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The transfer function of a phase lead controller is (1 + 3Ts) / (1 + Ts). The maximum value of phase provided by this controller is;
- a)90
- b)60
- c)45
- d)30
Correct answer is option 'D'. Can you explain this answer?
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The transfer function of a phase lead controller is (1 + 3Ts) / (1 + ...
Introduction:
The transfer function of a phase lead controller is given as (1 + 3Ts) / (1 + Ts), where T is the time constant of the controller. The maximum value of phase provided by this controller can be determined by analyzing the transfer function.
Analysis:
To find the maximum phase value, we need to determine the frequency at which the phase shift is maximum. This can be done by analyzing the transfer function in the frequency domain.
Frequency domain analysis:
The transfer function can be expressed as H(s) = (1 + 3Ts) / (1 + Ts), where s is the complex frequency variable.
Gain:
The magnitude of the transfer function can be written as |H(s)| = |(1 + 3Ts) / (1 + Ts)|.
Phase:
The phase of the transfer function can be written as φ(s) = arg((1 + 3Ts) / (1 + Ts)).
Maximum phase:
To find the maximum phase, we need to find the frequency at which the phase shift is maximum. This can be done by finding the frequency at which the derivative of the phase with respect to frequency is zero.
Derivative of phase:
Taking the derivative of the phase shift equation with respect to frequency, we get:
dφ(s) / ds = d/ds(arg((1 + 3Ts) / (1 + Ts)))
Setting the derivative to zero:
Setting the derivative to zero, we get:
dφ(s) / ds = 0
Solving for s:
Solving the equation for s, we can determine the frequency at which the phase shift is maximum.
Maximum phase value:
Once we have the frequency at which the phase shift is maximum, we can substitute this frequency into the transfer function to get the maximum phase value.
Conclusion:
The maximum value of phase provided by the given phase lead controller is 30 degrees.
The transfer function of a phase lead controller is given as (1 + 3Ts) / (1 + Ts), where T is the time constant of the controller. The maximum value of phase provided by this controller can be determined by analyzing the transfer function.
Analysis:
To find the maximum phase value, we need to determine the frequency at which the phase shift is maximum. This can be done by analyzing the transfer function in the frequency domain.
Frequency domain analysis:
The transfer function can be expressed as H(s) = (1 + 3Ts) / (1 + Ts), where s is the complex frequency variable.
Gain:
The magnitude of the transfer function can be written as |H(s)| = |(1 + 3Ts) / (1 + Ts)|.
Phase:
The phase of the transfer function can be written as φ(s) = arg((1 + 3Ts) / (1 + Ts)).
Maximum phase:
To find the maximum phase, we need to find the frequency at which the phase shift is maximum. This can be done by finding the frequency at which the derivative of the phase with respect to frequency is zero.
Derivative of phase:
Taking the derivative of the phase shift equation with respect to frequency, we get:
dφ(s) / ds = d/ds(arg((1 + 3Ts) / (1 + Ts)))
Setting the derivative to zero:
Setting the derivative to zero, we get:
dφ(s) / ds = 0
Solving for s:
Solving the equation for s, we can determine the frequency at which the phase shift is maximum.
Maximum phase value:
Once we have the frequency at which the phase shift is maximum, we can substitute this frequency into the transfer function to get the maximum phase value.
Conclusion:
The maximum value of phase provided by the given phase lead controller is 30 degrees.
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Community Answer
The transfer function of a phase lead controller is (1 + 3Ts) / (1 + ...
Given transfer function -
α = Z/P
Here, Z = 1/3T and P = 1/T
Maximum phase shift -
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The transfer function of a phase lead controller is (1 + 3Ts) / (1 + Ts). The maximum value of phase provided by this controller is;a)90b)60c)45d)30Correct answer is option 'D'. Can you explain this answer?
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