Page 95 - Mechatronics with Experiments

CLOSED LOOP CONTROL 81
Bode Plots of Standard Elements of a Transfer Function

1. Constant Gain, K : A constant gain will have a constant logarithmic magnitude as a
0
function of frequency, and a zero phase. If the sign of the gain is negative, the phase
◦
will be −180 ,
20 log | ⋅ | = 20 log K (2.57)
10 10 0
Im(K ) −1 0
0
−1
∠(.) =tan =tan = 0 (2.58)
Re(K ) Re(K )
0
0
Im(K ) 0
◦
0
=tan −1 =tan −1 =−180 ; for K < 0 (2.59)
0
Re(K ) Re(K )
0
0
2. Pole/zero at the origin: Pole at the origin
1 1 1 1 −jN90
| s=jw = = = ⋅ e (2.60)
N jN90
s N (jw) N w e w N
The magnitude and phase in Bode plots is given by
| 1 | 1
20 log | | = 20 log =−20 N log w (2.61)
10 | N | 10 w N 10
|(jw) |
( )
1 ◦
∠ =−N ⋅ 90 (2.62)
(jw) N
Similarly, for zero(s) at the origin, the Bode plot is
N N N jN90 N jN90
s | s=jw = (jw) = w e = w e (2.63)
and
N
N
20 log |(jw) | = 20 log w = 20 N log w (2.64)
10
10
10
N
∠(jw) = N ⋅ 90 ◦ (2.65)
The Bode plots of gain, pole(s), and zero(s) at the origin are shown in Figure 2.30.
3. First-order pole and zero: Let us consider a pole on a real axis,
1 1
| s=jw = (2.66)
(s∕p + 1) (jw∕p + 1)
i
i
1
= −1 (2.67)
2 1∕2 j tan
[1 + (w∕p ) ] e (w∕p i )
i
1 −j tan −1 (w∕p i )
= e (2.68)
2 1∕2
[1 + (w∕p ) ]
i
The magnitude and phase as a function of frequency are given by
| 1 | 1 2 1∕2
20 log | | = 20 log =−20 log [1 + (w∕p ) ] (2.69)
i
2 1∕2
10 | | [1 + (w∕p ) ]
i
| jw∕p + 1 | i
≈−20 log 1 = 0; for w∕p ≪ 1 (2.70)
i
≈−20 log(w∕p ); for w∕p ≫ 1 (2.71)
i
i
1 −1
∠ =− tan (w∕p ) (2.72)
i
(jw∕p + 1)
i
Similiar algebraic calculations can be carried out for a zero on the real axis, and the
Bode plots are as follows,