Bode plot why 20

Instead of using a simple logarithm, we will use a deciBel named for Alexander Graham Bell. Originally, the Bel named after Alexander Graham Bell was developed for measuring power. A factor of 10 change in power typically measured in Watts would be a 1 Bel change. Or a factor of 10 change in power would be a 10 deciBel change. A deciBel is one tenth of a Bel. However, the Bel is also commonly used for measuring voltage. Since power is proportional to voltage squared, a factor of 10 change in voltage is a factor of change in power - this corresponds to 20 decibels.

The fact that the deciBel is a logarithmic term transforms the multiplications and divisions of the individual terms to additions and subtsractions. Plotting the constant term is trivial, however the other terms are not so straightforward. These plots will be discussed below. However, once these plots are drawn for the individual terms, they can simply be added together to get a plot for H s.

If we look at the phase of the transfer function, we see much the same thing: The phase plot is easy to draw if we take our lead from the magnitude plot. First note that the transfer function is made up of four terms. If we want. Plotting the constant term is trivial; the other terms are discussed below.

The discussion above dealt with only a single transfer function. Another derivation that is more general, but a little more complicated mathematically is here. Following the discussion above, the way to make a Bode Diagram is to split the function up into its constituent parts, plot the magnitude and phase of each part, and then add them up.

The following gives a derivation of the plots for each type of constituent part. Examples, including rules for making the plots follow in the next document , which is more of a "How to" description of Bode diagrams. The phase is also constant. Let's consider three cases for the value of the frequency, and determine the magnitude in each case. We can write an approximation for the magnitude of the transfer function:. The high frequency approximation is at shown in green on the diagram below.

To draw a piecewise linear approximation, use the low frequency asymptote up to the break frequency, and the high frequency asymptote thereafter. The resulting asymptotic approximation is shown highlighted in transparent magenta. The maximum error between the asymptotic approximation and the exact magnitude function occurs at the break frequency and is approximately -3 dB.

Magnitude of a real pole: The piecewise linear asymptotic Bode plot for magnitude is at 0 dB until the break frequency and then drops at 20 dB per decade as frequency increases i. At these frequencies We can write an approximation for the phase of the transfer function. We can write an approximation for the phase of the transfer function.

A piecewise linear approximation is not as easy in this case because the high and low frequency asymptotes don't intersect. Instead we use a rule that follows the exact function fairly closely, but is also somewhat arbitrary. Its main advantage is that it is easy to remember. This line is shown above.

Note that there is no error at the break frequency and about 5. The second example shows a double pole at 30 radians per second. There is another approximation for phase that is occasionally used. The latter is shown on the diagram below. The development of the magnitude plot for a zero follows that for a pole. Refer to the previous section for details.

The magnitude of the zero is given by. Magnitude of a Real Zero: For a simple real zero the piecewise linear asymptotic Bode plot for magnitude is at 0 dB until the break frequency and then increases at 20 dB per decade i.

The phase of a single real zero also has three cases which can be derived similarly to those for the real pole, given above :. This example shows a simple zero at 30 radians per second. The asymptotic approximation is magenta, the exact function is the dotted black line.

In this case there is no need for approximate functions and asymptotes, we can plot the exact funtion. It also goes through 20 dB at 0. Since there are no parameters i. This example shows a simple pole at the origin.

The exact dotted black line is the same as the approximation magenta. No interactive demo is provided because the plots are always drawn in the same way. This example shows a simple zero at the origin. The magnitude and phase plots of a complex conjugate underdamped pair of poles is more complicated than those for a simple pole. This is the low frequency case. We can write an approximation for the magnitude of the transfer function.

This is the high frequency case. That is, for every factor of 10 increase in frequency, the magnitude drops by 40 dB. It can be shown that a peak occurs in the magnitude plot near the break frequency. Whereas, yaxis represents the magnitude linear scale of open loop transfer function in the magnitude plot and the phase angle linear scale of the open loop transfer function in the phase plot.

The following table shows the slope, magnitude and the phase angle values of the terms present in the open loop transfer function. This data is useful while drawing the Bode plots. The magnitude plot is a horizontal line, which is independent of frequency.

The 0 dB line itself is the magnitude plot when the value of K is one. The Zero degrees line itself is the phase plot for all the positive values of K.

In this case, the phase plot is 90 0 line. This Bode plot is called the asymptotic Bode plot. As the magnitude and the phase plots are represented with straight lines, the Exact Bode plots resemble the asymptotic Bode plots.