P3.9 A speed control system using fluid flow components is to be designed. The system is a pure fluid control system because it does not have any moving mechanical parts. The fluid may be a gas or a liquid. A system is desired that maintains the speed within 0.5% of the desired speed by using a tuning fork reference and a valve actuator. Fluid control systems are insensitive and reliable over a wide range of temperature, electromagnetic and nuclear radiation, acceleration, and vibration. The amplification within the system is achieved by using a fluid jet deflection amplifier.

The system can be designed for a 500-kW steam turbine with a speed of 12,000 rpm. The block diagram of the system is shown in Figure P3.9. In dimensionless units, we have b = 0.1, J = 1, and K1 = 0.5. (a)

Determine the closed-loop transfer function

w(s) T(s) == R(s) (b) Determine a state variable representation. (c) Determine the characteristic equation obtained from the A matrix.