AND8054 Datasheet, PDF (7/28 Pages) ON Semiconductor – Designing RC Oscillator Circuits with Low Voltage Operational Amplifiers and Comparators for Precision Sensor Applications

AND8054/D
OSCILLATOR DESIGN PROCEDURE
Listed below is a procedure to design RC active
oscillators:
Step 1: Find LG and ∆s
Step 2: Solve ∆s = 0 for s = jω_o using methods I, II or III
Method
I:
Solve
remainder
of
N(s)
s2 ) w_o2
=0
Method II: Solve N(jω_ο)_REAL = N(jω_o)_IMAG = 0
Method III: Routh’s stability test
Step 3: Form sub–circuit design equations
Step 4: Verify LG ≥ 1
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,
shown in Appendix I, provides a method of determining the
transfer equation from a signal flow diagram. Mason’s
Theorem, listed below, 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.
ǒ Ǔ ^T(s)
+
1
A
* LG
+
A
D(s)
+
A
N(s)
D(s)
Step 2: Solve ∆s
The second step in the procedure determines the zeroes of
N(s). Several different control theory techniques such as the
Bode or Nyquist stability tests can be used, or one of the
three methods that are listed below. Examples of the
application of the three different methods listed below will
be provided.
Method I: ^N(s)
s² ) w^2o
An equation is established for the oscillation frequency ω_o
when
N(s)
is
divided
by
s²
+
ω_ο²
(i.e.
N(s) ₎
s² ) w^2o
and
the
remainder is solved to be equal to zero. 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
described in [12] and is based on factoring the characteristic
equation to have a s² + ω_o2 term. For example, when a third
order system can be factored in the form (s + β)(s² + ω_o2),
the pole locations are at s = ± jω_o and s = –β. Method I will
be demonstrated by analyzing 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 + ω_o2, this method can be used for higher order
circuits such as the 4^th order ratio oscillator.
Method II: Solve N(jw_o)_REAL = N(jw_o)_IMAGINARY = 0
The oscillation equation sometimes can be determined
directly from the characteristic equation by substituting
s = jω_o into N(s) and arranging N(jω_o) 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(jω_o)_REAL and
N(jω_o)_IMAGINARY. If N(s) = jω_o = 0, the poles of the
characteristic equation will be on the imaginary axis at ±jω_ο
with an oscillation frequency of ω_o. A summary of the
oscillation equations for 2^nd and 3^rd order oscillators
obtained using Method II [13] is shown in Appendix II. The
application of Method II is shown for the 3^rd order absolute
oscillator with the inverter capacitor C₄.
Method III: Routh Stability Test
The Routh stability criterion [12] 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, shown
in Appendix III, 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 = jω_o for s, and the summation of the row is set to zero. If
the row equation produces a nontrivial solution for ω_o, the
procedure is complete and the frequency of oscillation is
equal to ω_o. If the row equation does not yield an equation
that can be solved for ω_o, 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 ω_o. Method III
will be demonstrated by analyzing the ratio oscillator.
Step 3: Sub–circuit Design Equations
The third step in the design procedure is to form the design
equations for the sub–circuits formed by each operational
amplifier. The oscillation equation can be simplified by
selecting the R’s and C’s with the assumptions shown in the
“Design Equation” section. The amplifier gain and
pole/zero locations for the absolute and ratio oscillator are
also shown.
Step 4: Verify LG ≥ 1
The final step in the procedure verifies that the loop gain
is equal to or greater than one after the R’s and C’s
component values have been chosen. 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.
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