This document is a compilation of all of the Bode plot pages in one document for convenient printing.

Contents Introduction The Frequency Domain: What do Bode plots represent?

The Bode Plot: Why do we use the magnitude and phase plots?

The Asymptotic Plot: Defining the rules for making sketches.

The Method: Applying the rules to make sketches.

Examples: A series of Examples.

Rules Redux: A compact representation of the rules (including a pdf).

BodePlotGui: A MatLab GUI that helps to explain the method.

BodePaper: A MatLab function that will create the plots necessary for making sketches by hand.

Bode Plots Overview

Bode plots are a very useful way to represent the gain and phase of a system as a function of frequency. This is referred to as the frequency domain behavior of a system. This web page attempts to demystify the process. The various parts are more-or-less stand alone, so if you want to skip one or more, that should not be a problem. If you are only interested in a quick lesson on how to make Bode diagrams, go topart 4, "Making Plots." A MatLab program to make piecewise linear Bode plots is described inpart 6.

The documents are:

1. What is the frequency domain response? In other words, "What does a Bode Plot represent?"

2. Why is the Bode representation used? There are many ways to represent frequency response. Why are magnitude and phase plots used? This includes some animations.

3. How are the piecewise linear asymptotic approximations derived?4. Rules for making Bode plots.This is a quick "How to" lesson for drawing Bode plots.

5. Some examples (1,2,3,4,5,6) - (combined into one file).

6. BodePlotGui: A software tool for generating asymptotic Bode plots.7. A MatLab program for making semi-logarithmic paper for drawing your own Bode plots.8. A table summarizing Bode rules9. The MATLAB files discussed in these documents. Also available is acompilation of items 1 through 8, for easy printing.

What Bode Plots Represent: The Frequency Domain

Why Sine Waves?

One of the most commonly used test functions for a circuit or system is the sine wave. This is not because sine waves are a particularly common signal. They are in fact quite rare - the transmission of electricity (a 60 Hz sine wave in the U.S., 50 Hz in much of the rest of the world) is one example. The reason sine waves are important is complex and involve a branch of Mathematics called Fourier Theory. Briefly put: any signal going into a circuit can be represented by a sum of sine waves of varying frequency and amplitude (often an infinite sum). As a simple example, consider the graphs shown below. The top graph shows a triangle wave. The middle graph shows a number of sine waves of varying frequency and amplitude. The bottom graph shows the sum of the sine waves (red) and the original triangle wave (dotted black).

Clearly even just a few sine waves are sufficient in this case to closely approximate the original function. Fourier states that any function (with some very minor restrictions that won't concern us), can be represented in this way.

This is why sine waves are important. Not because they are common, but because we can represent arbitrarily complex functions using only these very simple function.

Determining system output given input and transfer function

Given that sine waves are important, how can we analyze the response of a circuit or system to sinusoidal inputs? There are many ways to do this, depending on your mathematical sophistication. Let's use a fairly basic explanation that uses phasors. If you are unfamiliar with phasors, you can find a description in almost any circuits or systems textbook.

Using complex impedances it is possible to find the transfer function of a circuit. For example, the circuit below is described by the transfer function, H(s), where s=j.

CircuitTransfer Function

Consider the case where R=1 and C=0.1. In that case:

Generally we know the input Viand want to find the output Vo. We can do this by simple multiplication

If we have a phasor representation for the input and the transfer function, the multiplication is simple (multiply magnitudes and add phases). Finding the output becomes easy. Let's look at some examples:

Example 1

and the transfer function evaluates to

The output is just the product of the input and the transfer function (evaluated as phasors)

Note that Vohas an amplitude of 0.95 and lags Viby 72. It is a phase "lag" because the output lags, or follows, the input (the input goes up before the output so the output is following the input).

Example 2

Change input phase

Both input and output have shifted 40 from those in Example 1.

Example 3

Change input frequency

Frequency has changed, magnitude of output has increased, but phase lag has decreased (to 45).

Example 4

Cosine Input

Note: all angles are given in degrees. They should be changed to radians before evaluation by calculator or computer.

Key Concept: Sine waves can be used to represent other functions

This document explainedbrieflywhy sine waves are important and how to find the output of a system given a sinusoidal input (i.e., represent input and transfer function as phasors and multiply to determine output as a phasor).

Why Use Magnitude and Phase Plots?

Theprevious documentmade the case that the study of sinusoidal inputs is important, and showed how phasor representations of the input and the system transfer function can be used to easily determine system output. This document will explain why the standard way of representing the transfer function is with two plots: magnitude vs. frequency and phase vs. frequency. At thebottom of this page there is an animationto help develop an intuitive understanding of the concepts describe in this page.

The difficulty in representing the transfer function comes about because we need to plot a complex number, H(s) or H(j), as a function of frequency. Consider the transfer function

To graph this, the most straightforward way (with a computer) might be to plot the value of H(s) as the frequency changes. This yields the blue line in the three-dimensional plot shown below.

It would obviously be hard to get accurate information about the real and imaginary parts of H(s) from such a plot. It is easier if we plot the real and imaginary parts as a function of frequency (the red and green projections of the blue line). Clearly, in this case, two 2-dimensional graphs (one for real and one for imaginary) are superior to a single 3-dimensional graph.

However, in the last document we showed that to easily determine the output given the input, we would like to have the transfer function in phasor notation. This means that we should make a plot of magnitude and phase. Again, we could make a single 3-dimensional plot (the blue line), but it would be easier to interpret the results if we make two 2-dimensional plots (the magenta and cyan lines).

To clarify further, lets make separate plots of the magnitude and phase.

Note: Standard Bode plots are logarithmic on the frequency axis, and plot the magnitude in dB's (deciBels). We'll explore that in thenext installment.

Consider the examples from theprevious document.

Example 1

and the transfer function evaluates to

The output is just the product of the input and the transfer function (evaluated as phasors)

Note that Vohas an amplitude of 0.95 and lags Viby 72.

Example 2

Change input phase

Both input and output have shifted 40 from those in Example 1.

Example 3

Change input frequency

Frequency has changed, magnitude of output has increased, but phase lag has decreased (to 45).

Example 4

Cosine input (i.e, a change of phase).

The magnitude and phase plots determine the phasor representation of the transfer function at any frequency. On the graphs below we can see that at 10 rad/sec the phasor representation of the transfer function has a magnitude of 0.707 and a phase of -45. This means that at 10 rad/sec the magnitude of the output will be 0.707 times the magnitude of the input and the output will lag the input by 45.

Key Concept: The frequency response is shown with two plots, one for magnitude and one for phase.

The frequency response of a system is presented as two graphs: one showing magnitude and one showing phase. The phasor representation of the transfer function can then be easily determined at any frequency. The magnitude of the output is the magnitude of the phasor representation of the transfer function (at a given frequency) multiplied by the magnitude of the input. The phase of the output is the phase of the transfer function added to the phase of the input.

An animation

To get a more intuitive idea of what the frequency response represents, consider the system below. (Hit start button to show animation)

For an animation of an analogous electrical system, gohere.

Animation byAmes BielenbergThe transfer function of the system is given by (with m=1, b=0.5, k=1.6, u=input to system, y=output (the position of the mass):

You can see by the animation that at low frequencies the input and output are equal, and in phase. At intermediate frequencies the system is somewhat resonant, and the output actually gets larger than the input (but there is a growing phase lag). As frequency increases further, the output decreases. The outline of the peaks of the output plot is similar to the magnitude plots above (the phase plot is not obvious, but it obviously starts at 0 and then decreases - if you type "ctrl +" you can zoom in to see the phase shift). In this case, the magnitude plot would start at one (output=input) at low frequencies, it would then increase, followed by a decrease.

Bode plots (introduced next) formalize a particular method for drawing magnitude and phase plots (as a function of frequency) associated with a given transfer function.

The Asymptotic Bode Diagram: Derivation of Approximations

Contents: Asymptotic approximation for... A Constant Real Pole Real Zero Pole at Origin Zero at Origin Complex Pole Complex Zero

Introduction

Given a transfer function, such as

the question naturally arises: "How can we display this function?" In theprevious documentthe argument was made that the most useful way to display this function is with two plots, the first showing the magnitude of the transfer function and the second showing its phase. One way to do this is by simply entering many values for the frequency, calculating the magnitude and phase at each frequency and displaying them. This is what a computer would naturally do. For example if you use MATLAB and enter the commands

>> MySys=tf(100*[1 1],[1 110 1000])

Transfer function:

100 s + 100

------------------------------

s^2 + 110 s + 1000

>> bode(MySys)

you get a plot like the one shown below. The asymptotic solution is given elsewhere.

However, there are reasons to develop a method for drawing Bode diagrams manually. By drawing the plots by hand you develop an understanding about how the locations of poles and zeros effect the shape of the plots. With this knowledge you can predict how a system behaves in the frequency domain by simply examining its transfer function. On the other hand, if you know the shape of transfer function that you want, you can use your knowledge of Bode diagrams to generate the transfer function.

The first task when drawing a Bode diagram by hand is to rewrite the transfer function so that all the poles and zeros are written in the form (1+s/0). The reasons for this will become apparent when deriving therules for a real pole. A derivation will be done using the transfer function from above, but it is also possible to doa more generic derivation. Let's rewrite the transfer function from above.

Now lets examine how we can easily draw the magnitude and phase of this function when s=j.

First note that this expression is made up of four terms, a constant (0.1), a zero (at s=-1), and two poles (at s=-10 and s=-100). We can rewrite the function (with s=j) as four individual phasors.

