BODE PLOTS

4.9 BODE PLOTS

Sketching frequency response plots (|H(j ω)| and ∠H(jω) versus ω) is considerably facilitated by the use of logarithmic scales. The amplitude and phase response plots as a function of ω on a logarithmic scale are known as Bode plots. By using the asymptotic behavior of the amplitude and the phase responses, we can sketch these plots with remarkable ease, even for higher-order transfer functions.

Let us consider a system with the transfer function

where the second-order factor (s 2 +b 2 s+b 3 ) is assumed to have complex conjugate roots. [ † ] We shall rearrange Eq. (4.81a) in the form

and

This equation shows that H(j ω) is a complex function of ω. The amplitude response |H(jω)) and the phase response ∠H(jω) are given by

and

From Eq. (4.82b) we see that the phase function consists of the addition of only three kinds of term: (i) the phase of j ω), which is 90° for all values of ω, (ii) the phase for the first-order term of the form 1 + jω/a, and (iii) the phase of the second-order term

We can plot these three basic phase functions for ω in the range 0 to ∞ and then, using these plots, we can construct the phase function of any transfer function by properly adding these basic responses. Note that if a particular term is in the numerator, its phase is added, but if the term is in the denominator, its phase is subtracted. This makes it easy to plot the phase function ∠H(jω) as a function of ω. Computation of |H(jω)|, unlike that of the phase function, however, involves the multiplication and division of various terms. This is a formidable task, especially when, we have to plot this function for the entire range of ω (0 to ∞).

We know that a log operation converts multiplication and division to addition and subtraction. So, instead of plotting |H(j ω)|, why not plot log |H(jω)| to simplify our task? We can take advantage of the fact that logarithmic units are desirable in several applications, where the variables considered have a very large range of variation. This is particularly true in frequency response plots, where we may have to plot frequency response over a

range from a very low frequency, near 0, to a very high frequency, in the range of 10 10 or higher. A plot on a linear scale of frequencies for such a range from a very low frequency, near 0, to a very high frequency, in the range of 10 10 or higher. A plot on a linear scale of frequencies for such a

There is another important reason for using logarithmic scale. The Weber-Fechner law (first observed by Weber in 1834) states that human senses (sight, touch, hearing, etc.) generally respond in logarithmic way. For instance, when we hear sound at two different power levels, we judge one sound twice as loud when the ratio of the two sound powers is 10. Human senses respond to equal ratios of power, not equal increments in

power. [ 11 ] This is clearly a logarithmic response. [ † ] The logarithmic unit is the decibel and is equal to 20 times the logarithm of the quantity (log to the base 10). Therefore, 20 log 10 |H(j ω)| is simply the

log amplitude in decibels (dB). [ ‡ ] Thus, instead of plotting |H(j ω)|, we shall plot 20 log 10 |H(j ω)| as a function of ω. These plots (log amplitude and phase) are called Bode plots. For the transfer function in Eq. (4.82a) , the log amplitude is

The term 20 log(Ka 1 a 2 /b 1 b 3 ) is a constant. We observe that the log amplitude is a sum of four basic terms corresponding to a constant, a pole or zero at the origin (20 log |j ω|), a first-order pole or zero (20 log |1 + jω/a|), and complex-conjugate poles or zeros (20 log |1 + jωb 2 /b 3 + (j ω) 2 /b 3 |). We can sketch these four basic terms as functions of ω and use them to construct the log-amplitude plot of any desired transfer function. Let us

discuss each of the terms.

4.9-1 Constant ka 1 a 2 /b 1 b 3

The log amplitude of the constant ka 1 a 2 /b 1 b 2 term is also a constant, 20 log(Ka 1 a 2 /b 1 b 3 ). The phase contribution from this term is zero for positive value and π for negative value of the constant.

4.9-2 Pole (or Zero) at the Origin

LOG MAGNITUDE A pole at the origin gives rise to the term −20 log |jω|, which can be expressed as

This function can be plotted as a function of ω. However, we can effect further simplification by using the logarithmic scale for the variable ω itself. Let us define a new variable u such that

Hence The log-amplitude function −20u is plotted as a function of u in Fig. 4.38a . This is a straight line with a slope of −20. It crosses the u axis at u = 0.

The ω-scale (u = log ω) also appears in Fig. 4.38a . Semilog graphs can be conveniently used for plotting, and we can directly plot ω on semilog paper. A ratio of 10 is a decade, and a ratio of 2 is known as an octave. Furthermore, a decade along the ω-scale is equivalent to 1 unit along the

u-scale. We can also show that a ratio of 2 (an octave) along the ω-scale equals to 0.3010 (which is log 10 2) along the u-scale. [ † ]

Figure 4.38: (a) Amplitude and (b) phase responses of a pole or a zero at the origin. Note that equal increments in u are equivalent to equal ratios on the ω-scale. Thus, one unit along the u-scale is the same as one decade along the

ω-scale. This means that the amplitude plot has a slope of −20 dB/decade or −20(0.3010) = −6.02 dB/octave (commonly stated as −6 dB/octave).

Moreover, the amplitude plot crosses the ω axis at ω = 1, since u = log 10 ω = 0 when ω = 1.

For the case of a zero at the origin, the log-amplitude term is 20 log ω. This is a straight line passing through ω = 1 and having a slope of 20 For the case of a zero at the origin, the log-amplitude term is 20 log ω. This is a straight line passing through ω = 1 and having a slope of 20

The phase is constant ( −90°) for all values of ω, as depicted in Fig. 4.38b . For a zero at the origin, the phase is ∠jω = 90°. This is a mirror image of the phase plot for a pole at the origin and is shown dotted in Fig. 4.38b .