2. Lecture 6
• Lecture topics
1. First order transfer functions
2. Analogous first order elements
a. Fluidic
b. Electrical
c. Mechanical
3. Transfer functions for first order
element
• What we have already learnt
• The transfer function G(s) of an element is defined as the ratio of
the Laplace transform of the output to the Laplace transform of
the input, provided the initial conditions are zero.
𝐺 𝑠 =
∆𝑇 𝑠
∆𝑇𝐹 𝑠
=
1
1 + 𝜏𝑠
1
1 + 𝜏𝑠
∆𝑇𝐹 𝑠 ∆𝑇 𝑠
First order transfer function block
4. Transfer functions for first order
element
• For our example of temperature sensor, transfer function only relate
changes in sensor temperature to the changes in its environment
temperature.
• The overall relationship between changes in sensor output signal O
and environment temperature will be steady state sensitivity times the
transfer function
∆𝑂 𝑠
∆𝑇𝐹 𝑠
=
∆𝑂
∆𝑇
∙
∆𝑇 𝑠
∆𝑇𝐹 𝑠
Steady state sensitivity
5. Transfer functions for first order
element
• The steady state sensitivity for an ideal sensor is equal to the slope K
of ideal straight line.
• If the temperature sensor is non-linear and subject to small
temperature fluctuations, then
Δ𝑂
Δ𝑇
=
𝑑𝑂
𝑑𝑇
• The derivative being evaluated at the steady-state temperature T(0−)
around which the fluctuations are taking place.
6. Transfer functions for first order
element
• Example:
• For a copper–constantan thermocouple measuring small
fluctuations in temperature around 100 °C, ΔE/ΔT is found by
evaluating dE/dT at 100 °C to give ΔE/ΔT = 35 μV °C−1.
• If the time constant of the thermocouple is 10s the overall
dynamic relationship between changes in e.m.f. and fluid
temperature is:
∆𝑂 𝑠
∆𝑇𝐹 𝑠
=
∆𝐸 𝑠
∆𝑇𝐹 𝑠
= 35 ×
1
1 + 10𝑠
8. Analogous first order elements
• Fluidic element cont.
• Volume flow rate can be given by,
𝑃𝐼𝑁 = ℎ𝐼𝑁 ∙ 𝜌 ∙ 𝑔 𝑃 = ℎ ∙ 𝜌 ∙ 𝑔
Where, and
𝑄 =
𝜌 ∙ 𝑔
𝑅𝐹
× ℎ𝐼𝑁 − ℎ ...(1.3)
𝑄 =
1
𝑅𝐹
× 𝑃𝐼𝑁 − 𝑃 ...(1.1)
Therefore, 𝑄 =
1
𝑅𝐹
× ℎ𝐼𝑁 ∙ 𝜌 ∙ 𝑔 − ℎ ∙ 𝜌 ∙ 𝑔 ...(1.2)
9. Analogous first order elements
• Fluidic element cont.
Again, Q can be written by
Now using equation (1.3) and (1.4) we can write,
Therefore,
𝐴𝐹 ∙ 𝑅𝐹
𝜌 ∙ 𝑔
∙
𝑑ℎ
𝑑𝑡
+ 𝒉 = ℎ𝐼𝑁 ...(1.6)
𝑄 = 𝐴𝐹
𝑑ℎ
𝑑𝑡
...(1.4)
𝑄 = 𝐴𝐹
𝑑ℎ
𝑑𝑡
=
𝜌 ∙ 𝑔
𝑅𝐹
× ℎ𝐼𝑁 − ℎ ...(1.5)
10. Analogous first order elements
• Fluidic element cont.