We will show (below) that drawing the magnitude and phase of each individual phasor is fairly straightforward. The difficulty lies in trying to draw the magnitude and phase of H(j). We can write H(j) as a single phasor:

Drawing the phase is fairly simple. We can draw each phase term separately, and then simply add them. The magnitude term is not so straightforward because of the fact that the magnitude terms aremultiplied, it would be much easier if they were added - then we could draw each term on a graph and justaddthem. A method for doing this is outlined below.

The Magnitude Plot

One way to transform multiplication into addition is by using the logarithm. Instead of using a simple logarithm, we will use a deciBel (named for Alexander Graham Bell).(Note: Why the deciBel?)The relationship between a quantity, Q, and its deciBel representation, X, is given by:

So if Q=100 then X=40; Q=0.01 gives X=-40; X=3 gives Q=1.41; and so on.

If we represent the magnitude of H(s) in deciBels we get

The advantage of using deciBels (and of writing poles and zeros in the form (1+s/0)) are now revealed. The fact that the deciBel is a logarithmic term transforms the multiplication of the individual terms to additions. Another benefit is apparent in the last line that reveals just two types of terms, a constant term and terms of the form 20log10(|1+j/0|). Plotting the constant term is trivial, however the other terms are not so straightforward. These plots will be discussedbelow. However, once these plots are drawn for the individual terms, they can simply be added together to get a plot for H(s).

The Phase Plot

If we look at the phase of the transfer function, we see much the same thing: The phase plot is easy to draw if we take our lead from the magnitude plot. First note that the transfer function is made up of four terms. If we want

Again there are just two types of terms, a constant term and terms of the form (1+j/0). Plotting the constant term is trivial; the other terms are discussedbelow.

A more generic derivation

The discussion above dealt with only a single transfer function. Another derivation that is more general, but a little more complicated mathematically ishere.

Making a Bode Diagram

Following the discussion above, the way to make a Bode Diagram is to split the function up into its constituent parts, plot the magnitude and phase of each part, and then add them up. The following gives a derivation of the plots for each type of constituent part. Examples, including rules for making the plots follow inthe next document, which is more of a "How to" description of Bode diagrams.

A Constant Term

Consider a constant term,

Magnitude

Clearly the magnitude is constant

Phase

The phase is also constant. If K is positive, the phase is 0 (or any even multiple of 180). If K is negative the phase is -180, or any odd multiple of 180. We will use -180 because that is what MATLAB uses. Expressed in radians we can say that if K is positive the phase is 0 radians, if K is negative the phase is - radians.

Example: Bode Plot of Gain Term

Key Concept: Bode Plot of Gain Term

For a constant term, the magnitude plot is a straight line.

The phase plot is also a straight line, either at 0 (for a positive constant) or 180 (for a negative constant).

A Real Pole

Consider a simple real pole

The frequency 0is called the break frequency, the corner frequency or the 3 dB frequency (more on this last name later).

Magnitude

The magnitude is given by

Let's consider three cases for the value of the frequency:

Case 1)0. This is the high frequency case. We can write an approximation for the magnitude of the transfer function

The high frequency approximation is at shown in green on the diagram below. It is a straight line with a slope of -20 dB/decade going through the break frequency at 0 dB. That is, for every factor of 10 increase in frequency, the magnitude drops by 20 dB.

Case 3)=0. The break frequency. At this frequency

This point is shown as a red circle on the diagram.

To draw a piecewise linear approximation, use the low frequency asymptote up to the break frequency, and the high frequency asymptote thereafter.

The resulting asymptotic approximation is shown highlighted in pink. The maximum error between the asymptotic approximation and the exact magnitude function occurs at the break frequency and is approximately 3 dB.

The rule for drawing the piecewise linear approximation for a real pole can be stated thus:

For a simple real pole the piecewise linear asymptotic Bode plot for magnitude is at 0 dB until the break frequency and then drops at 20 dB per decade (i.e., the slope is -20 dB/decade).Phase

The phase of a single real pole is given by is given by

Let us again consider three cases for the value of the frequency:

Case 1)0. This is the high frequency case. We can write an approximation for the phase of the transfer function

The high frequency approximation is at shown in green on the diagram below. It is a straight line with a slope at -90.

Case 3)=0. The break frequency. At this frequency

This point is shown as a red circle on the diagram.

A piecewise linear approximation is not as easy in this case because the high and low frequency asymptotes don't intersect. Instead we use a rule that follows the exact function fairly closely, but is also arbitrary. Its main advantage is that it is easy to remember. The rule can be stated as

Follow the low frequency asymptote until one tenth the break frequency (0.1 0) then decrease linearly to meet the high frequency asymptote at ten times the break frequency (10 0).This line is shown above. Note that there is no error at the break frequency and about 5.7 of error at one tenth and ten times the break frequency.

