Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Electrical Engineering SSC JE (Technical)

Electrical Engineering (EE) : Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

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TIME DOMAIN ANALYSIS

  • The time response of a control system is usually divided into tow parts the transient response and the steady state response. Let y(t) denote the time response of a continuous time system. then, in general, is can be written as,

y(t) = yt(t) + yss(t)
where,

yt(t) denotes the transient response and
yss(t) denotes the steady state response. 

  • In control system, transient response is defined as the time response that goes to zero as time becomes very large. Thus yt(t) has the property.

Lim yt(t) = 0
 t → ∝

  • The steady-state response is simply that part of the total response that remains after teh transient has died out. Thus the steady-state response can still very in a fixed pattern, such as a sine wave, or a temp function that increases with time.

STANDARD TEST SIGNALS

  • The various inputs of disturbances affecting the performance of a system are mathematically represented as standard test signals.
    (a) Sudden input: step signal
    (b) Velocity Type of Input: Ramp signal
    (c) Acceleration type of Input: Parabolic signal
    (d) Sudden Shocks: Impulse signal
    ⇒ signals (a) and (d) are bounded input signals.
    ⇒ signals (b) and (c) are unbounded input signals.
    ⇒ Signals (a), (b) and (c) are for time domain analysis.
    ⇒Signal (d) is important for steady state analysis.
  • Every transfer function representing the control system if of particular type and order.
  • The steady state analysis depends on the type of the system.
  • The type of the system is determined from open loop transfer function G(s)H(s)

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

If P = 0 ⇒ type-0 system

If P = 1 ⇒ type-1 system

If P = 2 ⇒ type-2 system

......................................

If P = n ⇒ type-n system

  • The number of open loop poles occurring at origin determines the type of the system.
  • The transient state analysis depends on order of the system.
  • The order of the system is obtained from closed loop transfer function Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev 
  • The highest power of [1+G(s) H(s)] determines the order of the system.

STEADY STATE RESPONSE ANALYSIS

  • To obtain an expression for error:

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev
E(s) = R(s) - B(s)
E(s) = R(s) - C(s) H(s)
E(s) = R(s) - E(s) G(s) H(s)
E(s) [1 + G(s) H (s)] = R(s)
Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev
Applying the final value theorem,
Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev
Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev
Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Steady State Error for Different types of Inputs

  • Step input,

               R(s) = A/s

   essChapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRevChapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev 

 ess  =  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

K= Position error constant

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

 Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  • Ramp input.

              R(s) = A/s2

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

 Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Kv = Velocity error constant

  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  • Parabolic input,

R(s) = A/s3

ess = Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

 Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Ka = Acceleration error constant

  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Steady State Error for Different Types of Systems
Type-0 system

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  • Step input

           R(s) = A/s

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

 Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  • Ramp input,

           R(s) = A/s2

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

 Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  • Parabolic input,

R(s) = A/s3

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

 =∞

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Observations
 (a) essμ1/K; where K is system gain It means with the in cease of system gain, the value of steady state error will decrease.
 (b) The maximum type number of a liner control system is 2. Beyond type 2, the system exhibits non linear behaviour.

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

TRANSIENT STATE ANALYSIS

  • It deals with the nature of response of a system and depends on order of a system.

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Zero Order System

  • In equation (4.3) if all the terms except a0, b0 are made 0, then the resulting equation describes a zero order system.

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

 Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  • Example: Sensors and transducers

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  • We do not need any time domain analysis because the input, output characteristics are linearly dependent.

First Order System

  • In equation (4.3) if all the terms except a1, a0 and b0 are zero then the resulting expression describes a first order system.

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Let,

K= gain = b0/a0
 T= time constant = a1/a0

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  • Example: RC-filter

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  • Transient analysis

Let,

Xi(s) = 1/s (unit step)

X0 =   Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

 Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

 Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  • The time constant is defined as time taken by the system response to reach 63% of the final value.
  • Example: Thermal system, liquid level system, pneumatic system etc.

Second Order system

  • The response of second order system exhibits continuous and sustained oscillations about the steady state value of the input, with the frequency known as undamped natural frequency wn rad/ sec.
  • These oscillations are damped to the steady state value of the input using appropriate damping methods:
  • The damping is represented as damping factor or damping ratio (x) .
  • The standard transfer function of a 2nd order system is therefore expressed in terms of x and wn ; where, wn = undamped natural frequency in rad/sec
  • Example: All indicating instruments, RLC network

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

 Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Comparing:

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  • Effect of damping on closed loop poles and nature of response:

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Closed loop poles

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

 Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Case-1: Undamped

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

case-2: Under damped

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

case-3: Critical damped

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

case-4: Over damped

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  • Most of the control system are designed for damping less than 1 because the response is neither too fast ( ξ = 1) nor too slow (ξ >1) in reaching the steady state value of the input.

CHARACTERISTICS OF UNDERDAMPED SYSTEMS

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

 Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

(b) Damping coefficient Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev
(c) Time constant of underdamped response (T) 

T= Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

(d) Damped natural frequency (ω d)

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

TRANSIENT ANALYSIS (UNDERDAMPED RESPONSE)

Let,

R(s) = 1/S

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  • The error is given as,

e(t) = r(t) –c(t)
 r(t) = 1

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  • The steady-state error,

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

 

  • The time response expression given by equation (4.4) indicates that for values of ξ < 1, the response presents exponentially decaying Oscillations having a frequency Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev and the time constant of exponential decay is 1/ξωn

TIME RESPONSE SPECIFICATION

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Delay Time (td)

  • The deley time is the time required for the response to reach half the final value the very first time.

           Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Rise Time (tr)

  • The rise time is the time required for the response to rise from 10% to 90%, 5% to 95% or 0% to 100% if its final value. For underdamped second order system, the 0% to 100% is normally used

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

 

peak Time (tp)

  • The peak time is the time required for the response to reach the first peak of overshoot.

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

For

1st peak, n = 1
 2nd peak, n =3

maximum peak Overshoot /maximum percent overshoot

  • The maximum overshoot is the maximum peak value of the response curve measured form unity, and is given by

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  • If the final state value of the response differs from unity, then it is common to use the maximum percentage overshoot. It is defined by Maximum

percentave overshoot =  Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  • The amount of the maximum percentage overshoot indicates the relative stability of the system.

Settling Time (ts)

  • The settling time is the time required for the response curve to reach and stay within a range about the final value of size specified by absolute percentage of the final value (usually 2% or 5%).
  • Settling time for the 2% tolerance band - 4T = Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev
  • Settling time for the 5% tolerance band = 3T = Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

TIME RESPONSE ANALTYSIS OF HIGHER ORDER SYSTEMS

  • Consider a third order polynomial (Characteristic equation)

1 + G(s) H(s) = 0
 s3 + P s2 +q S + K = 0
 (s+P1) (s2 + qs + K1)=0 

Chapter 4 - Time Domain Analysis - Notes, Control System, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  • Time response of a higher order system is done by approximating it to second order system with respect to dominant poles. The number of closed loop poles lying in the dominant region must be greater than or equal to number of poles lying in insignificant region


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