Module 10
Rules for Plotting the Root Locus
(4 hours)
(P173) In this section the concepts outlined previously will
be developed further into some straightforward guidelines
for plotting more complex root loci,which will be
illustrated by focusing on a specific example.
The locus starts at the open-loop poles ( the
closed-loop poles for K = 0 ),and finishes at the
open-loop zeros (the closed-loop zeros for K= ),
The number of segments going to infinity is n-m.
(根轨迹始于开环极点，终于开环零点。趋于无穷大的线段条数为 n-m。若 n>m,则有 n-m条根轨迹终止于无穷远处；若 m>n，则有 m-n条根轨迹起始于无穷远处。 )
Rule #1 The Starting Points and the End Points
of the Root Locus (根轨迹的起点和终点 ）
[ Proving ]
)(
)(
)(
1
1
i
n
i
j
m
j
ps
zsK
sG
p o l el o o po p e np
z e r ol o o po p e nz
i
j
0)()(0)(1
11
j
m
ji
n
i
zsKpssG
At the starting point of the root locus,K=0
),,2,1(;,0)( nipsps ii
At the end point of the root locus,K
and the characteristic equation can be written as
0)()(1
11
j
m
ji
n
i
zspsK
whenK
jzs?
),,2,1( mj
Segment of the real axis to the left of on an odd number
of poles or zeros are segments of the root locus,
remembering that complex poles or zeros have no effect.
(实轴上的根轨迹，是其右侧的开环零、极点数之和为奇数的所在线段。或者说，实轴上，对应零、极点数之和为奇数的左边线段为根轨迹。复数零、极点对该线段没有影响。 )
Rule #2 The Segments of the Root Locus on the
Real Axis(实轴上的根轨迹 ）
[ Proving ]
j
1p2p3p
j
1p
2p
3p
1p
11 ps?
21 ps?
31 ps?
1s
180
1s
180
The loci are symmetrical about the real axis since
complex roots are always in conjugate pairs.（ 根轨迹关于实轴对称，因为复数根总是成对出现的。 ）
The angle between adjacent asymptotes is 360o/(n-m),
and to obey the symmetry rule,the negative real axis is
one asymptote when n-m is odd.（ 相邻的渐近线之间的夹角是
360o/(n-m),并同样服从对称规律。当 n-m 是奇数时，负实轴也是一个渐近线。 ）
The Angle of the asymptotes and real axis is,(渐近线与实轴正向的夹角是 ）
Rule #3 The Symmetry and the Asymptotes of
the Root Locus (根轨迹的对称性和渐近线 ）
)1,,1,0(12 mnkmn ka
The asymptotes intersect the real axis at,
where is the sum of the real parts of the open-loop
poles (including complex roots) and is the sum
of the real parts of the open-loop zeros (also including
complex zeros).
（渐近线与实轴的交点是,式中是开环极点的实部的和（包括复数极点）；
是开环零点的实部的和（包括复数零点）。）
Rule #4 The Real Axis intercept of the
Asymptotes (渐近线和实轴的交点 ）
a?
mn
zp ji
a?
ip
jz
jz
ipmn zp jia
SP.
j
01?2?
1
03
)2()1(0
mn
zp ji
a?
2;3/5/
1;
0;3/
)1,,1,0(
12
k
k
k
mnk
mn
k
a
60
The angle of emergence from complex poles is given by 180o –
Σ(angles of the vectors from all other open-loop poles to the poles
in question) + Σ(angles of the vectors from the open-loop zeros to
the complex pole in question).
Rule #5 The Angle of Emergence from Complex
Poles and The Angle of Entry into Complex Zeros (根轨迹的出射角和入射角）
n
ij
j
m
j
jijip zpppi
1 1
)()(180?
The angle of entry into a complex zero may be found from the
same rule and then the sign changed to produce the final result.
m
ij
j
n
j
jijiz pzzzi
1 1
)()(1 8 0?
SP.
