TIME RESPONSE

OF

FIRST ORDER SYSTEM

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1 SYED HASAN SAEED

RESPONSE OF FIRST ORDER SYSTEM WITH UNIT STEP INPUT:

For first order system

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sT

T

ssC

sTssC

ssR

sRsT

sC

sTsR

sC

1

1)(

)1(

1)(

1)(

)(1

1)(

1

1

)(

)(

Input is unit step

After partial fraction

Take inverse Laplace

Where ‘T’ is known as ‘time constant’ and defined as the time required for the signal to attain 63.2% of final or steady state value.

Time constant indicates how fast the system reaches the final value.

Smaller the time constant, faster is the system response.

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632.011)(

1)(

1/

/

eetc

etc

TT

Tt

When t=T

RESPONSE OF FIRST ORDER SYSTEM WITH UNIT RAMP FUNCTION:

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Ts

Ts

T

ssC

sTssC

ssR

sRsT

sC

sTsR

sC

1

11)(

)1(

1)(

1)(

)(1

1)(

1

1

)(

)(

2

2

2

Input is unit Ramp

After partial fraction

We know that

Take inverse Laplace, we get

The steady state error is equal to ‘T’, where ‘T’ is the time constant of the system.

For smaller time constant steady state error will be small and speed of the response will increase.

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TTeTLimit

eTte

TeTttte

tctrte

TeTttc

Tt

t

Tt

Tt

Tt

)(

)1()(

)(

)()()(

)(

/

/

/

/

Error signal

Steady state error

RESPONSE OF THE FIRST ORDER SYSTEM WITH UNIT IMPULSE FUNCTION:

Input is unit impulse function R(s)=1

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TteT

tc

TsTsC

sTsC

sRsT

sC

/1)(

/1

11)(

1.1

1)(

)(1

1)(

Inverse Laplace transform

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1)(

1)(

1)(

2

sR

ssR

ssRFor unit Ramp Input

For Unit Step Input

For Unit Impulse Input

TtTeTttc /)(

Ttetc /1)(

TteT

tc /1)(

It is clear that, unit step input is the derivative of unit ramp input and unit impulse input is the derivative of unit step input. This is the property of LTI system.

Compare all three responses:

THANK YOU

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