Bode magnitude plots: sketching frequency response given H(s)

Summary

This video explains the concept of Bodé magnitude plots in signals and systems, focusing on how to sketch the frequency response of a linear time-invariant system. The presenter demonstrates how to generate Bodé plots for different system functions, such as constants, zeros, and poles. Using step-by-step examples, including expressions like H(s) = 10s / (s + 100000), the video shows how to break down complex systems into simpler components and combine them into an overall Bodé plot. The content offers valuable insights into constructing and interpreting these plots for analyzing system behavior in the frequency domain.

Takeaways

• 😀 A Bode plot is used to sketch the frequency response of a linear time-invariant system, given its system function.
• 😀 The Bode magnitude plot is a straight-line approximation of 20 log10 of the frequency response magnitude.
• 😀 A constant term in the system function results in a Bode plot that is flat at a certain level, determined by the value of the constant.
• 😀 A system function of the form H(s) = s leads to a Bode plot with a slope of +20 dB/decade that crosses 0 dB at ω = 1.
• 😀 A system function with a zero at -a, like H(s) = 1/(s + a), results in a Bode plot that is flat at 0 dB up to frequency ω = a, then increases at +20 dB/decade.
• 😀 A system function with a pole at -b, like H(s) = 1/(s + b), results in a Bode plot that is flat at 0 dB up to frequency ω = b, then decreases at -20 dB/decade.
• 😀 The frequency response for more complex systems can be obtained by combining the individual Bode plots of each component.
• 😀 In the example H(s) = 10s / (s + 1000), the Bode plot consists of three components: a constant factor, a term with s, and a term with a pole at -1000.
• 😀 The overall Bode plot is the result of summing the individual curves from the components, shifting and adding them appropriately.
• 😀 A Bode plot with a zero at 0 dB starts with a +20 dB/decade slope from the lowest frequency, while a zero at a finite frequency, like -10, starts with a flat line until it reaches that frequency and then increases at +20 dB/decade.

Q & A

What is a Bode magnitude plot?

A Bode magnitude plot is a graphical representation of the frequency response of a linear time-invariant system, often approximated using a straight-line method for the magnitude of the frequency response, typically expressed as 20 log10 |H(jω)|.

How do you calculate the frequency response magnitude for a system with the transfer function H(s)?

The frequency response magnitude is calculated by evaluating the transfer function H(s) at s = jω (where ω is the frequency), then taking 20 log10 of the absolute value of the result, i.e., 20 log10 |H(jω)|.

What does the slope of a Bode plot represent?

The slope of a Bode plot represents the rate of change of the magnitude of the frequency response with respect to frequency. For example, a positive slope of +20 dB per decade indicates that the magnitude increases by 20 dB for every tenfold increase in frequency.

What is the effect of a constant K in the system function H(s) on the Bode plot?

If the system function H(s) is a constant K, the Bode plot will be a flat line at 20 log10 |K|, representing a constant magnitude response at all frequencies. The value of K determines the height of the line, with K > 1 giving a positive value in dB and K < 1 giving a negative value.

How does the Bode plot change when H(s) is equal to s?

For H(s) = s, the frequency response magnitude is 20 log10 |jω|, which is a straight line with a slope of +20 dB per decade, crossing the 0 dB axis at ω = 1.

What is the impact of a pole or zero on the Bode plot?

A zero at -a causes the Bode plot to be flat (0 dB) until the frequency reaches a = ω, at which point the magnitude increases at a rate of +20 dB per decade. Conversely, a pole at -b causes the Bode plot to be flat at 0 dB until ω = b, after which the magnitude decreases at a rate of -20 dB per decade.

How do you sketch a Bode plot for a system with multiple components?

To sketch a Bode plot for a system with multiple components, you first sketch the Bode plot for each individual term (constant, zero, pole, etc.), then add the magnitude contributions from each term. This involves summing the individual plots (adding or subtracting dB values at each frequency).

What happens to the Bode plot when you have a system with a combination of poles and zeros?

In a system with a combination of poles and zeros, the Bode plot is formed by combining the individual contributions of each zero (which increases the magnitude at +20 dB per decade after the zero's frequency) and each pole (which decreases the magnitude at -20 dB per decade after the pole's frequency). These effects are summed together to form the overall plot.

How do you find the intersection points on the Bode plot?

The intersection points on a Bode plot can be found by analyzing the contribution of each term. For instance, a zero at ω = a will cause the plot to cross the 0 dB line at this frequency, and a pole at ω = b will cause the plot to flatten out and decrease after this point. These points mark the transitions in slope or magnitude.

What is the significance of the 'minus 40 dB' value in the example with the transfer function H(s) = 10s/(s + 1000)?

In the example H(s) = 10s/(s + 1000), the constant factor contributes a value of 20 log10(1/100) = -40 dB. This shifts the Bode plot downward by 40 dB, affecting the overall shape of the plot before considering the effects of the zeros and poles.