# BODE DIAGRAMS AND FREQUENCY RESPONSE

This is a section in the chapter on classical feedback analysis techniques. Use the left hand toolbar to access the other chapters and themes.

It is implicit in several of these chapters that students have core competence in some mathematical topics such as polynomials, roots, complex numbers, exponentials and Laplace. More information on these can be found in the Mathematics theme on the left hand toobar. This section contains the following topics. Under each topic there are hardcopy (pdf) notes, a video talk through of key derivations with example problems and also occasional tutorial sheets for users to test themselves. |

- Bode diagrams 1 - basic concepts and illustration of frequency response.
- Bode diagrams 2 - frequency response gain and phase for transfer functions.
- Bode diagram 3 - efficient computation of frequency response.
- Bode diagram 4 - frequency response with RHP poles and zeros.
- Bode diagram 5 - tutorial sheets on frequency response.
- Bode diagrams 6 - plotting frequency response.
- Bode diagrams 7 - what is a Bode diagram?.
- Bode diagrams 8 - sketching for single simple factors.
- Bode diagrams 9 - sketching for multiple simple factors.
- Bode diagrams 10 - sketching with asymptotic information.
- Bode diagrams 11 - tutorial sheet on sketching with asymptotic methods and MATLAB.
- Bode diagrams 12 - lag compensator.
- Bode diagrams 13 - impact of lag compensator.
- Bode diagrams 14 - lead compensator.
- Bode diagrams 15 - impact of lead compensator.
- Bode diagrams 16 - lead-lag compensator.
- Bode diagrams 17- quadratic factors and resonance.
- Bode diagrams 18 - bandwidth.
- TUTORIAL SHEET on Bode diagrams

## Bode diagrams 1 - basic concepts and illustration of frequency response

Introduces the concept of frequency response and uses examples to demonstrate how the gain and phase of the output change as the frequency of the input is changed. Gives definitions for gain and phase in terms of frequency response.

- A summary of key facts and derivations is given in the pdf file.
- A talk through video is on youtube.

Frequency response describes what? |

## Bode diagrams 2 - frequency response gain and phase for transfer functions

Demonstrates how to solve for the frequency response parameters of a system from a transfer function model and hence shows that the gain and phase have simple analytic dependence upon the system parameters.

- A summary of key facts and derivations is given in the pdf file.
- A talk through video is on youtube.

Frequency response parameters have a simple link to transfer function parameters? |

## Bode diagram 3 - efficient computation of frequency response

Building on the definition for system gain and phase in terms of transfer function parameters, this video shows how using a factorised version of the transfer function enables the user to write down insightful expressions for gain and phase by inspection. Focus is on factors with LHP roots.

- A summary of key facts and derivations is given in the pdf file.
- A talk through video is on youtube.

Frequency response, by hand, is computed using which technique? |

## Bode diagram 4 - frequency response with RHP poles and zeros

Students often make silly mistakes when computing the frequency response of systems with RHP factors. This video presents a simple approach for avoiding simple errors and getting the answer right first time.

- A summary of key facts and derivations is given in the pdf file.
- A talk through video is on youtube.

For a system with RHP zeros, whihc of the following is not advisable when determining the frequency response. |

## Bode diagram 5 - tutorial sheet on frequency response

Gives a number of tutorial questions on finding the frequency response for a number of alternative transfer functions for students to try. Also provides quick worked solutions.

- A summary of key facts and derivations is given in the pdf file.
- A TUTORIAL sheet with typical questions.
- A talk through video is on youtube.

## Bode diagrams 6 - plotting frequency response

Introduces the plotting of frequency response information and illustrates the use of MATLAB to do so. Indicates the weaknesses of using linear graph scales for these plots.

- A summary of key facts and derivations is given in the pdf file.
- A talk through video is on youtube.

## Bode diagrams 7 - what is a Bode diagram?

Tackles the weaknesses of simple graphical displays of frequency response information and thus introduces the definition of a Bode diagram which uses logarithmic scales. Discusses some key logarithmic values which help with Bode diagram interpretation.

- A summary of key facts and derivations is given in the pdf file.
- A talk through video is on youtube.

Which statement is incorrect? |

## Bode diagrams 8 - sketching for single simple factors

Develops Bode diagrams for simple poles, zeros and integrators from first principles. Introduces the concept of approximation and known values at key frequencies.

