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 Home How To Signals & Systems Bode plot

 Bode plot

# Bode plot

A Bode plot, named for Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase plot:

A Bode magnitude plot is a graph of log magnitude against log frequency often used in signal processing to show the transfer function or frequency response of an LTI system.

It makes multiplication of magnitudes a simple matter of adding distances on the graph, since

$\log(a \cdot b) = \log(a) + \log(b)\,$

The Bode plot describes the output response of a frequency-dependent system for a normalised input. The magnitude axis of the Bode plot is often converted directly to decibels.

A Bode phase plot is a graph of phase against log frequency, usually used in conjunction with the magnitude plot, to evaluate how much a frequency will be phase-shifted. For example a signal described by: Asin(ωt) may be attenuated but also phase-shifted. If the system attenuates it by a factor x and phase shifts it by −Φ the signal out of the system will be (A/x) sin(ωt − Φ). The phase shift Φ is generally a function of frequency.

The magnitude and phase Bode plots can seldom be changed independently of each other — if you change the amplitude response of the system you will most likely change the phase characteristics as well and vice versa. For minimum-phase systems the phase and amplitude characteristics can be obtained from each other with the use of the Hilbert Transform.

If the transfer function is a rational function, then the Bode plot can be approximated with straight lines. These asymptotic approximations are called straight line Bode plots or uncorrected Bode plots and are useful because they can be drawn by hand following a few simple rules. Simple plots can even be predicted without drawing them.

The approximation can be taken further by correcting the value at each cutoff frequency. The plot is then called a corrected Bode plot.

## Rules for hand-made Bode plot

The main idea about Bode plots is that one can think of the log of a function in the form:

$f(x) = A \prod (x + c_n)^{a_n}$

as a sum of the logs of its poles and zeros:

$\log(f(x)) = \log(A) + \sum a_n log(x + c_n)$

This idea is used explicitly in the method for drawing phase diagrams. The method for drawing amplitude plots implicitly uses this idea, but since the log of the amplitude of each pole or zero always starts at zero and only has one asymptote change (the straight lines), the method can be simplified.

### Straight-line amplitude plot

Amplitude decibels is usually done using the 20Log10(X) version. Given a transfer function in the form

$H(s) = A \prod \frac{(s + x_n)^{a_n}}{(s + y_n)^{b_n}}$
where s = jω, xn and yn are constants, and H is the transfer function:
• at every value of s where ω = xn (a zero), increase the slope of the line by $20 \cdot a_n dB$ per decade.
• at every value of s where ω = yn (a pole), decrease the slope of the line by $20 \cdot a_n dB$ per decade.
• The initial value of the graph depends on your boundaries. The initial point is found by putting the initial angular frequency ω into the function and finding |H(jω)|.
• The initial slope of the function at the initial value depends on the number and order of zeros and poles that are at values below the initial value, and are found using the first two rules.

To handle irreducible 2nd order polynomials, $ax^2 + bx + c \$ can, in many cases, be approximated as $(\sqrt{a}x + \sqrt{c})^2$.

Note that zeros and poles happen when ω is equal to a certain xn or yn. This is because the function in question is the magnitude of H(jω), and since it is a complex function, $|H(j\omega)| = \sqrt{H \cdot H^* }$. Thus at any place where there is a zero or pole involving the term (s + xn), the magnitude of that term is $\sqrt{(x_n + j\omega) \cdot (x_n - j\omega)}= \sqrt{x_n^2-\omega^2}$.

### Corrected amplitude plot

To correct a straight-line amplitude plot:

• at every zero, put a point $3 \cdot a_n\ \mathrm{dB}$ above the line,
• at every pole, put a point $3 \cdot b_n\ \mathrm{dB}$ below the line,
• draw a smooth line through those points using the straight lines as asymptotes (lines which the curve approaches).

Note that this correction method does not incorporate how to handle complex values of xn or yn. In the case of an irreducible polynomial, the best way to correct the plot is to actually calculate the magnitude of the transfer funcition at the pole or zero corresponding to the irreducible polynomial, and put that dot over or under the line at that pole or zero.

### Straight-line phase plot

Given a transfer function in the same form as above:

$H(s) = A \prod \frac{(s + x_n)^{a_n}}{(s + y_n)^{b_n}}$

the idea is to draw separate plots for each pole and zero, then add them up. The actual phase curve is given by $- \mathbf{arctan}\bigg(\frac{\mathbf{im}[H(s)]}{\mathbf{re}[H(s)]}\bigg)$

To draw the phase plot, for each pole and zero:

• if A is positive, start line (with zero slope) at 0 degrees,
• if A is negative, start line (with zero slope) at 180 degrees,
• for a zero, slope the line up at $45 \cdot a_n$ degrees per decade when $\omega = \frac{x_n}{10}$,
• for a pole, slope the line down at $45 \cdot b_n$ degrees per decade when $\omega = \frac{y_n}{10}$,
• flatten the slope again when the phase has changed by $90 \cdot a_n$ degrees (for a zero) or $90 \cdot b_n$ degrees (for a pole),
• After plotting one line for each pole or zero, add the lines together.

## Example

A lowpass RC filter, for instance has the following frequency response:

$H(f) = \frac{1}{1+j2\pi f R C}$

The cutoff frequency point fc (in hertz) is at the frequency

$f_\mathrm{c} = {1 \over {2\pi RC}}$.

The line approximation of the Bode plot consists of two lines:

• for frequencies below fc it is a horizontal line at 0 dB,
• for frequencies above fc it is a line with a slope of −20 dB per decade.

These two lines meet at the cutoff frequency. From the plot it can be seen that for frequencies well below the cutoff frequency the circuit has an attenuation of 0dB, the filter does not change the amplitude. Frequencies above the cutoff frequency are attenuated - the higher the frequency, the higher the attenuation.