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The complete Nyquist plot of the open loop transfer function G(s) H (s) of a feedback control system shown in figure,
If G(s) H (s) has one zero in right half of the s-plane, the number of poles that, the closed loop system will have in right half of s-plane is
- a)0
- b)1
- c)4
- d)3
Correct answer is option 'D'. Can you explain this answer?
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The complete Nyquist plot of the open loop transfer function G(s) H (...
Given Nyquist plot of G(s)H(s) is shown below
Case 1 :
Here, G(s)H(s) has one zero in the right half of s-plane i.e. Z = 1 .
According to principle of argument
N =P− Z (Anti clockwise direction) …(i)
Where,
N = Number of encirclements of Nyquist plot of G(s)H(s) about origin in anti-clockwise direction P = Number of poles of G(s)H(s) in right half of s-plane.
Z = Number of zeros of G(s)H(s) in right half of s-plane.
Here, N =−2 (anti clockwise direction)
Z = 1
From equation (i)
−2= P − 1
P = –1 …(ii)
So, poles cannot be negative in numbers so no more further discussion on this.
Case 2 : If we assume Nyquist contour in anti clockwise direction then according to principle of argument
N =P− Z (Clockwise direction) …(iii)
Where,
N = Number of encirclements of Nyquist plot of G(s)H(s) about origin in clockwise direction
P = Number of poles of G(s)H(s) in right half of s-plane.
Z = Number of zeros of G(s)H(s) in right half of s-plane.
Here, N = 2 (Clockwise direction)
Z = 1
From equation (iii)
2 = P−1
P = 3 …(iv)
According to Nyquist stability criteria
N =P− Z (Anti clockwise direction) …(v)
Where, N = Number of encirclements about (−1+ 0 j ) in clockwise direction
Z = Number of closed loop poles in right half of s-plane.
P = Number of open loop poles or poles of G(s)H(s) in right half of s-plane.
From given Nyquist plot number of encirclements about (−1+ 0 j ) is
N = 1−1= 0
Using equation (v)
0 = P−Z
Z = P
Z = 3
So here number of poles in right half of s-plane is 3.
Hence, the correct option is (D).
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Community Answer
The complete Nyquist plot of the open loop transfer function G(s) H (...
Given Nyquist plot of G(s)H(s) is shown below
Case 1 :
Here, G(s)H(s) has one zero in the right half of s-plane i.e. Z = 1 .
According to principle of argument
N =P− Z (Anti clockwise direction) …(i)
Where,
N = Number of encirclements of Nyquist plot of G(s)H(s) about origin in anti-clockwise direction P = Number of poles of G(s)H(s) in right half of s-plane.
Z = Number of zeros of G(s)H(s) in right half of s-plane.
Here, N =−2 (anti clockwise direction)
Z = 1
From equation (i)
−2= P − 1
P = –1 …(ii)
So, poles cannot be negative in numbers so no more further discussion on this.
Case 2 : If we assume Nyquist contour in anti clockwise direction then according to principle of argument
N =P− Z (Clockwise direction) …(iii)
Where,
N = Number of encirclements of Nyquist plot of G(s)H(s) about origin in clockwise direction
P = Number of poles of G(s)H(s) in right half of s-plane.
Z = Number of zeros of G(s)H(s) in right half of s-plane.
Here, N = 2 (Clockwise direction)
Z = 1
From equation (iii)
2 = P−1
P = 3 …(iv)
According to Nyquist stability criteria
N =P− Z (Anti clockwise direction) …(v)
Where, N = Number of encirclements about (−1+ 0 j ) in clockwise direction
Z = Number of closed loop poles in right half of s-plane.
P = Number of open loop poles or poles of G(s)H(s) in right half of s-plane.
From given Nyquist plot number of encirclements about (−1+ 0 j ) is
N = 1−1= 0
Using equation (v)
0 = P−Z
Z = P
Z = 3
So here number of poles in right half of s-plane is 3.
Hence, the correct option is (D).
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The complete Nyquist plot of the open loop transfer function G(s) H (s) of a feedback control system shown in figure,If G(s) H (s) has one zero in right half of the s-plane, the number of poles that, the closed loop system will have in right half of s-plane isa)0b)1c)4d)3Correct answer is option 'D'. Can you explain this answer?
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