The transfer function of a phase lead compensator is found to be the form (s+z₁ ) / (s + p₁ ) and that of a lag compensator to be of the form (s+z₂ ) / (s + p₂ ). Then which of the following conditions must be satisfied?

Explanation:

Phase Lead and Lag Compensators

The transfer functions of phase lead and lag compensators are given as:

• Phase lead compensator: (s + z₁) / (s + p₁)
• Phase lag compensator: (s + z₂) / (s + p₂)

To determine the correct option, we analyze the conditions for these compensators:

Phase Lead Compensator:

• In a phase lead compensator, the purpose is to advance the phase of the system by introducing a zero z₁ and a pole p₁.
• The zero z₁ is designed to be larger than the pole p₁ (z₁ > p₁).
• This arrangement ensures that the phase contribution is positive in the frequency range of interest, which improves the transient response of the system.

Phase Lag Compensator:

• In a phase lag compensator, the purpose is to delay the phase of the system by introducing a zero z₂ and a pole p₂.
• The pole p₂ is designed to be larger than the zero z₂ (z₂ < p₂).
• This arrangement ensures that the phase contribution is negative in the frequency range of interest, which improves the steady-state accuracy of the system.

Correct Option Analysis:

The correct option is:

Option 1: z₁ > p₁ and z₂ < p₂

This option satisfies the conditions for both the phase lead and lag compensators. In the phase lead compensator, the zero is greater than the pole (z₁ > p₁), and in the phase lag compensator, the zero is smaller than the pole (z₂ < p₂). These arrangements ensure proper phase advancement and delay for their respective compensators.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: z₁ > p₁ and z₂ > p₂

This option is incorrect as it assumes that the zero is greater than the pole for both the lead and lag compensators. While this is true for the phase lead compensator (z₁ > p₁), it contradicts the condition for the phase lag compensator, where the pole must be greater than the zero (z₂ < p₂).

Option 3: z₁ < p₁ and z₂ < p₂

This option is incorrect as it assumes that the zero is smaller than the pole for both the lead and lag compensators. While this is true for the phase lag compensator (z₂ < p₂), it contradicts the condition for the phase lead compensator, where the zero must be greater than the pole (z₁ > p₁).

Option 4: z₁ < p₁ and z₂ > p₂

This option is incorrect as it assumes that the zero is smaller than the pole for the lead compensator and larger for the lag compensator. This contradicts the correct conditions for both compensators. In the phase lead compensator, the zero must be greater than the pole (z₁ > p₁), and in the phase lag compensator, the pole must be greater than the zero (z₂ < p₂).

Conclusion:

Understanding the conditions for phase lead and lag compensators is essential for ensuring proper system performance. A phase lead compensator requires the zero to be greater than the pole, while a phase lag compensator requires the pole to be greater than the zero. The correct option, Option 1, satisfies these conditions, making it the appropriate choice.