ERT 350 - Week 3 SEM 1 2016/2017 1

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Instrumentation, Measurement and Control

in Biosystems(ERT 350/3)

Introduction to Typical Application of Instrumentation Systems

- Part 3-

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Dynamic Performance Characteristic of Instruments

ERT 350 - Week 3 SEM 1 2016/2017 2

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Dynamic Characteristic

• For time dependent signal / time varying quantities the dynamic characteristic need to be considered.

• Situation where the desired input is not constant but varies rapidly with the time.

• They are generally represented by the relations between input and output parameters that are governed by the relevant differential equations applicable in the given situation.

ERT 350 - Week 3 SEM 1 2016/2017 3

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Dynamic Characteristic

The dynamic inputs to an instrument may be of the following types:

1. Periodic input –Varying cyclically with time or repeating itself after a constant interval.

ERT 350 - Week 3 SEM 1 2016/2017 4

Periodic signal with time period T

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Dynamic Characteristic

The dynamic inputs to an instrument may be of the following types:

2. Transient input –Varying non-cyclically with time. The signal is of a definite duration and becomes zero after a certain period of time.

ERT 350 - Week 3 SEM 1 2016/2017 5

Transient signal

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Dynamic Characteristic

The dynamic inputs to an instrument may be of the following types:

3. Random input –Varying randomly with time, with no definite amplitude and period.

ERT 350 - Week 3 SEM 1 2016/2017 6

Random signal

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Dynamic Characteristic

Examples of dynamic signal:

1. Vibration excitation due to unbalance of a rotating body (periodic-harmonic).

2. Pressure variation in an internal combustion engine (periodic).

3. Forces due to an explosion (transient).

4. Pressure fluctuations in fluid flow due to turbulence (random)

ERT 350 - Week 3 SEM 1 2016/2017 7

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Dynamic Characteristic

• For studying the dynamic characteristics of an instrument or a combination of instruments, it is necessary to represent each instrument by its mathematical model which is obtained from the governing relation between its input and output signal.

ERT 350 - Week 3 SEM 1 2016/2017 8

INSTRUMENTInput signal Output signal

Xi (t) Xo (t)

𝐾

1 + 𝜏𝐷

Input signal Output signal

Xi (t) Xo (t)

Block diagram of a first order instrument

of a temperature measuring system.

Xi (t) = Input signal

Xo (t) = Output signal

K = Static sensitivity constant

= Time constant

D = Time derivative operator

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Dynamic Response

Dynamic input signals;

1. Periodic input – harmonic type

2. General periodic input – non-harmonic type

3. Transient type input

4. Random input

ERT 350 - Week 3 SEM 1 2016/2017 9

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Periodic Input – Harmonic Signal

ERT 350 - Week 3 SEM 1 2016/2017 10

T = Period = 1/f (frequency)

Xi

Xi

t

Xi

The harmonic input signal can be represented as xi (t) = Xi sin t.

Where Xi is the amplitude and the circular frequency.

= 2/T = 2f

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Periodic Input – Harmonic Signal

• First order system –The governing equation of a first-order system is:

𝑎1𝑑𝑥𝑜

𝑑𝑡+ 𝑎𝑜𝑥𝑜 = 𝑏𝑜𝑥𝑖 𝑜𝑟 𝜏𝐷 + 1 𝑥𝑜 = 𝐾𝑥𝑖(𝑡) (eq. 1)

Where K= bo/ao (static sensitivity),

𝜏 = a1/ao (time constant)

D= d/dt (time derivative operator).

ERT 350 - Week 3 SEM 1 2016/2017 11

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Periodic Input – Harmonic Signal

The steady state solution xo(t) is obtained for

𝜏𝐷 + 1 𝑥𝑜 = 𝐾𝑥𝑖(𝑡)

by putting D=j, where j is complex operator = −1.

𝑥𝑜

𝑥𝑖=

𝐾

1+𝑗𝜔𝜏, giving amplitude ratio;

𝑋𝑜𝑋𝑖

=𝑥𝑜𝑥𝑖

=𝐾

1 + 𝜔2𝜏2

and 𝜑 = 𝑡𝑎𝑛−1(−𝜔𝜏)

𝑥𝑜 = 𝑋𝑜 sin(𝜔𝜏 + 𝜑)

ERT 350 - Week 3 SEM 1 2016/2017 12

Xo = amplitude of

output

= phase, negative

indicates that the output

signal lags behind the

input.

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Periodic Input – Harmonic Signal

• Second order system –The governing equation of a second-order system is:

𝑎2𝑑2𝑥𝑜𝑑𝑡2

+ 𝑎1𝑑𝑥𝑜𝑑𝑡

+ 𝑎𝑜𝑥𝑜 = 𝑏𝑜𝑥𝑖

• May be written in a dimensionless form given by

𝐷2

𝜔𝑛2 +

2𝜉

𝜔𝑛𝐷 + 1 𝑥𝑜 = 𝐾𝑥𝑖(𝑡)

ERT 350 - Week 3 SEM 1 2016/2017 13

K= bo/ao (static sensitivity),

𝜔𝑛 = 𝑎𝑜/𝑎2 (undamped natural frequency)

𝜉 = 𝑎1/2 𝑎𝑜𝑎2 (damping ratio)

D= d/dt (time derivative operator).

