Appunti Industrial Measurements + Domande Scritto, Prof. Palermo

Introduction

The measurement of a physical quantity is composed by a number, an uncertainty range and a measurement unit.

X = (x ± U) g

• x = best value
• U = uncertainty range
• g = measurement unit

The element that provides in measuring the physical quantity is the transducer.

Passive transducers: The energy necessary to do the conversion is provided by the measurand.

Active transducers: an external power source.

Static Characteristics of Measurement Systems

Measurement Range

Is the range of measurand values, the instrument can measure holding the other metrological characteristics (Sensitivity, Stiffness, Rapidity, Precision).

The measurement range is limited by the upper limit and the lower limit.

Nominal Load: is the name of the upper limit if the lower limit is 0.

Limit Load: is the upper limit admitted to avoid permanent damages in the transducer.

Nominal Load Measurement Range Service Load Service Range Limit Load | Breaking Load Max Loading Range Destruction Range

Sensibility

Is the attitude of the transducer to detect "small" variations of the measurand.

Sensitivity S' is the derivative of the graduation curve.

S = dσ/di

i ----_transducer----- σ

Accuracy and Precision

Accuracy

Is the function of systematic errors; it quantifies the closeness of the measure to the true value of the measured.

Inaccuracy: E_a = |(x̄ - x_t)| / x_t x 100

Precision

Is the function of random errors. An high precision means to obtain the same result doing many measures of the same quantity (under unchanged conditions).

• Highly Precise
• Poorly Accurate

• Systematic errors -> Non Negligible
• Random errors -> Negligible

• Poorly Precise
• Highly Accurate

• Systematic errors -> Negligible
• Random errors -> Non Negligible

• Highly Precise
• Highly Accurate

• Systematic errors -> Negligible
• Random errors -> Negligible

Real Conditions

In real conditions, when the input is fast (f→∞), the instrument is not able to follow it so the given output is null. This because mechanical parts cannot have infinite accelerations and electrical parts cannot have _{ ²

The bandpass is the range of frequencies in which the response of the instrument respects a fixed dynamic error Edyn.

It's often used Edyn ≈ 0.293 expressed in decibels

K dB ≄ 20 log₁₀ (1-Edyn)

so 20 log₁₀ 0.707 ≄ -3 dB

The limit frequency of the passband (ft) is named cutoff frequency.

Linear Instruments

An instrument is dynamically linear if it's possible to describe the motion of his parts with a linear differential equation.

An n^th-order instrument is described by an n^th order equation; by a physical point of view, m is the number of energy forms in the instrument.

• 1^st-Order Instruments

a y(t) = b x(t) so x(t) = y(t)

The measure is directly proportional to the input and the response of the instrument is instantaneous.

SIGNAL PROCESSING

AMPLIFIERS

Electrical signals are attenuated in any transduction/processing stage, so it's necessary to use amplifiers to restore the magnitude.

The core of the modern signal amplifier is the transistor, composed by a semiconductor with 3 terminals that connect it with the external circuit.

The application of a voltage at 2 terminals allows the regulation of the current through the transistor, so the signal can be amplified.

The functioning of the device is based on the P-N junction.

• Transistor BJT (Bipolar Junction Transistor)
• Transistor MOS (Metal Oxide Semiconductor)

Transistors can be properly combined to enhance the amplification (gain) A; the result is an object called operational amplifier (Op Amp).

The Op Amp is represented by a triangle with two inputs (V₊, V_-) and an output V_out.

The device is powered by 2 pins (V_s+, V_s-).

The Op Amp has gain and input impedance very high.

Ideal Characteristics

• Amplification: A → ∞ (10⁷)
• Input Impedance: Z_in → ∞ (10¹⁰ Ω)
• Output Impedance: Z_out → 0 (10 Ω)
• Bandwidth: BW → ∞

Example 2

For the capacitors we have

i(t) = C _d(u(t))/_dt - ωCA sin(ωt+φ)

let's use phasors

I = jωCV = jωC A e^jωt e^jφ = ωCA [j cos(ωt+φ) - sin(ωt+φ)]

i(t) = Re[ I e^jωt ] = ωCA sin(ωt+φ)

Phasors allow us to define the general ELECTRICAL IMPEDANCE Z

• Resistors : Z_R = R
• Capacitors : Z_C = 1/jωC
• Inductors : Z_L = jωL

So we can generalize the Ohm's Law : V = Z ⋅ I

Passive Low-Pass Filter

A passive low-pass filter is composed by a resistor and a capacitor.

The input signal will be attenuated and phase-shifted by the filter, depending on its frequency.

Using the voltage divider principle we obtain

V_OUT = Z_C / (Z_R+Z_C) V_IN = 1/(1+jωRC) V_IN = 1/(1+jωλ) V_IN

λ = RC time constant of 1^st order system

GAIN G = |V_OUT/V_IN| = 1/√(1+(ωλ)²)

• ω = 0 → G = 1 (0 dB) φ = 0
• ω = 1/λ = ω_C → G = 1/√2 (-3 dB) φ_C = -π/4
• ω >> 1/λ → G → 0 φ → -π/2

So we have that high frequencies are attenuated and the CUTOFF FREQUENCY is

ω_C = 1/λ = 1/RC

Measurement Systems

The measurement system is represented as an operational amplifier.

In the real case, output is affected by common mode voltage.

V_out = A_dV_d + A_cV_cm

V_d = V₊-V_-

A_d = ^{A₊ + A_-}/₂

V_cm = ^{V₊ + V_-}/₂

A_c = A₊ - A_-

Common Mode Rejection Ratio

C M R R = 20 log₁₀ (^A_d/_Ac)

• Differential Measurement

Differential measurement system requires two inputs, no one ground-referred.

An ideal differential system only responds to the voltage difference between (+) and (-) pins.

So the system amplifies the voltages with the same gain coefficient A and, trough the subtraction, allows to reject the noise on the inputs. In the real case, the common mode voltage introduces an error.

The system needs 2N cables (N = number of acquisition channels).

• Single-Ended Measurement

In single-ended systems, it's necessary only one input channel. All channels use the negative pin of the instrumentation amplifier as common reference.

Single-ended systems need (N + 1) cables (N = number of acquisition channels), so they are more economic (of the same number of channels) or allow the double of the measures (in case of some number of cables).

They have a lower accuracy, due to the presence of a ground loop current circuit.