Solution 2.39a: Integrators vs Differentiators

[stanigator]

For the question (problem 2.39a) stated in the picture attached to this message, I don't understand why integrators should be used in favour of differentiator for the block diagram representing the differential equation system. Is this b/c of the nature of the differentiator electronics too susceptible to environmental noise?

 

[ranger]

In addition to what you mention of the susceptibility to noise, another point comes to mind. The differentiator would represent an unstable system with regards to certain types of bounded inputs. An example of this would be a step input. This input would result in the unbounded output with the characteristics of an impulse. That is, the derivative of a unit step, $u(t)$, is in fact the unit impulse function, $\delta(t)$:

$$\frac{du}{dt} = \delta(t)$$

 

[oswaler]

The choice between integrators and differentiators depends on the specific application and system being modeled. In general, integrators are used when the output of the system needs to be a cumulative sum of the input over time. This is useful for modeling systems that involve accumulation, such as in finance or population growth.

On the other hand, differentiators are used when the output of the system needs to be the rate of change of the input over time. This is useful for modeling systems that involve rates of change, such as in physics or engineering.

In the case of the block diagram representing a differential equation system, the use of integrators may be more appropriate because the output of the system is often a cumulative sum of the input over time. Additionally, the use of differentiators may introduce noise and instability in the system due to the amplification of high-frequency components in the input signal.

However, it is important to note that the choice between integrators and differentiators also depends on the accuracy and resolution required for the specific application. In some cases, a combination of both integrators and differentiators may be necessary to accurately model a system. Ultimately, the decision should be based on the specific requirements and characteristics of the system being modeled.