A process temperature is known to fluctuate with a frequency of 0.1 Hz. The engineer needs to monitor this process using a first-order thermocouple and wishes to have no more than 2% amplitude reduction.

a. What time constant of the thermocouple is needed?

t ≤ 0.31s

Explanation:

Given

Frequency = 0.1Hz

Dynamic Error:

σ = 1 - M (σ)

From the question

σ = 1 - M (σ) ≤ 2%

1 - M (σ) ≤ 0.02

The above corresponds to

M (σ) = 1 / (1 + (wt)) ² ≥ 100% - 0.02

M (σ) = 1 / (1 + (wt)) ² ≥ 1 - 0.02

M (σ) = 1 / (1 + (wt)) ² ≥ 0.98

Since frequency = 0.1Hz

w = 2πf =

w = 0.628571428571428

w = 0.628rads

So, M (0.628) = 1 / (1 + (0.628t) ²) ≥ 0.98

Solving 1 / (1 + (0.628t) ²) ≥ 0.98 we get that

t ≤ 0.31s