The slope of the frequency response of an integrator is linear with a negative slope. To understand why this is the case, let's first define what an integrator is and how it works.

An integrator is a circuit that performs mathematical integration on an input signal. Mathematically, integration is the process of finding the integral of a function, which represents the area under the curve of the function. In the context of an integrator circuit, this means that the output voltage is proportional to the accumulated sum of the input voltage over time.

Now, let's consider the frequency response of an integrator. The frequency response is a plot that shows how the output voltage of a circuit varies with different input frequencies. In the case of an integrator, the input frequency refers to the rate at which the input voltage changes.

- The Ideal Integrator Response:

In an ideal integrator, the output voltage is directly proportional to the integral of the input voltage. This means that as the input frequency increases, the rate of change of the input voltage also increases, resulting in a higher output voltage. Similarly, as the input frequency decreases, the rate of change of the input voltage decreases, leading to a lower output voltage.

- The Slope of the Frequency Response:

The slope of the frequency response of an integrator is a measure of how the output voltage changes with respect to the input frequency. In the case of an integrator, this slope is linear and negative.

- Explanation:

The negative slope of the frequency response can be understood by considering the behavior of an integrator at different frequencies. At low frequencies, where the input voltage changes slowly, the integrator has enough time to accumulate and integrate the input voltage, resulting in a large output voltage. As the input frequency increases, the rate of change of the input voltage becomes faster, and the integrator has less time to accumulate the voltage. This leads to a decrease in the output voltage.

- Conclusion:

In summary, the slope of the frequency response of an integrator is linear with a negative slope because the output voltage decreases as the input frequency increases. This behavior is due to the nature of integration, where the output voltage is proportional to the accumulated sum of the input voltage over time.