)22)(22(
)5(
)84(
)5()(
2 jsjss
sK
sss
sKsGH
123390135180
3
2
0
4
2
2
180
))5()22((
))22()22(()022((180
111
22
tgtgtg
j
jjj
j
The point where the locus crosses the imaginary axis
may be obtained by sunbstituting s = jω into the
characteristic equation and solving for ω.
Rule #6 The Root Locus Crossing with the Imaginary
Axis (根轨迹与虚轴的交点）
0])(1[Im
0])(1R e [
0)(10)(1
jGH
jGH
jGHsGH
05)8(4 23 KKsss
)84(
)5()(
2
sss
sKsGHSP.
0])8[()45( 32 KjK
0)8(
045
2
2
K
K
32.6
32
K
The point at which the locus leaves a real-axis segment
is found by determining a local maximum value of K,
while the point at which the locus enters a real-axis
segment is found by determining a local minimum value
of K.(根轨迹离开实轴区段的点（分离点）由该区段的最大 K值来确定；而根轨迹进入实轴区段的点（分离点）由该区段的最小 K值来确定。 )
Rule #7 The Breakaway Point of the Root Locus (根轨迹的分离点）
j
01?2?
60
j
01?2?
Assume the breakaway point s = d:
)
)(
)(;1
)(
)(
)((0)1(
1
1
1
1
j
m
j
i
n
i
i
n
i
j
m
j
ds
zs
ps
K
ps
zsK
sGHK
ds
d
n
i i
m
j j pdzd 11
11)2(
SP.
)1()( ss
KsGH
5.0
0)12()]1([)1(
d
dss
ds
dK
ds
d
dsds
5.001
1
111
0
1
)2(
11
ddd
ddpdzd
n
i i
m
j j
j
0-1 -0.5
SP.
)2)(1()( sss
KsGH
SP.
)1(
)2()(
ss
sKsGH
)i n t(423.0
)(577.1
2
1
pob r e a k a w a yd
upg i v e nbetod
)i n t(414.3
)i n t(586.0
2
1
pob r e a k a w a yd
pob r e a k a w a yd
Rule #8 The angle between the direction of
emergence (or entry) of q coincident poles (or zeros) on
the real axis (根轨迹离开或进入实轴上 q重极点（或零点）方向之间的夹角）
q
3 6 0
SP.
)1()( 2 ss
KsGH
SP.
3)1()( s
KsGH
j
0-1
j
0-1
3 poles
The gain at a selected point s1 on the locus is obtained
by joining the point to all open-loop poles and zeros
and measuring the length of each line,
The gain is given by
Rule #9 The gain at a selected point s1 on the locus
(在某特定点上的根轨迹增益 K）
jtit zsps,
j
m
j
i
n
i
zs
ps
K
1
1
At the breakaway point s =
-2.6,Gain K is
SP.
2473.0
216.2216.2
6.236.226.2
2
6.2
1
1
jj
zs
ps
K
s
j
m
j
i
n
i
If there are at least two more open-
loop poles than open-loop zeros,the
sum of the closed-loop poles is
constant,independent of K,and
equal to the sum of the real parts of
the open-loop poles.
(如果开环极点比开环零点至少多 2个，闭环极点的和为一不依赖于 K的常数，且等于开环极点的实部的和。 ）
Rule #10 The sum of the closed-loop poles (闭环极点之和）
)22)(22(
)5()(
jsjss
sKsGH
1,32,1 sjsif
2201
5.1
The number of branches of the root locus is equal
to the maximum in the number N of poles and the
number M of zeros of the open-loop transfer
function.
（根轨迹的分支数等于传递函数中极点数 M和零点数 N
中的最大数）
Rule #11 The number of branches of the root locus
(根轨迹的分支数）
MNb,m a x?
3
5
1223
,,321
jjps
sjsjs
ii
Instructional objectives:
At the end of this lecture students
should be able to
Find the directions of the pole asymptotes
Find the values of K for which the poled
cross the imaginary axis
Determine plot the root locus and select
“good” values for K
Rules for Plotting the Root Locus
(4 hours)
(P173) In this section the concepts outlined previously will
be developed further into some straightforward guidelines
for plotting more complex root loci,which will be
illustrated by focusing on a specific example.
