Measurements & Instrumentation Formula Book

11.1

Lissajous Patterns: Phase Measurement

Pattern Shape vs Phase Angle

φ = 0° or 360°: Straight line (Q I & III)

0° < φ < 90° or 270° < φ < 360°: Ellipse (Q I & III)

φ = 90° or 270°: Circle

90° < φ < 180° or 180° < φ < 270°: Ellipse (Q II & IV)

φ = 180°: Straight line (Q II & IV)

0°→180°: Clockwise | 180°→360°: Anti-clockwise

Phase Measurement:

\boxed{\phi = \sin^{-1}\!\left(\frac{X_0}{X_m}\right) = \sin^{-1}\!\left(\frac{Y_0}{Y_m}\right)} \quad \text{(Q I,III)}

\phi = 180° - \sin^{-1}\!\left(\frac{X_0}{X_m}\right) \quad \text{(Q II,IV)}

Second possibility: φ₂ = 360° − φ

φ 0°

Line (Q I,III) φ = sin⁻¹(X₀/Xm) φ = 0°

🌀 Lissajous Phase Calculator

X₀: X_m:

Quadrant:

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11.2

Digital Voltmeters (DVM)

DVM Resolution

\boxed{\text{Resolution} = \frac{1}{10^N}} \quad \text{Sensitivity} = \text{Resolution}\times\text{Range}

N = number of full digits | 3½ digit: N=3, Res=0.1%

Dual Slope Integrating DVM

V_x \cdot T_1 = V_{ref} \cdot T_2 \;\Rightarrow\; \boxed{V_x = V_{ref}\cdot\frac{T_2}{T_1}}

V_x = \frac{V_{ref}}{10^N}\cdot X \quad (X = \text{counter reading})

T₁ = integration | T₂ = de-integration | T_conv = T₁+T₂

Noise rejection: T₁ = n·T₀ where T₀ = 1/50Hz = 20ms

Advantages: Noise rejection, high accuracy (independent of RC), auto-zero

V_x 2.0 V

0000

T₂ ∝ V_x Count = —

Digits (N): Res = 0.01%

📟 DVM Resolution & Dual Slope

N digits: Range (V):

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Dual Slope

V_ref (V): T₁ (ms): T₂ (ms):

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11.3

Q-Meter: Quality Factor Measurement

Principle: Series Resonance | At resonance: V_C = Q × V_source

\boxed{Q = \frac{\omega L}{R}} \qquad Q_{true} = Q_m\!\left[1+\frac{R_s}{R}\right]

Distributed Capacitance

\boxed{C_d = \frac{C_1 n^2 - C_2}{n^2 - 1}} \quad f_2 = n\cdot f_1

L = \frac{1}{4\pi^2 f_s^2(C+C_d)} \qquad R = \frac{\omega L}{Q}

Applications: Q factor, BW, L, C_d, Capacitance

%err(Q) = −R_s/(R+R_s)×100 | %err(C_d) = −C_d/(C+C_d)×100

f (kHz) 500

Q = — V_C = —

🔬 Q-Meter Calculator

ω (rad/s): L (mH): R (Ω):

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Distributed Capacitance

C₁ (pF): C₂ (pF): n (f₂/f₁):

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True Q

Q_m: R_s (Ω): R (Ω):

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11.4

Galvanometer & Transducer Overview

D'Arsonval Galvanometer

T_d = BINA \quad T_c = K\theta \quad \boxed{\theta = \frac{NBA}{K}\cdot I = \frac{G}{K}\cdot I}

S_I = \frac{\theta}{I} = \frac{NBA}{K} = \frac{G}{K}\;\text{(rad/A)}

Types: Tangent, Astatic, Mirror, Ballistic, D'Arsonval/Weston

Transducer Classification

Converts one form of energy to another (also: pick-up)

Active (self-generating): Piezo, Photovoltaic, Thermocouple

Passive (needs power): LVDT, Thermistor, RTD, Capacitive

By function — Primary: Bellows, Bourdon, Thermistor | Secondary: LVDT

I (mA) 0

Transducer Classification

Active
Piezo · Photovoltaic
Thermocouple

Passive
LVDT · Thermistor
RTD · Capacitive

Analog
LVDT · Strain
gauge · Thermistor

Digital
Encoder · Digital
tachometer

📡 Galvanometer Deflection Calculator

B (T): N: A (cm²): K (N·m/rad): I (mA):

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11.5

Strain Gauge & Thermistor

Strain Gauge

\boxed{G_f = \frac{\Delta R/R}{\Delta L/L} = \frac{\Delta R/R}{\varepsilon}} = 1+2\gamma+\frac{\delta\rho/\rho}{\varepsilon}

\text{Metal: } G_f \approx 1+2\gamma \approx 2\text{–}4 \quad \text{Semi: } G_f > 100

