Page 97 - Mechatronics with Experiments

CLOSED LOOP CONTROL 83
2 1∕2 j tan −1 (w∕z i )
= (jw∕z + 1) = [1 + (w∕z ) ] e (2.73)
(s∕z + 1)|
i s=jw i i
2 1∕2
2 1∕2
20 log |jw∕z + 1| = 20 log [1 + (w∕z ) ] = 20 log [1 + (w∕z ) ] (2.74)
i
i
i
=≈ 20 log 1 = 0; for w∕z ≪ 1 (2.75)
i
=≈ 20 log(w∕z ); for w∕z ≫ 1 (2.76)
i
i
−1
∠(jw∕z + 1) =tan (w∕z ) (2.77)
i
i
The Bode plots of first-order (real) poles and zeros are also shown in Figure 2.30.
4. Second-order (complex conjugate) poles and zeros: Consider a complex conjugate
pole pair, and its frequency response
1 1
| s=jw = ( )
2
(s∕w ) + 2 (s∕w ) + 1 j tan −1 2 w∕w i
i
i
2
2 1∕2
2
2 2
[(1 − w ∕w ) + (2 w∕w ) ] e 1−(w 2 ∕w )
i
i i
(2.78)
The magnitude and phase as function of frequency are given by
2 2
2 1∕2
2
20 log | ⋅ | =−20 log[(1 − w ∕w ) + (2 w∕w ) ] (2.79)
i i
≈ 0; w∕w ≪ 1 (2.80)
i
≈−40 log(w∕w ); w∕w ≫ 1 (2.81)
i i
( )
2 (w∕w )
−1 i
∠(⋅) =− tan (2.82)
2
2
(1 − w ∕w )
i
Similarly, the Bode plot of a complex conjugate zero and its asymptotic plot can be
found as (Figure 2.30).
( )
j tan −1 2 w∕w i
2 2 2 2 2 1∕2 1−(w 2 ∕w )
2
(s∕w ) + 2 (s∕w ) + 1| = [(1 − (w ∕w )) + (2 w∕w ) ] e i
i i s=jw i i
(2.83)
2 2
2 1∕2
2
20 log | ⋅ | = 20 log[(1 − (w ∕w )) + (2 w∕w ) ] (2.84)
i i
≈ 0; w∕w ≪ 1 (2.85)
i
≈ 40 log(w∕w ); w∕w ≫ 1 (2.86)
i
i
( )
2 (w∕w )
i
−1
∠(.) =tan (2.87)
2
2
1 − (w ∕w )
i
Nyquist (Polar) Plots of Standard Elements of a Transfer Function
Nyquist plots (also called polar plots) are the graphical representation of the frequency
response data on a complex plane where the y-axis is the imaginary part, and the x-axis is
the real part of the frequency response. The frequency is parameterized along the curve.
= Re(G(jw)) + jIm(G(jw)) = X(w) + jY(w) (2.88)
G(jw) = G(s)| s=jw
For every point along the imaginary axis of the s-plane, s = jw, there is a point on the
curve plotted on the (G(jw))-plane. Nyquist plots of various standard elements are shown
in Figure 2.31.
Log Magnitude versus Phase Plots of Standard Elements of a Trans-
fer Function Frequency response is conveniently represented by a complex function.
Graphical representation of a complex function must convey the real and imaginary part or
magnitude and phase information. Another possible way of plotting the frequency response