The angle of emergence from complex poles is given by

Qpl?180????(pl?pj)???(pl?zj)

j?lj

?180??90??14??120???46?

The characteristic equation is s(s?8)(s2?2s?4)?7K?0 Substituting s?j? yields the following equations: ?4?20?2?7K?0 and 32??10?3?0

Solving the equations yields: K?7.68 and ??1.79.

1K??(s4?10s3?20s2?32s)

7dK1??(4s3?30s2?40s?32) ds7The point at which the locus leaves the real axis is given by Rule 7,which yields the result s??6.1-6.07 for the break point.

Finally, we can plot the root locus shown in Fig.1

Fig.1

(2)

There are two complex closed-loop poles and two real closed-loop poles for K?20.

The root to the left of the pole at s?0and the right of the point at s??6.1 is best determined.

We will use a trial point st.

This root is found to be at approximately s??3.4.

Next, the root to the left of the point at s??6.1 and the right of the pole at s??8 is best determined.

We will also use a trial point st.

This root is found to be approximately s??7.6. Rule 10 may now be used to find the remaining root.

?3.4?7.6??????10 ???0.5s?0.5?2.26jand the points at s??3.4 and s??7.6.

Module11

Problem 11.1

Using the methed outlined in the previous section for multiloop systems, the inner loop will be closed first using the root locus methed, and these closeloop poles will become the open-loop poles of the outer loop.

kG(s)in?2 when k2=10

s?1the inner closed-loop pole is s=-11

G(s)out?2k1k2 wherek1=10

(s?2)(s?11)the outer closed-loop poles are

s??6.5?719j

Problem 11.2

1.GROUP 4 finished Solution：

The open-loop transfer function of the system is :

0.01125 G(s)?2500?1?0.5s??0.237KVss(1?0.5s?0.237Kv)1?1?0.5sThe closed-loop transfer function of the system is: