# Module 7.1: Frequency Response

## Introduction

The classical methods for analysing control loops and designing controllers make considerable use of what control engineers call the `frequency domain'. The idea of considering the `frequency response' of a process is that instead of, for example, seeing how long it takes to respond to, e.g. a sudden step change in valve position (its response in the `time domain'), we consider how it behaves when the valve is made to open and shut so as to follow a sine wave at different frequencies.

This module starts by describing the Sine Wave and it's definition. Then we look at how the output from a process can tell us how the process will act under control by looking at the Bode Diagram.

A sine wave:

is characterised by 3 parameters, viz, its:

• amplitude a ,
• period P or frequency f , and
• phase p
The phase p of a sinewave is a relative quantity; since the sine function can take any argument and has no absolute starting point. Mathematically, a sinewave y(t) varying with time t is decribed by:

y(t) = a sin (w t + p)

Here a is the amplitude and has whatever whatever units the quantity y represents in its physical form.

The phase p has units of angle, either degrees or radians.

In the equation as written w has units of angle per unit time , typically radians/minute for process control situations, because the quantity w t must have units of angle. The `true' frequency f is 2 pi times the angular frequency, i.e.

w =

The reciprocal of frequency f is the period P , i.e. the time occupied by a complete cycle of the sinewave.

## The Frequency Response Diagram

Consider a process whose input, e.g. a valve, is made to move sinusoidally, and whose output, i.e. some measurement is observed. If input and output are examined together, e.g as shown below, then a number of points may be noted.

• The output will be a sinusoid of the same frequency as the input.
• The ratio of the output amplitude to that of the input amplitude (which will have units of process gain) will in general vary with the frequency of the sine wave input.
• The difference in phase between the input and output sine waves will also depend on the frequency.

The first of the above statements is strictly speaking only true if the process is described by linear differential and algebraic equations, but even for `real' nonlinear processes the output will be approximately sinusoidal and will cycle with the input frequency.

We call the ratio of the output to input amplitudes the Amplitude Ratio for the process for which we will use the symbol

AR(w)

(Note that control literature use the symbol |G(w)|)

The difference between input and output phases is called the Phase angle:

PA(w)

for the process. (The literature uses G(w))

For a process we may plot AR and PA as functions of frequency together in what is called a Bode Diagram.

The Bode diagram comprises two graphs, one of Amplitude Ratio (AR) versus frequency and one of Phase Angle (PA) versus frequency. These are somtimes combined into a single diagram with a common frequency scale, but are here shown as separate graphs. Each of these terms will be described in turn below and then the Bode Diagram and how to intepret it, discussed in a later section.

The frequency response of most processes without controllers is such that both AR and PA normally decrease monotonically with increasing frequency.

## Amplitude ratio

Amplitude ratio (AR) may be thought of as the frequency dependent process gain. It may be expressed in dimensional or dimensionless form. The latter is prefered. In practice the interesting range of AR may cover several orders of magnitude. For this reason it is often expressed on a logarithmic scale in decibels (dB). This can only be used when the AR has been made dimensionless, and the relevant definition is:

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

When the AR or gain is unity, this thus corresponds to 0dB. AR is given in dB on the diagram below.

Zero frequency corresponds to a situation where the input is changing infinitely slowly, and thus a change occurs over an infinite length of time. Zero frequency thus corresponds to steady state and so the value of the AR at zero frequency is just the process steady state gain.

We will not here discuss the theory which shows why amplitude ratio normally decreases with frequency, but the following qualitative explanation should serve as a justification, coupled with experimental observations in the Virtual Control Lab.

A large frequency corresponds to a short timescale, and vice versa. So a very short distubance corresponds to a high frequency disturbance.

Consider what happens if we introduce say, concentration disturbances of shorter and shorter duration into the input of a continuous stirred tank of given size. Intuitively, we can see that as the distubance time is reduced, the `damping out' effect of a the tank will increase. Thus a very short disturbance will be almost completely attenuated, while a very long one will have a larger effect.

Thus processes which exhibit a `stirred tank like' or lag type of response will show a decrease of amplitude ratio with frequency.

Conversely, a pure time delay type of process such as a perfect pipe (without backmixing) or a conveyer belt, should pass any disturbance completely unchanged, regardless of its length or frequency. Thus the AR versus frequency diagram for a time delay will show a constant value.

It has already been noted that if two process elements having steady state gains G 1 and G 2 are placed in series then the the overall gain is their product G 1 G 2 . It is intuitively obvious that this should apply to the gain or AR at any frequency. We can thus produce the AR diagram for two processes or process elements in series by multiplying the ARs at each frequency. (If AR is given in dB then the db values are of course added.)

## Phase Angle

Note that phase angle or phase lag is always negative. This implies that disturbance passing through a process is inevitably delayed rather than accelerated by the process. (If it were otherwise, it would imply the reversal of time!)

The effect of frequency on phase angle is most readily understood by analysing a pure time delay. Consider a process which represents a pure time delay of 1 minute. If its input is a sine wave of period 1 minute, then the output will be delayed by exactly 1 period or 360 degrees.

Similarly, if the sine wave had a period of 2 minutes it would be delayed by 1/2 or half a period or 180 degrees, and so on. Thus the negative PA is directly proportional to frequency for a delay, and becomes minus infinity at infinite frequency.

Qualitatively similar arguments apply to a lag, but a single lag can be shown to produce a maximum PA of -90 degrees at infinite frequency. You can verify this in the lab experiments.

If two process elements are connected in series, one of them causes a given sine wave to have a phase change of 10 degrees, the a change of 20 degrees, then the combined phase shift or PA is clearly (10+20)=30 degrees. PAs for elements in series are thus simply additive.