Test #5 Control Systems

Question 1

Write the Time Domain Equations for the five processes we studied in class.

Answer 1

INTEGRAL: Output(t) = k(Input(t)-Constant)*t

FIRST ORDER LAG: tau(dOutput(t) / dInput(t))+Output(t) = G * Input(t)

SECOND ORDER LAG: A₂(d²Output(t)/dt²)+A₁(dOutput(t)/dt)+A₀ * Output(t) = G * Input(t)

DEAD TIME: Output(t) = G * Input(t-t_d)

FIRST ORDER LAG + DEAD TIME:

tau(dOutput(t) / dInput(t))+Output(t) = G * Input(t-t_d)

Question 2

Sketch and correctly label the step responses for each of the five processes we discussed in class.

Answer 2

graphs

Question 3

What information does the frequency response curve give about the process ?

Answer 3

The frequency response curve tells the gain and phase relationship between the input and output signals.

Question 4

If you were to determine the frequency response of an electrically based black box with a set of input signal leads and a set of output signal leads what test equipment would you need?

Answer 4

A signal generator and an oscilloscope is all that would be required.

Question 5

Briefly describe the procedure you would utilize using the test equipment in part b) above, to determine the frequency response of the black box.

Answer 5

I would subject the input of the proces to sinusoidal signals of a wide rande of frequencies. Using the oscilloscope I would record the resultant outputs and from that determine the gain and phase relationships for each frequency. These I would plot to display the frequency response.

Question 6

For the integral process shown in figure 1 determine the time it takes the output to rise to 2m in height given the following data:

Q_out = 63x10^-3 m³/s

Q_input = 126x10^-3 m³/s

R = 2m

Repeat above using R = 4m

Repeat above using R = 6m

Plot time vs. R for R = 2m

Answer 6

graph

Question 7

The following terms are associated with the second order lag process. Give the name for each term and explain the information it gives you about the second order lag process.

Answer 7

1. The damping coefficient tells us how quickly any oscillations will decay in amplitude.
2. The resonant frequency tell us the frequency that the system would naturally desire to oscillate at.
3. The damped frequency tell us the frequency of the oscillations in an underdamped response.
4. The damping ratio tell us whether the process is underdamped p<1, critically damped p=1, or overdamped p>1.

Question 8

If you wish to make an underdamped mass-spring-dashpot system overdamped which component would increase in strenght?

Answer 8

The strenght of the dashpot or the capacitance of the spring.

Question 9

If you wish to make an overdamped R-L-C circuit underdamped which component would you increase in strenght?

Answer 9

You would increase the strenght of the inductance

Question 10

The roots of the characteristic equation of a second order lag process will identify the type of second order lag process. State all the types of roots and what type of second order lag process they identify.

Answer 10

Real and unequal - Overdamped process

Real and equal - Critically damped process

Complex conjugates - Underdamped process

Question 11

Given the following process and the associated equations

A) Sketch the step response of the system (fluid temperature vs. time)

B) Sketch the step response of the system (fluid temperature vs. time) after the wall’s thickness has been reduced.

GRAPH

Answer 11

Since the wall’s thickness is reduced, the overall thermal resistance has dropped. As a result the thermal time constant has dropped and the fluid temperature will rise to the water jacket temperature faster.

GRAPH

Question 12

Identify by circling all the energy storage elements shown in the figures and indicate the capacitive, inertial and resistive elements.

GRAPH

Answer 12

GRAPH

Question 13

Explain the frequency response graph shown for the process shown below.

GRAPH

Answer 13

At low frequencies there is greater time for the voltage in the capacitor or the height of the fluid to build up and break down causing greater variances in the output and thus a higher gain. The phase difference is locked at -90 degrees as the height of the fluid and the voltage across the capacitor hit their maximum until the energy flow switches direction. So the output always lags the input by 90 degrees.

Question 14

The raidus of the tank is increased from 1m to 3m. Comment on the effect this will have on the 63.2% response time. Also, what is another name for the 63.2% response time?

Answer 14

If the radius of the tank is increased then the capacitance is increased. As a result the time constant is increased and the 63.2% response time is increased. Another word for 63.2% response time is time constant.