Root Locus Diagram MCQ Quiz - Objective Question with Answer for Root Locus Diagram - Download Free PDF

Explanation:

Root Locus Analysis

Definition: Root locus analysis is a graphical method used in control systems to study how the roots of a system's characteristic equation change with variations in a system parameter, typically the gain (K). It provides critical insights into the stability and behavior of the system as the gain is adjusted. The root locus plot helps determine the locations of poles and zeros of the closed-loop transfer function as a function of the gain.

Correct Option Analysis:

The correct option is:

Option 1: The root loci start at the open loop poles.

This statement is always true with respect to root locus analysis. Root locus plots are constructed based on the principle that the loci of the roots (poles) of the closed-loop transfer function start at the poles of the open-loop transfer function when the gain (K) is zero. This is a fundamental concept in root locus theory, as the open-loop poles represent the initial locations of the system's poles before any feedback is applied.

Reasoning:

To understand why this statement is correct, consider the characteristic equation of the closed-loop transfer function:

1 + KG(s) = 0

Here, G(s) is the open-loop transfer function, and K is the gain. When K = 0, the characteristic equation reduces to:

G(s) = 0

This equation indicates that the roots of the closed-loop system are the same as the poles of the open-loop transfer function. As the gain K increases, the roots (poles) move along the root locus paths, but they always start at the open-loop poles.

 Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: Number of separate root loci is the number of open-loop poles.

This statement is incorrect. The number of separate root loci is determined by the total number of poles and zeros of the open-loop transfer function, not just the number of poles. If there are more poles than zeros, the number of root loci corresponds to the total number of poles. However, if the number of poles equals the number of zeros, the paths will merge and terminate at the zeros.

Option 3: The root loci terminate at open-loop poles.

This statement is incorrect. The root loci do not terminate at the open-loop poles. Instead, the root loci terminate at the open-loop zeros or go to infinity if the system has more poles than zeros. This is a key feature of root locus analysis, as the termination points depend on the locations of the open-loop zeros and the relative number of poles and zeros.

Option 4: The root loci break away at open-loop poles.

This statement is incorrect. Root loci do not break away at the open-loop poles. Breakaway points occur at locations along the root locus paths where multiple branches of the root locus diverge or converge. These points are calculated based on the derivative of the characteristic equation and do not necessarily coincide with the open-loop poles.

Explanation:

Number of Loci, Starting Points, Ending Points, and Asymptotes

Understanding the Parameters:

The question involves analyzing a system based on certain parameters: the number of loci, starting points, ending points, and asymptotes. Let us define each of these terms:

• Number of Loci: This refers to the number of branches or paths traced by the roots of the characteristic equation in the complex plane as a specific parameter (such as gain) is varied.
• Starting Points: These are the initial positions of the roots of the characteristic equation (typically the poles of the transfer function) in the complex plane when the gain is zero.
• Ending Points: These are the final positions of the roots as the gain approaches infinity. These points can be either finite (zeros of the transfer function) or infinite, depending on the system configuration.
• Number of Asymptotes: Asymptotes describe the directions in which the loci tend toward infinity in the complex plane. The number of asymptotes is determined by the difference between the number of poles and zeros of the system.

Correct Option Analysis:

The correct option is:

Option 2: 2 loci, 2 starting points (-2, -4), ending points (2 + j4, 2 - j4), and 0 asymptotes.

Let us analyze this in detail:

Step 1: Determining the Number of Loci

The number of loci is equal to the total number of poles in the system, as each pole corresponds to a branch in the root locus. From the given data, there are 2 poles at (-2, -4). Hence, the number of loci is 2.

Step 2: Identifying Starting Points

The starting points are the locations of the poles of the system when the gain is zero. From the given system, the poles are located at (-2, -4). Therefore, the starting points are -2 and -4.

Step 3: Identifying Ending Points

As the gain increases to infinity, the loci of the poles move toward either the zeros of the system or infinity. From the given data, the ending points are specified as (2 + j4, 2 - j4), which are complex conjugate zeros. Hence, the ending points are (2 + j4, 2 - j4).

Step 4: Determining the Number of Asymptotes

The number of asymptotes is given by the formula:

Number of Asymptotes = Number of Poles - Number of Zeros

From the given data:

• Number of poles = 2 (located at -2 and -4).
• Number of zeros = 2 (located at 2 + j4 and 2 - j4).