γ = Poisson's ratio | ε = ΔL/L (strain) | Lateral strain = −γ·ε

Thermistor (NTC)

\boxed{R_{T_1} = R_{T_2}\exp\!\left[\beta\!\left(\frac{1}{T_1}-\frac{1}{T_2}\right)\right]}

NTC: T↑→R↓ | β=2000–5000K | Range: −100°C to 300°C | 0.5Ω to 75MΩ

ε (μstrain) 1000

Thermistor R vs T (NTC)

β 3500 K

📏 Strain Gauge Calculator

ΔR/R: ΔL/L (ε):

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🌡️ Thermistor Resistance

R_T2 (kΩ): β (K): T₁ (°C): T₂ (°C):

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11.6

Thermocouple & LVDT

Thermocouple (Seebeck Effect)

\boxed{E = a(\Delta\theta) + b(\Delta\theta)^2}

(+) Metal	(−) Metal	Range
Copper	Constantan	−250 to +400 °C
Iron	Constantan	−200 to +850 °C
Pt-Rh	Platinum	0 to +1400 °C
Rhodium	Iridium	0 to +2100 °C

LVDT — Linear Variable Differential Transformer

V_{out}=V_{S1}-V_{S2} \qquad \boxed{S_{LVDT}=\frac{V_{out}}{\text{displacement}}\;\text{(V/mm)}}

Center: V_out=0 | Right: +ve | Left: −ve | Range: ±125–250mm

Advantages: Frictionless, infinite resolution, high sensitivity, linear

Thermocouple — Seebeck Effect

ΔT (°C) 100

Type:

LVDT Cross-Section

Core position 0%

V_out = 0 (Null)

🌡️ Thermocouple EMF Calculator

a (mV/°C): b (mV/°C²): Δθ (°C):

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11.7

Piezoelectric Transducer & Measurements

Piezoelectric Effect — Dynamic Pressure/Force

\boxed{d = \frac{Q}{F}\;\text{(C/N)}} \quad \boxed{g = \frac{E}{p} = \frac{V}{t\cdot p}\;\text{(V·m/N)}}

\boxed{E_0 = g\cdot t\cdot p}

Natural: Quartz, Rochelle salt | Synthetic: BaTiO₃, KDP

Temperature Sensor Ranges

Bimetallic ~27°C | Thermistor −100→300°C | RTD →600°C | TC →1400°C | Pyrometer 1200→3500°C

Low-Pressure Gauges

Pirani 10⁻³–10⁻¹ | TC vacuum ≤10⁻² | McLeod ~10⁻⁴ | Ionization 10⁻⁷–10⁻⁸ mmHg

Mechanical: Bourdon > Bellow > Diaphragm | High: Bridgman

F (N) 100

Q = — pC E₀ = — V

Temperature Sensor Ranges

Vacuum Gauge Ranges (mmHg)

⚡ Piezoelectric Calculator

d (pC/N): F (N): g (mV·m/N): t (mm): p (kPa):

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11.8

Module 11: Complete Reference

Topic	Key Formula
Lissajous	φ = sin⁻¹(X₀/X_m) (Q I,III) \| 180°−sin⁻¹(…) (Q II,IV)
DVM	Res = 1/10^N \| V_x=V_ref·T₂/T₁
Q-Meter	Q = ωL/R \| C_d = (C₁n²−C₂)/(n²−1)
Galvanometer	θ = (NBA/K)·I
Strain Gauge	G_f = (ΔR/R)/ε = 1+2γ (metal)
Thermistor	R_T1=R_T2·exp[β(1/T₁−1/T₂)]
Thermocouple	E = a(Δθ)+b(Δθ)²
LVDT	S = V_out/displacement (V/mm)
Piezoelectric	d=Q/F \| g=V/(t·p) \| E₀=g·t·p

Click to reveal formula

Lissajous Phase

DVM Resolution

Q Factor

Galvanometer

Gauge Factor

Thermistor

Thermocouple

Piezoelectric

Sensor Selector Quiz

Measure temperature up to 2000°C?

Module 11 — Lissajous Phase, DVM, Q-Meter, Transducers & Sensors

Lissajous Patterns: Phase Measurement

🌀 Lissajous Phase Calculator

Digital Voltmeters (DVM)

📟 DVM Resolution & Dual Slope

Dual Slope

Q-Meter: Quality Factor Measurement

🔬 Q-Meter Calculator

Distributed Capacitance

True Q

Galvanometer & Transducer Overview

📡 Galvanometer Deflection Calculator

Strain Gauge & Thermistor

📏 Strain Gauge Calculator

🌡️ Thermistor Resistance

Thermocouple & LVDT

🌡️ Thermocouple EMF Calculator

Piezoelectric Transducer & Measurements

⚡ Piezoelectric Calculator

Module 11: Complete Reference