1.First order system

1.Step response for first order system

τ = time constant (sec) is defined as ratio of amplitude to initial rate of

t y(t)
τ      63.2% of its ultimate value
2τ    86.5% of its ultimate value
3τ    95% of its ultimate value
4τ    98% of its ultimate value

2.First order response

The unit step response of a first order system with time constant τ and steady state gain kp

2.Second order system

1.Overshoot of second order function

2.For second order process control system

3.Stability of a system

1.Stability at cross over frequency
For stability at cross over frequency AR ≤ 1

2.Process involving series of transfer function
Calculations involving bode stability criteria – for process involving series of transfer functions

4.Laplace Transformation
Y(t) = y(t) – y(s) and X(t) = x(t) – x(s)

1.Laplace transformation of few common functions
Laplace transformation of Δs, sin(at), cosh(at), t^-(1/2), t^1/2, e^-u2, tf(t), e^at

2.Laplace transformation of periodic function
Laplace transfer function f(t) of periodic function with period T

3.Unit delayed function
Laplace of unit delayed function (t-to)

5.Transfer function of proportional derivative controller

6.Amplitude ratio

7.Temperature sensing through thermowell
Temperature sensing through thermowell is a first order process

8.Transfer function of feed forward controller
Transfer function of feed forward controller for perfect disturbance rejection

9.Time constant
1. Time constant for thermometer

2.Time constant for flow of liquid through container
Time constant for flow of liquids through container of volume v and flow rate of liquid q

3.System in series
Equivalent time constant for two systems in series

Offset is defined as difference between setpoint at t tending to infinity and output at t tending to infinity

11.Initial value theoram

12.Final value theoram

13. Steady state gain

14. Half life of first order system

Linearization / linearization of tank model of process flow

16. Output of an impulse function

17.Integral time and proportional gain

Equation for resistor

19.Amplitude ratio and gain margin
Amplitude ratio is inverse of gain margin

20.Equation for control valves

1.Equal percentage


               m = x

21.Transfer function of pure dead time system
Transfer function of pure dead time system with dead time τD is

22.Transfer function of first order system

23.Transfer function of feedback control system
In feedback control system G and H denote open loop and closed loop transfer function then overall transfer function of feedback control system is

24.Descartes rule of sign change for transfer function

Maximum number of positive roots is equal to number of sign changes in f(x).