Lag Compensators MCQ Quiz - Objective Question with Answer for Lag Compensators - Download Free PDF

Lag compensator:

Transfer function:

If it is in the form of , then a

If it is in the form of , then a > b

Maximum phase lag frequency: 

Maximum phase lag: 

ϕm is negative

Pole zero plot:

The pole is nearer to the origin.

Filter: It is a low pass filter (LPF)

Effect on the system:

• Rise time and settling time increases and Bandwidth decreases
• The transient response becomes slower
• The steady-state response is improved
• Stability decreases

Application:

The equivalent Laplace transform network for the given network is,

By applying voltage division,

The above system to be lag compensator, β > 1

1\)

⇒ R₁ > 0

Therefore, at any value of R₂ the given system acts as lag compensator.

Concept

The slope in dB/decade of the asymptotic Bode magnitude plot is decided by open loop poles and open loop zeroes.

One pole provides a slope of -20 dB/decade.

One zero provides a slope of +20 dB/decade.

Calculation

Given, 

and 

When they are connected in cascade, the modified open loop transfer function is:

Before the cut-off frequency ω = 3, poles at s = 0, 0.1, and 0.5 will exist only.

Total no.of open-loop poles = 3

Total slope provided by poles = -20 × 3 = -60 dB/decade

Lag compensator:

Transfer function:

If it is in the form of , then a

If it is in the form of , then a > b

Maximum phase lag frequency: 

Maximum phase lag: 

ϕm is negative

Pole zero plot:

The pole is nearer to the origin.

Filter: It is a low pass filter (LPF)

Effect on the system:

• Rise time and settling time increases and Bandwidth decreases
• The transient response becomes slower
• The steady-state response is improved
• Stability decreases

Concept:

In general, the lead and lag compensator is represented by the below transfer function

If a > b then that is lag compensator because pole comes first.

If a  then that is the lead compensator since zero comes first.

Analysis:

Lead compensator:

1) When sinusoidal input applied to this it produces sinusoidal output with the phase lead input.

2) It speeds up the Transient response and increases the margin for stability.

A circuit diagram is as shown:

Response is:

Lead constant  

Important Points

Sink node:

• A local sink is a node of a directed graph with no exiting edges, also called a terminal.
• It is the output node in the signal flow graph. It is a node, which has only incoming branches.
   

Lag Compensator: 

• Phase lag network offers high gain at low frequency.
• Thus, it performs the function of a low pass filter.
• The introduction of this network increases the steady-state performance of the system.

Damping Ratio:

• The damping ratio gives the level of damping in the control system related to critical damping.
• The damping ratio is defined as the ratio of actual damping to the critical damping of the system.
• It is the ratio of the damping coefficient of a differential equation of a system to the damping coefficient of critical damping.
  ζ = actual damping / critical damping
   

Cut-off rate:  It is the slope of the log-magnitude curve near the cut-­off region of the Bode-plot.

Concept:

In general, the lead and lag compensator is represented by the below transfer function

If a > b then that is lag compensator because pole comes first.

If a  then that is the lead compensator since zero comes first.

Analysis:

Lead compensator:

1) When sinusoidal input applied to this it produces sinusoidal output with the phase lead input.

2) It speeds up the Transient response and increases the margin for stability.

A circuit diagram is as shown:

Response is:

Lead constant  

Important Points

Concept:

Lag compensator:

Transfer function:

If it is in the form of , then a

If it is in the form of , then a > b

Maximum phase lag frequency: 

Maximum phase lag: 

ϕm is negative

Pole zero plot:

The pole is nearer to the origin.

Given:

Zero = -2

Pole = -1

Analysis:

The pole-zero plot of T(s) is as shown:

Since the pole is closer to the origin than zero.

It is a lag compensator. 

Lead compensator:

Transfer function:

If it is in the form of , then a > 1

If it is in the form of , then a

Maximum phase lead frequency: 

Maximum phase lead: 

ϕm is positive

Pole zero plot:

The zero is nearer to the origin.

Lead compensator:

Transfer function:

If it is in the form of , then a > 1

If it is in the form of , then a

Maximum phase lead frequency: 

Maximum phase lead: 

ϕm is positive

Pole zero plot:

The zero is nearer to the origin.

Filter: It is a high pass filter (HPF)

Lag compensator:

Transfer function:

If it is in the form of , then a

If it is in the form of , then a > b

Maximum phase lag frequency:

Maximum phase lag::

ϕm is negative

Pole zero plot:

The pole is nearer to the origin.

