Process control tutorial memo

1. Process Control II (PSC260_1S)
Tutorial 2: Memo
Tshemese-Mvandaba

2. Tutorial 2 Solution
Tshemese-Mvandaba 2
2.1 Derive the transfer function that links the input and the output.
Hint: Mathematical Model the Resistive – Inductive (RL) network
2.1.1 Model Figure 1 in the time
domain
2.1.2 Model Figure 1 in Laplace (s-
domain)
2.1.3 Derive the Transfer Function
in the s-domain

3. Tutorial 2 Solution
Tshemese-Mvandaba 3
2.1 Derive the transfer function that links the input and the output.
Hint: Mathematical Model the Resistive – Inductive (RL) network
2.1.1 Model Figure 1 in the time domain
2.1.2 Model Figure 1 in Laplace (s-domain)
2.1.3 Derive the Transfer Function in the s-domain
Solution:
The input voltage is the sum of the voltage across the resistor and inductor:

4. Tutorial 2 Solution
Tshemese-Mvandaba 4
The current through the inductor is given by:
Solution:
Substituting for the current gives the Mathematical model in the TIME DOMAIN:

5. Tutorial 2 Solution
Tshemese-Mvandaba 5
LAPLACE (s-domain):
Taking out the common factor:
The TRANSFER FUNCTION in the s-domain:

6. Tutorial 2 Solution
Tshemese-Mvandaba 6
The INVERSE LAPLACE:
From the LAPLACE TRANSFORMS TABLE the OUTPUT VOLTAGE in the TIME DOMAIN:

7. Tutorial 2 Solution 2
Tshemese-Mvandaba 7
2.2 Derive the transfer function that links the input and the output.
Hint: Mathematical Model the Resistive – Inductive (RL) network
2.1.1 Model Figure 2 in the time domain
2.1.2 Model Figure 2 in Laplace (s-domain)
2.1.3 Derive the Transfer Function in the s-domain
Solution:
The output voltage is taken across the inductor.
+
Vin
R
L
C
V
+
Vout
The current through the inductor is given by:

8. Tutorial 2 Solution
Tshemese-Mvandaba 8
Therefore substituting in Eq2 we have:
+
Vin
R
L
C
V
+
Vout

9. Tutorial 2 Solution
Tshemese-Mvandaba 9
Solution
+
Vin
R
L
C
V
+
Vout

10. Tutorial 2 Solution
Tshemese-Mvandaba 10
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Question 2
2.1 Find the transfer function that corresponding to the differential equation shown below:
LAPLACE (s-domain) the above:

11. Tutorial 2 Solution
Tshemese-Mvandaba 11
Question 2
2.1 Find the transfer function that corresponding to the differential equation shown below:
Find the poles & zero’s to evaluate the stability: