The Bode Diagrams – Plotting of the frequency response of a control system.

The Bode Diagrams is the plotting of the frequency response of a system with separate magnitude and phase plots. The log-magnitude and phase frequency response curves as functions of log ω  are called Bode plots or Bode diagrams. Sketching Bode plots can be simplified because they can be approximated as a sequence of straight lines. Straight-line approximations simplify the evaluation of the magnitude and phase frequency response.

When plotting separate magnitude and phase plots, the magnitude curve can be plotted in decibels (dB) vs. log ω, where dB = 20 log M. Meanwhile, the phase curve is plotted as phase angle vs. log ω.

Example

Plot The Bode Diagram for the frequency response of the characterized by the system Transfer Function G(s):

Basic Factors of G(jω)H(jω)

The main advantage of using a logarithmic trace is the relative ease of plotting the frequency response curves. The basic factors that are frequent in an arbitrary transfer function G(jω)H(jω) are:

1. The gain K
2. Integral and derivative factors ,
3. First order factors ,
4. The quadratic factors .

Once we become familiar with the logarithmic traces of these basic factors, it is possible to use them in order to construct a composite logarithmic trace for any form of G(jω)H(jω), plotting the curves for each factor and adding individual curves in graphical form, since adding the logarithms of the gains corresponds to multiplying them among themselves.

The process of obtaining the logarithmic trace is further simplified by asymptotic approximations for the curves of each factor.

The gain K. A number greater than the unit has a positive value in decibels, while a number smaller than the unit has a negative value.

The logarithmic magnitude curve for a constant gain K is a horizontal line whose magnitude is 20 log K decibels. The phase angle of the gain K is zero. The effect of varying the gain K in the transfer function is that the logarithmic magnitude curve of the transfer function is raised or lowered by the corresponding constant amount, but does not affect the phase curve.

Integral and derivative factors(Poles and Zeros in the origin). The logarithmic magnitude of l/jω in decibels is:

The phase angle of l/jω is constant and equal to -90 °.

In Bode’s Diagram, the frequency ratios are expressed in terms of octaves or decades. An octave is a frequency band from ω₁ to 2ω₁, where ω₁ is any frequency. A decade is a frequency band from ω₁ to 10ω₁, where, again, ω₁ is any frequency. (On the logarithmic scale of the semi-logarithmic paper, any given frequency ratio is represented by the same horizontal distance. For example, the horizontal distance of ω=1  to ω=10 is equal to that of ω=3  to ω=30.

If the logarithmic magnitude of -20 log ω dB is plotted against ω on a logarithmic scale, a line is obtained. To draw this line, we need to locate a point (0 dB, ω= 1) on it. Given that:

The slope m of the line for l/jω is:

The phase angle of the factor l/jω is constant and equal to -90^°.

Smilarly:

The slope m of the line forjω is:

The phase angle of the factor jω is constant and equal to -90^°.

The following figure shows frequency response curves for l/jω and jω, respectively.

Note that both logarithmic quantities become equal to 0 dB at ω=1.

Therefore, if the transfer function contains the factor (l/jω)ⁿ or (jω)ⁿ, the logarithmic magnitude becomes, respectively, in:

Or well:

Therefore, the slopes of the logarithmic magnitude curves for the factors (l/jω)ⁿ and (jω)ⁿ are -20n dB/decade und 20n dB/decade, respectively.

The phase angle of (l/jω)ⁿ is equal to -90^°n over the entire frequency range, while the phase angle of (jω)ⁿ is equal to 90^°n over the entire frequency range. The magnitude curves will pass through the point (0 dB,  ω= 1).

First order Factors. The logarithmic magnitude of l/(1+jωT) in decibels is:

For low frequencies, such that ω<<1/T, the logarithmic magnitude is approximated by:

Therefore, the logarithmic magnitude curve for low frequencies in this factor is the constant 0 dB line. For high frequencies, such that:

The latter is an approximate expression for the high frequency range. At ω=1/T, the logarithmic magnitude is equal to 0 dB; at ω=10/T, the logarithmic magnitude is -20 dB. Therefore, the value of -20 log ωT dB decreases by 20 dB for all decades of ω. Thus, for ω>>1/T, the logarithmic magnitude curve is a straight line with a slope of -20dB/decade (or -6 dB/octave).

Our analysis shows that the logarithmic representation of the frequency response curve of the factor l/(1+jωT) is approximated by two asymptotes (straight lines), one of which is a straight line of 0 dB for the frequency range 0<ω<1/T and the other is a straight line with a slope of -20 dB/decade (or -6 dB/octave) for the frequency range 1/T<ω<∞. The frequency in which the two asymptotes meet is called the corner frequency or cutoff frequency. For the factor l/(1+jωT), the frequency ω=1/T is the corner frequency, since at that point both asymptotes have the same value.

An advantage of Bode Diagrams is that, for reciprocal factors, for example, the factor 1+jωT, the logarithmic magnitude and phase angle curves only need to change sign. Therefore, the slope of the high frequency asymptote of 1+jωT is 20 dB/decade, and the phase angle varies from 0^° to 90^° as the frequency ω increases from zero to infinity, as can be seen in the following figure:

Quadratic Factors. Control systems usually have quadratic factors of the form:

If ζ>1, this quadratic factor is expressed as a product of two first-order factors with real poles. If 0<ζ<1, this quadratic factor is the product of two complex conjugate factors.

The asymptotic frequency response curve for is obtained as follows. Given that:

For low frequencies such that ω<<ω_n, the logarithmic magnitude becomes:

Therefore, the low frequency asymptote is a horizontal line at 0 dB. For high frequencies such that ω>>ω_n, the logarithmic magnitude becomes:

The equation for the high frequency asymptote is a line with a slope of -40dB/decade, given that:

The high frequency asymptote intersects the low frequency at ω=ω_n, since at this frequency:

This frequency ω_n is the corner frequency for the quadratic factor considered.

Sources:

1. Modern_Control_Engineering, Ogata 4t
2. Control Systems Engineering, Nise
3. Sistemas de Control Automatico, Kuo

Literature review by:

Prof. Larry Francis Obando – Technical Specialist – Educational Content Writer

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