Lecture 16 Time Domain Analysis

8/9/2019 Lecture 16 Time Domain Analysis

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Time Response Analysis

Winston Netto

Assistant Professor

Department of Instrumentation and Control Engineering,

Manipal Institute of Technology

Manipal


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Time response of the system is the output of the system as a function of time. It is

denoted by c(t).

Time response can be obtained by finding the solution of differential equationgoverning the system or from the transfer function of the system with the input to

the system.

The time response of a control system consist of two parts: transient response

and steady state response.

Transient response (or natural response): Response of a system to a change from

equilibrium.

Steady state response (or forced response): Response of the system as t

approaches infinity.

Winston Netto, MIT, Manipal


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, ()() =()

1 ()

, = ()1 ()

, = − = − ()1 ()

Pole of a transfer function : The value of the Laplace transform variable, s,

that cause the transfer function to become infinite.

Zero of a transfer function: The value of the Laplace transform variable, s,

that cause the transfer function to become zero.

Roots of characteristic equation (denominator polynomial of T.F) is Pole. ‘X’

Roots of numerator polynomial of T.F is zero. ‘O’



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Test Signals



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Name of the signal Time domain equation Laplace Transform

Step A A/s

Ramp At A/S2

Parabolic At2 /2 A/S3

Impulse δ(t) 1

Sinusoid Sin ωt ω/(s2

+ω2

)Winston Netto, MIT, Manipal


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Order of a system

The order of the system is given by the order of the differential equation

governing the system. If the system is governed by nth order differentialequation, then the system is called nth order system.

The order of the system can be also determined from the transfer function –

Maximum power of s in the denominator polynomial gives the order of system.

The value of n gives the number of poles in the transfer function



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First Order System Response

Transfer function of a first order system is given by,

()() =

+

Where, T is the time - constant

Consider a first order system with transfer function

Step response for the given transfer function is given by,

Parameter a describes the transient response.

For t = 1/a, c(t) = 0.632



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Time Constant

Time taken by step response to rise

to 63% of final value.

Rise Time (T r )

Time for the waveform to go from

0.1 to 0.9 of its final value.

T r =

.3

.5 =

.

Settling Time (Ts)

Time for the response to reach and

stay within 2 % of its final value.

=4

Note: Since the pole of transfer function isat –a, we can say the pole located at the

reciprocal of the time constant, and farther

the pole from the imaginary axis, the faster

the transient response.



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Response of First order system for unit ramp input

Consider a first order system with transfer function

()() = =

Response, = =

( ) (, = ; =)

= − = − [ ()] By Partial fraction,

( ) =

( )

∴ = ( )

= −


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Second Order System

The standard form of transfer function of second order system is given by,

()() =

ζ

Where, is the undamped natural frequency, rad/secζ is the damping ratio

The response c(t) of second order system depends on the value of damping ratio.

Depending on the value of ζ, the system can be classified into four cases,Case 1: Undamped system ζ = 0Case 2: Under damped system 0 < ζ < Case 3: Critically damped system ζ = 1Case 4: Over damped system ζ >

The characteristic equation of second order system is,

ζ = 0Winston Netto, MIT, Manipal


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1. Step Response of Undamped second order system


()() =

ζ

for undamped system, ζ = 0()

() =

=

=

Unit step response,

By partial fraction expansion,

=

( ) =

=

+ ; = − =

= −

=



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2. Step Response of Under damped second order system


()() =

ζ

for under damped system, 0 < ζ < 1,

The roots of the characteristic equation are, s = ζ ± n ( 1 ζ2) = ζ ±

Unit step response, =

++ =

++


=

ζ =

(ζ) ζ



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Adding and subtracting ζ to the denominator of second term,

()=

(ζ)

ζ ζ ζ

= (ζ)

(ζ ζ) ( ζ)

=

(ζ)( ζ ) ( ζ)

= (ζ)

( ζ )

=

( ζ )( ζ )

ζ( ζ )

Multiplying and dividing by in the third term of the equation,

=

( ζ )

( ζ ) ζ

[(ζ) ]



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taking inverse Laplace transform

= − −

= −( − )

= −( − )

= − (ζ ζ ) Note: On constructing a right angled triangle with ζ and ζ, we get

= ζ

; = ζ ; = ζ

ζ

( ) = − ( )

= −

ζ ( )



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3. Step Response of critically damped second order system


()() =

ζ for critical damped system, ζ = 1

()() =

=

Unit step response, = =


= ; =

Response in time domain, = − − = −( )



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4. Step Response of over damped second order system


()() = ζ

for over damped system, ζ >1Roots of the characteristic equation; sa, sb = ζ ± ζ2 1 = (ζ ∓ ζ2 1)

Let s1 = -sa and s2 = -sb ; therefore s1 = ζ ζ2 1 and s2 = ζ ζ2 1

()() =

( ) ( )

Unit step response, = ( ) ( ) = ( ) ( )

Applying partial fraction, =



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=

ζ

ζ

Response in time domain,

= ζ

− ζ

−

= ζ ( − − )



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Time domain specification

Transient response characteristics of a control system to a unit step input is specified

in terms of time domain specifications.

1. Delay time, td2. Rise time, tf3. Peak time, tp4. Maximum overshoot, Mp

5. Settling time, ts



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• Peak time (tp): It is the time taken for the response to reach the peak value for

the very first time.

•

Peak Overshoot (Mp): It is defined as the ratio of the maximum peak valuemeasured from the final value to the final value.

• Settling time (ts): It is defined as the time taken by the response to reach and

stay within a specified error. The usual tolerable error is 2% or 5% of the final

value.

• Rise time (tr): It is the time taken for the response to raise from 0% to 100% for

the very first time.

i. For under damped system, the rise time is calculated from 0% to 100%.

ii. For over damped system it is the time taken by the response to raise from 10%

to 90%.

iii. For critically damped system, it is the time taken for response to raise from 5%

to 95%.

• Delay time (td): It is the time taken for the response to reach 50% of the final

value, for the very first time.



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Expression for time domain specification

1. Rise time (tr): =

;,=− ζ

ζ , = ζ

2. Peak time (tp): =

3. Peak Overshoot (Mp): = (∞)

(∞)

% = (∞)(∞)

% = −−

4. Settling time (ts): % , =

ζ

% , =

ζ



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Rise time (tr)

Unit step response of second order system for under damped case is given by

= − ζ ( )

At t=tr, c(t) = c(tr) = 1, therefore, = − ζ ( )

= −

ζ ; =

therefore, =

,=



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=

− ζ (( ) )

= − ζ ()

= , = ; ∴ −

ζ =

=

,=

=



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Peak Overshoot (Mp)

Unit step response of second order system for under damped case is given by

= − ζ ( )

, % = (∞)(∞)

= ∞, = ∞ =

= , = = −

ζ ( )

,= , ∴ = −

ζ ( )

, = ζ, ∴ = − −

ζ ( )



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=

− () ; = −

−

ζ ζ

% = − −

, % = −−



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Settling time(ts)

The settling time is the time it takes for the amplitude of the decaying sinusoid to

reach 0.02

% = , − ζ = .

− = . ζ

ζ =(. ζ)

= (. −)

−

For all values of ζ (under damped case), the settling time can be approximated as,

, =


Lecture 16 Time Domain Analysis

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