
Concept explainers
(a)
The root locus of the given characteristic equation

Explanation of Solution
Given:
Concept Used:
Root Locus technique.
Calculation:
The characteristic equation is defined as
On solving we get,
Where,
Therefore, Poles are
And zero is
Total number of branches is
Centroid is calculated as:
Where poles are
Angle of asymptotes is calculated as:
Angle of asymptotes
Where P = no. of poles
Z = No. of zeros
On replacing the values
The
Breakaway point is calculated as:
To calculate breakaway point, replace
So,
Breakaway point is
Calculating the value of
For Stability
Auxiliary equation is defined as:
Replace the value of
This is the point on the imaginary axis
Root locus plot of the characteristic equation is
Fig.1
Conclusion:
Root has been plotted for the given characteristic equation is shown in Fig.1.
(b)
The root locus of the given characteristic equation

Explanation of Solution
Given:
Concept Used:
Root Locus technique.
Calculation:
The characteristic equation is defined as
Where,
Therefore, Poles are
And zero is
Total number of branches is
Centroid is calculated as:
Where poles are
Angle of asymptotes is calculated as:
Angle of asymptotes
Where P = no. of poles
Z = No. of zeros
On replacing the values
The
Breakaway point is calculated as:
To calculate breakaway point, replace
So,
Calculating the value of
For Stability
Auxiliary equation is defined as:
Replace the value of
For
For
This is the point on the imaginary axis
Angle of departure is calculated where there are either poles or zero is imaginary.
Root locus plot of the characteristic equation is
Fig.2
Conclusion:
Root has been plotted for the given characteristic equation is shown in Fig.2.
(c)
The root locus of the given characteristic equation

Explanation of Solution
Given:
Concept Used:
Root Locus technique.
Calculation:
The characteristic equation is defined as
Where,
Therefore, Poles are
And zero is
Total number of branches is
Centroid is calculated as:
Where poles are
Angle of asymptotes is calculated as:
Angle of asymptotes
Where P = no. of poles
Z = No. of zeros
On replacing the values
The
Breakaway point is calculated as:
To calculate breakaway point, replace
So,
Breakaway point is
Calculating the value of
For Stability
Auxiliary equation is defined as:
Replace the value of
This is the point on the imaginary axis
Root locus plot of the characteristic equation is
Fig.3
Conclusion:
Root has been plotted for the given characteristic equation is shown in Fig.3.
(d)
The root locus of the given characteristic equation

Explanation of Solution
Given:
Concept Used:
Root Locus technique.
Calculation:
The characteristic equation is defined as
On solving we get,
Therefore, Poles are
And zero is
Total number of branches is
Centroid is calculated as:
Where poles are
Angle of asymptotes is calculated as:
Angle of asymptotes
Where P = no. of poles
Z = No. of zeros
On replacing the values
The
Breakaway point is calculated as:
To calculate breakaway point, replace
So,
Breakaway point is
Calculating the value of
For Stability
Auxiliary equation is defined as:
Replace the value of
This is the point on the imaginary axis
Root locus plot of the characteristic equation is
Fig.4
Conclusion:
Root has been plotted for the given characteristic equation is shown in Fig4.
(e)
The root locus of the given characteristic equation

Explanation of Solution
Given:
Concept Used:
Root Locus technique.
Calculation:
The characteristic equation is defined as
Where,
Therefore, Poles are
And zero is
Total number of branches is
Centroid is calculated as:
Where poles are
Angle of asymptotes is calculated as:
Angle of asymptotes
Where P = no. of poles
Z = No. of zeros
On replacing the values
The
Breakaway point is calculated as:
To calculate breakaway point, replace
So,
Breakaway point is
Calculating the value of
For Stability
Auxiliary equation is defined as:
Replace the value of
This is the point on the imaginary axis
Root locus plot of the characteristic equation is
Fig.5
Conclusion:
Root has been plotted for the given characteristic equation is shown in Fig.5.
(f)
The root locus of the given characteristic equation

Explanation of Solution
Given:
Concept Used:
Root Locus technique.
Calculation:
The characteristic equation is defined as
Where,
Therefore, Poles are
And zero is
Total number of branches is
Centroid is calculated as:
Where poles are
Angle of asymptotes is calculated as:
Angle of asymptotes
Where P = no. of poles
Z = No. of zeros
On replacing the values
The
Breakaway point is calculated as:
To calculate breakaway point, replace
So,
Breakaway point is
Calculating the value of
For Stability
Auxiliary equation is defined as:
Replace the value of
For
For
Root locus plot of the characteristic equation is
Fig.6
Conclusion:
Root has been plotted for the given characteristic equation is shown in Fig6.
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