Substituting these values:

Number of Asymptotes = 2 - 2 = 0

Therefore, there are 0 asymptotes.

Conclusion for Correct Option:

Based on the analysis, the system has:

• Number of Loci: 2
• Starting Points: (-2, -4)
• Ending Points: (2 + j4, 2 - j4)
• Number of Asymptotes: 0

This matches the description provided in Option 2, making it the correct choice.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: 1 locus, starting points (2, 4), ending points (2 + j4, 2 - j4), and 0 asymptotes.

This option is incorrect because the number of loci is mismatched. The system has 2 poles, which means it should have 2 loci, not 1 as stated in this option. Additionally, the starting points are incorrectly given as (2, 4) instead of (-2, -4).

Option 3: 1 locus, starting points (-2, -4), ending points (2 + j4, 2 - j4), and 0 asymptotes.

This option is incorrect because, similar to Option 1, it states only 1 locus instead of 2. While the starting points and ending points are correct, the discrepancy in the number of loci makes this option invalid.

Option 4: 2 loci, starting points (2, 4), ending points (2 + j4, 2 - j4), and 0 asymptotes.

This option is incorrect because, although the number of loci and asymptotes are accurate, the starting points are incorrectly given as (2, 4) instead of (-2, -4). This error invalidates the option.

Conclusion:

Option 2 is the correct choice as it accurately describes the system's parameters: 2 loci, starting points at (-2, -4), ending points at (2 + j4, 2 - j4), and 0 asymptotes. The analysis of other options highlights the importance of correctly identifying the number of loci, starting points, ending points, and asymptotes to evaluate the system's behavior.

Explanation:

To determine the centroid of the asymptotes in the root locus of the given loop transfer function, we need to first understand the parameters involved. The given loop transfer function of the closed-loop control system is:

G(s) H(s) = k(s + 1) / [s(s + 2)(s + 3)]

The root locus method involves plotting the roots of the characteristic equation of the closed-loop system as the gain "k" varies from 0 to ∞. The centroid of the asymptotes is a point on the real axis where the asymptotes of the root locus intersect the real axis. This point is important as it provides insights into the system's stability and behavior.

The formula for finding the centroid (σ) of the asymptotes is given by:

σ = (sum of poles - sum of zeros) / (number of poles - number of zeros)

From the given transfer function:

Poles: s = 0, s = -2, s = -3

Zeros: s = -1

Number of poles (P) = 3

Number of zeros (Z) = 1

Sum of poles = 0 + (-2) + (-3) = -5

Sum of zeros = -1

Substituting these values into the formula for the centroid:

σ = (-5 - (-1)) / (3 - 1)

σ = (-5 + 1) / 2

σ = -4 / 2

σ = -2

Therefore, the centroid of the asymptotes in the root locus is at (-2, 0).

Correct Option Analysis:

The correct option is:

Option 3: (-2, 0)

This option correctly identifies the centroid of the asymptotes for the given loop transfer function

Explanation:

To determine the gain at the breakaway point for the open-loop transfer function \( G(s)H(s) = \frac{K}{s(s+1)} \), we need to follow the steps involved in root locus analysis. The breakaway point is the point on the real axis where the root locus leaves or enters the real axis.

Step-by-Step Solution:

Step 1: Identify the poles and zeros of the open-loop transfer function.

The open-loop transfer function is given by:

\( G(s)H(s) = \frac{K}{s(s+1)} \)

From this transfer function, we can see that there are two poles:

• Poles: \( s = 0 \) and \( s = -1 \)
• Zeros: None

Step 2: Determine the characteristic equation.

The characteristic equation is obtained by setting the denominator of the closed-loop transfer function to zero:

\( 1 + G(s)H(s) = 0 \)

Substitute \( G(s)H(s) \):

\( 1 + \frac{K}{s(s+1)} = 0 \)

Multiply through by \( s(s+1) \) to clear the fraction:

\( s(s+1) + K = 0 \)

Or:

\( s^2 + s + K = 0 \)

This is the characteristic equation of the system.

Step 3: Find the breakaway point.

The breakaway point occurs where the root locus intersects the real axis between the poles. To find the breakaway point, we need to find the values of \( s \) where the derivative of the characteristic equation with respect to \( s \) is zero:

\( \frac{d(s^2 + s + K)}{ds} = 0 \)

Differentiate the characteristic equation with respect to \( s \):

\( 2s + 1 = 0 \)

Solve for \( s \):

\( 2s + 1 = 0 \)

\( 2s = -1 \)

\( s = -\frac{1}{2} \)

The breakaway point is at \( s = -\frac{1}{2} \).

