Equation 6.7a (Bode amplitude ratio) shows that the attenuation of a measurement oscillation
decreases as the measurement time constant ( τ f ) and frequency of the oscillation ( wo )
decreases. For a measurement time constant greater than the period, Equation 6.7a simplifies to
Equation 6.7c via the use of Equation 6.7b to get from the frequency domain to the time domain
for better visualization on trend charts. Equation 6.7c shows the measurement oscillation
amplitude is the process amplitude attenuated by a factor that is proportional to the oscillation
period and inversely proportional to the time constant.
Am = A p ∗
1
1 + (τ m ∗ wo ) 2
(6.7a)
Converting the oscillation from the frequency domain to the time domain:
ωo =
2 ∗π
to
(6.7b)
For a measurement time constant greater than the oscillation period ( t m > t o ):
Am = Ap ∗
to
2 ∗ p ∗t m
(6.7c)
Where:
A p = process oscillation amplitude (sec)
A m = measurement oscillation amplitude (sec)
to = period of oscillation (sec)
τ m = measurement time constant (sec)
wo = frequency of oscillation (radians/sec)
The above equations can be used to show the beneficial effect of a primary process time constant
in attenuating disturbance oscillations at the input of a continuous process. For a well-mixed
volume, the process time constant can be approximated as the residence time (volume/flow).