Example 1: Real Pole

The first example is a simple pole at 10 radians per second. The low frequency asymptote is the dashed blue line, the exact function is the solid black line, the cyan line represents 0.

Example 2: Repeated Real Pole

The second example shows a double pole at 30 radians per second. Note that the slope of the asymptote is -40 dB/decade and the phase goes from 0 to -180.

Key Concept: Bode Plot for Real Pole

For a simple real pole the piecewise linear asymptotic Bode plot for magnitude is at 0 dB until the break frequency and then drops at 20 dB per decade (i.e., the slope is -20 dB/decade). An nthorder pole has a slope of -20n dB/decade.

The phase plot is at 0 until one tenth the break frequency and then drops linearly to -90 at ten times the break frequency. An nthorder pole drops to -90n.

A Real Zero

The piecewise linear approximation for a zero is much like that for a pole Consider a simple zero:

Magnitude

The development of the magnitude plot for a zero follows that for a pole. Refer tothe previous sectionfor details. The magnitude of the zero is given by

1. Again there are three cases:

2. At low frequencies, 0, the gain increases at 20 dB/decade and goes through the break frequency at 0 dB.

4. At the break frequency, =0, the gain is about 3 dB.

The rule for drawing the piecewise linear approximation for a real zero can be stated thus:

For a simple real zero the piecewise linear asymptotic Bode plot for magnitude is at 0 dB until the break frequency and then increases at 20 dB per decade (i.e., the slope is +20 dB/decade).Phase

The phase of a simple zero is given by:

1. The phase of a single real zero also has three cases:

2. At low frequencies, 0, the phase is 90.

4. At the break frequency, =0, the phase is 45.

The rule for drawing the phase plot can be stated thus:

Follow the low frequency asymptote until one tenth the break frequency (0.1 0) then increase linearly to meet the high frequency asymptote at ten times the break frequency (10 0).Examples

This example shows a simple zero at 30 radians per second. The low frequency asymptote is the dashed blue line, the exact function is the solid black line, the cyan line represents 0.

Key Concept: Bode Plot of Real Zero:

For a simple realzerothe piecewise linear asymptotic Bode plot for magnitude is at 0 dB until the break frequency and thenrisesat +20 dB per decade (i.e., the slope is +20 dB/decade). An nthorderzerohas a slope of +20n dB/decade.

The phase plot is at 0 until one tenth the break frequency and thenriseslinearly to +90 at ten times the break frequency. An nthorderzerorisesto +90n.

A Pole at the OriginA pole at the origin is easily drawn exactly. Consider

Magnitude

The magnitude is given by

This function is represented by a straight line on a Bode plot with a slope of -20 dB per decade and going through 0 dB at 1 rad/ sec. It also goes through 20 dB at 0.1 rad/sec, -20 dB at 10 rad/sec...

The rule for drawing the magnitude for a pole at the origin can be thus:

For a pole at the origin draw a line with a slope of -20 dB/decade that goes through 0 dB at 1 rad/sec.Phase

The phase of a simple zero is given by:

The rule for drawing the phase plot for a pole at the origin an be stated thus:

The phase for a pole at the origin is -90.Example: Real Pole at Origin

This example shows a simple pole at the origin. The black line is the Bode plot, the cyan line indicates a zero reference (dB or ).

Key Concept: Bode Plot for Pole at Origin

For a simple pole at the origin draw a straight line with a slope of -20 dB per decade and going through 0 dB at 1 rad/ sec. An nthorder pole has a slope of -20n dB/decade.

The phase plot is at -90. An nthorder pole is at -90n.

A Zero at the OriginA zero at the origin is just like a pole at the origin but the magnitude increases, and the phase is positive.

Key Concept: Bode Plot for Zero at Origin

For a simplezeroat the origin draw a straight line with a slope of +20 dB per decade and going through 0 dB at 1 rad/ sec. An nthorderzerohas a slope of +20n dB/decade.

The phase plot is at +90. An nthorderzerois at +90n.

A Complex Conjugate Pair of PolesThe magnitude and phase plots of a complex conjugate (underdamped) pair of poles is more complicated than those for a simple pole. Consider the transfer function:

Magnitude

The magnitude is given by

Let's consider three cases for the value of the frequency:

Case 1)0. This is the high frequency case. We can write an approximation for the magnitude of the transfer function

The high frequency approximation is at shown in green on the diagram below. It is a straight line with a slope of -40 dB/decade going through the break frequency at 0 dB. That is, for every factor of 10 increase in frequency, the magnitude drops by 40 dB.

Case 3)0. It can be shown that a peak occurs in the magnitude plot near the break frequency. The derivation of the approximate amplitude and location of the peak are givenhere. We make the approximation that a peak exists only when

0