Test: Compensators - 2 - Electrical Engineering (EE) MCQ

Lead compensator:
Transfer function:

If it is in the form of  then a < 1

If it is in the form of then a > b
Maximum phase lag frequency: ω_m = 1√Ta
Maximum phase lag::

ϕ_m 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
• 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

In general, there are two situations in which compensation is required.

• In the first case, the system is absolutely unstable, and the compensation is required to stabilize it as well as to achieve a specified performance.
• In the second case, the system is stable, but the compensation is required to obtain the desired performance.
   

The systems which are of type-2 or higher are usually absolutely unstable. For type-2 or higher systems, only the lead compensator is required because only the lead compensator improves the margin of stability.
Both Statement I and Statement II are individually true and Statement II is the correct explanation of Statement I
Note: In type-1 and type-0 systems, stable operation is always possible if the gain is sufficiently reduced. In such cases, any of the three compensators, lead, lag, lag-lead may be used to obtain the desired performance.

Lead compensator:
Transfer function:

If it is in the form of  then a < 1

If it is in the form of then a > b
Maximum phase lag frequency: ω_m = 1√Ta
Maximum phase lag::

ϕ_m is positive

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

⇒ R₁ > 0
Therefore, at any value of R₂ the given system acts as lag compensator.

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

b < a
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 = 
The system is still unstable.

Concept:
We know:
H_c(s) = s+as+b;
when α = zero/pole = α/b is greater than 1 i.e., α > 1 then it is lag compensator.
else when α < 1, then it is lead compensator.
Calculation:
For lead compensator: H_c(s) = s+a/s+b

ϕ = tan⁻¹(ω/1) − tan⁻¹(ω/2)
i.e., phase angle will be positive for ω > 0
similarly, for lag compensator:  H_c(s) = s+2/s+1

ϕ = tan⁻¹(ω/2) −tan⁻¹(ω)
i.e. phase angle is negative for ω > 0.
Now,
For lead compensator, gain crossover frequency (ωgc) increase which causes an increase in B.W. and hence the speed of the system improves.
For lag compensator; Gain crossover frequency (ωpc) decreases which causes decreases in B.W. and hence speed of the system decreases.

Advantages of Lag Compensator:

• A 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.
• The lag network offers a reduction in bandwidth and this provides longer rise time and settling time and so the transient response.
• The angular contribution of the pole is more than that of the compensator zero because the pole dominates the zero in the lag compensator.

 
Advantages of Lead Compensator:

• It improves the damping of the overall system.
• The enhanced damping of the system supports less overshoot along with less rise time and settling time. Therefore, the transient response gets improved.
• The addition of a lead network improves the phase margin.
• A system with a lead network provides a quick response as it increases bandwidth thereby providing a faster response.
• Lead networks do not disturb the steady-state error of the system.
• It maximizes the velocity constant of the system.

Concept:
Lag compensator:
Transfer function:
If it is in the form of then a < 1
If it is in the form ofthen a > b
Maximum phase lag frequency: 
 ω_m = 1/T√a
Maximum phase lag: ϕ_m = sin⁻¹(α−1/α+1)
ϕ_m 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.

Concept:
The standard phase lead compensating network is given as:

Maximum phase Occurs at
ω_m = 1/T√α
Maximum phase (ϕm)
sinϕ_m = 1−α/1+α

Calculation:
Given:

a = 3 and T = 0.1
∵ a > 1 
It is a Lead compensator.
Maximum phase lead is given as:

Concept:
The speed of response and, the steady-state error can be simultaneously improved if both phase-lag and phase-lead compensation networks are used. However, instead of using two separate lag and lead networks, a single network combining both can be used.

Lag-lead compensator:

In the lag-lead compensator network, the lag compensator is nearer to the origin.

Lead-lag compensator:

In the lead-lag compensator network, the lead compensator is nearer to the origin.

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.

Pole zero diagram of the above transfer function is

It represents a standard lead lag compensator. For a lead lag compensator, maximum lead occurs at initial frequency.
So, maximum lead occurs at 0.1 < ω < 1.

Lead compensator:
Transfer function:

If it is in the form of  then a < 1

If it is in the form of then a > b
In the frequency domain, 

Phase angle, ∠G (jω) = tan⁻¹ωαT − tan⁻¹ωT

ϕ = tan^-1 ωaT – tan^-1 ωT
As a > 1 always (from the definition), ϕ is positive
Hence, it is clear that the phase of output voltage leads the phase of the input voltage.

Concept:
Lead and Lag compensators:
G_c(s) = (1 + aTs) / (1 + Ts) where a and T > 0, a > 1 (lead) & a < 1 (lag)
∠G_c(s) = ϕ = tan^-1 ωaT – tan^-1 ωT
ω_m is the geometric mean of the two corner frequencies 1/T and 1/aT
ωm = 1/T√α

Calculations:
Given Gc(s) = 
r = 1
By substituting r value we get G_c(s) = 
Here T = 1 and a = 0.1