Report Lab12

8/6/2019 Report Lab12

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1. Purpose of the Experiment: To evaluate the time and frequency characteristics of second

order under damped system. The experiment also involves Matlab and Pspice simulations

of the given second order system. After performing the simulations and verifying the results,

bread boarding was performed to verify and analyse the obtained results and to observe the

challenges in practical approach.

2. Equipment Used:

Type Model Serial No.

Oscilloscope 54652A US35030127Function Generator 33120A US34005597DC Power Supply E3630A KR34701916

3. Parts Used:

QTY Component Value

4 Resistor 1K1 Resistor 3.3K, 5.1K2 Capacitor 0.01UF3 Op-amp 741, LM318

4. S/W Used:

Matlab/Simulink 7.6.0.324 (R2010a)

Pspice 16.0.0.s001

5. Theory: Time and frequency behavior of a system is important. When designing a system, the time

behavior may well be the most important aspect of its behavior. Determining and analyzing the

behavior of the system is very important and some of the reasons supporting this concern are:

1. How quickly a system responds is important.

2. If the system is stable or not. Oscillations in a system are not usually desired.

3. Steady state response.

Second order behavior is part of the behavior of higher order systems and understanding second

order systems helps to understand higher order systems. They are simplest systems that exhibit

oscillations and overshoot. The system response can be over-damped, under-damped or critically

damped or un-damped as shown in figure below.


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Figure1

As, this experiment deals with the frequency response of the second order under-damped system thus

the time behavior was expected to look like this:

6. Procedure and Results:

6.1Second order system: The given block diagram of the second order system was:

Figure2

The general block diagram of a second order with the overall transfer function, system is shown below.


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Figure3: General Block diagram of second order system

Thus, from the given block diagram of the second order system the Total transfer function of the system

was obtained as:

T(s) =C(s)/R(s) = 2*109/(s2+3*104+2*109)-------------------- (1)

From the obtained transfer function, various other values were calculated manually, by comparing the

obtained transfer function with the general second order transfer function given as:

T(s) = 222

2nn

n

wsws

w

++ -----------------------------(2)Comparing equation (1) and (2), we got:

Natural angular velocity, Wn = 4.472*104

Damping, = .335

Step ResponseNow, the obtained values ofWnand from the mathematical calculations done manually, were verified

by the step response of Matlab simulation of the transfer function given by equation (1). The result

obtained from Matlab step response was:


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Figure4: Matlab output response for given second

order system

From, Matlab simulation result, Maximum Peak or Amplitude= 1.32V

Percentage overshoot, P.O. = 1.32(max. peak) - 1.0(desired value)

= 0.32

Maximum Peak time, Tp = 7.3*10-5 sec

Using the formula for P.O. =2

1

e , value of damping, was obtained and verified, which was

calculated and found out to be, = 0.33

Also, using the formula for Maximum Peak time, and the above obtained value of damping, natural

angular velocity, was then calculated and verified.

21

=

n

p

w

T

Thus, natural angular velocity was found out to be, wn = 4.49*104 Hz

Thus, Matlab verifies the manual calculations.

Then, Pspice simulations were performed by designing equivalent electrical circuit for the given transfer

function and its step response was observed. The circuit and the simulated result are shown below:


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Figure5: Pspice Schematic of given second order system

using op-amps


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T i m e

0 s 5 0 u s 1 0 0 u s 1 5 0 u s 2 0 0 u s 2 5 0 u s 3 0 0 u s 3 5 0 u s 4 0 0 u s 4 5 0 u s

V ( V 3 : + ) V ( R 7 : 1 )

- 0 . 8 V

- 0 . 4 V

0 V

0 . 4 V

0 . 8 V

1 . 2 V

1 . 6 V

Figure6: Pspice simulation result for second order system

DC response

Results from Pspice Simulation, Maximum Peak or Amplitude= 1.34V

Percentage overshoot, P.O. = 1.34(max. peak) - 1.0(desired value)

= 0.34

Maximum Peak time, Tp = 7.7*10-5 sec


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Frequency Characteristics

Also, from the obtained value of Damping, Maximum amplitude of the system and its resonating

frequency were evaluated by using following formulas:

Thus, Operating or resonating frequency, fr= 6.337*103 Hz(Manual Calculation)

Maximum amplitude of the System, Mpw = 1.61 V(Manual Calculation)

The above obtained values were then verified by perfoming Matlab and Pspice Simulations.


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For obtaining the system frequency characteristics using Matlab, bode command was used to get the

plots for amplitude and phase.

