# Responses and Stability Study Notes for GATE EC

By BYJU'S Exam Prep

Updated on: September 25th, 2023

Responses and Stability Study Notes for GATE EC are essential resources for aspiring candidates preparing for the Graduate Aptitude Test in Engineering (GATE) in the Electronics and Communication (EC) stream. These study notes serve as comprehensive guides to help students understand the fundamental concepts and principles related to responses and stability in electronic systems. By delving into the intricacies of system responses and stability, these study notes enable GATE EC aspirants to strengthen their knowledge base and enhance their problem-solving abilities.

The Responses and Stability Study Notes for GATE EC provide a systematic approach to studying the various types of system responses and stability criteria encountered in electronic circuits and systems. They cover topics such as transient and steady-state responses, frequency response analysis, and stability analysis of feedback systems. With clear explanations, illustrative examples, and practice questions, these study notes offer a well-rounded understanding of the subject matter, allowing students to grasp the nuances of responses and stability in electronic systems.

## Responses and Stability

Responses and stability are crucial aspects in the field of electronics and communication (EC). The study of responses focuses on understanding how electronic systems behave in different situations, such as transient and steady-state conditions. Stability analysis, on the other hand, examines the ability of a system to maintain its desired performance over time. Mastering these concepts is essential for GATE EC aspirants, as they form the foundation for designing and analyzing electronic circuits and systems. By studying responses and stability, candidates can gain insights into the behaviour of electronic systems and develop the skills necessary to ensure their reliable and efficient operation.

Transients and steady-state responses are fundamental concepts in the study of electronic systems. These responses describe how a system behaves over time under different conditions. In this article, we will delve into the intricacies of transient and steady-state responses, exploring their significance and applications in electronic circuits and systems.

1. The Basics of Transient Responses:

• Definition of transient response and its significance.
• Analysis of system behaviour during the transient period.
• Time constants and their role in determining transient response characteristics.

• Definition and importance of steady-state response.
• Analyzing system behaviour when it reaches a stable state.
• Frequency and amplitude considerations in steady-state response analysis.
3. Time-Domain Analysis of Transient Responses:

• Techniques and methods for analyzing transient responses in the time domain.
• Response to step inputs and their impact on system behaviour.
• Understanding overshoot, settling time, and rise time in transient response analysis.
4. Frequency-Domain Analysis of Steady-State Responses:

• Introduction to frequency response analysis.
• Bode plots and their role in characterizing steady-state response.
• Transfer function analysis for steady-state frequency response evaluation.
5. Practical Applications and Implications:

• Real-world examples highlight the importance of transient and steady-state responses.
• Impact of response characteristics on circuit and system performance.
• Design considerations for achieving desired transient and steady-state response behaviours.

By gaining a comprehensive understanding of transient and steady-state responses, electronics and communication engineers can effectively analyze and design electronic systems. Whether it’s designing filters, amplifiers, or control systems, mastering these response types is essential for ensuring reliable and optimized performance. So, let’s dive into the fascinating world of transient and steady-state responses and unlock their secrets in the realm of electronic systems.

## Impulse Response and Location of Poles of Transfer Function in s-plane

Understanding the impulse response and the location of poles of a transfer function in the s-plane are crucial aspects of system analysis and design. These concepts provide insights into the system’s time-domain behaviour and stability characteristics, allowing engineers to make informed decisions.

• Impulse Response h(t):

h(t) Ae-at u(t); a>0

### Transfer Function H(s) = L{h(t)}: Pole at s = -a ROC isσ > -a Where σ is the real part of s.

Location of Poles in s-plane and ROC

The location of poles in the s-plane and the Region of Convergence (ROC) are key concepts in the field of signal processing. They provide insights into the stability, behaviour, and frequency response characteristics of systems represented by Laplace transforms.

