class 4 dynamic performance. the dynamic characteristics of a measuring instrument describe its...

Class 4

Dynamic Performance

Dynamic Performance

The dynamic characteristics of a measuring instrument

describe its behavior between the time a measured quantity

changes value and the time when the instrument output

attains a steady value in response.

Because dynamic signals vary with time, the measurement

system must be able to respond fast enough to keep up with

the input signal. Further, we need to understand how the

input signal is applied to the sensor because that plays a role

in system response.

Mechanical Zero-Order Systems

The simplest model of a measurement system is the zero-order system

model. This is represented by the zero-order differential equation:

K is the static sensitivity or steady gain of the system. It is a measure of

the amount of change in the output in response to the change in the

input.

)()(1

)(

tKftfa

x

tfxa

o

o

l1 l

2

f(t)x(t)

x(t)/l2

= f(t)/l

1

K = l2

/l1

Mechanical Zero-Order Systems

In a zero-order system, the output responds to

the input signal instantaneously.

If an input signal of magnitude f(t) = A were

applied, the instrument would indicate KA. The

scale of the measuring device would normally

be calibrated to indicate A directly.

)()(1

)(

tKftfa

x

tfxa

o

o

l1 l

2

f(t)x(t)

x(t)/l2

= f(t)/l

1

K = l2

/l1

Electrical Zero-Order Systems

In a zero-order system, the output responds to

the input signal instantaneously.

vi

R1

R2

Vo

io

io

vRR

Rv

RRR

viRv

21

1

121

1

A Unity Gain Zero-Order system

l

xi

xo

xo

(t) = xi(t)

K = 1

Non-zero Order Systems Measurement systems that contain storage or dissipative elements

do not respond instantaneously to changes in input. In the bulb

thermometer, when the ambient temperature changes, the liquid

inside the bulb will need to store a certain amount of energy in

order for it to reach the temperature of the environment. The

temperature of the bulb sensor changes with time until this

equilibrium is reached, which accounts physically for its lag in

response.

In general, systems with a storage or dissipative capability but

negligible inertial forces may be modeled using a first-order

differential equation.

m

k

c

xi

xo

Non-zero Order Systems Measurement systems that contain storage or dissipative elements

do not respond instantaneously to changes in input.

In the bulb thermometer, for example, when the ambient

temperature changes, the liquid inside the bulb will need to store

a certain amount of energy in order for it to reach the temperature

of the environment. The temperature of the bulb sensor changes

with time until this equilibrium is reached, which accounts

physically for its lag in response.

First Order Systems Consider the time response of a bulb thermometer for measuring body

temperature. The thermometer, initially at room temperature, is placed under

the tongue. Body temperature itself is constant (static) during the

measurement, but the input signal to the thermometer is suddenly changed

from room temperature to body temperature. This, is a step change in the

measured signal.

The thermometer must gain energy from its new environment to reach

thermal equilibrium, and this takes a finite amount of time. The ability of any

measurement system to follow dynamic signals is a characteristic of the

measuring system components.

Body Temperature

Room Temperature

t

time

First Order Systems Suppose a bulb thermometer originally at temperature T

o is suddenly

exposed to a fluid temperature T∞

. Develop a model that simulates the

thermometer output response.

Rate of energy stored = Rate of energy in

The ratio mcp/hA has a units of seconds and is called the time constant

Body Temperature

Room Temperature

t

time

TTdt

dT

hA

mc

hAThATdt

dTmc

TThAdt

dTmc

QE

p

p

p

instored

First Order Systems

The ratio mcp/hA has a units of seconds and is called the time constant, τ. If

the time constant is much less then 1 second, the system may be

approximated by a unity gain zero-order system.