The resulting first order differential equation for the system will be,
The time constant for fluidic element can be given by
𝝉𝑭 =
𝐴𝐹 ∙ 𝑅𝐹
𝜌 ∙ 𝑔
...(1.9)
𝐴𝐹 ∙ 𝑅𝐹
𝜌 ∙ 𝑔
∙
𝑑ℎ
𝑑𝑡
+ 𝒉 = ℎ𝐼𝑁 ...(1.7)
Or, 𝝉𝑭 ∙
𝑑ℎ
𝑑𝑡
+ 𝒉 = ℎ𝐼𝑁 ...(1.8)
11. Transfer functions for first order
element
• A simple problem,
• Two overhead water tanks of 1 m diameter each are connected at
the bottom with a cylindrical cross section pipe of 1 cm diameter
and 10 cm length. There is a valve connected in the pipe to
control the flow. One of the tank is full with water level of 1.5 m
above the connecting pipe centreline. If the valve is opened, what
will be the water level in the second tank after 5 sec,10 sec and
15 sec. Dynamic viscosity of water at 25 C = 0.89 mPa-S
12. Transfer functions for first order
element
• Solution,
• Algorithm:
• Step 1: Determine the governing differential equation.
• Step 2: Determine time constant
• Step 3: Solve the differential equation at t = 5 sec, 10 sec and 15 sec
• Step 1:
From equation (1.6) we can write,
Where, RF is the fluidic resistance =
8𝜇𝐿
𝜋𝑅4
=
8×0.89×10−3×0.1
3.14×
0.01
2
4 = 362802.55
𝐴𝐹 ∙ 𝑅𝐹
𝜌 ∙ 𝑔
∙
𝑑ℎ
𝑑𝑡
+ 𝒉 = ℎ𝐼𝑁 ...(1.6)
13. Transfer functions for first order
element
• Solution,
• Step 2:
Time constant can be given by
• Step 3:
Solve the following differential equation
𝐴𝐹 ∙ 𝑅𝐹
𝜌 ∙ 𝑔
=
𝜋
4
∙ 12
× 362802.55
1000 × 9.81
= 29.03 𝑠𝑒𝑐
29.03 ∙
𝑑ℎ
𝑑𝑡
+ 𝒉 = 1.5
Solution
T, sec h, m
5
10
15
h = 3/2 - (3*exp(-(100t/2903))/2
15. Analogous first order elements
• Electrical element cont.
• Voltage difference across the resistor is,
𝑉𝐼𝑁 − 𝑉 = 𝑖 ∙ 𝑅 ...(2.1)
Charge stored = 𝒒 = 𝑪 ∙ 𝑽 ...(2.2)
Current = 𝒊 =
𝒅𝒒
𝒅𝒕
= 𝑪 ∙
𝒅𝑽
𝒅𝒕
...(2.3)
16. Analogous first order elements
• Electrical element cont.
Now we can rewrite equation 1 as,
The time constant for electrical element can be given by
𝑖 ∙ 𝑅 + 𝑉 = 𝑉𝐼𝑁 ...(2.4)
Or, 𝑹 ∙ 𝑪 ∙
𝑑𝑉
𝑑𝑡
+ 𝑽 = 𝑽𝐼𝑁 ...(2.5)
Or, 𝝉𝑬 ∙
𝑑𝑉
𝑑𝑡
+ 𝑽 = 𝑽𝐼𝑁 ...(2.6)
𝝉𝑬 = 𝑹 ∙ 𝑪 ...(2.7)
18. Analogous first order elements
• Mechanical element cont.
• Displacement of the system,
𝑥 =
𝐹
𝑘
...(3.1)
Or,
𝒅𝒙
𝒅𝒕
=
𝒅
𝒅𝒕
𝐹
𝑘
=
1
𝑘
𝑑𝐹
𝑑𝑡
...(3.2)
Again, 𝑭𝑰𝑵 − 𝑭 = 𝝀 ∙
𝒅𝒙
𝒅𝒕
...(3.3)
19. Analogous first order elements
• Mechanical element cont.
Using equation 3.2 and 3.3,
The time constant for mechanical element can be given by
𝑭𝑰𝑵 − 𝑭 = 𝝀 ∙
1
𝑘
𝑑𝐹
𝑑𝑡
...(3.4)
Or,
𝜆
𝑘
𝑑𝐹
𝑑𝑡
+ 𝑭 = 𝑭𝐼𝑁 ...(3.5)
𝝉𝑴 =
𝜆
𝑘
...(3.6)