Step 4: Determine the gain \( K \) at the breakaway point.

Substitute \( s = -\frac{1}{2} \) into the characteristic equation to find the gain \( K \):

\( \left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right) + K = 0 \)

Simplify the equation:

\( \frac{1}{4} - \frac{1}{2} + K = 0 \)

\( \frac{1}{4} - \frac{2}{4} + K = 0 \)

\( -\frac{1}{4} + K = 0 \)

Solve for \( K \):

\( K = \frac{1}{4} \)

Therefore, the gain at the breakaway point is \( K = 0.25 \).

Correct Option:

The correct option is:

Option 1: 0.25

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: -0.25

This option is incorrect because the gain \( K \) at the breakaway point cannot be negative. The calculation shows that the correct value is \( 0.25 \), not \( -0.25 \).

Option 3: -0.5

Similar to option 2, this option is incorrect because the gain \( K \) at the breakaway point cannot be negative. The correct value is \( 0.25 \), not \( -0.5 \).

Option 4: 0.5

This option is incorrect because the gain \( K \) at the breakaway point is \( 0.25 \), not \( 0.5 \). The calculation clearly shows that the value is \( 0.25 \).

Conclusion:

Understanding the root locus method and the process of finding the breakaway point is essential for control system analysis. The breakaway point for the given open-loop transfer function \( G(s)H(s) = \frac{K}{s(s+1)} \) was found to be at \( s = -\frac{1}{2} \), and the corresponding gain \( K \) at this point was determined to be \( 0.25 \). This analysis confirms that the correct option is indeed option 1.

Explanation:

Correct Option Analysis:

The correct option is:

Option 2: Exponential and oscillatory increase.

This option correctly describes the behavior of a system whose poles are complex with positive real parts. In control systems and signal processing, the poles of a system determine its stability and response characteristics. The presence of complex poles indicates that the system will exhibit oscillatory behavior, while positive real parts of the poles indicate that the amplitude of these oscillations will grow exponentially over time, leading to an unstable system with increasing oscillatory output.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: Exponential decay.

This option is incorrect because exponential decay occurs when the poles of the system have negative real parts. In such a case, the system's response diminishes over time, leading to a stable system with decreasing amplitude. Since the poles in the given scenario have positive real parts, this option does not apply.

Option 3: Decaying and oscillatory.

This option is incorrect because decaying oscillations occur when the poles are complex with negative real parts. This results in an exponentially decaying amplitude of the oscillatory response, leading to a stable system. However, the given scenario describes poles with positive real parts, which leads to increasing, not decreasing, oscillations.

Option 4: Exponential increase.

This option is partially correct but incomplete. Exponential increase describes the growth of the system's response over time due to positive real parts of the poles. However, it does not account for the oscillatory nature of the response indicated by the complex poles. Therefore, "exponential increase" alone does not fully capture the system's behavior.

Conclusion:

Understanding the implications of pole locations in the complex plane is crucial for analyzing system stability and response characteristics. Poles with positive real parts lead to an unstable system with an exponentially growing response, while complex poles introduce oscillations. The correct description for a system with complex poles and positive real parts is "exponential and oscillatory increase," highlighting the dual aspects of growth and oscillation in the system's behavior.

• The root locus is the strongest tool for determining stability and the transient response of the system as it gives the exact pole-zero location and also their effect on the response
• A Bode plot is a useful tool that shows the gain and phase response of a given LTI system for different frequencies
• The Nyquist plot in addition to providing absolute stability also gives information on the relative stability of stable systems and degree of instability of the unstable system
• Routh-Hurwitz criterion is used to find the range of the gain for stability and gives information regarding the location of poles

s² + 6Ks + 2s + 5 = 0

\(\Rightarrow {\rm{G}}\left( {\rm{s}} \right){\rm{H}}\left( {\rm{s}} \right) = \frac{{6{\rm{Ks}}}}{{{{\rm{s}}^2} + 2{\rm{s}} + 5}}\)

The point at which root locus enter real axis is break in point

\(\frac{{{\rm{dK}}}}{{{\rm{ds}}}} = 0\)

