
If we let x and y be the real and imaginary parts of lT, respectively, that is, lT ¼x þjy, then
x ¼ cos u 1, y ¼ sin u (8:151)
) (x þ 1)
2
þ y
2
¼ cos
2
u þ sin
2
u ¼ 1 (8:152)
and the AB-1 stability boundary is therefore a circle with center at (1, 0) and radius 1 in the lT or
x–y plane.
AB-1 integration is inappropriate for simulation of an undamped second-order system, a result
we observed earlier in Section 3.6. The charact eristic roots of a second-order system with z ¼0 are
l ¼jv
n
, and hence lT ¼jv
n
T, which corresponds to the imaginary axis in the lT plane. From
Figure 8.21, it is clear that the imaginary axis lies outside the AB-1 stability region ...