AIM: To study the Time Response, Root Locus and Bode Plot of the given function for different
values of ‘k’.
𝑘
𝑠 + 3𝑠 2 + 2𝑠
3
The root locus exists on real axis to
left of an odd number of poles and
zeros of open loop transfer function,
G(s)H(s), that are on the real axis.
These real pole and zero locations are
highlighted on diagram, along with
the portion of the locus that exists on
the real axis.
Root locus exists on real axis
between:
0 and -1, -2 and negative infinity
... because on the real axis, we have 3
poles at s = -1, -2, 0,and we have 0
zeros.
In the open loop transfer function,
G(s)H(s), we have n=3 finite poles,
and m=0 finite zeros, therefore we
have q=n-m=3 zeros at infinity.
Angle of asymptotes at odd multiples
of ±180°/q (i.e., ±60°, ±180°).
There exists 3 poles at s = 0, -2, -1,
...so sum of poles=-3.
There exists 0 zeros,
...so sum of zeros=0.
(Imaginary components of poles and
zeros, if any, cancel when
summed because they appear as
complex conjugate pairs.)
Asymptote intersect is at ( (sum of
poles)-(sum of zeros) )/q
Intersect is at ((-3)-(0))/3 = -3/3 = -1
(highlighted by five pointed star).
Break Out (or Break In) points occur
where N(s)D'(s)-N'(s)D(s)=0, or
3 s^2 + 6 s + 2 = 0.
This polynomial has 2 roots at s = -
1.6, -0.42.
From these 2 roots, there exists 2 real
roots at s = -1.6, -0.42.
These are highlighted on the diagram
above (with[Grab
squares or diamonds.)
your
Not all of these roots are on the
reader’s
locus. Of these 2 real roots, there
attention
exists 1 root at s = -0.42 on the locus
with a
(i.e., K>0).
great
quote
from the
document
or use this
Locus crosses imaginary axis at 2
space to
values of K.
emphasiz
These valuese a are
keynormally
determined by using
point. To Routh's method.
place this
This program does it numerically, and
text box
so is only an estimate.
anywhere
on the
Locus crosses where K = 0, 6,
page,tojust
corresponding crossing imaginary
drag
axis at s= 0, it.] respectively.
± 1.41j,
For K=2
The root locus exists on real axis to
left of an odd number of
poles and zeros of open loop transfer
function, G(s)H(s), that are
on the real axis.
These real pole and zero locations are
highlighted on diagram, along with
the portion of the locus that exists on
the real axis.
Root locus exists on real axis
between:
0 and -1 -2 and negative infinity
... because on the real axis,
we have 3 poles at s = -1, -2, 0,
and we have 0 zeros.
In the open loop transfer function,
G(s)H(s), we have n=3 finite poles,
and m=0 finite zeros, therefore we
have q=n-m=3 zeros at infinity.
Angle of asymptotes at odd multiples
of ±180°/q (i.e., ±60°, ±180°).
There exists 3 poles at s = 0, -2, -1,
...so sum of poles=-3.
There exists 0 zeros,
...so sum of zeros=0.
Asymptote intersect is at ( (sum of
poles)-(sum of zeros) )/q
Intersect is at ((-3)-(0))/3 = -3/3 = -1
(highlighted by five pointed star).
Break Out (or Break In) points occur
where N(s)D'(s)-N'(s)D(s)=0, or
6 s^2 + 12 s + 4 = 0
This polynomial has 2 roots at s = -
1.6, -0.42.
From these 2 roots, there exists 2 real
roots at s = -1.6, -0.42.
These are highlighted on the diagram
above (with squares or diamonds.)
Not all of these roots are on the
locus. Of these 2 real roots, there
exists 1 root at s = -0.42 on the locus
(i.e., K>0).
Break-away (or break-in) points on
the locus are shown by squares.
Locus crosses imaginary axis at 2
values of K.
These values are normally
determined by using Routh's method.
This program does it numerically, and
so is only an estimate.
Locus crosses where K = 0, 3,
corresponding to crossing imaginary
axis at s= 0, ± 1.41j, respectively.
For k=3
The root locus exists on real axis to
left of an odd number of
poles and zeros of open loop transfer
function, G(s)H(s), that are
on the real axis.
These real pole and zero locations are
highlighted on diagram,
along with the portion of the locus
that exists on the real axis.
Root locus exists on real axis
between:
0 and -1
-2 and negative infinity
... because on the real axis,
we have 3 poles at s = -1, -2, 0,
and we have 0 zeros.
In the open loop transfer function,
G(s)H(s), we have n=3 finite poles,
and m=0 finite zeros, therefore we
have q=n-m=3 zeros at infinity.
Angle of asymptotes at odd multiples
of ±180°/q (i.e., ±60°, ±180°).
There exists 3 poles at s = 0, -2, -1,
...so sum of poles=-3.
