8.4 Complete Response to Sinusoidal Excitation

Second-order circuits respond to sinusoidal inputs with a combo of steady-state and transient components. The steady-state part oscillates at the input frequency, while the transient part decays over time. This complete response gives us the full picture of how the circuit behaves.

Understanding this response is key to analyzing filters, resonant circuits, and other systems. We'll look at how to calculate both parts, what factors influence them, and how to use this knowledge in real-world applications. It's all about getting the big picture of circuit behavior.

Sinusoidal Response of Second-Order Circuits

Components and Characteristics of Complete Response

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• Complete response of a second-order circuit to sinusoidal excitation comprises steady-state response and transient response
• Steady-state response oscillates at the same frequency as the input sinusoid representing long-term circuit behavior
• Transient response represents temporary circuit behavior immediately after sinusoidal excitation application
• Obtain complete response by solving second-order differential equation describing circuit behavior (typically using Laplace transform methods)
• Characteristic equation determines nature of transient response (overdamped, critically damped, or underdamped)
• Natural frequency and damping ratio influence complete response behavior
• Principle of superposition allows separate calculation of responses to different input components
  • Combine individual responses to obtain total response

Circuit Analysis and Response Calculation

• Solve complete response using second-order differential equation
• Use phasor analysis or complex impedance methods to find steady-state component at excitation frequency
• Determine transient component by solving homogeneous part of differential equation
• Initial conditions at excitation application crucial for transient component magnitude
• Transient component typically takes form of exponential decay terms (real or complex)
  • Underdamped systems include oscillatory terms at damped natural frequency
• Time constant influences rate of transient component decay
• Sum steady-state and transient components to obtain complete response
  • Ensure initial conditions are satisfied

Steady-State vs Transient Response

Steady-State Response Characteristics

• Represents long-term circuit behavior after transients die out
• Oscillates at same frequency as input sinusoid
• Analyze using phasor analysis or complex impedance methods
• Determined by circuit's transfer function at excitation frequency
• Magnitude depends on input sinusoid amplitude and transfer function
• Phase represents time shift between input and output waveforms
• Use Bode plots to graphically represent magnitude and phase response over frequency range

Transient Response Characteristics

• Temporary behavior immediately after sinusoidal excitation application
• Decays over time based on circuit's natural response characteristics
• Determined by solving homogeneous part of differential equation
• Depends on circuit's natural frequency and damping ratio
• Takes form of exponential decay terms (real or complex)
  • Underdamped systems show oscillatory behavior at damped natural frequency
• Initial conditions crucial for determining transient component magnitude
• Time constant influences decay rate of transient component

Magnitude and Phase of Steady-State Response

Calculating Magnitude and Phase

• Determine magnitude using input sinusoid amplitude and circuit's transfer function at excitation frequency
• Calculate phase using circuit's transfer function (represents time shift between input and output)
• Express transfer function as complex function of frequency derived from circuit's impedance characteristics
• Use Bode plots for graphical representation of magnitude and phase response
• Quality factor (Q) influences sharpness of resonance peak in magnitude response
• Bandwidth relates to frequency range where response magnitude within specified range of maximum value
• Utilize polar plots (Nyquist diagrams) for alternative visualization of magnitude and phase response

Factors Influencing Steady-State Response

• Circuit's transfer function shape affects magnitude and phase characteristics
• Resonance causes response magnitude to peak at specific frequency
• Damping ratio impacts resonance peak prominence (underdamped systems show more pronounced peak)
• Quality factor (Q) determines selectivity and sharpness of response curve
• Circuit topology (low-pass, high-pass, band-pass) influences overall response shape
• Component values (resistors, capacitors, inductors) directly affect transfer function
• Input frequency relative to circuit's natural frequency determines response behavior

Frequency Response in Second-Order Circuits

Frequency Response Fundamentals

• Describes how circuit's steady-state output varies in magnitude and phase with changing input frequency
• Transfer function H(jω) mathematically represents frequency response (relates output to input in frequency domain)
• Second-order circuits exhibit characteristic behaviors (resonance, bandwidth, cutoff frequencies)
• Resonance occurs when response magnitude peaks at specific frequency
• Cutoff frequencies (half-power frequencies) define circuit bandwidth
  • Points where output power is half of maximum value
• Damping ratio influences frequency response curve shape
  • Underdamped systems show more pronounced resonance peak
• Phase response typically exhibits total 180° shift from DC to infinity
  • Steepest rate of change near natural frequency

Applications and Design Considerations

• Design second-order low-pass, high-pass, and band-pass filters by manipulating frequency response characteristics
• Utilize frequency response analysis for circuit performance optimization
• Consider trade-offs between bandwidth, selectivity, and phase response in filter design
• Analyze stability of feedback systems using frequency response techniques
• Apply frequency response concepts in audio systems (equalizers, crossover networks)
• Use frequency response analysis in communication systems (channel equalization, signal filtering)
• Implement frequency response shaping in control systems for desired closed-loop behavior