The pole lies on the left half of the s-plane. ROC is a right-sided, causal system. Since the polelies on LHP and the imaginary axis is included in ROC, the system is stable.
• Impulse Response h(t):Ae-at u(-t); a>0

Transfer Function H(s) = L{h(t)} :
The Pole at s = -a ROC isσ < -a Where σ is the real part of s.
Location of Poles in s-plane and ROC
The pole at s = -a lies on the left half of the s-plane. The ROC is a not right-sided, non-causal system. Since the imaginary axis is not included in ROC, the system is unstable.
• Impulse Response h(t):h(t) Aeat u(t); a>0

Transfer Function H(s) = L{h(t)} :
The Pole at s = -a ROC isσ < a Where σ is a real part of s.
Location of Poles in s-plane and ROC:
The pole lies on the right half of the s-plane. ROC does not include the imaginary axis, or causal system. Since the pole lies on RHP and the imaginary axis, is not included in ROC, the system is unstable.
• Impulse Response h(t) :h(t) Aeat u(-t); a>0

Transfer Function H(s) = L{h(t)} :
The Pole at s = +a ROC isσ < +a Where σ is the real part of s.
Location of Poles in s-plane and ROC:
The pole lies on the right half of the s-plane. The ROC include the imaginary axis, a non-causal system. Since the imaginary axis is included in ROC, the system is stable.
• Impulse Response h(t) :h(t) Ae-a|t|; a>0

Transfer Function H(s) = L{h(t)} :
Pole at s = -a, +a ROC is -a <σ < + a Where σ is the real part of s.
Location of Poles in s-plane and ROC :
The pole at lies on RHP and the pole at s = – a, lie on LHP. ROC is a not right-sided, non-causal system. Since the imaginary axis is included in ROC, the system is stable.
• Impulse Response h(t) :

Transfer Function H(s) = L{h(t)} :
Pole at s = -a, -b ROC is -a <σ < +a Where σ is a real part of s.
Location of Poles in s-plane and ROC :
The pole lies on LHP. The ROC does include an imaginary axis, a non-causal system. Since the imaginary axis is not included in ROC, the system is unstable.
• Impulse Response h(t): h(t) A u(t)

Transfer Function H(s) = L{h(t)}:
Pole at s = 0 ROC isσ > 0 Where σ is a real part of s.
Location of Poles in s-plane and ROC:
The pole lies on the imaginary axis. The ROC is a right-sided, causal system. Since the imaginary axis is not included in ROC, the system is unstable.
• Impulse Response h(t): h(t) = At(u)t

Transfer Function H(s) = L{h(t)}:
Double Pole at s = 0 ROC isσ > 0 Where σ is a real part of s.
Location of Poles in s-plane and ROC:
s = -a – jb, +jb, The pole at lies on the imaginary axis. The ROC does not include the imaginary axis, or causal system. Since, the imaginary axis, is not included in ROC, the system is unstable.
• Impulse Response h(t):h(t) = 2A cos bt u(t)

Transfer Function H(s) = L{h(t)}:
Pole at s = -jb, +jb ROC isσ > 0 Where σ is a real part of s.
Location of Poles in s-plane and ROC:
The pole lies on the imaginary axis. The ROC does not include the imaginary axis, or causal system. Since the imaginary axis is not included in ROC, the system is unstable.
• Impulse Response h(t):h(t) = 2Aebt cos bt u(t) where a > 0

Transfer Function H(s) = L{h(t)}:
Pole at s = a-jb, -a+jb ROC isσ > -a Where σ is a real part of s.
Location of Poles in s-plane and ROC:
The pole lies on the right half of the s-plane. The ROC does not include an imaginary axis or causal system. Since poles lie on. RHP and the imaginary axis are included in ROC, the system is stable.
• Impulse Response h(t): h(t) = 2At cos bt u(t)

Transfer Function H(s) = L{h(t)}:
Double Pole a s = -jb, +jb ROC isσ > a Where, σ is a real part of s.
Location of Poles in s-plane and ROC:
The pole at s = -jb, + jb, lies on the imaginary axis. The ROC does not include an imaginary axis or causal system. Since the imaginary axis is included in ROC, the system is unstable.

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