- A summary of key facts and derivations is given in the pdf file.
- A talk through video is on youtube.
- The MATLAB file bodeasymptote.m

## Bode diagrams 9 - sketching for multiple simple factors

Develops Bode diagrams for systems comprising mutliple simple poles, zeros and integrators from first principles. Demonstrates how rules of logarithms allow simple insights into the construction of Bode diagrams, but recognises that albeit conceptually simple, the method is cumbersome.

- A summary of key facts and derivations is given in the pdf file.
- Talk through video is on youtube.
- The MATLAB file bodeasymptote.m

## Bode diagrams 10 - sketching with asymptotic information

Builds on the previous video by showing how some asymptotic information in the Bode plot can be obtained with minimal or no computation. This asymptotic information can be used as the basis for suprisingly accurate Bode diagram sketching for systems with multiple simple poles and zeros and requires minimal extra computations.

Which statement is incorrect? |

## Bode diagrams 11 - tutorial sheet on sketching with asymptotic methods and MATLAB

Demonstrates, through several examples, how simple asymptotic information and a few explicit computations can capture a fairly accurate bode diagram which thus is useful for insight into any subsequent design. Also demonstrates the use of MATLAB to form exact plots and shows how these compare to the hand drawn sketches.

- A summary of key facts, derivations and tutorial questions is given in the pdf file.
- More extensive tutorial questions.
- Talk through video is on youtube.

How can you initialise the aysmptotes of a gain plot when the system has an integrator? |

## Bode diagrams 12 - lag compensator

Gives a detailed analysis of the bode diagram of a lag compensator. Core information is the ratio of pole to zero. [Warning: includes a minor typo on slide 9 - high frequency gain should be K]

- A summary of key facts and derivations is given in the pdf file.
- Talk through video 1 is on youtube.

## Bode diagrams 13 - impact of lag compensator

Builds on analysis of the bode diagram of a lag compensator and properties of Bode diagrams to show how compensation with a Lag affects the Bode diagram of a system, that is compares the Bode diagrams of G(s) and G(s)M(s). Uses very simple observations and computations so that the compensated sketch can be done by inspection.

- A summary of key facts and derivations is given in the pdf file.
- Talk through video 1 is on youtube. ([Warning: includes a minor typo on slide 6 - high frequency gain should be K])

Which statement is incorrect? |

## Bode diagrams 14 - lead compensator

Gives a detailed analysis of the bode diagram of a lead compensator and how this is affected by the pole/zero ratio.

- A summary of key facts and derivations is given in the pdf file.
- Talk through video 1 is on youtube. ([Warning: includes a minor verbal typo on penultimate slide where geometric mean is described as sqrt(2) rather than sqrt(1.5). ])

## Bode diagrams 15 - impact of lead compensator

Builds on analysis of the bode diagram of a lead compensator and properties of Bode diagrams to show how compensation with a Lead affects the Bode diagram ofa system, that is compares the Bode diagrams of G(s) and G(s)M(s). Uses very simple observations and computations so that the compensated sketch can be done by inspection.

- A summary of key facts and derivations is given in the pdf file.
- Talk through video 1 is on youtube.

Which statement is incorrect? |

## Bode diagrams 16 - lead-lag compensator

Gives a detailed analysis of the bode diagram of a lead-lag compensator and emphasises key attributes and thus differences with a lead compensator. Also illustrates that a good sketch can be produced using just a few elementary observations at key corner frequencies.

- A summary of key facts and derivations is given in the pdf file.
- Talk through video 1 is on youtube.

## Bode diagrams 17- quadratic factors and resonance

Considers transfer functions which include complex poles, that is under-damped modes, and investigates the associated Bode diagrams. Shows that under-damped modes can lead to peaks in the gain plot; these peaks are evidence of resonance, that is frequencies where the gain is disproportionately high.

- A summary of key facts and derivations is given in the pdf file.
- Talk through video 1 is on
*youtube*.

## Bode diagrams 18 - bandwidth

Introduces possible definitions and interpretations of bandwidth and illustrates how this can be estimated from Bode gain plots. Also, illustrates links between open-loop bandwidth and the expected bandwidth of the same system when connected with unity negative feedback.

- A summary of key facts and derivations is given in the pdf file.
- Talk through video 1 is on youtube.

Which definition of bandwidth is correct? |