\({\rm{K}} = - \frac{{\left( {{{\rm{s}}^2} + 2{\rm{s}} + 5} \right)}}{{6{\rm{s}}}}\)

\(\frac{{{\rm{dK}}}}{{{\rm{ds}}}} = \frac{{ - 6{\rm{s}}\left( {2{\rm{s}} + 2} \right) + \left( {{{\rm{s}}^2} + 2{\rm{s}} + 5} \right)\left( 6 \right)}}{{{{\left( {6{\rm{s}}} \right)}^2}}} = 0\)

⇒ -12s² – 12s + 6s² + 12s + 30 = 0

\(\Rightarrow {\rm{s}} = \pm \sqrt 5\)

Break-in point always exist on root locus plot,

\(\Rightarrow {\rm{s}} = - \sqrt 5\)

Concept:

1. Every branch of a root locus diagram starts at a pole (K = 0) and terminates at a zero (K = ∞) of the open-loop transfer function.

2. The root locus diagram is symmetrical with respect to the real axis.

3. A number of branches of the root locus diagram are:

N = P if P ≥ Z

= Z, if P ≤ Z

4. Number of asymptotes in a root locus diagram = |P – Z|

5. Centroid: It is the intersection of the asymptotes and always lies on the real axis. It is denoted by σ.

\(\sigma = \frac{{\sum {P_i} - \sum {Z_i}}}{{\left| {P - Z} \right|}}\)

ΣPi is the sum of real parts of finite poles of G(s)H(s)

ΣZi is the sum of real parts of finite zeros of G(s)H(s)

6. The angle of asymptotes: \({\theta _l} = \frac{{\left( {2l + 1} \right)\pi }}{{P - Z}}\)

l = 0, 1, 2, … |P – Z| – 1

7. On the real axis to the right side of any section, if the sum of the total number of poles and zeros are odd, the root locus diagram exists in that section.

8. Break-in/away points: These exist when there are multiple roots on the root locus diagram.

At the breakpoints gain K is either maximum and/or minimum.

So, the roots of \(\frac{{dK}}{{ds}}\) are the breakpoints.

Observation:

n root-locus plot, the breakaway points, must lie on the root loci.

Concept:

The characteristic equation for a given open-loop transfer function G(s) is

1 + G(s) H(s) = 0

According to the Routh tabulation method,

The system is said to be stable if there are no sign changes in the first column of Routh array

The number of poles lies on the right half of s plane = number of sign changes

Calculation:

Root locus plot intersects the jω axis, i.e. system has roots on the jω axis.

Open loop transfer function \(G\left( s \right)H\left( s \right) = \frac{K}{{s\left( {s + 4} \right)\left( {s + 16} \right)}}\)

Characteristic equation: \(1 + \frac{K}{{s\left( {s + 4} \right)\left( {s + 16} \right)}} = 0\)

⇒ s (s + 4) (s + 16) + K = 0

⇒ s³ + 20s² + 64s + K = 0

By applying Routh tabulation method,

\(\begin{array}{*{20}{c}} {{s^3}}\\ {{s^2}}\\ {{s^1}}\\ {{s^0}} \end{array}\left| {\begin{array}{*{20}{c}} 1&{64}\\ {20}&K\\ {\frac{{20\left( {64} \right) - K}}{{20}}}&{}\\ {}&{} \end{array}} \right.\)

To have roots on jω axis, s¹ row must be equal to zero.

⇒ K – 20(64) = 0

⇒ K = 1280

Now, 20 s² + 1280 = 0

s = jω

⇒ ω² = 64 ⇒ ω = 8 rad/sec

Effect of addition of zero to the system:

• The angle of asymptotes increases and this root locus shift towards the left side of the s-plane slightly more
• Breakaway point shifts towards the left in the s plane
• The system becomes more stable i.e. less oscillatory
• The range of ‘K’ value of stability increases
• The relative stability increases

Effect of addition of pole to the system:

• The angle of asymptotes decreases and this root locus shift towards the right side of the s-plane slightly more
• Breakaway point shifts towards the right in the s plane
• The system becomes less stable i.e. more oscillatory
• The range of ‘K’ value of stability decreases
• The relative stability decreases

Concept:

1. Every branch of a root locus diagram starts at a pole (K = 0) and terminates at a zero (K = ∞) of the open-loop transfer function.

