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Lecture #21 EGR 272 – Circuit Theory II

Lecture #21 EGR 272 – Circuit Theory II. Read : Chapter 14 in Electric Circuits, 6 th Edition by Nilsson. Bode Plots

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Lecture #21 EGR 272 – Circuit Theory II

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  1. Lecture #21 EGR 272 – Circuit Theory II Read: Chapter 14 in Electric Circuits, 6th Edition by Nilsson Bode Plots We have seen that determining the frequency response for 1st and 2nd order circuits involved a significant amount of work. Using the same methods for higher order circuits would become very difficult. A new method will be introduced here, called the Bode plot, which will allow us to form accurate “straight-line” approximations for the log-magnitude and phase responses quite easily for even high-order transfer functions. This technique will also show how various types of terms in a transfer function affect the log-magnitude and phase responses Illustration - A Bode plot is used to make a good estimate of the actual response. 20log|H(jw)| Bode “straight-line” approximation Actual log-magnitude response w(rad/s) (w on a log scale)

  2. Lecture #21 EGR 272 – Circuit Theory II Decibels However, when scaled logs of the quantities are taken, the unit of decibels (dB), is assigned. • There are two types of Bode plots: • The Bode straight-line approximation to the log-magnitude (LM) plot, LM versus w (with w on a log scale) • The Bode straight-line approximation to the phase plot, (w) versus w (with w on a log scale)

  3. Lecture #21 EGR 272 – Circuit Theory II Standard form for H(jw): Before drawing a Bode plot, it is necessary to find H(jw) and put it in “standard form.” Show the “standard form” for H(jw) below:

  4. Lecture #21 EGR 272 – Circuit Theory II Example: Find H(jw) for H(s) shown below and put H(jw) in “standard form.”

  5. Lecture #21 EGR 272 – Circuit Theory II The additive effect of terms in H(jw): The reason that Bode plot approximations are used with the log-magnitude is due to the fact that this makes individual terms in the LM additive. The phase is also additive. Show how the LM and phase of each term in 20log|H(jw)| is additive (or acts separately). Drawing Bode plots: To draw a Bode plot for any H(s), we need to: 1) Recognize the different types of terms that can occur in H(s) (or H(jw)) 2) Learn how to draw the log-magnitude and phase plots for each type of term.

  6. Lecture #21 EGR 272 – Circuit Theory II 5 types of terms in H(jw) 1) K (a constant) 2) (a zero) or (a pole) 3) jw (a zero) or 1/jw (a pole) 4) 5) Any of the terms raised to a positive integer power. Each term is now examined in detail.

  7. Lecture #21 EGR 272 – Circuit Theory II 1. Constant term in H(jw) If H(jw) = K = K/0 Then LM = 20log(K) and (w) = 0 , so the LM and phase responses are: • Summary: A constant in H(jw): • Adds a constant value to the LM graph (shifts the entire graph up or down) • Has no effect on the phase

  8. Lecture #21 EGR 272 – Circuit Theory II 2. A) 1 + jw/w1 (a zero): The straight-line approximations are: To determine the LM and phase responses, consider 3 ranges for w: 1) w << w1 2) w >> w1 3) w = w1

  9. Lecture #21 EGR 272 – Circuit Theory II So the Bode approximations (LM and phase) for 1 + jw/w1 are shown below. Discuss the amount of error between the actual responses and the Bode approximations. • Summary: A 1 + jw/w1 (zero) term in H(jw): • Causes an upward break at w = w1 in the LM plot. There is a 0dB effect before the break and a slope of +20dB/dec or +6dB/oct after the break. • Adds 90 to the phase plot over a 2 decade range beginning a decade before w1 and ending a decade after w1 .

  10. Lecture #21 EGR 272 – Circuit Theory II 2. B) (a pole): The straight-line approximations are: To determine the LM and phase responses, consider 3 ranges for w: 1) w << w1 2) w >> w1 3) w = w1

  11. Lecture #21 EGR 272 – Circuit Theory II So the Bode approximations (LM and phase) for are shown below. Discuss the amount of error between the actual responses and the Bode approximations. • Summary: A 1 + jw/w1 (zero) term in H(jw): • Causes an downward break at w = w1 in the LM plot. There is a 0dB effect before the break and a slope of -20dB/dec or -6dB/oct after the break. • Adds -90 to the phase plot over a 2 decade range beginning a decade before w1 and ending a decade after w1 .

  12. Lecture #21 EGR 272 – Circuit Theory II Example: Sketch the LM and phase plots for the following transfer function.

  13. Lecture #21 EGR 272 – Circuit Theory II Example: Sketch the LM and phase plots for the following transfer function.

  14. Lecture #21 EGR 272 – Circuit Theory II Example: Sketch the LM and phase plots for the following transfer function.

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