Module Directory

This module provides a basic understanding of the analysis of linear systems and introduction to filter design techniques for analogue signal processing. The Laplace transform and its application in circuit and system theory are introduced, together with the concepts of system transfer function and impulse response, and techniques for deriving the transfer function of a circuit.

The steady-state response of systems to sinusoidal inputs is presented. Bode plotting techniques are covered, and the effects of feedback are investigated, and techniques for ensuring stability are discussed. Butterworth and Chebyshev filter approximations are introduced. After covering the concepts of frequency and impedance transformations, selected standard analysis and design techniques applied to low-pass, high-pass, band-pass and band-stop filters of both passive and active types are examined.

The aim of this module is:

• To introduce techniques for circuit and linear system analysis including time-domain and Laplace domain techniques as well as filter design techniques for analogue signal processing.

By the end of this module, students will be expected to be able to:

1. Demonstrate an understanding of time-domain differential equations relating to circuits, and their Laplace transforms.


2. Conduct basic analysis of circuits in the Laplace and frequency domains.


3. Make a Bode approximation to the steady state response of a circuit and assess the stability of a closed loop system.


4. Apply Butterworth and Chebyshev approximations to ideal filter characteristics.


5. Perform frequency and impedance transformation to derive circuits of low-pass, band-pass and high-pass filters.


6. Realise simple passive and active filters.

Outline Syllabus

• The Laplace Transform
  
  
  
  • The differential equations arising from simple electric circuits. The Laplace transform of impulse, step and decaying exponential waveforms. Laplace impedances of the basic circuit elements. Circuit analysis via Laplace impedances.
  
  
  
  


• Poles and Zeros, Partial Fractions
  
  
  
  • Poles and zeros of transfer functions.  Inverse Laplace transforms for first- and second-order systems.  Use of partial fractions for higher order systems.
  
  
  
  


• Frequency Response
  
  
  
  • Steady-state response to sinusoidal inputs.  Magnitude, phase and group delay.  Bode straight line gain approximation.  From magnitude response to poles and zeros.
  
  
  
  


• Feedback, Gain and Phase Margins
  
  
  
  • Open- and closed-loop systems.  The sensitivity of a closed-loop system to disturbances.  The s-plane and stability.  Gain and phase margins.
  
  
  
  


• Filter Approximations
  
  
  
  • One port, two port and n-port networks. Basic properties of realisable transfer functions.  Constructing transfer function from amplitude response. Minimum phase filter. Common (e.g. Butterworth and Chebyshev) filter approximations. Transfer function of basic low-pass passive filters. 
  
  
  
  


• Transformations in Filter Design
  
  
  
  • Frequency transformation: transformation from low-pass response to high-pass, band-pass, band-stop responses.  Impedance scaling: real and complex scaling.
  
  
  
  


• Passive Filter Analysis and Design 
  
  
  
  • Lossless ladder networks and their properties. Design of passive filters from tables of element values. 
  
  
  
  


• Active Filter Analysis and Design 
  
  
  
  • Introduction to active filters.  Cascade and direct methods. Second-order active blocks (e.g. Sallen & Key). Cascade against direct method in active filter design.
  
  
  
  

This module will be delivered via:

• Lectures.
• Lab groups.
• Classes