Step 1: Determine the characteristic equation of the closed-loop system as 1+GH(s)=0 (4.11) and rearrange the equation, if necessary so that the parameter of interest appears as the multiplying factor in the form 1+KP(s)=0 (4.12) where is a ratio of two polynomials in 2022-2-3
2022-2-3 11 Step 1: Determine the characteristic equation of the closed-loop system as 1 GH s 0 and rearrange the equation, if necessary, so that the parameter of interest, , appears as the multiplying factor in the form 1 KP s 0 where is a ratio of two polynomials in . (4.11) (4.12)
Step 2: Write in pole-zero form and write the standard root locus form 1+K (4.13) I(s+P, The magnitude condition is Ⅱ(+x)mK S+Z K K (4.14 s十 st p or (4.15) K s Z 2022-2-3 12
2022-2-3 12 Step 2: Write in pole-zero form and write the standard root locus form. 1 1 1 0 zp n i i n j j s z K s p The magnitude condition is 1 1 1 1 1 z z p p n n i i i i n n j j j j s z s z K K s p s p 1 1 p z n j j n i i s p K s z or (4.13) (4.14) (4.15)
The angle condition is II(s+zi ∠K=1 =±180°(2k+ 1) S+p or (+4s)08±=(+)了-(+)了 2022-2-3 13
2022-2-3 13 The angle condition is : 1 1 180 2 1 z p n i i n j j s z K k s p 1 1 1 2 081 p z n n i i j i k p s z s or (4.16)
Step 3: Locate the finite poles and finite zeros of on the s-plane with selected symbols (Use symbol for the poles and o for zeros). We are usually interested e in tracing the root locus as varies in the range 0≤K I(+)+k(s+)=0 (4.18) This result shows that when, the roots of the characteristic equation are simply the poles of II+p)+(+)=0 2022-2-3 14
2022-2-3 14 Step 3: Locate the finite poles and finite zeros of on the s -plane with selected symbols (Use symbol for the poles and o for zeros). We are usually interested in tracing the root locus as varies in the range 0 K 1 1 0 p z n n j i j i s p K s z This result shows that when , the roots of the characteristic equation are simply the poles of . 1 1 1 0 p z n n j i j i s p s z K (4.18) (4.19)
shows that K as approaches infinity the roots of the characteristic equation are simply the zeros of P(s). Therefore, we conclude that the root locus of the characteristic equation 1+KP(s)=o begins at the poles of P(s) and ends at the zeros of P(s as K increases from o to infinity. note that the number of P(s)the finite zeros of P(s)is usually less than or equal to itheanmber of finite poles of. This is because in most of n the cases in Egn. (4.13). This implies that there are branches of root locus approaching to the zeros at infinity 2022-2-3 15
2022-2-3 15 shows that K as approaches infinity, the roots of the characteristic equation are simply the zeros of P(s). Therefore, we conclude that the root locus of the characteristic equation 1+KP(s)=0 begins at the poles of P(s) and ends at the zeros of P(s) as K increases from 0 to infinity. Note that the number of P(s) the finite zeros of P(s) is usually less than or equal to the number of finite poles of . This is because in most of the cases in Eqn. (4.13). This implies that there are branches of root locus approaching to the zeros at infinity. n p n z nbprannczhes of root locus approaching to the zeros at infinity. n p n z n p n z