Bode plots solve frequency domain problems

Vance J. VanDoren, consulting editor

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Every child who has ever held a spring upright knows that tugging on the top end causes the bottom end to start bouncing and that repeated tugging keeps those oscillations going. Some may notice that even though both ends always oscillate at the same frequency, the bottom end bounces higher at some frequencies than at others. Truly gifted children might even notice that the bottom end oscillates out of sync with the top end and lagsfurther and further behind as the frequency increases.

Engineers know that many mechanical, electrical, and chemical processes with energy-storing components behave the same way. A step change at the input end causes decaying oscillations at the output. A sinusoidal input, on the other hand, causes a sinusoidal output with an amplitude and a lag time that depend on the frequency of the oscillations. Not coincidentally, these two phenomena are related and form the basis of the frequency domain techniques that are fundamental to the analysis of feed-back controllers.

Two ways to calculate
There are basically two ways to analyze the behavior of a process in the frequency domain. The direct method is to drive it with a series of sinusoidal inputs, each with the same amplitude but different frequency. The amplitude and lag time of the sinusoidal outputs that result in each case can be plotted against frequency to produce a Bode plot for the process. The sample Bode plot in the figure shows how high the bottom end of the spring will bounce and how much it will lag the top end when the top end is set oscillating at various frequencies.

The second frequency domain analysis method uses Fourier's Theorem to compute the process' Bode plot indirectly. Fourier's Theorem states that any signal which is not itself a sine wave can be expressed as a sum of sine waves. A step input, for example, results when a few high-amplitude, low-frequency sine waves are added to a larger collection of low-amplitude, high-frequency sine waves. Fourier's Theorem also gives a formula for computing the amplitude of each component sine wave plus its lag time relative to the lowest frequency compo-nent. Note that plotting the amplitude and lag time for each component against its frequency yields a Bode plot of the signal just like theBode plot for an entire process.

In fact, the Bode plot for a process can be derived from the Bode plots of its input and output signals. Simply divide each amplitudein the output's Bode plot by the corresponding amplitude in the input's Bode plot. The result-ing quotient is the amplitude for the process' Bode plot at that frequency. To get the lagtimes for the process' Bode plot, simply subtract each input lag time from the correspondingoutput lag time.

Conversely, multiplying the amplitudes of an arbitrary input signal with the amplitudes of the process will give the amplitudes resulting in the output when that input is actually applied to that process. The output's lag times are computed by adding the lag times of the input to the lag times of the process. With this mathematical 'trick,' control engineers can predict the effects of a controller's actions on any process with a known Bode plot.

Consulting Editor Vance J. VanDoren, Ph.D, P.E., president of VanDoren Industries, West Lafayette, Ind.