Amplitude Ratio

We can define amplitude as the maximum displacement made by a wave from the point of equilibrium. When we throw a stone in the water, we can see a disturbance in the water, which produces ripples or waves. Radio waves, microwaves, and x-rays are electromagnetic waves that do not need any medium of propagation. Every wave has amplitude, irrespective of its kind. The Amplitude ratio of waves is the ratio of the Amplitude output of wave signal and amplitude input of wave signal. The value of the lowest amplitude could be negative. When we talk about waves, the number of times a wave cycle passes through a certain point is known as frequency.

About sine waves

We can characterise a sine wave via three parameters i.e.

• Amplitude a

• Frequency f

• Phase φ

When we talk about sine waves, phase p is a relative quantity. As the sine wave has no starting point, it can take any argument. Mathematically, we can describe a sine wave that varies with time by:

            y(t) = a sin (ωt + φ)

Here a denotes the amplitude, and it can have any unit. The quantity y is represented in its physical form.

The phase φ has the unit of angle, which could be degrees or radians.

In the above equation, ω will have an angle per unit of time, usually radians/minute situations of process control, because the quantities ω and t must have units of angles. The f is considered a true frequency, and it is 2π times bigger than the angular frequency.

W = 2π f

Period φ is the reciprocal of frequency f, i.e. the time is taken by a complete cycle of the sinewave.

Amplitude Ratio

We can think of the Amplitude ratio as a process gain dependent on frequency. We can express the amplitude ratio as a dimension or dimensionless form. However, it is preferred to express it in dimensionless form. In practice, the range of amplitude ratio can cover many orders of magnitude. Due to this reason, it is frequently expressed on a scale of logarithm in decibels; however, it can be used when the amplitude ratio is dimensionless. The amplitude ratio formula is:

Amplitude Ratio (dB) = 20 log 10 AR (dimensionless)

Zero frequency is a situation in which the changes coming in the input are infinitely slow, and hence it takes an infinite length of time for the changes to happen. Therefore, zero frequency relates to a stable state, and that is why the value of amplitude ratio at zero frequency is nothing but a process of steady-state gain.

It will not be discussed why the amplitude ratio decreases with frequency. A large frequency would correspond to a shorter timescale. Therefore, even a short disturbance will cause a high-frequency disturbance if we consider what would happen if we introduce a concentration of disturbance with a shorter duration into the input of a continuously stirred tank. It could be seen that the damping out the effect of the tank would decrease with the reduction in disturbance of time. Therefore, even a short disturbance would be almost totally weakened, whereas a very long one may have a larger effect.

Hence, the processes that show a stirred tank-like or lag-type response would show a reduction of amplitude ratio with frequency. On the contrary, a pure time delay kind of process, e.g. a perfect pipe or a conveyer belt, would pass through any disturbance without any change, irrespective of its frequency and length. Therefore, a constant value would be shown by amplitude ratio in competition with the frequency diagram for a time delay.

It is noted that when two process elements with steady gains G1 and G2 are positioned in series, the total gain would be their product, i.e. G1G2. Understandably, this should be applied to the gain or atomic ratio at any frequency. Hence, we can produce the diagram of the atomic ratio for two processes by multiplying the atomic ratio at each frequency.

Conclusion

Amplitude is the maximum displacement made by a point in a wave to the point of equilibrium. Every wave has amplitude, irrespective of what type of wave it is. We can define the Amplitude ratio as the ratio of amplitude output of wave signal and amplitude input of wave signal. Amplitude ratio formula is – Amplitude Ratio (dB) = 20 log 10 AR. Amplitude ratio is a kind of process gain that depends on the frequency and can be expressed in both dimension or dimensionless form, but mostly, the dimensionless form is preferred.