There exists 0 zeros,
...so sum of zeros=0.
Asymptote intersect is at ( (sum of
poles)-(sum of zeros) )/q
Intersect is at ((-3)-(0))/3 = -3/3 = -1
(highlighted by five pointed star).
Break Out (or Break In) points occur
where N(s)D'(s)-N'(s)D(s)=0, or
9 s^2 + 18 s + 6 = 0. (details below*)
This polynomial has 2 roots at s =-1.6,
-0.42.
From these 2 roots, there exists 2 real
roots at s = -1.6, -0.42.
These are highlighted on the diagram
above (with squares or diamonds.)
Not all of these roots are on the
locus. Of these 2 real roots,
there exists 1 root at s = -0.42 on the
locus (i.e., K>0).
Break-away (or break-in) points on
the locus are shown by squares.
Locus crosses imaginary axis at 2
values of K.
These values are normally
determined by using Routh's method.
This program does it numerically, and
so is only an estimate.
Locus crosses where K = 0, 2,
corresponding to crossing imaginary
axis at s= 0, ± 1.41j, respectively.
For k=4
The root locus exists on real axis to
left of an odd number of
poles and zeros of open loop transfer
function, G(s)H(s), that are
on the real axis.
These real pole and zero locations are
highlighted on diagram,
along with the portion of the locus
that exists on the real axis.
Root locus exists on real axis
between:
0 and -1
-2 and negative infinity
... because on the real axis,
we have 3 poles at s = -1, -2, 0,
and we have 0 zeros.
In the open loop transfer function,
G(s)H(s), we have n=3 finite poles,
and m=0 finite zeros, therefore we
have q=n-m=3 zeros at infinity.
Angle of asymptotes at odd multiples
of ±180°/q (i.e., ±60°, ±180°).
There exists 3 poles at s = 0, -2, -1,
...so sum of poles=-3.
There exists 0 zeros,
...so sum of zeros=0.
Asymptote intersect is at ( (sum of
poles)-(sum of zeros) )/q
Intersect is at ((-3)-(0))/3 = -3/3 = -1
(highlighted by five pointed star).
Break Out (or Break In) points occur
where N(s)D'(s)-N'(s)D(s)=0, or
12 s^2 + 24 s + 8 = 0. (details below*)
This polynomial has 2 roots at s = -
1.6, -0.42.
From these 2 roots, there exists 2 real
roots at s = -1.6, -0.42.
These are highlighted on the diagram
above (with squares or diamonds.)
Not all of these roots are on the
locus. Of these 2 real roots, there
exists 1 root at s = -0.42 on the locus
(i.e., K>0).
Break-away (or break-in) points on
the locus are shown by squares.
Locus crosses imaginary axis at 2
values of K.
These values are normally
determined by using Routh's method.
This program does it numerically, and
so is only an estimate.
Locus crosses where K = 0, 1.5,
corresponding to crossing imaginary
axis at s= 0, ± 1.41j, respectively.
For k=5
The root locus exists on real axis to
left of an odd number of poles and
zeros of open loop transfer function,
G(s)H(s), that are on the real axis.
These real pole and zero locations are
highlighted on diagram, along with
the portion of the locus that exists on
the real axis.
Root locus exists on real axis
between:
0 and -1 -2 and negative infinity
... because on the real axis, we have 3
poles at s = -1, -2, 0, and we have 0
zeros.
In the open loop transfer function,
G(s)H(s), we have n=3 finite poles,
and m=0 finite zeros, therefore we
have q=n-m=3 zeros at infinity.
Angle of asymptotes at odd multiples
of ±180°/q (i.e., ±60°, ±180°).
There exists 3 poles at s = 0, -2, -1,
...so sum of poles=-3.
There exists 0 zeros,
...so sum of zeros=0.
Asymptote intersect is at ( (sum of
poles)-(sum of zeros) )/q
Intersect is at ((-3)-(0))/3 = -3/3 = -1
(highlighted by five pointed star).
reak Out (or Break In) points occur
where N(s)D'(s)-N'(s)D(s)=0, or
15 s^2 + 30 s + 10 = 0.
This polynomial has 2 roots at s = -
1.6, -0.42.
From these 2 roots, there exists 2 real
roots at s = -1.6, -0.42.
These are highlighted on the diagram
above (with squares or diamonds.)
Not all of these roots are on the
locus. Of these 2 real roots, there
exists 1 root at s = -0.42 on the locus
(i.e., K>0).
Break-away (or break-in) points on
the locus are shown by squares.
Locus crosses imaginary axis at 2
values of K.
These values are normally
determined by using Routh's method.
This program does it numerically, and
so is only an estimate.
Locus crosses where K = 0, 1.2,
corresponding to crossing imaginary
axis at s= 0, ± 1.41j, respectively.
