 Hello everyone, I am Mrs. Meenakshi Shrigandhi from Vulturend Institute of Technology, Sholapur. Welcome to the video lecture on Transient Response Analysis. Learning outcome, at the end of this session, student will be able to explain the time response of a control system in terms of transient response and steady state response. Let's understand what is time response of a control system means. Most of the control system use time as its independent variable, so it is important to analyze the response given by the system for the applied excitation which is function of time. Analysis of response means to see the variations of output with respect to time. The output behavior with respect to time should be within specified limits to have a satisfactory performance of the system. The complete base of stability analysis lies in the time response analysis. The system stability, system accuracy and system evaluation is always based on the time response analysis and its corresponding results. Definition of time response. The response given by the system which is function of the time to the applied excitation is called time response of a control system. In any practical system, output of the system takes some finite time to reach to its final value. This time varies from system to system and is dependent on many factors. The final value achieved by the output also depends on the different factors like friction, mass or inertia of moving elements, some non-linearities present, etc. Let's see an example for time response. Consider a simple ammeter as a system. It is connected in a system so as to measure current of magnitude of 30 ampere. The ammeter pointer hence must deflect to show us 30 ampere reading on it which is the ideal value that it must show. Now pointer will take some finite time to stabilize to indicate some reading and after stabilizing also it depends on various factors like friction, pointer, inertia, etc. And even after this, it is not guaranteed whether it will show us accurate 30 ampere or not. Based on the example, we can classify the total output response into two parts. First is the part of output during the time it takes to reach to its final value which is called as transient response. And second is the final value attained by the output which will be near towards desired value if system is stable and accurate which is called as steady state response. Transient response. The output variation during the time it takes to achieve its final value is called as transient response. The time required to achieve the final value is called as the transient rate. This can also be defined as that part of the time response which decays to zero after some time as system output reaches to its final value. The transient response can be exponential or oscillatory. As shown in the figure, this is the exponential transient response and this is the oscillatory transient response. Symbolically, it is denoted as C of T of T. To get the desired output, the system must pass satisfactorily through transient rate. Transient response must vanish after some time to get the final value which is closer to the desired value. Such system in which the transient response dies after some time are called as stable systems. Mathematically, for stable operating system the equation is given as C of T is equal to zero. Try to think and answer what information about the system we can get from transient response. Pause the video for some time and note down the answer in your book. From transient response, we can get the following information about the system. When the system has started showing its response to the applied excitation, what is the rate of rise of output? This can indicate about the speed of the system whether output is increasing exponentially or it is oscillating. If output is oscillating, whether it is overshooting its final value, what is the settling time? When it is settling down to its final value. So this information is achieved from transient response. Let's see study state response. It is that part of the time response which remains after complete transient response vanishes from the system output. The study state response is generally the final value achieved by the system output. Its significance is that it tells us how far the actual output is from its desired value. The study state response indicates the accuracy of the system. The symbol for study state output is C of S is. Try to think and answer what information about the system we can get from study state response. Pause the video for some time and note down the answer in your book. From study state response we can get the following information about the system. How much away the system output is from its desired value which indicates error. Whether this error is constant or varying with time. So the entire information about the system performance can be obtained from transient and study state response. Hence the total time response C of T is given as C of T is equal to C of S is plus C of T of T. Where C of S is representing the study state output where C of T of T is giving the transient state output. Difference between the desired output and the actual output of the system is called as study state error which is denoted as E of S is. This error indicates the accuracy and plays an important role in designing the system. This is indicated by the overall time response containing the exponential transient output and study state output when the input applied is the step input. As shown in the figure the difference between the desired output and the actual output is the study state error. This is the time response containing the oscillatory transient output and the study state output when the input applied is the step input. Again you can see that the difference between the desired output and the actual output is nothing but the study state error. Let's see the standard test inputs. In practice many signals are available which are the functions of the time and can be used as the reference inputs for the various control system. These signals are the step, ramp, sawtooth type, square wave, triangular, etc. But while analyzing the system it is highly impossible to consider each one of it as an input and study the response. Hence from the analysis point of view these signals which are most commonly used as reference inputs are defined as standard test inputs. The evaluation of the system can be done on the basis of the response given by the system to the standard test inputs. Once system behaves satisfactorily to a test input its time response to actual input is assumed to be up to the mark. These standard test signals are step input, ramp input, parabolic input, and impulse input. The standard test input is the step input which is a position function. It is a sudden application of input at a specified time as shown in the figure. Mathematically it can be defined as r of t equal to a for t greater than or equal to 0 and it is equal to 0 when t is less than 0. If a is equal to 1 which is indicating magnitude then it is called as unit step function and it is denoted by u of t. Laplace transform of such input is a of s the next standard test input is the ramp input which is the velocity function. It is a constant rate of change in input that is the gradual application of input as shown in the figure. Mathematically it can be defined as r of t is equal to a of t for t greater than or equal to 0 and it is equal to 0 for t less than 0. If a is equal to 1 then it is called as unit ramp input. Laplace transform of such input is a by s square. The next standard test input is the impulse input. It is the input applied instantaneously for short duration of time of very high amplitude as shown in the figure. Mathematically it can be denoted as r of t is equal to a for t equal to 0 and it is equal to 0 for t not equal to 0. It is the pulse whose magnitude is infinite while its pulse tends to 0. Area of the impulse is nothing but its magnitude. If a is equal to 1 then it is called as unit impulse response denoted as delta of t. Laplace transform of such input is 1. The next standard test input is the parabolic input which is the acceleration function. This is the input which is 1 degree faster than a ramp type of input as shown in the figure. Mathematically it can be denoted as r of t is equal to a t square by 2 for t greater than or equal to 0 and it is equal to 0 for t less than 0. If a is equal to 1 then it is called as unit parabolic input. Laplace transform of such input is a by s cube. These are my references. Thank you.