The locus starts at the open-loop poles ( the
closed-loop poles for K = 0 ),and finishes at the
open-loop zeros (the closed-loop zeros for K= ),
The number of segments going to infinity is n-m.
(根轨迹始于开环极点，终于开环零点。趋于无穷大的线段条数为 n-m。若 n>m,则有 n-m条根轨迹终止于无穷远处；若 m>n，则有 m-n条根轨迹起始于无穷远处。 )
Rule #1 The Starting Points and the End Points
of the Root Locus (根轨迹的起点和终点 ）
[ Proving ]
)(
)(
)(
1
1
i
n
i
j
m
j
ps
zsK
sG
p o l el o o po p e np
z e r ol o o po p e nz
i
j
0)()(0)(1
11
j
m
ji
n
i
zsKpssG
At the starting point of the root locus,K=0
),,2,1(;,0)( nipsps ii
At the end point of the root locus,K
and the characteristic equation can be written as
0)()(1
11
j
m
ji
n
i
zspsK
whenK
jzs?
),,2,1( mj
Segment of the real axis to the left of on an odd number
of poles or zeros are segments of the root locus,
remembering that complex poles or zeros have no effect.
(实轴上的根轨迹，是其右侧的开环零、极点数之和为奇数的所在线段。或者说，实轴上，对应零、极点数之和为奇数的左边线段为根轨迹。复数零、极点对该线段没有影响。 )
Rule #2 The Segments of the Root Locus on the
Real Axis(实轴上的根轨迹 ）
[ Proving ]
j
1p2p3p
j
1p
2p
3p
1p
11 ps?
21 ps?
31 ps?
1s
180
1s
180
The loci are symmetrical about the real axis since
complex roots are always in conjugate pairs.（ 根轨迹关于实轴对称，因为复数根总是成对出现的。 ）
The angle between adjacent asymptotes is 360o/(n-m),
and to obey the symmetry rule,the negative real axis is
one asymptote when n-m is odd.（ 相邻的渐近线之间的夹角是
360o/(n-m),并同样服从对称规律。当 n-m 是奇数时，负实轴也是一个渐近线。 ）
The Angle of the asymptotes and real axis is,(渐近线与实轴正向的夹角是 ）
Rule #3 The Symmetry and the Asymptotes of
the Root Locus (根轨迹的对称性和渐近线 ）
)1,,1,0(12 mnkmn ka
The asymptotes intersect the real axis at,
where is the sum of the real parts of the open-loop
poles (including complex roots) and is the sum
of the real parts of the open-loop zeros (also including
complex zeros).
（渐近线与实轴的交点是,式中是开环极点的实部的和（包括复数极点）；
是开环零点的实部的和（包括复数零点）。）
Rule #4 The Real Axis intercept of the
Asymptotes (渐近线和实轴的交点 ）
a?
mn
zp ji
a?
ip
jz
jz
ipmn zp jia
SP.
j
01?2?
1
03
)2()1(0
mn
zp ji
a?
2;3/5/
1;
0;3/
)1,,1,0(
12
k
k
k
mnk
mn
k
a
60
The angle of emergence from complex poles is given by 180o –
Σ(angles of the vectors from all other open-loop poles to the poles
in question) + Σ(angles of the vectors from the open-loop zeros to
the complex pole in question).
Rule #5 The Angle of Emergence from Complex
Poles and The Angle of Entry into Complex Zeros (根轨迹的出射角和入射角）
n
ij
j
m
j
jijip zpppi
1 1
)()(180?
The angle of entry into a complex zero may be found from the
same rule and then the sign changed to produce the final result.
m
ij
j
n
j
jijiz pzzzi
1 1
)()(1 8 0?
SP.
