# Physics

1. A 5 to 20 bar reverse acting proportional pressure controller has an output

of 4 to 20 mA. The set point is 11 bar. Determine:

(a) the measured value pressure which gives an output of 15 mA when

the proportional band setting of the controller is 40%

(b) the proportional band setting which will give an output of 8 mA when

the measured value is 14 bar and the desired value is 11 bar.

2. FIGURE 1(a) shows a flow control system whose output Qo is regulated

by using a proportional controller to control Q1. (Note that Q2 is not

controlled). The control system can be represented by the block diagram

shown in FIGURE 1(b).

(a) Derive a relationship between Qo and Q2, B, C and DV, where C is

the gain of the controller.

(b) If Q2 is 3000 m 3 h–1 when the bias B is 1000 m3 h–1, the PB setting of

the controller is 40% and the desired value of the controller is

4000 m3 h–1, determine the resultant value of Qo.

(c) If Q2 now changes to 2500 m 3 h–1, determine the new bias figure

required to ensure that Qo is maintained at its original flow rate

value.

2

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FIG. 1

Q1

Q2

FC

F

Dv

Mv

Qo

ε Dv C

B

Proportional controller

+ –

+ +

+ +

1.0

Qo

F = Flowmeter

Q2

Q1

(a)

(b)

3

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3. (a) With the aid of a sketch explain how proportional action is produced

in a pneumatic controller whose output is 0.2 to 1.0 bar. Assume that

the controller is direct acting.

(b) Show, mathematically, that the output is dependent on the difference

between the measured and desired values.

(c) With the aid of a well annotated sketch describe the construction and

operation of a P + I + D controller having a pneumatic output.

4. (a) FIGURE 2 shows an electronic ‘black box’ whose output is ten times

the difference between its two input signals. Show how the ‘black

box’ could be realised using just two operational amplifiers and five

resistors. Give the relative values of the resistors.

FIG. 2

(b) With the aid of circuit diagrams show diagrammatically, and prove

mathematically, how op-amps are utilized to produce:

(i) integral action

(ii) derivative action.

(c) Show how generation of the above actions are combined with

proportional action generation to produce a three term electronic

output controller.

Vo = 10(V2 – V1)

V1

V2

4

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5. The proportional control system of FIGURE 3(a) has an input, θ1, of 10 units. The uncontrolled input, θ2, has a value of 50 units, prior to a step change down to 40 units. The result of this disturbance upon the

output, θo, is shown in FIGURE 3(b).

(a) Calculate the change in offset in the output produced by the step

change.

(b) Draw a modified block diagram to show how the offset could be

minimised by the inclusion of another control action. Also, show by

means of a sketch how the modification might be expected to affect

the output response.

(c) Show, by drawing a modified block diagram, how the magnitude of

the disturbance could be minimised by the inclusion of a third type of

control action.

FIG. 3(a)

G 9

+ –

θ1 + +

θo

θ2

H 0.1

+ –

+ +

5

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FIG. 3(b)

6. FIGURE 4(a) shows a flow control system that is controlled by a P + I

controller. The control objective is to maintain a constant flow rate, Qo,

for varying values of input flow rate, Q2. The system can be represented

by the block diagram shown in FIGURE 4(b).

Assume that the following initial conditions apply.

DV = 2000 ε = 0 I = 1000 C = 0.2

Q2 = 1000 Ki = 0.8

Show that following a permanent step disturbance in Q2 from 1000 m 3 h–1

to 1200 m3 h–1, the resulting offset is eliminated.

You should continue your calculations until Qo is within 4 m 3 h–1 of its

final value for two successive calculations.* Derive any formulae used.

*A spreadsheet can be used to advantage here!

0 10 20 30 40 50 60

Time (minutes)

θo

6

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FIG. 4

Q1

Q2

Qo

DVFC

F

(a)

+ –

Dv + +C Qo

Q1

Q2

I

εCε

Ki ∫

++

1.0 Qo

(b)

7

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7. (a) FIGURE 5 shows the input and output waveforms for a proportional

plus integral controller. State:

(i) the controller’s proportional gain

(ii) the controller’s integral action time.

FIG. 5

0

Input

Output

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Time (minutes)

8

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(b) FIGURE 6 shows a proportional plus derivative controller that has a

proportional band of 20% and a derivative action time of 0.1 minutes.

Construct the shape of the output waveform for the triangular

input waveform shown, if the input rises and falls at the rate of 4 units

per minute.

FIG. 6

8. (a) FIGURE 7 shows the closed-loop response of a plant to a step input

when the proportional only gain was set to 4. Use the ‘Quarter

Amplitude Response Method’ to estimate the required settings of a P

+ I + D controller.

FIG. 7 Plant Response to Step Input

(b) If the same plant was ‘tuned’ using the ‘Ultimate Cycle Method’,

estimate the P + I + D controller settings if a proportional only gain

of 6 was required to produce steady oscillations.

Time (seconds)

P ro

ce ss

o ut

pu t

+5

–5

10 20 30 40 50 600

P+Dθi θo

1 minute

9

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