# Transient Response in Control System

By Deepak Yadav|Updated : March 26th, 2023

Transient Response is an important concept in the analysis of dynamic systems. In engineering and physics, dynamic systems are systems that change over time, often in response to an external stimulus or disturbance. Understanding how a system responds to such stimuli is critical in many fields, including control systems, signal processing, and electronics.

Transient Response refers to the behavior of a system during the time it takes to reach a stable state after a disturbance. In other words, it's the system's initial response to a sudden change in its inputs. On the other hand, steady-state response refers to the behavior of a system when it has reached a stable state and is no longer changing in response to the original disturbance. In this article, we will explore these concepts in more detail, including the factors that affect the transient and steady-state response, and how they are used in practice.

## Transient Response

Transient response refers to the initial behavior of a dynamic system following a sudden change or disturbance in its input. This response is often characterized by changes in the system's output that occur during the time it takes for the system to reach a new, stable state. Understanding transient response is essential in the analysis and design of many dynamic systems, including control systems, signal processing, and electronics. By studying the factors that affect the transient response, engineers and scientists can better predict and control the behavior of dynamic systems, leading to more efficient and effective designs.

Presence or Absence of Transients: Transients occur in the response due to instant change in the sources that are applied to the electric circuit and or due to switching action in the circuit. The two possible switching actions are:- Opening the switch and Closing the switch. There are 3 basic components in Linear Electrical Circuits:

• Resistance (R)
• Capacitance (C)
• Inductance (L)

The transient part will be absent in the response of an electrical circuit or network if the circuit contains only resistances. This is because the resistor is having the ability to adjust any amount of voltage and current in it.

The transient part is present in the response of an electrical circuit or network due to the presence of energy-storing elements like inductors and or capacitors. This is Because they can’t change the energy stored in those elements instantly.

## Inductor Behavior

Let us assume the switching action takes place at time t = 0. The Inductor current does not change instantaneously (due to inductor property) when the switching action takes place. That means, the value of the inductor current just after the switching action as well as just before the switching action will same.

Mathematically, Inductor current can be represented as IL(0-) = IL(0+)

## Capacitor Behavior

The capacitor voltage does not change instantaneously similar to that of the inductor current when the switching takes place. That means, the value of capacitor voltage just after the switching action as well as just before the switching action will same.

Mathematically, Capacitor Voltage can be represented as VC(0-) = VC(0+)

Voltage-Current Relationships for Different Passive Elements:-

Note: VR, VL, and VC are the voltages across R, L, and C elements respectively while iR, iL, and iC Represent the current through R, L, and C elements respectively.

Now, Steps for Finding a Transient Response:
• Step 1: Identify the variable of interest (Inductor current for R-L circuit, Capacitor voltage for R-C circuit).
• Step 2: Determine the initial value of the variable according to the circuit.
• Step 3: Calculate the final value of the variable according to the circuit.
• Step 4: Calculate the time constant for the circuit (Circuit after switching).
Resistive Circuits will have NO Transient

Consider the resistive circuit as shown below

When the switch is ON, the voltage across R becomes V volts immediately (in zero time).

Instantaneous Voltage v(t) = V = iR for t ≥ 0

Instantaneous Voltage v(t) = 0for t <0

First Order RC Circuit:

Using a loop, the sum of the voltage will be zero.

VTh = RTh · i (t) + v(t)

... Eq. (1)

Substitute in the capacitor current.

... Eq. (2)

This simplifies the differential equation,

... Eq. (3)

Move the second term to the right-hand side and then divide by the numerator

... Eq. (4)

The indefinite integral resolves to the following form

... Eq. (5)

D is a constant of integration. Removing the natural log and solving for v(t) shows

... Eq. (6)

The constant eD represented by A can be found at time t = 0

eDA=v(0)- VTH ... Eq. (7)

We can also solve for the final steady state response.

... Eq. (8)

Substitute eq. (8) and (7) into eq. (6).

... Eq. (9)

the time constant from the product in the exponential term.

... Eq. (10)

Therefore, the final form of the complete response is:-

## Inductor and Thevenin’s Equivalent Circuit

Below is an inductor element connected to a circuit that has been reduced to its Thevenin equivalent. Now we will use Thevenin theorem to solve this circuit:

Applying KVL to the loop of this circuit

VTh = RTh · i (t) + v(t) ... Eq. (11)

The voltage across an inductor is given by:-

... Eq. (12)

Use this in eq. (11).

... Eq. (13)

Rearrange the equation into a form that is easier to integrate.

... Eq. (14)

on Divide by the term in brackets, and integrate.

... Eq. (15)

The integral becomes,

... Eq. (16)

... Eq. (17)

At time t = 0, the constant eD = A is revealed.

... Eq. (25)

As time moves to infinity, the steady-state or forced response is found.

... Eq. (18)

The time constant is,

... Eq. (19)

Therefore the complete response of the current through an inductor element connected to a Thevenin equivalent circuit is given by

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## FAQs about Transient Response

• Transient response refers to the behaviour of a system immediately after a change in its input, while steady-state response refers to the behaviour of the system once it has settled into a stable state after the transient response has decayed.

• The transient response of a system is determined by the system's natural response and forced response. The natural response depends on the system's inherent properties, while the forced response depends on the input applied to the system.

• The steady-state response of a system is directly proportional to the input signal once the transient response has decayed. In other words, if the input signal changes, the steady-state response of the system will also change.

• Understanding the transient and steady-state response of a control system is crucial for designing effective controllers and ensuring stable operation. The transient response determines how quickly the system responds to changes in the input, while the steady-state response determines the accuracy of the system's output.

• No, a system cannot have a steady-state response without a transient response. The transient response is necessary for the system to reach a stable state and achieve a steady-state response.

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