Body Temperature

Room Temperature

t

time

TTdt

dT

TTdt

dT

hA

mcp

1st Order Systems Examples:

Bulb Thermometer

RC Circuits

Terminal velocity

Mathematical Model:

𝜏𝑑𝑥𝑑𝑡 + 𝑥= 𝑓ሺ𝑡ሻ τ: Time constant 𝑓ሺ𝑡ሻ: Input (quantity to be measured) 𝑥: Output (instrument response)

1st Order Systems with Step Input

𝑓ሺ𝑡ሻ= 𝐾𝑢(𝑡)

𝑢ሺ𝑡ሻ= ቄ0 𝑡 < 01 𝑡 ≥ 0

ds

𝜏𝑑𝑥𝑑𝑡 + 𝑥= 𝐾𝑢(𝑡)

𝑥ሺ0ሻ= 𝑥0

Second Order Systems In the system shown, the input displacement, x

i, will

cause a deflection in the spring, and some time will be

needed for the output displacement xo

to reach the

input displacement.

m

k

c

xi

xo

Second Order Systems

If m/k << 1 s2 and c/k << 1 s, the system may be approximated as a zero order system with unity gain.

If, on the other hand, m/k << 1 s2 , but c/k is not, the system may be approximated by a first order system. Systems with a

storage and dissipative capability but negligible inertial may be modeled using a first-order differential equation.

m

k

c

xi x

o

iiooo

iiooo

ooioi

o

xxk

cxx

k

cx

k

m

kxxckxxcxm

xmxxcxxk

xmF

Example – Automobile Accelerometer Consider the accelerometer used in seismic and vibration engineering to

determine the motion of large bodies to which the accelerometer is

attached.

The acceleration of the large body places the piezoelectric crystal into

compression or tension, causing a surface charge to develop on the

crystal. The charge is proportional to the motion. As the large body

moves, the mass of the accelerometer will move with an inertial

response. The stiffness of the spring, k, provides a restoring force to

move the accelerometer mass back to equilibrium while internal

frictional damping, c, opposes any displacement away from equilibrium.

m

k

c

xi x

o

Piezoelectric crystal

Zero-Order systems Can we model the system below as a zero-order system? If the mass, stiffness, and damping coefficient satisfy certain

conditions, we may.

m

k

c

xi

xo

i

ioioioi

ooioi

o

xmmck

xmxxmxxcxxk

xmxxcxxk

xmF

First Order Systems Measurement systems that contain storage elements do not respond

instantaneously to changes in input. The bulb thermometer is a good

example. When the ambient temperature changes, the liquid inside the

bulb will need to store a certain amount of energy in order for it to reach

the temperature of the environment. The temperature of the bulb

sensor changes with time until this equilibrium is reached, which

accounts physically for its lag in response.

In general, systems with a storage or dissipative capability but negligible

inertial forces may be modeled using a first-order differential equation.

𝜏𝑑𝑥𝑑𝑡 + 𝑥= 𝐾𝑢(𝑡)

𝑥ሺ0ሻ= 𝑥0 𝑥ሺ𝑡ሻ= 𝐾+ሺ𝑥0 − 𝐾ሻ𝑒−𝑡/𝜏 𝑥ሺ𝑡ሻ− 𝐾𝑥0 − 𝐾 = 𝑒−𝑡/𝜏

𝑥ሺ𝑡ሻ− 𝑥0𝐾− 𝑥0 = 1− 𝑒−𝑡/𝜏

1st Order Systems with Step Input

Error Ratio

Excitation Ratio

Note that the excitation ratio also represents the system response in case of

x0=0 and K=1

error ratio = current errorstarting error = current deviation from final valuestarting deviation from final value

excitation ratio = current excitationdesired (input) excitation= current deviation from initial valueinput deviation from initial value

Excitation ratio may also be called response ratio = current response / desired

response

Example 1

A bulb thermometer with a time constant τ =100 s. is subjected to a step

change in the input temperature. Find the time needed for the response

ratio to reach 90%

Example 1 Solution

A bulb thermometer with a time constant τ =100 s. is subjected to a step change in the input temperature. Find the time

needed for the response ratio to reach 90%

𝑥ሺ𝑡ሻ− 𝑥0𝐾− 𝑥0 = 1− 𝑒−𝑡/𝜏 = 0.9

𝑥ሺ𝑡ሻ− 𝐾𝑥0 − 𝐾 = 𝑒−𝑡/𝜏 = 0.1

𝑡 𝜏= lnሺ10ሻ= 2.3Τ

𝑡 = 2.3× 𝜏= 230 𝑠.≈ 4 minutes

1st Order Systems with Ramp Input excitation ratio = current excitationdesired (input) excitation

excitation ratio = current deviation from initial valueinput deviation from initial value

𝜏𝑑𝑥𝑑𝑡 + 𝑥= 𝑥0 + 𝐾𝑟𝑡𝑢(𝑡)

𝑥ሺ0ሻ= 𝑥0 𝑥ሺ𝑡ሻ= 𝑥0 + 𝐾𝑟𝑡− 𝐾𝑟𝜏(1− 𝑒−𝑡/𝜏)

𝑆.𝑆.𝐸= lim𝑡→∞ሺ𝑥(𝑡) − 𝑓(𝑡)ሻ 𝑆.𝑆.𝐸= 𝐾𝑟𝜏

𝐸𝑥𝑐𝑖𝑡𝑎𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑖𝑜 = 𝑥ሺ𝑡ሻ− 𝑥0𝐾𝑟𝑡

𝐸𝑥𝑐𝑖𝑡𝑎𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑖𝑜 = 1− (1− 𝑒−𝑡/𝜏)𝑡 𝜏Τ

limሺ𝑡 𝜏Τ ሻ→0ቆ1− (1− 𝑒−𝑡/𝜏)𝑡 𝜏Τ ቇ

= limሺ𝑡 𝜏Τ ሻ→0ሺ1ሻ− lim

ሺ𝑡 𝜏Τ ሻ→0൫1− 𝑒−𝑡/𝜏൯𝑡 𝜏Τ

= 1− limሺ𝑡 𝜏Τ ሻ→0𝑒−𝑡/𝜏1 = 1− 1 = 0

Error = 𝑥ሺ𝑡ሻ− 𝑓ሺ𝑡ሻ= −𝐾𝑟𝜏(1− 𝑒−𝑡/𝜏)

Steady State Error

Note that using L’Hospital rule

1st Order Systems with Harmonic Input

𝜏𝑑𝑥𝑑𝑡 + 𝑥= 𝐹 cos (ωt) 𝑥ሺ𝑡ሻ= 𝐶𝑒−𝑡/𝜏 + 𝑋cos (ωt − φ)

𝑋= 𝐹ඥ1+ሺ𝜏𝜔ሻ2

φ = tan−1ሺ𝜏𝜔ሻ

C depends on the initial conditions and the exponential term will vanish with time. We are interested in the particular steady solution 𝑋cos (ωt − φ). Solving for 𝑋 and φ, we find


𝐴𝑟 = 𝑋𝐹= 1ඥ1+ሺ𝜏𝜔ሻ2 = 1

ඥ1+ 4𝜋2𝑇𝑟2

φ = tan−1ሺ𝜏𝜔ሻ=tan−1ሺ2𝜋𝑇𝑟ሻ

Define the amplitude ratio 𝐴𝑟 = 𝑋 𝐹Τ and the time ratio 𝑇𝑟 = 𝜏 𝑇Τ where 𝑇= 2𝜋 𝜔Τ is the period of the excitation function,


The amplitude ratio, Ar(ω), and the corresponding phase shift,

ϕ, are plotted vs. ωτ. The effects of τ and ω on frequency

response are shown.

For those values of ωτ for which the system responds with Ar

near unity, the measurement system transfers all or nearly all

of the input signal amplitude to the output and with very little

time delay; that is, X will be nearly equal to F in magnitude

and ϕ will be near zero degrees.


At large values of ωτ the measurement system filters out any frequency information of the input signal by responding with very small

amplitudes, which is seen by the small Ar(ω) , and by large time delays, as evidenced by increasingly nonzero ϕ.


Any equal product of ω and τ produces

the same results. If we wanted to measure

signals with high-frequency content, then

we would need a system having a small τ.