Filter: It is a low pass filter (LPF)

Effect on the system(Lag compensator):

• Rise time and settling time increases  and Bandwidth increases
• The transient response becomes faster
• The steady-state response is not affected
• Improves the stability

Effect on the system(Lead compensator):

• Rise time and settling time decrease and Bandwidth increases
• The transient response becomes faster
• The steady-state response is not affected
• Improves the stability
• The velocity constant is usually increased
• Helps to increase the system error constant though to a limited extent
• The slope of the magnitude curve is reduced at the gain crossover frequency, as a result, relative stability improves
• The margin of stability of a system (phase margin) increased

Lead compensator:

Transfer function:

If it is in the form of , then a > 1

If it is in the form of , then a

Maximum phase lead frequency: 

Maximum phase lead: 

ϕm is positive

Pole zero plot:

The zero is nearer to the origin.

Filter: It is a high pass filter (HPF)

Effect on the system:

• Rise time and settling time decreases and Bandwidth increases
• The transient response becomes faster
• The steady-state response is not affected
• Improves the stability

Lag compensator:

Transfer function:

If it is in the form of , then a

If it is in the form of , then a > b

Maximum phase lag frequency: 

Maximum phase lag: 

ϕm is negative

Pole zero plot:

The pole is nearer to the origin.

Filter: It is a low pass filter (LPF)

Effect on the system:

• Rise time and settling time increases and Bandwidth decreases
• The transient response becomes slower
• The steady-state response is improved
• Stability decreases

For all ω Numerator is less than Denominator maximum value of |G(s)| occurs at ω = 0

In dB, 20 log (1) = 0 dB

Let H(f) be TF of phase-lag controller

For lag Network ∠H(b) = -ve

b

Pole is near to j ω axis than zero 

Concept:

• Lag Compensation adds a pole at origin (or for low frequencies).
• Lag Compensation to reduce the steady-state error of the system.
• An unstable system is a system that has at least one pole at the right side of the s-plane.
• So, even if we add a lag compensator to an unstable system, it will remain unstable.

Explanation:

If an already unstable system has a CLTP =

After using a lag compensator the CLTP =

Lag compensator:

Transfer function:

If it is in the form of , then a

If it is in the form of , then a > b

Maximum phase lag frequency: 

Maximum phase lag: 

ϕm is negative

Pole zero plot:

The pole is nearer to the origin.

Filter: It is a low pass filter (LPF)

Effect on the system:

• Rise time and settling time increases and Bandwidth decreases
• The transient response becomes slower
• The steady-state response is improved
• Stability decreases

Application:

The equivalent Laplace transform network for the given network is,

By applying voltage division,

The above system to be lag compensator, β > 1

1\)

⇒ R₁ > 0

Therefore, at any value of R₂ the given system acts as lag compensator.

Lag compensator:

The transfer function of a lag compensator is given by

Where,  1\)

• Both pole and zero lie in LHS of real plane but PC C i.e. zero is farther away from origin
• Lag Compensation adds a pole at origin (or for low frequencies).
• It helps to reduce the steady-state error of the system.
• An unstable system is a system which has at least one pole at the right side of the s-plane,
• Even if we add a lag compensator to an unstable system, it will remain unstable.

Effects of phase-lag compensator:

• It improves the steady-state response
• It makes the system dynamic response slower. Reduces steady-state error
• Increases the rise time
• Decreases the bandwidth
• Reduces stability margin i.e. the system becomes less stable

Explanation:

Phase lag compensation is an integrator, as it reduces the steady-state error.

We know that

Velocity constant =1/steady state error

The effect of phase-lag compensation on the servo system is that for given relative stability, the velocity constant is increased.

Sink node:

• A local sink is a node of a directed graph with no exiting edges, also called a terminal.
• It is the output node in the signal flow graph. It is a node, which has only incoming branches.
   

Lag Compensator: 

• Phase lag network offers high gain at low frequency.
• Thus, it performs the function of a low pass filter.
• The introduction of this network increases the steady-state performance of the system.

Damping Ratio:

• The damping ratio gives the level of damping in the control system related to critical damping.
• The damping ratio is defined as the ratio of actual damping to the critical damping of the system.
• It is the ratio of the damping coefficient of a differential equation of a system to the damping coefficient of critical damping.
  ζ = actual damping / critical damping
   

Cut-off rate:  It is the slope of the log-magnitude curve near the cut-­off region of the Bode-plot.