# Designing Operational Amplifier Oscillator Circuits for Precision Resistive and Capacitive Sensor Applications

The design of RC operational amplifier oscillators requires the
use of a formal design procedure. In general, the design equations
for these oscillators are not available; therefore, it is necessary
to derive the design equations symbolically to select the RC
components and to determine the influence of each component on the
frequency of oscillation. This article describes a design procedure
for two different variable oscillator circuits that you can use in
precision capacitive-sensor applications. These two oscillators
have an output frequency proportional to the product of two
capacitors (C1 *C2 ) and the ratio of two capacitors
(C1 /C2 ).

Transducer System

Many different circuits are available to accurately measure
capacitive sensors. The design choices include switched capacitor
circuits, analog multivibrators, AC bridges, digital logic ICs, and
RC operational amplifier oscillators. The requirements for a
precision sensor circuit include high accuracy, reliable start-up,
good long-term stability, low sensitivity to stray capacitance, and
a minimal component count. State-variable RC operational amplifier
oscillators meet all of these requirements and form the basis for
this study.

A block diagram of a capacitive sensor system is shown in Figure 1. Counting the number of clock pulses in a time window
formed by the square-wave oscillator output of a comparator circuit
gives the oscillation frequency. A digital-logic counter circuit or
the Time Processing Unit (TPU) channel of a microprocessor
implements the counter circuit. Using a curve-fitting routine with
calibration data from the sensor over the operating range can
correct for temperature effects. You can use an analog IC sensor to
monitor the sensor temperature or, for very precise applications,
use a second oscillator with platinum resistive-temperature-device
(RTD) sensors.

It is often important for the sensor system to compute the ratio
of two capacitors. Calculating the ratio of the capacitors reduces
the transducer's sensitivity to dielectric errors from factors such
as temperature. In other cases, such as in airdata quartz P pressure sensors, the desired measurement is
equal to the ratio of two capacitances (CMEAS / CREF ). Furthermore, the dual sensors
are typically designed to double the CMEAS in capacitance, while CREF may vary less than one percent.
Thus, the transducer's accuracy is increased if the electronic
circuitry can directly detect the CMEAS / CREF ratio.

Sensor Applications

RC operational amplifier oscillators can accurately detect both
resistive and capacitive sensors; however, this paper will only
analyze capacitive applications. The three basic configurations of
capacitive sensors and their attributes are shown in Figure 2. The absolute and dual capacitive sensors will be used
with the absolute and ratio state-variable oscillator circuits.
Differential capacitive sensors typically are not used in precision
applications; therefore, they will not be analyzed in this
paper.

Absolute State Variable Oscillator

Use the absolute state-variable oscillator when the measurement
is proportional to either one or two capacitors (in other words,
frequency is proportional to C1 *C2 ). The block diagram and schematic
of the absolute circuit are shown in Figures 3 and 4. This circuit consists of two integrators and an inverter
circuit. Each integrator has a phase shift of 90° and the
inverter adds an additional 180° phase shift; thus, a total
phase shift of 360° is fed into the input of the first
integrator, which produces an oscillator. The first integrator
stage consists of amplifier A1 ,
resistor R1 and sensor
capacitance C1 . The second
integrator consists of amplifier A2 , resistor R2 and sensor capacitance C2 . Resistor-capacitor combinations
R1 and C1 , and R2 and C2 , set the gain of each integrator
stage, in addition to setting the oscillation frequency. The
inverter stage consists of amplifier A3 , resistors R3 and R4 and capacitor C4 . Capacitor C4 is not essential for normal
operation; however, it ensures oscillator startup under extreme
ambient temperature conditions.

Ratio State Variable Oscillator

Use the ratio state-variable oscillator for dual capacitive
sensors when the oscillation frequency is proportional to the ratio
of sensor capacitances C3 and
C4 (in other words, frequency
is proportional to C3 / C4 ). The block diagram and schematic
of the ratio circuit are shown in Figures 3 and 5. This circuit consists of two integrators and a
differentiator circuit. The integrators formed by amplifier A1 and A2 are identical to the integrators
used in the absolute circuit. Amplifier A3 , resistors R3 , R4 and R5 , and the sensor capacitors C3 and C4 form the differentiator stage which
provides a 180° phase shift. The values of resistors R3 , R4 and R5 are selected to set the break
frequencies of the differentiator stage, so that the gain of the
stage is equal to -C3 /C4 . Resistor R5 provides a DC current path through
capacitor C3 in order to
initiate oscillation at power-up. Because R4 and R5 are relatively large (M ), they may be replaced with a three resistor
“Tee” network in order to use readily available resistors.

Oscillator Theory

An oscillator is a positive-feedback control system, which does
not have an external input signal, but will generate an output
signal if certain conditions are met. In practice, a small input is
applied to the feedback system from factors such as noise pick-up,
or power-supply transients, initiating the feedback process and
resulting in sustained oscillations. A block diagram of an
oscillator is shown in Figure 6.