2. Root locus diagram is symmetrical with respect to the real axis.

3. Number of branches of the root locus diagram are:

N = P if P ≥ Z

= Z, if P ≤ Z

4. Number of asymptotes in a root locus diagram = |P – Z|

5. Centroid: It is the intersection of the asymptotes and always lies on the real axis. It is denoted by σ.

\(\sigma = \frac{{\sum {P_i} - \sum {Z_i}}}{{\left| {P - Z} \right|}}\)

ΣPi is the sum of real parts of finite poles of G(s)H(s)

ΣZi is the sum of real parts of finite zeros of G(s)H(s)

6. Angle of asymptotes: \({\theta _l} = \frac{{\left( {2l + 1} \right)\pi }}{{P - Z}}\)

l = 0, 1, 2, … |P – Z| – 1

7. On the real axis to the right side of any section, if the sum of the total number of poles and zeros are odd, the root locus diagram exists in that section.

8. Break in/away points: These exist when there are multiple roots on the root locus diagram.

At the breakpoints gain K is either maximum and/or minimum.

So, the roots of \(\frac{{dK}}{{ds}}\) are the break points.

Application:

The loop transfer function of a system is given by: \(G\left( s \right) = \frac{k}{{\left( {s + 3} \right)\left( {{s^2} + 3s + 2} \right)}}\)

\(G\left( s \right) = \frac{k}{{\left( {s + 3} \right)\left( {s + 1} \right)\left( {s + 2} \right)}}\)

Number of open loop poles = 3

Number of open loop zeros = 0

Number of Asymptotes = |3 – 0| = 3

Centroid, \(\sigma = \frac{{\left( {-1 - 2 - 3} \right) - \left( { 0} \right)}}{3} = - 2\)

Centroid \( = \left( { - 2,\;0} \right)\)

Concept:

1. Every branch of a root locus diagram starts at a pole (K = 0) and terminates at a zero (K = ∞) of the open-loop transfer function.

2. Root locus diagram is symmetrical with respect to the real axis.

3. Number of branches of the root locus diagram are:

N = P if P ≥ Z

= Z, if P ≤ Z

4. Number of asymptotes in a root locus diagram = |P – Z|

5. Centroid: It is the intersection of the asymptotes and always lies on the real axis. It is denoted by σ.

\(\sigma = \frac{{\sum {P_i} - \sum {Z_i}}}{{\left| {P - Z} \right|}}\)

ΣPi is the sum of real parts of finite poles of G(s)H(s)

ΣZi is the sum of real parts of finite zeros of G(s)H(s)

6. Angle of asymptotes: \({\theta _l} = \frac{{\left( {2l + 1} \right)\pi }}{{P - Z}}\)

l = 0, 1, 2, … |P – Z| – 1

7. On the real axis to the right side of any section, if the sum of the total number of poles and zeros are odd, the root locus diagram exists in that section.

8. Break-in/away points: These exist when there are multiple roots on the root locus diagram.

At the breakpoints gain K is either maximum and/or minimum.

So, the roots of \(\frac{{dK}}{{ds}}\) are the break points.

Calculation:

\(G\left( s \right) = \frac{K}{{s\left( {s + 4} \right)}}\)

Number of poles (P) = 2

Number of zeros (Z) = 0

P – Z = 2

Angle of asymptotes,\({\theta _l} = \frac{{\left( {2l + 1} \right)\pi }}{{P - Z}}\), l = 0, 1

Given \({\rm{G}}\left( {\rm{s}} \right) = \frac{{{\rm{Ks}}}}{{\left( {{\rm{s}} - 1} \right)\left( {{\rm{s}} - 4} \right)}}\)

Characteristic equation is 1 + G(s) H(s) = 0

\({1+\rm{G}}\left( {\rm{s}} \right)H(s) = 1+ \frac{{{\rm{Ks}}}}{{\left( {{\rm{s}} - 1} \right)\left( {{\rm{s}} - 4} \right)}}\)

⇒ (s - 1)(s - 4) + Ks = 0

∴ \( {\rm{K}} = \frac{{ - \left( {{\rm{s}} - 1} \right)\left( {{\rm{s}} - 4} \right)}}{{\rm{s}}}\)

For finding the breakaway point \( \frac{{{\rm{dK}}}}{{{\rm{ds}}}} = 0\)

\(⇒ \frac{{{\rm{dK}}}}{{{\rm{ds}}}}=- \left[ {\frac{{{\rm{s}}\left( {2{\rm{s}} - 5} \right) - \left( {{{\rm{s}}^2} - 5{\rm{s}} + 4} \right)}}{{{{\rm{s}}^2}}}} \right] = 0\)

⇒ s² - 4 = 0

⇒ s = ± 2

By plotting on s plane 

Centroid = \(\frac{{\left( {1 + 4} \right) - 0}}{1} = 5\)

∴ The valid break-away point is between 1 and 4.

