Notes: Steady State Response | Network Theory (Electric Circuits) - Electrical Engineering (EE) PDF Download

Steady state response is the behavior of a system after it has reached a stable condition in response to a constant input or disturbance. It represents the system's behavior once all transient effects have dissipated, and the system has stabilized.

Characteristics:

• Occurs when the system's output remains constant or oscillates at a constant amplitude and frequency.
• Reflects the long-term behavior of the system under a constant or periodic input or disturbance.

Importance:

• The steady state response is typically easier to analyze and predict compared to the transient response.
• Steady-state analysis is crucial in engineering and science for understanding system performance over the long term.

Applications in Control Theory:

• Commonly applied to servomechanisms, which are systems designed to use feedback to ensure the output follows the input.
• Essential for studying the time response of systems.

Components of Time Response:

• The time response of a system is divided into two parts:
  1. Transient Response: The initial reaction of the system before reaching a steady state.
  2. Steady State Response: The behavior after the transient effects have subsided, and the system is stable.

Utility:

• Analyzing the steady state response helps in predicting system performance and ensuring desired outcomes in practical applications.

Transient Response

After applying an input to an electric circuit, the output takes certain time to reach steady state. So, the output will be in transient state till it goes to a steady state. Therefore, the response of the electric circuit during the transient state is known as transient response.
The transient response will be zero for large values of ‘t’. Ideally, this value of ‘t’ should be infinity. But, practically five time constants are sufficient.

Presence or Absence of Transients

Transients occur in the response due to sudden change in the sources that are applied to the electric circuit and / or due to switching action. There are two possible switching actions. Those are opening switch and closing switch.

• The transient part will not present in the response of an electrical circuit or network, if it contains only resistances. Because resistor is having the ability to adjust any amount of voltage and current.
• The transient part occurs in the response of an electrical circuit or network due to the presence of energy storing elements such as inductor and capacitor. Because they can’t change the energy stored in those elements instantly.

Inductor Behavior

Assume the switching action takes place at t = 0. Inductor current does not change instantaneously, when the switching action takes place. That means, the value of inductor current just after the switching action will be same as that of just before the switching action.
Mathematically, it can be represented as

Capacitor Behavior

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

Mechanical System of second order

Steady state Response

The part of the time response that remains even after the transient response has become zero value for large values of ‘t’ is known as steady state response. This means, there won’t be any transient part in the response during steady state.

Inductor Behavior

If the independent source is connected to the electric circuit or network having one or more inductors and resistors (optional) for a long time, then that electric circuit or network is said to be in steady state. Therefore, the energy stored in the inductor(s) of that electric circuit is of maximum and constant.

Mathematically, it can be represented as

Therefore, inductor acts as a constant current source in steady state.
The voltage across inductor will be

So, the inductor acts as a short circuit in steady state.

Capacitor Behavior

If the independent source is connected to the electric circuit or network having one or more capacitors and resistors (optional) for a long time, then that electric circuit or network is said to be in steady state. Therefore, the energy stored in the capacitor(s) of that electric circuit is of maximum and constant.

Mathematically, it can be represented as

Therefore, capacitor acts as a constant voltage source in steady state.

The current flowing through the capacitor will be

So, the capacitor acts as an open circuit in steady state.

Finding the Response of Series RL Circuit

Consider the following series RL circuit diagram.
In the above circuit, the switch was kept open up to t = 0 and it was closed at t = 0. So, the DC voltage source having V volts is not connected to the series RL circuit up to this instant. Therefore, there is no initial current flows through inductor.
The circuit diagram, when the switch is in closed position is shown in the following figure.
Now, the current i flows in the entire circuit, since the DC voltage source having V volts is connected to the series RL circuit.
Now, apply KVL around the loop.
The above equation is a first order differential equation and it is in the form of
By comparing Equation 1 and Equation 2, we will get the following relations.
The solution of Equation 2 will be

Where, k is the constant.
Substitute, the values of x, y, P & Q in Equation 3.
We know that there is no initial current in the circuit. Hence, substitute, t = 0 and 𝑖 = 0 in Equation 4 in order to find the value of the constant k.
Substitute, the value of k in Equation 4. 
Therefore, the current flowing through the circuit is 
So, the response of the series RL circuit, when it is excited by a DC voltage source, has the following two terms.

We can re-write the Equation 5 as follows −
Where, τ is the time constant and its value is equal to L/R.
Both Equation 5 and Equation 6 are same. But, we can easily understand the above waveform of current flowing through the circuit from Equation 6 by substituting a few values of t like 0, τ, 2τ, 5τ, etc.

Finding the Response of Series RL Circuit

Consider the following series RL circuit diagram.

In the above circuit, the switch was kept open up to t = 0 and it was closed at t = 0. So, the AC voltage source having a peak voltage of Vm volts is not connected to the series RL circuit up to this instant. Therefore, there is no initial current flows through the inductor.
The circuit diagram, when the switch is in closed position, is shown in the following figure.

Now, the current i(t) flows in the entire circuit, since the AC voltage source having a peak voltage of Vm volts is connected to the series RL circuit.

We know that the current i(t) flowing through the above circuit will have two terms, one that represents the transient part and other term represents the steady state.

Mathematically, it can be represented as
In the previous chapter, we got the transient response of the current flowing through the series RL circuit. It is in the form of

Calculation of Steady State Current

If a sinusoidal signal is applied as an input to a Linear electric circuit, then it produces a steady state output, which is also a sinusoidal signal. Both the input and output sinusoidal signals will be having the same frequency, but different amplitudes and phase angles.
We can calculate the steady state response of an electric circuit, when it is excited by a sinusoidal voltage source using Laplace Transform approach.
The s-domain circuit diagram, when the switch is in closed position, is shown in the following figure.

In the above circuit, all the quantities and parameters are represented in s-domain. These are the Laplace transforms of time-domain quantities and parameters.

The Transfer function of the above circuit is

Substitute s=jω in the above equation.
Magnitude of H(jω) is

Phase angle of H(jω) is
We will get the steady state current i_ss(t) by doing the following two steps −
The steady state current i_ss(t) will be
Substitute the value of i_ss(t) in Equation 2.
We know that there is no initial current in the circuit. Hence, substitute t = 0 & i(t) = 0 in Equation 3 in order to find the value of constant, K.
Substitute the value of K in Equation 3.

Equation 4 represents the current flowing through the series RL circuit, when it is excited by a sinusoidal voltage source. It is having two terms. The first and second terms represent the transient response and steady state response of the current respectively.
We can neglect the first term of Equation 4 because its value will be very much less than one. So, the resultant current flowing through the circuit will be