On the other hand, systems of large τ may

be adequate to measure signals of low-

frequency content. Often the trade-offs

compete available technology against

cost.

dB = 20 log Ar(ω)


The dynamic error,δ(ω), of a system is defined as

δ(ω) = (X(ω) – F)/F

δ(ω) = Ar(ω) –1

It is a measure of the inability of a system to adequately reconstruct the amplitude of the input signal for a particular input frequency.

We normally want measurement systems to have an amplitude ratio at or near unity over the anticipated frequency band of the input

signal to minimize δ(ω) .

As perfect reproduction of the input signal is not possible, some dynamic error is inevitable. We need some way to quantify this. For a

first-order system, we define a frequency bandwidth as the frequency band over which Ar(ω) > 0.707; in terms of the decibel defined as

dB = 20 log Ar(ω)

This is the band of frequencies within which Ar(ω) remains above 3 dB

Example 2

A temperature sensor is to be selected to measure temperature within a reaction vessel. It is suspected that the temperature will

behave as a simple periodic waveform with a frequency somewhere between 1 and 5 Hz. Sensors of several sizes are available, each

with a known time constant. Based on time constant, select a suitable sensor, assuming that a dynamic error of 2% is acceptable.

Example 2. Solution

A temperature sensor is to be selected to measure

temperature within a reaction vessel. It is suspected that the

temperature will behave as a simple periodic waveform with a

frequency somewhere between 1 and 5 Hz. Sensors of several

sizes are available, each with a known time constant. Based

on time constant, select a suitable sensor, assuming that an

absolute value for the dynamic error of 2% is acceptable.

Accordingly, a sensor having a time constant of 6.4 ms or less

will work.

aC𝛿ሺωሻaC≤ 0.02 −0.02 ≤ 𝛿ሺωሻ≤ 0.02 −0.02 ≤ 𝐴𝑟 − 1 ≤ 0.02 0.98 ≤ 𝐴𝑟 ≤ 1.02

0.98 ≤ 1ඥ1+ሺ𝜏𝜔ሻ2 ≤ 1.0

0 ≤ 𝜏𝜔≤ 0.2

0 ≤ 2𝜏𝜋(5) ≤ 0.2

The smallest value of 𝐴𝑟 will occur at the largest frequency

𝜔= 2𝜋𝑓= 2𝜋(5)

𝜏≤ 6.4× 10−3s.

2nd Order Systems Example:

Spring – mass damper

RLC Circuits

Accelerometers

Mathematical Model:

𝑑2𝑥𝑑𝑡2 + 2𝜁𝜔𝑛 𝑑𝑥𝑑𝑡 + 𝜔𝑛2𝑥= 𝑓ሺ𝑡ሻ 𝜁 Damping ratio (dimensionless) 𝜔𝑛 Natural frequency (1/s) 𝑓ሺ𝑡ሻ: Input (quantity to be measured) 𝑥: Output (instrument response)

2nd Order Systems with step input

𝑓ሺ𝑡ሻ= 𝐾𝑢(𝑡)

𝑢ሺ𝑡ሻ= ቄ0 𝑡 < 01 𝑡 ≥ 0

ds

𝑑2𝑥𝑑𝑡2 + 2𝜁𝜔𝑛 𝑑𝑥𝑑𝑡 + 𝜔𝑛2𝑥= 𝐴𝑓ሺ𝑡ሻ 𝜁 Damping ratio (dimensionless) 𝜔𝑛 Natural frequency (1/s) 𝑓ሺ𝑡ሻ: Input (quantity to be measured) 𝑥: Output (instrument response) 𝐴: Arbitrary constant

Correction to Figliola’s Book

Theory and Design for Mechanical Measurements

The sign should be – not +

012

12

12

1

12

1)0(

2

20

2

20

2

2

KAKAeeKAKAy

2nd Order Systems with periodic input

class 4 dynamic performance. the dynamic characteristics of a measuring instrument describe its...

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