1 Lecture #21 EGR 272 – Circuit Theory II Bode Plots We have seen that determining the frequency response for 1 st and 2 nd 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. Read: Chapter 14 in Electric Circuits, 6 th Edition by Nilsson w(rad/s) Actual log-magnitude response Bode “straight-line” approximation (w on a log scale) 20log|H(jw)|

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 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. 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.

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/w 1 (a zero): The straight-line approximations are: To determine the LM and phase responses, consider 3 ranges for w: 1)w << w 1 2)w >> w 1 3)w = w 1

9 Lecture #21 EGR 272 – Circuit Theory II So the Bode approximations (LM and phase) for 1 + jw/w 1 are shown below. Summary: A 1 + jw/w 1 (zero) term in H(jw): Causes an upward break at w = w 1 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 w 1 and ending a decade after w 1. Discuss the amount of error between the actual responses and the Bode approximations.

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 << w 1 2)w >> w 1 3)w = w 1

11 Lecture #21 EGR 272 – Circuit Theory II So the Bode approximations (LM and phase) for are shown below. Summary: A 1 + jw/w 1 (zero) term in H(jw): Causes an downward break at w = w 1 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 w 1 and ending a decade after w 1. Discuss the amount of error between the actual responses and the Bode approximations.

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.