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Periodic Input – Harmonic Signal

The steady state solution 𝑥𝑜(𝑡) is obtained for

𝐷2

𝜔𝑛2 +

2𝜉

𝜔𝑛𝐷 + 1 𝑥𝑜 = 𝐾𝑥𝑖(𝑡)

by putting D=j.𝑥𝑜 = 𝑋𝑜 sin(𝜔𝜏 + 𝜑)

𝑋𝑜 =𝐾𝑋𝑖

(1 − 𝑟2)2+(2𝜉𝑟)2

𝜑 = − 𝑡𝑎𝑛−12𝜉𝑟

1 − 𝑟2

𝑟 =𝜔

𝜔𝑛

ERT 350 - Week 3 SEM 1 2016/2017 14

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Example 1

A first order instrument is to measure signal with frequency content up to 100 Hz with an amplitude inaccuracy of 5 %. What is the maximum allowable time constant ? What will be the phase shift at 50 Hz ?.

ERT 350 - Week 3 SEM 1 2016/2017 15

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Example 1 - Solution

𝑋𝑜

𝑋𝑖=

𝐾

1+𝜔2𝜏2and 𝜑 = 𝑡𝑎𝑛−1(−𝜔𝜏)

𝑋𝑜𝐾𝑋𝑖

=1

1 + 𝜔2𝜏2= 0.95

𝜔 = 2𝜋𝑓 = 2 × 3.142 × 100 = 628.4𝑟𝑎𝑑

𝑠

𝜏 =

10.95

2

− 1

628.42= 5.23 × 10−4

Phase angle at 50Hz, 𝜑 = 𝑡𝑎𝑛−1(−𝜔𝜏)

𝜑 = 𝑡𝑎𝑛−1(−2 × 𝜋 × 50 × 5.23 × 10−4)

𝜑 = −9.33

ERT 350 - Week 3 SEM 1 2016/2017 16

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Example 2

A second-order instrument is subjected to a sinusoidal input. Undamped natural frequency is 3 Hz and damping ratio is 0.5. Calculate the amplitude ratio and phase angle for an input frequency of 2 Hz.

ERT 350 - Week 3 SEM 1 2016/2017 17

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Example 2 - Solution

Amplitude ratio = 𝑋𝑜

𝑋𝑖=

𝐾

(1−𝑟2)2+(2𝜉𝑟)2and 𝜑 = − 𝑡𝑎𝑛−1

2𝜉𝑟

1−𝑟2

It is assumed that xo and xi are of the same form and the static sensitivity

is equal to 1.

Amplitude ratio = 𝑋𝑜

𝑋𝑖=

1

(1−𝑟2)2+(2𝜉𝑟)2

𝑟 =𝜔

𝜔𝑛=2𝜋 × 2

2𝜋 × 3= 0.67 𝑎𝑛𝑑 𝜉 = 0.5

𝑋𝑜𝑋𝑖

=1

(1 − 0.672)2+(2 × 0.5 × 0.67)2= 1.153

𝜑 = − 𝑡𝑎𝑛−12𝜉𝑟

1 − 𝑟2= − 𝑡𝑎𝑛−1

2 × 0.5 × 0.67

1 − 0.672= −50.6

ERT 350 - Week 3 SEM 1 2016/2017 18

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Example 3

A temperature measuring system, with a time constant of 2 s is used to measure temperature of a heating medium, which changes sinusoidally between 350 and 300 °C with a periodic time of 20 s. Find the maximum and minimum values of temperature, as indicated by the measuring system and the time lag between the output and input signals.

ERT 350 - Week 3 SEM 1 2016/2017 19

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Example 3 – Solution

Taking the system as a first order, the governing equation is 𝜏𝐷 + 1 𝑥𝑜 = 𝑥𝑖 𝑡 , where K is assumed as 1.

The medium has a mean temperature of 325°C and amplitude of 25°C.

Using 𝑋𝑜

𝑋𝑖=

1

1+𝜔2𝜏2

with = 2 s and 𝜔 = 2 × 𝜋 ×1

20= 0.314 𝑟𝑎𝑑/𝑠

𝑋𝑜𝑋𝑖

= 0.847

For input amplitude 𝑋𝑖=25°C, 𝑋𝑜 =21.2°C.

Thus the indicated temperature will vary between

346.2 ~303.8°C

ERT 350 - Week 3 SEM 1 2016/2017 20

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Example 3 - Solutions

Phase angle

𝜑 = 𝑡𝑎𝑛−1 −𝜔𝜏 = 𝑡𝑎𝑛−1 −0.314 × 2 = −32.1°

Time Lag

𝑇𝐿 =32.1°

360°× 20 𝑠 = 1.78𝑠

ERT 350 - Week 3 SEM 1 2016/2017 21

Firdaus Muttalib

School of Bioprocess

Engineering

20/9/2016

Example 3 - Solutions

ERT 350 - Week 3 SEM 1 2016/2017 22

Input and output signal