Condition for a Differentiator
A differentiator is a circuit that performs differentiation operation on an input signal. It is commonly used in signal processing and control systems. The condition that a differentiator should satisfy is that it should have zero response at zero frequency.

Explanation
To understand why a differentiator should have zero response at zero frequency, let's first understand the concept of differentiation. Differentiation is a mathematical operation that calculates the rate of change of a function with respect to its independent variable. In the context of a differentiator circuit, it means that the output signal represents the rate of change of the input signal.

When a differentiator is subjected to a constant input signal (zero frequency), it should ideally produce zero output because a constant signal does not change over time. However, in practice, a differentiator circuit may have some response at zero frequency due to imperfections and limitations of the circuit components.

Importance of Zero Response at Zero Frequency
Having zero response at zero frequency is important for a differentiator for the following reasons:
1. Eliminating DC Offset: A differentiator circuit with zero response at zero frequency prevents the output signal from having a DC offset. A DC offset is a non-zero voltage level that can distort the desired output and introduce errors in the differentiation process.

2. Accurate Differentiation: A differentiator is primarily used to calculate the derivative of a signal. By having zero response at zero frequency, the differentiator accurately represents the high-frequency components of the input signal, which is essential for precise differentiation.

3. Stability of the Circuit: A differentiator with finite response at zero frequency can cause stability issues in the circuit. A non-zero response at zero frequency can lead to amplification of noise and unwanted low-frequency components, which may cause instability and oscillations in the circuit.

Conclusion
In conclusion, a differentiator should have zero response at zero frequency to ensure accurate differentiation, eliminate DC offset, and maintain stability in the circuit. This condition allows the differentiator to effectively calculate the rate of change of an input signal and produce the desired output.