Figure7: Matlab System Freqeuncy characteristic for given second order system(Amp and Pahse)

showing the 3db point

From the above plot, Maximum Amplitude and Resonating Frequency was calculated as:

20 log(Vo/Vi) = 3.8 db

Log Vo = 0.19 (Vi = 1)

Maximum Amplitude, Vo = 1.55 V

Resonating Frequency, fr= 6.2 Khz

After that, Pspice simulations were obtained by applying the AC input to the same circuit shown in Figure

5, thus the new obtained circuit and the results for AC response are:


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U 1

u A 7 4 1

+3

-2

V +7

V

-

4

O U T6

O S 11

O S 25

U 2

u A 7 4 1

+3

-2

V +7

V

-

4

O U T6

O S 11

O S 25

U 3

u A 7 4 1

+3

-2

V +7

V

-

4

O U T6

O S 11

O S 25

R 1

1 k

R 2

3 . 3 k

R 3

5 . 1 k

R 4

1 k

R 5

1 k

R 6

1 k

C 1

0 . 0 1 u f

C 2

0 . 0 1 u f

0

- V C C

- V C C

- V C C

V C C

V C C

V C C

0

U 4

u A 7 4 1

+3

-2

V +7

V

-

4

O U T6

O S 11

O S 25

R 7

1 k

R 8

1 k

0

- V C C

V C C

V 3

1 V a c

0 V d c

V

V

Figure8: Pspice Schematic of given second order system using

op-amps with AC input

F r e q u e n c y

1. 0K H z 3.0 K Hz 1 0K Hz 3 0K Hz 10 0 KHz 3 0 0KHz 1. 0MH z

V ( V 3: + ) V ( R7 : 1 )

0V

0.4V

0.8V

1.2V

1.6V

2.0V

Figure9: Pspice simulation result for AC input showing the Maximum Amplitude of system

Results from Pspice Simulation, Maximum Amplitude = 1.61 V

As, the lab deals with the second order under-damped system as it is clear

from the results obtained, the roots of the characteristic equation should be

imaginary. And this was confirmed by obtaining the root locus plot and the

roots from the Matlab.

Characteristic Equation : s2 + 3*104s + 2*109


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Figure10: Matlab root locus plot showing the roots for the characteristic equation


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Figure11: Pole-Zero Map showing the roots for the characteristic equation

Also, Polar plot was obtained for the total transfer function using Nyquist command in Matlab, which is

shown below.

Figure12: Nyquist plot for the given second order transfer function


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The circuit was also designed and constructed by bread boarding and the system response wasobtained at a very low frequency of 30 Hz.

Figure13: Bread boarding observation for 1V AC input at Maximum Amplitude

Maximum Amplitude = 1.65V

Resonating Frequency = 6.6KHz


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7. Conclusion: The overall work done in this lab gave a lucid idea about how to observe and measure

various characteristics of a given system. Given only a transfer function and from there on, calculating

various factors like, wn, damping, Mpw, fr etc. of the system, built a confidence in dealing and

measuring these with ease in future. Firstly, these factors were obtained manually, and then were

verified by the simulations obtained in both Matlab and Pspice. Working on the softwares like Matlab

and Pspice shows the power and strength of these tools in handling complexities of various types of

system. In particular Matlab, various commands in Matlab, made big calculations too easy.

The frequency response of the given Transfer function for a second order system is evaluated. TheValues for Mpw, fr, wn, P.O are calculated and the circuit is constructed and the result was verifiedexperimentally.

Parameter Formula(calculated) Frequency Plot Unit Step Plot

Experimental

Value

n

(Angular Velocity)4.472*10^4 4.487*10^4 4.4*10^4

fr

( Resonant Frequency)

6.337KHz 6.2KHz 6.28KHz 6.66KHz

Mp

(Maximum Amplitude)

1.58V 1.55V, 1.61V

(Matlab) (Pspice)

1.32V, 1.34V

(Matlab) (Pspice)

1.65V

From, bread boarding it was observed that when natural frequency starts coming equals to system

frequency, the gain starts increasing and the oscillations show up.

Lab eperiment also depicts the fact that evaluating various parameters associated with secon order

system(or any order system) before-hand helps to adjust the final practical circuit or real time devices

according to their extreme limits of tolerating the unstability and fluctuations (oscillations) occuring at

the initial stages of input value. By this, one can achieve stability to ones choice or systems

capability of bearing the unstability.

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Report Lab12