THE BODE PLOT

The Bode Plot

When we perform the preceding experiment the output we measure is a voltage

V_{x} (t)

which is related to the cart position X(t) by the equation:

V_{x} (t) = K_{x} X (T) or K_{x} = \frac{V_{x}}{X} = \frac{Volts}{meter}

Thus our steady state output for an input

V_{i} (t) = A \cos (ω t)

V_{x} (t) = \frac{A K_{c} K_{x}}{ω (j ω + B)} \sin (ω t)

We now wish to make a Bode plot of the magnitude (in decibels) of the output voltage corresponding to the cart position versus the frequency (in radians) of the input. The magnitude in decibels of

V_{x} (t)

\begin{aligned} D B & = 20 \log {| V_{x} (t) |}_{max} \\ = 20 \log_{10} \frac{| A K c K x |}{ω (j ω + B)} \\ = 20 \log_{10} | \frac{A K c K x \frac{1}{B}}{ω (\frac{ω}{B} + 1)} | \\ = 20 \log_{10} | A K c K x \frac{1}{B} | - 20 \log_{10} | ω | - 20 \log | j \frac{ω}{B} + 1 | \end{aligned}

The Bode plot thus consists of three parts. First there is a constant gain term of

20 \log_{10} | A K c K x \frac{1}{B} |

This is a horizontal line on the Bode plot. Second, there is a pole at the origin with a gain of

- 20 \log_{10} | ω |

This is a line with slope of -20 DB/decade with a value of 0 DB at ω = 1 rad/sec. Third, there is a pole on the real axis at -B given by the term

- 20 \log | j \frac{ω}{B} + 1 |

The magnitude of this at low frequencies is:

- 20 \log_{10} | j \frac{ω}{B} + 1 | \approx - 20 \log_{10} | 1 | = 0 D B (ω » B)

and at high frequencies the magnitude is:

- 20 \log_{10} | j \frac{ω}{B} + 1 | \approx - 20 \log_{10} | j \frac{ω}{B} | (ω » B)

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