Question

Level (\(h\)) in a steam boiler is controlled by manipulating the flow rate (\(F\)) of the make-up water using a proportional (P) controller. The transfer function is \[ \frac{h(s)}{F(s)} = \frac{0.25(1-s)}{s(2s+1)}. \] The measurement and valve transfer functions are both 1. A process engineer wants the closed-loop response to give decaying oscillations under servo mode. Which one of the following is the CORRECT value of the controller gain?

• A. 0.25
• B. 2
• C. 4
• D. 6

Hint:

For stability, check sign of coefficients and discriminant.
Decaying oscillations require underdamped roots (complex with negative real part).
Servo mode means reference tracking response.

Answer 1

The Correct Option is B

Step 1: Open-loop transfer function with proportional controller gain \(K_c\): \[ G(s) = K_c \cdot \frac{0.25(1-s)}{s(2s+1)}. \]

Step 2: Closed-loop characteristic equation: \[ 1+G(s)=0 \;\;\Rightarrow\;\; 1+ \frac{0.25K_c(1-s)}{s(2s+1)}=0. \] Multiply through: \[ s(2s+1) + 0.25K_c(1-s) = 0. \]

Step 3: Simplify: \[ 2s^2 + s + 0.25K_c -0.25K_c s = 0, \] \[ 2s^2 + \big(1-0.25K_c\big)s + 0.25K_c=0. \]

Step 4: For decaying oscillations, the system must be underdamped (complex roots with negative real parts). Condition: \[ \Delta = b^2 - 4ac < 0, \] where \(a=2,\; b=(1-0.25K_c),\; c=0.25K_c\). \[ \Delta=(1-0.25K_c)^2 - 2K_c. \]

Step 5: Test given options:

• \(K_c=0.25:\;\Delta=(1-0.0625)^2-0.5=0.8789-0.5>0\) → no oscillations.
• \(K_c=2:\;\Delta=(1-0.5)^2-4=0.25-4=-3.75<0\) → decaying oscillations.
• \(K_c=4:\;\Delta=(1-1)^2-8=-8<0\). But coefficient \(b=(1-1)=0\) → real part = 0 → sustained oscillations (not decaying).
• \(K_c=6:\;\Delta=(1-1.5)^2-12=0.25-12=-11.75<0\), coefficient \(b=(1-1.5)=-0.5\). With \(a=2>0\), roots have positive real parts → unstable.

Final Answer: \[ \boxed{K_c = 2} \]