Test: Compensators - 1 - Electrical Engineering (EE) MCQ

Lag 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/T√α
Maximum phase lag: ϕ_m = sin⁻¹(a−1/a+1)
ϕ_m 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

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 < b 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 α = 

Phase Lead Compensator:

• A lead compensator provides a positive phase shift for increasing the value of frequencies from 0 to ∞.
• It is also known as a differentiator circuit.
• For a lead network, zero is nearer to the origin.
• It is used to improve the transient response of the system.
• It increases the damping of the system.

Phase Lag Compensator:

• A lead compensator provides a negative phase shift for increasing the value of frequencies from 0 to ∞.
• It is also known as an integrator circuit.
• For a lag network, pole is nearer to the origin.
• It is used to improve the steady state response of the system.
• It decreases the steady-state error of the system.

The standard T/F of the compensator is 

Maximum phase lead

Maximum phase lead frequency, 
ωm = 1/T√a
Calculation:
The given transfer function is,
By comparing both transfer functions,
aT = 0.15
T = 0.05
a = 3
Maximum phase lead

= sin^-1 (0.5)
ϕ_m = 30° 

Application:

Poles: s = -2, -8
Zeros: s = -4, -6
The pole-zero plot of the above transfer function is shown below.

The above pole-zero represents that the given system is a lag-lead compensator.

Lag 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:
ωm = 1√Ta
Maximum phase lag::

ϕ_m 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

Given circuit is a lag compensator and transfer function is given as

On comparison we get, T = 1
βT = 10 ⇒ β = 10
The frequency at which maximum lead occurs is ωm = 1/T√β

For all ω Numerator is less than Denominator maximum value of |G(s)| occurs at ω = 0
|G(s)|_max = 1 
In dB, 20 log (1) = 0 dB

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

Phase lead occurs in phase lead compensator and it occurs at the high-frequency region.

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