)22)(22(
)5(
)84(
)5()(
2 jsjss
sK
sss
sKsGH
123390135180
3
2
0
4
2
2
180
))5()22((
))22()22(()022((180
111
22
tgtgtg
j
jjj
j
The point where the locus crosses the imaginary axis
may be obtained by sunbstituting s = jω into the
characteristic equation and solving for ω.
Rule #6 The Root Locus Crossing with the Imaginary
Axis (根轨迹与虚轴的交点）
0])(1[Im
0])(1R e [
0)(10)(1
jGH
jGH
jGHsGH
05)8(4 23 KKsss
)84(
)5()(
2
sss
sKsGHSP.
0])8[()45( 32 KjK
0)8(
045
2
2
K
K
32.6
32
K
The point at which the locus leaves a real-axis segment
is found by determining a local maximum value of K,
while the point at which the locus enters a real-axis
segment is found by determining a local minimum value
of K.(根轨迹离开实轴区段的点（分离点）由该区段的最大 K值来确定；而根轨迹进入实轴区段的点（分离点）由该区段的最小 K值来确定。 )
Rule #7 The Breakaway Point of the Root Locus (根轨迹的分离点）
j
01?2?
60
j
01?2?
Assume the breakaway point s = d:
)
)(
)(;1
)(
)(
)((0)1(
1
1
1
1
j
m
j
i
n
i
i
n
i
j
m
j
ds
zs
ps
K
ps
zsK
sGHK
ds
d
n
i i
m
j j pdzd 11
11)2(
SP.
)1()( ss
KsGH
5.0
0)12()]1([)1(
d
dss
ds
dK
ds
d
dsds
5.001
1
111
0
1
)2(
11
ddd
ddpdzd
n
i i
m
j j
j
0-1 -0.5
SP.
)2)(1()( sss
KsGH
SP.
)1(
)2()(
ss
sKsGH
)i n t(423.0
)(577.1
2
1
pob r e a k a w a yd
upg i v e nbetod
)i n t(414.3
)i n t(586.0
2
1
pob r e a k a w a yd
pob r e a k a w a yd
Rule #8 The angle between the direction of
emergence (or entry) of q coincident poles (or zeros) on
the real axis (根轨迹离开或进入实轴上 q重极点（或零点）方向之间的夹角）
q
3 6 0
SP.
)1()( 2 ss
KsGH
SP.
3)1()( s
KsGH
j
0-1
j
0-1
3 poles
The gain at a selected point s1 on the locus is obtained
by joining the point to all open-loop poles and zeros
and measuring the length of each line,
The gain is given by
Rule #9 The gain at a selected point s1 on the locus
(在某特定点上的根轨迹增益 K）
jtit zsps,
j
m
j
i
n
i
zs
ps
K
1
1
At the breakaway point s =
-2.6,Gain K is
SP.
2473.0
216.2216.2
6.236.226.2
2
6.2
1
1
jj
zs
ps
K
s
j
m
j
i
n
i
If there are at least two more open-
loop poles than open-loop zeros,the
sum of the closed-loop poles is
constant,independent of K,and
equal to the sum of the real parts of
the open-loop poles.
(如果开环极点比开环零点至少多 2个，闭环极点的和为一不依赖于 K的常数，且等于开环极点的实部的和。 ）
Rule #10 The sum of the closed-loop poles (闭环极点之和）
)22)(22(
)5()(
jsjss
sKsGH
1,32,1 sjsif
2201
5.1
The number of branches of the root locus is equal
to the maximum in the number N of poles and the
number M of zeros of the open-loop transfer
function.
（根轨迹的分支数等于传递函数中极点数 M和零点数 N
中的最大数）
Rule #11 The number of branches of the root locus
(根轨迹的分支数）
MNb,m a x?
3
5
1223
,,321
jjps
sjsjs
ii
Instructional objectives:
At the end of this lecture students
should be able to
Find the directions of the pole asymptotes
Find the values of K for which the poled
cross the imaginary axis
Determine plot the root locus and select
“good” values for K
