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 ).

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.

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.

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.

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.

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.

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 byand 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 4^{th}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

2^{nd}; and 3^{rd}order oscillators obtained using

Method II . The application of Method II is shown for the

3^{rd}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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