Modelling and control by Anthony Rossiter

# 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

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.

#### Quick test question

Frequency response describes what?
A. How the transient behaviour varies with a sinusoidal input..
B. Characterises the asymptotic output behaviour for a sinusoidal input.
C. How the amplitude of oscillation of an output signal depends upon the system input..
D. None of the above.

## 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.

#### Quick test question

Frequency response parameters have a simple link to transfer function parameters?
A. Yes.
B. No.

## 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.

#### Quick test question

Frequency response, by hand, is computed using which technique?
A. Substitute s=jw, multiply out long hand and reduce to a single complex number.
B. Factorise and find the gain and phase of each factor.
C. Factorise and sketch an argand diagram for all factors.
D. None of the above.

## 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.

#### Quick test question

For a system with RHP zeros, whihc of the following is not advisable when determining the frequency response.
A. Multiply our the transfer function long hand to form a single complex number.
B. Factorise and find the gain and phase of each factor.
C. Factorise and sketch an argand diagram for any factors with RHP roots.
D. None of the above.

## 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.

## 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.

#### Quick test question

Which statement is incorrect?
A. A bode diagram is a plot of 20log10(mod(G(jw))) against log10(w).
B. A bode diagram contains plots of gain and phase against frequency.
C. Bode diagrams use frequency on a logarithmic scale.
D. None of the above.

## 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.

## 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.

## 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.

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

#### Quick test question

Which statement is incorrect?
A. Asymptotes are often good enough for representing the gain bode diagram.
B. Asymptotes are a convenient mechanism for beginning a Bode diagram.
C. Asymptotes are often good enough for representing the phase bode diagram.
D. None of the above.

## 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.

#### Quick test question

How can you initialise the aysmptotes of a gain plot when the system has an integrator?
A. Find the exact value of G(jw) at the smallest corner frequency and use this.
B. Ignore the integrator and create the asymptotes without this.
C. Do an asymptotic computation at any corner frequency.
D. None of the above.

## 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])

#### Quick test question

Which statement is incorrect?
A. A lag compensator is a low gain strategy.
B. A lag compensator is a high gain strategy.
C. A lag compensator exploits the phase characteristic.
D. None of the above.

## 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.

#### Quick test question

Which statement is incorrect?
A. A lead compensator is a low gain strategy.
B. A lead compensator is a high gain strategy.
C. A lead compensator design exploits the gain characteristic.
D. None of the above.

## 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.

#### Quick test question

Which definition of bandwidth is correct?
A. Bandwidth is the range of frequencies for which gain is greater than 1.
B. Bandwidth is the range of frequencies for which gain is greater than 1/sqrt(2).
C. Bandwidth is the range of frequencies for which gain is large.
D. None of the above.