AND8054 Datasheet, PDF (27/28 Pages) ON Semiconductor – Designing RC Oscillator Circuits with Low Voltage Operational Amplifiers and Comparators for Precision Sensor Applications

AND8054/D
Appendix III: Routh’s Stability Test
Routh’s Stability Test [12] can be used to test the
characteristic equation to determine whether any of roots lie
on the imaginary axis. Routh’s 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. This technique arranges the numerator of
the characteristic equation (i.e. denominator of the transfer
equation) into the array listed below.
ǒ Ǔ ^T(s)
+
1
A
* LG
+
A
D(s)
+
A
N(Ds)
D(Ds)
N(Ds) + a0sn ) a1sn ^{* 1} ) a2sn ^{* 2} ) a3sn ^{* 3}
) ... ) an _{* 1s} ) an
sn
sn–1
sn–2
sn–3
.
.
.
s0
a0
a2
a4
AAA
a1
a3
a5
AAA
b1
b2
b3
AAA
c1
c2
c3
AAA
.
.
.
.
.
.
.
.
f1
an
an–1
bn–2
cn–3
where the coefficients b₁, b₂, b₃, etc., are evaluated as
follows:
b1
+
a1a2
*
a1
a0a3
b2
+
a1a4
*
a1
a0a5
b3
+
a1a6
*
a1
a0a7
The evaluation of the b’s is continued until the remaining
terms are equal to zero. The same pattern of cross
multiplying the coefficients of the two previous rows is
followed in evaluating the c’s, d’s, etc...
c1
+
b1a3
*
b1
b2a1
c2
+
b1a5
*
b1
b3a1
This process is continued until the n–th row has been
completed. The Routh stability criterion states:
1. A necessary and sufficient condition for stability is
that the first column of the array does not contain
sign changes.
2. The number of sign changes in the entries of the
first column of the array is equal to the number of
roots in the right half s–plane.
3. If the first element in a row is zero, it is replaced by
ε, and the sign changes when ε → 0 are counted
after completing the array.
4. The poles are located in the right half plane or on
the imaginary axis if all the elements in a row are
zero.
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