∴ Valid breakaway point is \(s = 2\)

∴ Gain at breakaway point \(= \frac{{{\rm{product\;of\;distance\;from\;poles}}}}{{{\rm{product\;of\;distance\;from\;zeros}}}}\)

\(= \frac{{\left( 1 \right)\left( 2 \right)}}{2} = 1\)

Concept:

1. Every branch of a root locus diagram starts at a pole (K = 0) and terminates at zero (K = ∞) of the open-loop transfer function.

2. Root locus diagram is symmetrical with respect to the real axis.

3. Number of branches of the root locus diagram are:

N = P if P ≥ Z

= Z, if P ≤ Z

4. Number of asymptotes in a root locus diagram = |P – Z|

5. Centroid: It is the intersection of the asymptotes and always lies on the real axis. It is denoted by σ.

\(\sigma = \frac{{\sum {P_i} - \sum {Z_i}}}{{\left| {P - Z} \right|}}\)

ΣPi is the sum of real parts of finite poles of G(s)H(s)

ΣZi is the sum of real parts of finite zeros of G(s)H(s)

6. Angle of asymptotes: \({\theta _l} = \frac{{\left( {2l + 1} \right)\pi }}{{P - Z}}\)

l = 0, 1, 2, … |P – Z| – 1

7. On the real axis to the right side of any section, if the sum of the total number of poles and zeros is odd, the root locus diagram exists in that section.

8. Break-in/away points: These exist when there are multiple roots on the root locus diagram.

At the breakpoints gain K is either maximum and/or minimum.

So, the roots of \(\frac{{dK}}{{ds}}\) are the breakpoints.

Application:

The loop transfer function of a system is given by: \(G\left( s \right)H\left( s \right) = \frac{{1}}{{s\left( {s + 1} \right)\left( {s + 2} \right)}}\)

Number of open loop poles = 3

Number of open loop zeros = 0

Number of Asymptotes = |3 – 0| = 3

Centroid, \(\sigma = \frac{{\left( {0 - 1 - 2} \right) - \left( { 0} \right)}}{3} = - 1\)

Angle of asymptotes,\({\theta _l} = \frac{{\left( {2l + 1} \right)\pi }}{{P - Z}}\),

where, l = 0, 1

θl = 60°, 180°, 300°

Concept:

1. Every branch of a root locus diagram starts at a pole (K = 0) and terminates at a zero (K = ∞) of the open-loop transfer function.

2. Root locus diagram is symmetrical with respect to the real axis.

3. Number of branches of the root locus diagram are:

N = P if P ≥ Z

= Z, if P ≤ Z

4. Number of asymptotes in a root locus diagram = |P – Z|

5. Centroid: It is the intersection of the asymptotes and always lies on the real axis. It is denoted by σ.

\(\sigma = \frac{{\sum {P_i} - \sum {Z_i}}}{{\left| {P - Z} \right|}}\)

ΣPi is the sum of real parts of finite poles of G(s)H(s)

ΣZi is the sum of real parts of finite zeros of G(s)H(s)

6. Angle of asymptotes: \({\theta _l} = \frac{{\left( {2l + 1} \right)\pi }}{{P - Z}}\)

l = 0, 1, 2, … |P – Z| – 1

7. On the real axis to the right side of any section, if the sum of the total number of poles and zeros are odd, the root locus diagram exists in that section.

8. Break-in/away points: These exist when there are multiple roots on the root locus diagram.

At the breakpoints gain K is either maximum and/or minimum.

So, the roots of \(\frac{{dK}}{{ds}}\) are the break points.

Calculation:

\(G\left( s \right) = \frac{K}{{s\left( {s + 4} \right)}}\)

Number of poles (P) = 2

Number of zeros (Z) = 0

P – Z = 2

Angle of asymptotes,\({\theta _l} = \frac{{\left( {2l + 1} \right)\pi }}{{P - Z}}\), l = 0, 1

θl = 90°, 270°

So option (2) is the correct answer.