Experiment #9 - Islamic University of Gaza

Linear Control Systems Lab EELE (3160-3161)

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Experiment #9

Implementing system using Op-Amp and Analog computer

1) Introduction:

The Op-Amp stand for Operational Amplifier, is basically a differential amplifier having a large voltage gain, very high input impedance and low output impedance. The op-amp has a "inverting" or (-) input and "non-inverting" or (+) input and a single output. The op-amp is usually powered by a dual polarity power supply in the range of (+/-) 5 volts to (+/-) 15 volts.

The circuit symbol for an op-amp is shown to the right, where:

𝑽+: non-inverting input

𝑽−: inverting input

𝑽𝒐𝒖𝒕 : output

𝑽𝑺+ : positive power supply

𝑽𝑺− : negative power supply

Operational amplifiers can be used to perform mathematical operations on voltage signals such as inversion, addition, subtraction, integration, differentiation, and multiplication by a constant and comparison.

2) Operational Amplifier (Op-amp) Applications:

A. The inverting op-amp: (It can be used as K controller)

1

2

R

R

V

VGain

i

o


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B. The non-inverting op-amp: (It can be used as K controller)

1

21R

R

V

VGain

i

o

C. The summer : (Adder)

)( 2

2

3

1

1

3 VR

RV

R

RVo 321 VVVVo

D. The differential amplifier: (Subtract)

If you set 𝑅1 = 𝑅2 then

)( 12

1

2 VVR

RVo

)( 12 VVVo


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E. The integrator:

F. The differentiator:

G. The comparator:

- 𝑽𝒓𝒆𝒇: Reference voltage used as reference to compare between it and the input

voltage.

dtVRC

V ino 1

t

VRCV in

o


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3) System to All Integral Block Diagram :

What is all integral block diagram?

It is another representation of a system but here integrators only used to represent it.

This method introduce the conversion between the mathematical systems to its corresponding physical one.

We first begin with the mathematical model of a system (Diff. Eq., T.F., and S.S) and convert it to state equations by some ways and represent it by using integrators and summers that can be implement by op-amp.

Represent the following systems by all integral block diagram:

Example 1:

Solution: )()0()( sXyssY

s

sX

s

ysY

)()0()(

ttxttxtuyty )()()().0()(

Example 2:

Solution:

ttyttyyty )(1)()0()(

x(t) y(t)

1)0()()(

yty

t

ty

0)0()()(

ytx

t

ty


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Example 3:

Solution:

Now we can convert the differential equation to all integral block diagram and as we can convert the transfer function and the state space model to these differential equations, we then can covert them to all integral model.

4) Physical Realization of a system by op-amp :

Section 1 talked about op-amp application and saw that op-amp can operate as inverter, summer and integrator. Section 2 talked about converting the system to All integral block diagram that depends on inversion, summing and integration. Then we can combine the two sections to implement any system by op-amps and it is the first step to transfer from theory to practice and from math to hardware component.


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Example 1:

Determine the analog diagram and the circuit to implement the following equation

Note the inversion operation of all inputs that is because the summer inverts the output as we introduced before and the inverter must have a unity gain.


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Example 2:

Determine the analog diagram and the circuit to implement the following system

From the above equation we know that we will use two integrators at least as follows

And by rearranging the equation, we conclude that

Don't forget that summer and integrators invert the output


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Because we know that integrator can do summation also, so we combine the two first blocks (summer, integrator) by one as follows, but we should invert the input to obtain the suitable output.

Note:

This is all integral block diagram, and we implement it by op-amp take care to invert the inputs.


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Example 3:

Determine the circuit to implement the following system

𝐺(𝑠) =1

𝑆 + 1


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5) Implementing Transfer Function Using OP-Amps:

To implement the transfer function by using operation amplifiers there is some steps to do this, we will explain it as shown in the next example

Example:

Given the T.F, implement it using by OP-AMPs

𝑮(𝒔) =𝒀(𝒔)

𝑼(𝒔)=

𝟔

𝑺𝟐 + 𝟐𝑺 + 𝟏𝟐

Firstly:

Convert the T.F to State Space the following MatLab code make this function:

The step response of the T.F is as shown in the following Figure

num=[6];

den=[1 2 12];

G=tf(num,den)

[a,b,c,d]=tf2ss(num,den)

Transfer function: 6 ----------------- s^2 + 2 s + 12 a = -2 -12 1 0 b = 1 0 c = 0 6 d = 0

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Step Response

Time (seconds)

Am

plit

ude


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Secondly:

Find the Block diagram of state space by MatLab Simulink, as shown in the following figure.

Step response for the block diagram is shown below

Thirdly:

Convert the block diagram to Op-Amps implementation


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6) Find the step response:

To get the step response of a system you should:

1. Implement the system on the analog computer.

2. Connect the output to the oscilloscope.

3. Apply a step input as 1 volt constant.

4. Rescale the graph of the oscilloscope to get a clear view.


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Exercise:

1. Determine the analog diagram and the circuit to implement the following system

a. 𝐻(𝑠) = (2

𝑠+5) (

5

𝑠+3)

b. 𝐻(𝑠) = (2

𝑠+5) + (

5

𝑠+3)

c. )(10)(

)(1

1)(

57

10)(

txty

tutxtx

Experiment #9 - Islamic University of Gaza

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