Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)

Electrical Engineering (EE): Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)

The document Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Network Theory (Electric Circuits).
All you need of Electrical Engineering (EE) at this link: Electrical Engineering (EE)

1. Introduction
Many applications of control theory are to servomechanisms which are systems using the feedback principle designed so that the output will follow the input. Hence there is a need for studying the time response of the system. The time response of a system may be considered in two parts:

  • Transient response: this part reduces to zero as t → ∞
  • Steady-state response: response of the system as t → ∞

2. Response of the first order systems

  • Consider the output of a linear system in the form Y(s) = G(s)U(s) where Y(s) : Laplace transform of the output, G(s) : transfer function of the system and U(s) : Laplace transform of the input.
  • Consider the first order system of the form ay + y = u , its transfer function is
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
  • For a transient response analysis it is customary to use a reference unit step function u(t) for which

Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)

  • It then follows that
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
  • On taking the inverse Laplace of equation, we obtain
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
  • The response has an exponential form. The constant 'a' is called the time constant of the system.
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
  • Notice that when t = a, then y(t) = y(a) = 1-e-1 = 0.63. The response is in two-parts, the transient part e-t/a, which approaches zero as t →∞ and the steady-state part 1, which is the output when t → ∞.
  • If the derivative of the input are involved in the differential equation of the system, that is Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE) then its transfer function is

Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)

  • where
    K = b / a
    z =1/ b : the zero of the system
    p =1/ a : the pole of the system
  • When U(s) =1/s , Equation can be written as
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
  • Hence,
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
  • With the assumption that z>p>0 , this response is shown in
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
  • We note that the responses to the systems have the same form, except for the constant terms K1 and K2. It appears that the role of the numerator of the transfer function is to determine these constants, that is, the size of y(t), but its form is determined by the denominator.

3. Response of second order systems 

  • An example of a second order system is a spring-dashpot arrangement, Applying Newton’s law, we find
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
  • where k is spring constant, µ is damping coefficient, y is the distance of the system from its position of equilibrium point, and it is assumed that  y(0) = y(0)' = 0.
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
  • Hence,  Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
  • On taking Laplace transforms, we obtain,

Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)

  • where K = 1/ M , a1 = µ/M , a2 = k/M. Applying a unit step input, we obtain
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
  • where Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE) are the poles of the transfer function that is, the zeros of the denominator of G(s).
  • There are there cases to be considered:

over-damped system:

  • In this case p1 and p2 are both real and unequal. Equation can be written as
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)

critically damped system:

  • In this case, the poles are equal: p1 = p2 = a1 / 2 = p , and
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
  • Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)

under-damped system:
In this case, the poles p1 and p2 are complex conjugate having the form p1,2 = α + iβ where α = a1/2 and Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
The three cases discussed above are plotted as:
Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
There are two important constants associated with each second order system:

  • The undamped natural frequency ωn of the system is the frequency of the response shown in Fig. Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
  • The damping ratio ξ of the system is the ratio of the actual damping µ(= a1M) to the value of the damping µc , which results in the system being critically damped.  Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
  • also,
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)

Some definitions:
Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)

  • Overshoot: defined as
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
  • Time delay τd: the time required for a system response to reach 50% of its final value
  • Rise time: the time required for the system response to rise from 10% to 90% of its final value
  • Settling time: the time required for the eventual settling down of the system response to be within (normally) 5% of its final value
  • Steady-state error ess: the difference between the steady state response and the input.

4. Steady state error 

  • Consider a unity feedback system
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
  • where
    r(t) : reference input
    c(t) : system output
    e(t) : error
  • We define the error function as
  • e(t) = r(t) − c(t)
  • hence, Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE) Since E(s) = R(s) − A(s)E(s) , it follows that Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE) and by the final value theorem
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
  • We now define three error coefficients which indicate the steady state error when the system is subjected to three different standard reference inputs r(s).

step input: r(t) = ku(t)
Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)

  • Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)  called the position error constant, then
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)

Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
Ramp input: r(t) = ktu(t)

  • In this case, Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE) is called the velocity error constant.
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)

Parabolic input: r(t) = 1/2 kt2 u(t)

  • In this case, Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE) where Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE) is called the acceleration error constant.
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
  • From the definition of the error coefficients, it is seen that ess depends on the number of poles at s = 0 of the transfer function. This leads to the following classification. A transfer function is said to be of type N if it has N poles at the origin. Thus if
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
  • At s = 0, Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE) K1 is called the gain of the transfer function. Hence the steady state error ess depends on j and r(t) as summarized in Table
    Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE)
The document Study Notes for Transient & Steady State Response Notes | Study Network Theory (Electric Circuits) - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Network Theory (Electric Circuits).
All you need of Electrical Engineering (EE) at this link: Electrical Engineering (EE)

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