Bode' Plots

Why Bode' Plots?

        Bode plots are the most widely used means of displaying and communicating frequency response information. There are many reasons for that.
 

Bode' plots are really log-log plots, so they collapse a wide range of frequencies (on the horizontal axis) and a wide range of gains (on thevertical axis) into a viewable whole.

In Bode' plots, commonly encountered frequency responses have a shape that is simple. That simple shape means that laboratory measurementscan easily be discerned to have the common factors that lead to those shapes. For example, first order systems have two straight line asymptotes and if you take data and plot a Bode' plot from the data, you can pick out first order factors in a transfer function from the straight line asymptotes.

        All in all, Bode' plots are widely used, not just to specify or show a frequency response, but they also give useful information for designing control systems. Stability criteria can be interpreted on Bode' plots and there are numerous design techniques based on Bode' plots.

        You need to know how to use Bode' plots when you encounter them in those situations, so this lesson will help you to understand the basics of Bode' plots.

        What do you need to learn about Bode' plots? Here is a short summary:

• What is a Bode' plot?
  • How is magnitude plotted? (dbs)
  • How is phase plotted? (degrees)
  • How is frequency plotted?  (on a logarithmic scale)
• Given a Transfer Function:
  • Be able to plot the Bode' plot, manually or with a math analysis program.
  • Know that the Bode' plot you generated "makes sense".
• Given a Bode Plot for a System:
  • Determine the transfer function of thesystem represented by the Bode' plot.

        Bode' plots are:

    Plots of frequency response. Gain and phase are displayed in separate plots.
    Logarithmic plots.
    The horizontal axis is frequency - plotted on a log scale. It can be either f or w.
    The vertical axis is gain, expressed in decibels - a logarithmic measure of gain.
    Sometimes, the vertical axis is simply a gain on a logarithmic scale.

• Given a Transfer Function:
  • Be able to plot the Bode' plot, manually or with a math analysis program.
    Know that the Bode' plot you generated "makes sense".

        That's what we will start with for first order systems.

        In this section we will work on that general goal for first order systems.  Let's look at an example Bode' plot for a first order system. Here's a plot for a sample transfer function.

        G(jw) = 1/(jwt+ 1) with t= .001

Here's the Bode' plot. Examine the following points for this plot.

 

The low frequency asymptote,

The high frequency asymptote,

The "mid point" where

wt = 1 That's at f = 159 Hz.

Let's look at the low frequency asymptote first. Here's the transfer function.

        If wis small, then the imaginary term in the denominator is small, and we have:

The low frequency behavior of the plot shows that the plot is flat at a value of 1.

Now, let's look at the high frequency asymptote. Here's the transfer function.

If wis large, then the imaginary term in the denominator dominates, and we have:

The magnitude of the gain is:

        The gain drops off inversely with frequency, but the Bode' plot drops off as a straight line. Hmmmm?  That's very interesting - that it is a straight line.  The straight line high frequency asymptote shouldn't be cause for consternation. If we have:

        Remember that the Bode' plot is log gain vs log frequency, so let's look at the logarithm of the magnitude of the gain.

log(|G(jw)|) = log(1/wt)= -log(wt) = -log(w) - log(t)

        So, log gain depends linearly upon the log of frequency (w) for higher frequencies. That's an important point to remember, and it is also a reason Bode' plots are used so much.  When the asymptotic behavior - both at high frequencies and low frequencies - is straight line behavior, it makes Bode' plots easier to sketch and easier to understand.

        Actually, we need to note that the slope of this plot - at high frequencies - is just -1. Look again at the asymptotic high frequency relationship between the gain and frequency.

log(|G(jw)|) = -log(w) - log(t)

When frequency increases by a factor of 10, log(w) increases by 1.

Therefore, when frequency increases by a factor of 10, log(|G(jw)|)decreases by 1.

Therefore, when frequency increases by a factor of 10, |G(jw)| decreases by a factor of 10.

           From this discussion, we need to draw a conclusion.

When frequency increases by a factor of 10, |G(jw)| decreases by a factor of 10.

        Check that conclusion on the plot to be sure you understand what it means.

        Here is a plot with the lower limit extended.

Check going from f = 300 to f = 3000.

        Does the gain decrease by a factor of 10 when the frequency increases by a factor of 10?

        The last point we need to examine is the behavior of the frequency response for frequencies between high frequency and low frequency - what we referred to as the mid-point earlier. If the frequency response function is given by:

G(jw) = 1/(jwt+ 1)

 If w= 1/t then (taking that frequency as the mid-point), we have:

G(jw) = 1/(j + 1)

The magnitude of the gain is:

|G(jw)| = 1/|j + 1| = 1/sqrt(2) ~= 0.707

This point is at w= 1000, or f =159Hz.

        There are some interesting things to note about this frequency response. Consider the interactive graph below. On that graph you can see the low frequency asymptote, the high frequency asymptote and the point where the gain is .707 of the low frequency gain.

Check the intersection of the two lines.

The intersection of the two lines occurs where w= 1/t.

For obvious reasons, this intersection is called the corner frequency.

        There's one last point to observe regarding first order systems. The general first order system has a transfer function of this form.

G(jw) = G_dc/(jwt+ 1)

        The point to note is that there is a DC gain term in the numerator.  This really is the DC gain. Let the frequency, w, be zero:

G(j0) = G_dc/(j0 + 1) = G_dc

        The effect of DC gain is to raise or lower the entire plot.  You need to understand the effect of a DC gain on a Bode' plot. Let's look at the entire transfer function.

G(jw) = G_dc/(jwt + 1)

log(|G(jw)|) = log(G_dc) - log(1/((wt)² + 1))

This really says that log(G_dc) is added at every frequency.

            Here is a movie where you can set the gain and see how the gain changes the Bode' plot.

Adding log(G_dc) at every frequency shifts the entire plot up by log(G_dc).

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Phase in 1st Order Bode' Plots

        We have looked exclusively at the magnitude portion of the Bode' plots we have examined. We need to look at the phase plot as well.

The transfer function is: G(jw) = G_dc/(jwt+ 1)

The phase angle at an angular frequency wis: Angle(G(jw)) = - tan^-1(jwt)

The phase plot - against frequency - is important in many systems.

We will plot the phase for this transfer function- the one used earlier in this section:

G(jw) = 1/(jwt+ 1) with t = .001

Note the following:

The phase starts at 0^o at low frequencies.

The phase goes to -90^o at high frequencies.

The phase is -45^o at a frequency of 159 Hz - the corner frequency.

There are several things to note at this point  

• Any transfer function is a ratio of polynomials - and those polynomials have real coefficients.

• Polynomials with real coefficients have real roots - first order factors - and complex conjugate pairs of roots - second order factors.

• Our discussion of this first order system model is really only addressing systems with one pole - one real root - in the denominator.

• More interesting systems have second order factors.