Page 843 - The Mechatronics Handbook

0066_frame_C27 Page 4 Wednesday, January 9, 2002 7:10 PM

of a high-order transfer function graphically. The basic types of factors that may occur in a transfer function
are as follows:

1. constant gain K,
±n
2. poles (or zeros) at the origin (jω) ,
±1
3. poles (or zeros) on the real axis (jωτ + 1) , and
±1
2
4. complex conjugate poles (or zeros) [(jω /ω n ) + 2ς( jω /ω n ) + 1] .
The curves of logarithmic magnitude and phase angle for these four factors can easily be drawn and then
added together graphically to obtain the curves for the complete transfer function. The process of drawing
the logarithmic plot can be further simplified by using asymptotic approximations to these curves and
obtaining the actual curves at specific important frequencies.
Constant Gain K
The logarithmic gain for the constant gain K is

 0°, if K > 0
20 log K = constant in decibel, ∠ K = 
 – 180°, if K < 0

The gain and phase curves are simply horizontal lines on the Bode diagram.
±n
Poles (or Zeros) at the Origin (jω)
Since

± n ± n
20 log jw = ± 20nlog w, ∠ ( jw) = ± n × 90°
±n
the slopes of the magnitude curves are ±20n dB/decade for the factor (jω) and the phase angles are
constants equal to ±n × 90°.
±1
Poles (or Zeros) on the Real Axis (jωτ + 1)
−1
For a pole factor (jωτ + 1) ,

1
1
------------------- = --------------------------
jwt + 1 w t + 1
2
2
The magnitude of the pole factor is 1 when ω << 1/τ, and 1/(ωτ) when ω >> 1/τ. Thus, there are two
asymptotic curves for the pole factor,

 0 dB, when w << -- 1 t

1
20 log ------------------- ≈ 
jwt + 1   1 1
--
 – 20 log wt = – 20 log w – log t , when w > > --

 t
The slope of the asymptotic curve when ω >> 1/τ is −20 dB/decade for the pole factor. The two asymptotes
intersect at ω = 1/τ, the break frequency or the corner frequency. The actual logarithmic gain at ω = 1/τ
−1
is −3 dB. The phase angle is φ(ω) = −tan ωτ.
The Bode diagram of a zero factor (jωτ + 1) is obtained in the same manner. However, the slope of
−1
the magnitude asymptotic curve when ω >> 1/τ is +20 dB/decade, and the phase angle is φ(ω) = +tan ωτ.
The Bode diagrams of first-order factors are shown in Fig. 27.4. Linear approximations to the phase angle
curves are also presented.

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