A proportional controller is to be used to control the pump output pressure in the system shown below
The steady state output torque of the engine is given by the equation
Tss = 1500 X - 25 N fit-lbf
where X is the throttle position in inches and N is the engine speed in rpm. The total combined rotational moment of inertia (I) of the engine and pump combined is 150 ft¬lbf-s2. The output of the pump is related to the engine speed by the equation
Q = 0.1 N gal/min
where N is in rpm. The torque required to drive the pump is determined by the pressure output from the pump
T = 0.5 P ft-lbf
Where P is in psi. The bypass circuit has a constant resistance R of 300 psi/(gal/min).
The voltage output of the pressure transducer is given by the equation
Vt = 0.0001 P Volts
Where P is in psi. The throttle servo is a first order system with a time constant of 1 second and a steady state responsegiven by the equation below.
Xss = 0.5 Vs inches
Where Vs is the voltage input to the servo.
The controller equation is
Vs = Kp(Vr Vt) Volts
where Vr is a reference input voltage and Kp is the controller constant
1. Draw the block diagram for this system showing all necessary transfer functions
2. Determine the single block transfer function with Vr and QL in the input function and P as the controlled variable.
3. Determine the value of Kp which will result in a fast and smooth response to changes in either Vr or QL.
4. Determine the equation for the steady state value of P for given values of Vr and QL using your value of Kp.
5. The system is operating at steady atate with Vr equal to 2 volts and QL equal to 0. QL suddenly changes to 25 gal/min and remains fixed at that value. Plot X, P, and N against time during the transition to a new steady state. These plots may be determined from an exact solution of the differential equation or a numerical solution based in the system equations. A listing of the program with comments explaining the solution must be included with the solution.