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E1 – ME2142-1 Frequency Response (EA-04-06) 12-Sept-2011

Name: ChenHaojin Matric Number: U096128E Group: 3C1

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Objective: 1. Learn how to use the frequency response method to identify the (linear) dynamics of a plant. 2. To perform a frequency response test on an aircraft electro-hydraulic servo-actuator and to determine the phase and gain margins of the servo. 3. To understand the relationship between the gain and instability. 4. Learn how to use Nichols Chart to find open loop gain and phase given close loop and phase.

Results: Table 1

2πƒ¦ P¦

Vi (V)

Vo (V)

Gain (dB) = 20 log(Vo/Vi)

Phase (deg) = Δtime x ƒ x 360°

0.5

3.142

5

4.640

-0.649

-0.1600

-28.800

1

6.283

5

4.160

-1.598

-0.1200

-43.200

2

12.566

5

3.040

-4.322

-0.0960

-69.120

3

18.850

5

2.320

-6.670

-0.0880

-95.040

4

25.133

5

1.700

-9.370

-0.0800

-115.200

5

31.416

5

1.340

-11.437

-0.0680

-122.400

6

37.699

5

1.080

-13.311

-0.0600

-129.600

7

43.982

5

0.856

-15.330

-0.0568

-143.136

8

50.265

5

0.696

-17.127

-0.0536

-154.368

9

56.549

5

0.576

-18.771

-0.0464

-150.336

10

62.832

5

0.480

-20.355

-0.0464

-167.040

11

69.115

5

0.416

-21.598

-0.0432

-171.072

12

75.398

5

0.348

-23.148

-0.0408

-176.256

13

81.681

5

0.296

-24.554

-0.0392

-183.456

14

87.965

5

0.264

-25.547

-0.0380

-191.520

15

94.248

5

0.220

-27.131

-0.0356

-192.240

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Table 2 w (rad/s)

Open Loop Gain (dB)

Open Loop Phase (Deg)

3.142

6

-95

6.283

1.4

-100

12.566

-4

-104

18.850

-7.6

-118

25.133

-10.6

-128

31.416

-11.5

-133

37.699

-14.2

-139

43.982

-16.1

-147

50.265

-17.8

-156

56.549

-19.3

-154

Sample Calculation: Table 1, The readings of , Vi, Vo and time are either given or obtained through the experiment. We need to calculate the Gain (dB) and the Phase (deg). Take the 1st row ( = 0.5 Hz, Vi = 5 V, Vo = 4.640 V and time = - 0.1600 s) for example: πƒ

ƒ

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Table 2, The measured gain K until the system has reached instability is 55.

The gain margin from Nichols Chart is 22 (dB). (See the Chart when the graph cut the -180°line)

The phase margin from Nichols Chart is - 80 (deg). (See the Chart when open gain = 0, the phase is -100°; then -180-(-100) = -80)

The theoretical value of open loop gain margin is:

From the bode graph,

From the bode plot, K = 13 dB

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Discussion: 1. As the frequency increase, the phase becomes larger. Reason: The time delays of the system become more obvious.

2. As the frequency increase, the gain of the system decrease. Reason: The output voltage is smaller than the input voltage due to the time delays of the system; the larger the time delays, the smaller the output voltage.

3. As the gain (K) increase, the damping ratio will decrease and the system response will be faster.

4. The gain margin: reading from Nichols Chart is 22 dB while the experiment value is 34.807 dB. (Can see sample calculation for detail, these two values are much difference) Reason: A. The unstable point for the system is difficult to determine simply by observation. B. The gain reading is just approximation, there should be certain error.

Conclusion: 1. Learnt how to use Nichols Chart to obtain the open loop gain and phase given the close loop gain and phase. 2. Know how the frequency response performs as the frequency changes. 3. Know how to find out the gain margin and phase margin through experimental method. As the gain increase, the time response of the system decrease, but the system becomes more unstable. In our case, when the gain reaches 55, the system reached instability. 4. Overall, this experiment helps us better understand the frequency response of a servo system.

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