AND8054 Datasheet, PDF (23/28 Pages) ON Semiconductor – Designing RC Oscillator Circuits with Low Voltage Operational Amplifiers and Comparators for Precision Sensor Applications

AND8054/D
Appendix I: Mason’s Reduction Theorem
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 theorem (12.) states
that the transfer function from input X to output Y is
T(s)
+
Y
X
+
S
i
PiDsi
Ds
where the terms are defined as:
Pi is the direct transmittance or path form input X
to output Y
Dsi is the system determinant.
Dsi + 1 if Pi touches all of the loops
Ds + 1 * SLj ) S^ȀLkLl * S^ȀLmLnLn )AAA
SLj is the sum of all loops (i.e. loop gains)
SLkLl is the sum of products of pairs of
non–touching loops
SLmLnLo is the sum of products of gains of
non–touching loops taken three at a
time
Mason’s Reduction Theorem should be used to determine
the transfer equation if the oscillator has more than one
feedback loop, such as the case for the circuit shown in
Figure 28. Obtaining T(s) also provides the additional
information required to complete a Bode plot of the
oscillator. In contrast, Step 1 of the design procedure only
provides the denominator of T(s) and will not provide the
numerator of the transfer equation. Mason’s equation can
be rewritten in the form listed below:
ǒ Ǔ ^T(s)
+
1
A
* LG
+
A
D(s)
+
A
N(s)
D(s)
The absolute and ratio oscillators only have a single
feedback loop, therefore, the calculation of
T(s) + ^V3
V11
is relatively easy because the path P₁ (equivalent to the
amplifier gain A) is defined as the voltage gain from node
V₁₁ to V₃ and will be equal to the loop gain LG₁. In order to
calculate the transfer equation, the intermediate voltage
node of V₁₁ is created by adding a “small” resistor R₁₁ in
series with resistor R₁ to the absolute and ratio circuits as
shown in Figures 29 and 30. Adding R₁₁ and V₁₁ is not
mathematically necessary; however, it greatly simplifies
the algebra in the transfer equations. Note, the numerator of
the transfer equation depends on the definition of the input
and outputs; however, the denominator (i.e. the oscillation
equation) is independent of the definition of T(s). If R₁ >>
R₁₁, then the gain of amplifier A₁ is a function only of R₁
and C₁.
ǒ Ǔ A1 +
–
s(R1
1
) R11)C1
^
–
1
sR1C1
Listed below are the calculation of T(s) for the absolute
oscillator with and without capacitor C₄ and the ratio
oscillator.
C1
R1
A1
–
V1 R2
+
R5
C2
A2
–
+
R3
V2
R4
A3
–
+
V3
Figure 28. Mason’s Theorem Provides a Method to Determine the Transfer Equation T(s) of an
Oscillator when there are Multiple Feedback Loops, as with the Modified Absolute Circuit
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