AND8054 Datasheet, PDF (26/28 Pages) ON Semiconductor – Designing RC Oscillator Circuits with Low Voltage Operational Amplifiers and Comparators for Precision Sensor Applications

AND8054/D
Appendix II: Method II:
Solve N(jω_o)_REAL = N(jω_o)_IMAGINARY = 0
The oscillation equation sometimes can be determined
directly from the characteristic equation by substituting
s = jω_ο into ∆s and arranging the N(jω_ο) into its real and
imaginary parts. However, this method is usually not
feasible for circuits which are 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 the
characteristic equation N(s) = jω_ο = 0, the poles of the
characteristic equation will be on the imaginary axis at ±jω_ο
with an oscillation frequency of ω_ο. The Method II
procedure is shown below for second and third order
oscillators [13].
Second–Order Circuits
ǒ Ǔ N2(s) + a0s2 ) a1s ) a2 + a0
s2
)
a1
a0
s
)
a2
a0
Let s = jω_ο be the frequency at which N₂(s) = 0. The
condition for oscillation is meet when the a₁ term is set to
zero, and the s–term is removed. The frequency of
oscillation is found from:
Ǹ w_o +
a2
a0
Third Order Circuits
N3(s) + a0s3 ) a1s2 ) a2s ) a3
Let s = jω_ο be the frequency at which N₃(s) = 0, and arrange
the equation into its real and imaginary parts:
N3(jw_o) + (–a1w^2o ) a3) ) jw_o(–a0w^2o ) a2) + 0
Thus, the real and imaginary parts equal zero when:
–a1w^2o ) a3 + 0 and –a0w^2o ) a2 + 0
Solving the above equations for w_o2 gives:
w
2
o
+
a3
a1
+
a2
a0
Summary of Method II Equations
Oscillator
Oscillation
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