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Bode (pronounced BO-dee) plots are graphs that show how the magnitude and phase of an electric circuit's system function vary for changing frequency. When done by hand, bode plots are best done as approximations to determine useful properties of the circuit. Mathematical programs like MATLAB can do perfectly accurate bode plots, and often have standard functions built in to do so, but this is hardly ever necessary.

The bode plot of the magnitude is very useful for determining what kind of filter a circuit is. Even more importantly, it can help to determine exactly what the frequencies are when the circuit crosses certain important voltage thresholds. For instance, when the magnitude is equal to 1, the input voltage is equal to output voltage. An interesting example of the bode plot of a system function's magnitude is that of the notch filter:

```
|
|H| |
|------     --------
|      \   /
|      |  |
|      |  |
|      |__|
|
|
---------------------
omega

```
For frequencies around the middle, the magnitude drops off sharply. For all other frequencies not near that middle value, the magnitude is high. Thus, the notch filter will filter out all values near that frequency. What frequency that is depends on the parameters (resistances, capacitances, etc.) of the circuit. You don't need a bode plot to tell you all of this information, as it can be done numerically. But it is nice to have and makes it easier to visualize.

How do humans make bode plots? There are key frequencies to mark down and solve for, and once that is done, it's easy. Simply solve for what happens at low frequencies, at high frequencies, and at the resonant frequency. You can draw sketches of the curve at those points, and connect the estimates to get a pretty good graph.

Many phase plots tend to look similar, having a shape like the function y= - x3:

```     |
phase|
angle|_______
|       \
|        |
|        |
|        |
|         \________
|
---------------------
omega
```

The time delay changes a lot around the middle of the plot, and flattens out at values not close to that resonant frequency.

Because of the varying values of frequency and the response they get from certain circuits, it is often prudent to use a logarithmic scale on at least one of the axes instead of a straight numerical scale. If not, you will end up with a long, stretched-out graph that won't offer much information.

Bode plots are used to asses the characteristics of feedback systems. These can come in many forms: electronic control systems, archaic mechanical systems such as Watt's Governor, even the human ear. Bode plots give valuable information about all of these systems.

A bode diagram consists of two log graphs. They are both based on a sine wave input and a sine wave output. The two graphs are 1) log loop Gain vs log Frequency and 2) Phase vs log Frequency. The two graphs have the same scale on the horizontal axis and so are normally shown on top of each other. For this write-up you can assume gain means loop gain.

• Assessing System Stability.

The most important point on a Nyquist diagram is the point (-1,j0). This corresponds to an absolute Gain of 1 and a phase of -180o. The reasons that absolute gain is used is that this is applicable to both positive feedback and negative feedback. These two horizontal lines are normally marked on Bode diagrams. The rule for stability is "If absolute gain is greater than or equal to one when absolute phase is greater than or equal to 180o" then the system will be unstable; notationally this is:
The system will be unstable if |Gain| >= 1 AND |Phase| >= 180

Assesing stability on a Bode diagram is easy: simply draw a line downwards from the intersection of the gain = 1 and the gain plot lines. Then draw a line up from the intersection of the phase = -180o and phase plot lines. If the line going up from phase = -180o is to the left of the line down from gain = 1 then the system will be unstable.

```              An Unstable System.

log Gain |_____
|     \___
|_______|_\________
1 |       |  |\_
|       |  |  \_
|       |  |    \___
|_______|__|________\__________
|  |
Phase |       ^  v
|_____  |  |
|     \ |  |
-180 |______\|__|______
|         \|_
|            \_______
|_____________________________
log frequency (log w)
```

If the lines are in the same place then the system will occilate forever and if they are the other way around the system will be stable and the occilations from a step input will die down to a steady state.

• Approximating Response.

Often when designing systems you will not be able to just plug your system into a Bode test rig and go. You will have to approximate its response by hand from its transfer function. The method used for this is called asymptotic approximation. It amounts to looking at the transfer function formula and figuring out which terms will be significant and when. This is very simple to do for Bode plots as the gradient of the gain curve and the magnitude of the phase are directly related to system's order.

```An example system is:         (1 + s/2)
(1 + s/0.3)(1 + s/0.8)(1 + s/4)

```

NB: "s" in this context is the Laplace operator. s = jw, s2 = w2. For these purposes we assume s = w because j can be easily cancelled. "s" is also the variable by which the order of the system is defined.

This system has one lead term and three lag terms i.e. one term that increases the order of the system and three that decrease it. For the purposes of this approximation "s" terms become significant when they are greater than one. The order they become significant is based on steadily increasing frequency. The points at which the various terms become significant are called corner frequencies for reasons that should become obvious.

At the beginning the only significant term is the 1 on the top. This dictates the initial gain of the system. The gradient of the gain curve is zero at this point becasue the overall order is zero. The phase at this point is also zero for the same reason.

The first "s" term to become significant is the (1 + s/0.3) term. It becomes significant at frequency = 0.3. After this point the order of the system is -1 thus the gradient of the approximate gain curve is -1. The phase also changes but in its case the order is directly related to its value. The approximation for this is 90o * order. Its graph therefore jumps directly to -90o.

The next most significant term is (1 + s/0.8). This becomes significant at s = 0.8. This term takes the system order to -2 because it is part of the denominator. Thus the gain gradient becomes -2 and the phase becomes -180o.

The next most significant term is (1 + s/2). This becomes significant at (you guessed it) s = 2. This term takes the system order back up to -1 becasue it is a denominator term. Thus the gain gradient becomes -1 and the phase becomes -90o.

The final term is left as an exercise for the reader. :-)

```The final approximation graph will look a little like:

|
|
log Gain  |___________
|          |\_ <-- Gradient = -1
|          |  \_
|          |   |\
|          |   | \
|          |   |  \  <-- Gradient = -2
|          |   |   \
|          |   |    \
|          |   |     \_
|          |   |      |\_
|          |   |      |  \_  <-- Gradient = -1
|          |   |      |    \_
|          |   |      |      \_
|          |   |      |       |\
|          |   |      |       | \
|          |   |      |       |  \  <-- Gradient = -2
|          |   |      |       |   \
|__________|___|______|_______|____\_________________
0.3   0.8    2       4                log frequency

|__________
phase    0|          |
|          |
|          |
|          |__        ________
-90|             |      |        |
|             |      |        |
|             |      |        |
|             |      |        |
|             |______|        |_______________
-180|
|___________________________________________________
log frequency
```

This information was gained while revising for my finals from the lecture notes provided by R.J.Mitchell, Dept. Cybernetics, Reading University