Question

What is the order of thermocouple system?

Answer 1

Measurement Lab
1
First-Order System: Transient Response of a
Thermocouple to a Step Temperature Change
Last Modified 9/6/06
Many mechanical, thermal, and electrical systems can be modeled using first-order
differential equations. In this experiment we will study the behavior of a first-order
system, the transient response of a thermocouple (TC).
First-Order Systems
A one-degree-of-freedom first-order system is governed by the first-order ordinary
differential equation
a1
dy
dt
+ a0y = F(t) , (1)
where y(t) is the response of the system (the output) to some forcing function F(t) (the
input). Eq. (1) may be rewritten as
τ dy
dt
+ y = kF(t), (2)
where τ=a1/a0 has the dimension of time and is the time constant for the system and k =
1/a0 is the gain.
Response of a First-order System to a Step Input
Consider a first-order system subjected to a constant force applied instantaneously at the
initial time t = 0
F(t)= 0, t < 0
A, t > 0


 . (3)
The initial condition is y(0) = 0. The solution to Eq. (2) with the step input Eq. (3) is
then
y(t) = kA 1− e−t τ ( ) . (4)
The response approaches the final value y∞= kA exponentially. By using the boundary
conditions equation (4) then may be rewritten as
() ()
kA
y t kA
e
y y
y t y t
−
− = = −
− −
∞
∞ τ
0
. (5)
The rate at which the response approaches the final value is determined by the time
constant. When t = τ, y has reached 63.2% of its final value as illustrated in Figure 1.
When t =5τ, y has reached 99.3% of its final value.