The poles of the denominator of the transfer equation, or
equivalently the zeroes of the characteristic equation, determine
the time-domain behavior of the system. If T(s) has all of its
poles located within the left-plane, the system is stable because
the corresponding terms are all exponentially decaying. In
contrast, if T(s) has one pole that lies within the right half
plane, the system is unstable because the corresponding term
exponentially increases in amplitude. An oscillator is on the
borderline between a stable and an unstable system and is formed
when a pair of poles is on the imaginary axis.

If the magnitude of the loop gain is greater than one and the
phase is zero, the amplitude of oscillation will increase
exponentially until a factor in the system such as the supply
voltage restricts the growth. In contrast, if the magnitude of the
loop gain is less than one and the phase is zero, the amplitude of
oscillation will exponentially decrease to zero.

Design Procedure

Listed below is a procedure to design RC
active oscillators: Step 1: Find LG and s
The oscillation frequency is determined by finding the poles of the
denominator of the transfer equation T(s), or equivalently the
zeroes of the numerator N(s) of the characteristic equation s. Mason's Reduction Theorem provides a method
of determining the transfer equation from a signal flow diagram.
Mason's Theorem shows that it is not necessary to obtain the
complete T(s) equation. The oscillation frequency can be determined
by analyzing the numerator N(s) of the s. s is found by obtaining the open loop gain (LG)
by breaking the feedback loop and applying a test voltage to the
circuit. Signal flow diagrams of the absolute oscillator and step 1
of the procedure are shown in Figures 7, 8 and 9. The signal-flow diagrams of the ratio oscillator and step 1
of the procedure are shown in Figures 10 and 11.

Step 2: Solve s
The second step in the procedure determines the zeroes of N(s). You
can use one of several different control theory techniques, such as
Bode or Nyquist stability tests, or use one of the following three
methods. Examples of the application of the three different methods
listed below will be provided.

Method I: When N(s) is divided by and the remainder is solved to be equal to
zero, an equation is established for the oscillation frequency . Method I is easy to implement for second- and
third-order systems, but with higher-order systems the algebra can
be tedious. Method I is based on factoring the characteristic equation
to have a s² + ² term. For example, when a third order
system can be factored in the form (s + )(s² + ²), the pole locations are at s = ±
j and s = – . Figure 12 demonstrates Method I using the absolute oscillator
without the inverter capacitor C4 . Although the analysis of this
second-order system is trivial because N(s) is already in the form
of s² + ², this method can be used for
higher-order circuits such as the 4th order ratio
oscillator.

Method II: Solve N(j )REAL =
N(j )IMAGINARY
= 0
The oscillation equation sometimes can be determined directly from
the characteristic equation by substituting s = j into N(s) and arranging the N(j ) into its real and imaginary parts. This
method is usually not feasible for fifth-order and higher
oscillators. This procedure is essentially a subset of the Routh
test, because the first two rows of the Routh array will correspond
to N(j )REAL and
N(j )IMAGINARY .
If N(s) = j = 0, the poles of the characteristic equation
will be on the imaginary axis at ±j with an oscillation frequency of . Figure 13 provides a summary of the oscillation equations for
2nd ; and 3rd order oscillators obtained using
Method II . The application of Method II is shown for the
3rd order absolute oscillator in Figure 14.

Method III: Routh Stability Test
The Routh stability criterion provides a method that determines the zeroes of
the characteristic equation directly from the characteristic
polynomial coefficients, without the necessity of factoring the
equation. The Routh test is the preferred method to use for
fourth-order and higher order oscillators. The Routh test consists
of forming a coefficient array. Next, the procedure substitutes s =
j for s, and the summation of the row is set to
zero. If the row equation produces a nontrivial solution for , the procedure is complete and the frequency
of oscillation is equal to . If the row equation does not yield an
equation that can be solved for , the procedure continues with the next row in
the Routh array. Usually, it is necessary only to complete the
first two or three rows of the Routh array to produce an equation
that can be solved for . The application of Method III is shown for
the ratio oscillator in Figure 15.

Step 3: Subcircuit Design Equations
The third step in the design procedure is to form the design
equations for the subcircuits formed by each operational amplifier.
The oscillation equation can be simplified by selecting the Rs and
Cs with the assumptions show in Figures 16 and 18. The amplifier gain and pole/zero locations for the absolute
and ratio oscillator are show in Figures 17 and 19. A Bode plot of the gain response of the ratio circuit's
differentiator amplifier is shown in Figure 20.

Step 4: Verify LG > 1
The final step in the procedure verifies that the loop gain is
equal to or greater than one after choosing the R and C component
values. This step is required to verify that the location and
clamping voltage of the limit circuit will not result in a LG < 1, or that the operational amplifiers will reach their saturation voltage. The limit circuit can be located across any of the three amplifiers as long as the LG > 1. Step 4 is demonstrated
by analyzing the limit circuit shown in Figure 21 for the ratio oscillator. This limit circuit is
suitable when the operational amplifier use dual power supplies
greater than 2V. Additional limit circuits for single power supply
applications and low voltage applications are given in , which also has reference designs of the absolute
and ratio oscillator circuits using ON Semiconductor's sub-1 volt
operational amplifiers and comparators.

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