-Pull OutputComparatorQ 1 ≡ 2N3906The measured and calculated oscillation frequency isshown in Table 2. Figure 9 shows the oscillation waveformwhen C 3 = 100 pF and C 4 = 47 pF. Note that thevoltage limit circuit adds distortion to the waveform ofamplifier A 2 . In most sensor applications, waveformdistortion is inconsequential because the measurementis proportional to frequency and not the amplitudeof the oscillation.TABLE 2:RATIO OSCILLATOR TEST RESULTSCapacitorValuesCalculated OscillationPeriod (Frequency)Measured OscillationPeriod (Frequency)C 3 = 47 pF C 4 = 47 pF 45.2 µs (22.1 kHz) 47.0 µs (21.3 kHz)C 3 = 47 pF C 4 = 100 pF 65.9 µs (15.2 kHz) 68.0 µs (14.7 kHz)C 3 = 47 pF C 4 = 220 pF 97.8 µs (10.2 kHz) 98.4 µs (10.2 kHz)C 3 = 56 pF C 4 = 47 pF 41.4 µs (24.2 kHz) 43.0 µs (23.3 kHz)C 3 = 56 pF C 4 = 220 pF 89.6 µs (11.2 kHz) 92.0 µs (10.9 kHz)C 3 = 100 pF C 4 = 47 pF 31.0 µs (32.3 kHz) 33.0 µs (30.3 kHz)C 3 = 100 pF C 4 = 220 pF 67.0 µs (14.9 kHz) 70.0 µs (14.3 kHz)Output of A 2 Amplifier (V 2 )Output of A 3 Amplifier (V 3 )FIGURE 9:Ratio Oscillator Test Results (C 3 = 100 pF, C 4 = 47 pF).© 2003 Microchip Technology Inc. DS00866A-page 11

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

AN866NOTES:© 2003 Microchip Technology Inc. DS00866A-page 14

Note the following details of the code protection feature on Microchip devices:• Microchip products meet the specification contained in their particular Microchip Data Sheet.• Microchip believes that its family of products is one of the most secure families of its kind on the market today, when used in theintended manner and under normal conditions.• There are dishonest and possibly illegal methods used to breach the code protection feature. All of these methods, to ourknowledge, require using the Microchip products in a manner outside the operating specifications contained in Microchip's DataSheets. Most likely, the person doing so is engaged in theft of intellectual property.• Microchip is willing to work with the customer who is concerned about the integrity of their code.• Neither Microchip nor any other semiconductor manufacturer can guarantee the security of their code. Code protection does notmean that we are guaranteeing the product as “unbreakable.”Code protection is constantly evolving. We at Microchip are committed to continuously improving the code protection features of ourproducts. Attempts to break microchip’s code protection feature may be a violation of the Digital Millennium Copyright Act. If such actsallow unauthorized access to your software or other copyrighted work, you may have a right to sue for relief under that Act.Information contained in this publication regarding deviceapplications and the like is intended through suggestion onlyand may be superseded by updates. It is your responsibility toensure that your application meets with your specifications.No representation or warranty is given and no liability isassumed by Microchip Technology Incorporated with respectto the accuracy or use of such information, or infringement ofpatents or other intellectual property rights arising from suchuse or otherwise. Use of Microchip’s products as criticalcomponents in life support systems is not authorized exceptwith express written approval by Microchip. No licenses areconveyed, implicitly or otherwise, under any intellectualproperty rights.TrademarksThe Microchip name and logo, the Microchip logo, dsPIC,KEELOQ, MPLAB, PIC, PICmicro, PICSTART, PRO MATE andPowerSmart are registered trademarks of MicrochipTechnology Incorporated in the U.S.A. and other countries.FilterLab, microID, MXDEV, MXLAB, PICMASTER, SEEVALand The Embedded Control Solutions Company areregistered trademarks of Microchip Technology Incorporatedin the U.S.A.Accuron, Application Maestro, dsPICDEM, dsPICDEM.net,ECONOMONITOR, FanSense, FlexROM, fuzzyLAB, In-Circuit Serial Programming, ICSP, ICEPIC, microPort,Migratable Memory, MPASM, MPLIB, MPLINK, MPSIM,PICC, PICkit, PICDEM, PICDEM.net, PowerCal, PowerInfo,PowerMate, PowerTool, rfLAB, rfPIC, Select Mode,SmartSensor, SmartShunt, SmartTel and Total Endurance aretrademarks of Microchip Technology Incorporated in theU.S.A. and other countries.Serialized Quick Turn Programming (SQTP) is a service markof Microchip Technology Incorporated in the U.S.A.All other trademarks mentioned herein are property of theirrespective companies.© 2003, Microchip Technology Incorporated, Printed in theU.S.A., All Rights Reserved.Printed on recycled paper.Microchip received QS-9000 quality systemcertification for its worldwide headquarters,design and wafer fabrication facilities inChandler and Tempe, Arizona in July 1999and Mountain View, California in March 2002.The Company’s quality system processes andprocedures are QS-9000 compliant for itsPICmicro ® 8-bit MCUs, KEELOQ ® code hoppingdevices, Serial EEPROMs, microperipherals,non-volatile memory and analog products. Inaddition, Microchip’s quality system for thedesign and manufacture of developmentsystems is ISO 9001 certified.DS00866A-page 15© 2003 Microchip Technology Inc.

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