INSTRUMENTATION & CONTROL (ME-402) - Unit 3 Short Notes

UNIT 3: INSTRUMENTATION & CONTROL - SHORT NOTES

I. FOUNDATIONS OF MEASUREMENT SYSTEMS

A. Elements of a Measurement System

A complete measurement system typically consists of:

1. Primary Sensing Element: Interfaces directly with the measurand (e.g., thermocouple, bourdon tube).

2. Signal Conditioner: Modifies the primary signal (e.g., amplifier, filter, bridge circuit) to a suitable form.

3. Display/Recorder: Presents the measured value (e.g., digital readout, chart recorder, computer).

4. Feedback Element: (In control systems) Compares input with reference.

Classification:

• By Function: Indicating (gauge), Recording (chart recorder), Controlling (controller).

• By Application: Industrial, laboratory, biomedical, etc.

[!TIP] Past papers frequently ask for examples. Link each element to a specific instrument (e.g., in a digital thermometer: thermocouple (sensing), cold-junction compensation & amplifier (conditioning), LCD display).

B. Errors and Uncertainties

Types of Errors:

• Systematic: Consistent, predictable bias (e.g., zero error, calibration error). Can be corrected.

• Random: Unpredictable fluctuations (e.g., electrical noise). Reduced by averaging.

• Gross: Mistakes (e.g., reading error, blunder). Should be eliminated.

Uncertainty Concepts:

• Absolute Uncertainty (±Δx): Margin of error in same units (e.g., ±0.02 g).

• Relative Uncertainty (%): $$\displaystyle \frac{\text{Absolute Uncertainty}}{\text{Measured Value}} \times 100\% $$.

• Accuracy vs. Precision: Accuracy = closeness to true value; Precision = repeatability/scatter.

• % of Full Scale (%FS) vs. % of True Value (%RD):

  • %FS Error: Constant absolute error across range. $$\displaystyle \text{Error} = \pm (\%FS) \times \text{Full Scale Value} $$.

  • %True Value Error: Proportional error. $$\displaystyle \text{Error} = \pm (\% \text{ of reading}) \times \text{Actual Reading} $$.

[!EXAMPLE] DEC 2024 Problem: Gauge range = 1000 kN/m², accuracy = ±1% FS.

Absolute error = $$\displaystyle 0.01 \times 1000 = \pm 10 \text{ kN/m}^2 $$.

At 100 kN/m², possible readings = $100 \pm 10$ → 90 to 110 kN/m².

If error is ±1% of true value at 100 kN/m²: Error = $$\displaystyle 0.01 \times 100 = \pm 1 \text{ kN/m}^2 $$. Readings = 99 to 101 kN/m².

C. Calibration and Standards

• Need: To establish relationship between instrument output and known standard, ensuring accuracy and traceability.

• Procedure: Compare instrument reading against a higher-accuracy standard at multiple points across its range. Generate calibration curve/equation.

• Traceability: Unbroken chain of comparisons linking measurement to national/international standards (e.g., NIST, BIPM).

• Calibration Tests:

  • Analysis: Fixed points (e.g., ice point, steam point).

  • Sequential: Increment/decrement through range to check hysteresis.

  • Random: Random order points to avoid drift bias.

• Dead Weight Tester (Pressure): Primary standard. Weights apply known pressure $$\displaystyle P = \frac{F}{A} = \frac{mg}{A} $$ to a piston-cylinder.

  DiagramSEARCH: dead weight tester diagram

D. Static Performance Characteristics

Defined for steady-state (constant) inputs.

[!TIP] JUN 2025 & DEC 2024 ask for impact on accuracy. Example: High hysteresis → inaccurate reading depending on history. Low resolution → cannot detect small changes.

E. Dynamic Performance Characteristics

Response to time-varying inputs.

• Fidelity: How accurately the output waveform reproduces the input waveform.

• Speed of Response: Quickness in following input changes (e.g., rise time, settling time).

• Measuring Lag: Delay in response (e.g., thermal mass in thermometer).

• Settling Time ($$\displaystyle t_s $$): Time to reach and stay within a tolerance band (e.g., ±2%) of final value.

• Rise Time ($$\displaystyle t_r $$): Time to go from 10% to 90% of final value for step input.

• Dynamic vs. Static: Static specs don't predict behavior for changing inputs. A system can be accurate statically but too slow dynamically.

[!EXAMPLE] JUN 2024: Servo motor $$\displaystyle H(s)=\frac{50}{s(s+10)} $$. Standard 2nd-order form: $$\displaystyle \frac{\omega_n^2}{s(s+2\zeta\omega_n)} $$. Comparing: $$\displaystyle 2\zeta\omega_n = 10 $$, $$\displaystyle \omega_n^2=50 \Rightarrow \omega_n = \sqrt{50} \approx 7.07 \text{ rad/s} $$. Time Constant (for dominant pole approximation) $$\displaystyle \tau \approx 1/10 = 0.1 \text{ s} $$. Settling Time (2%) $$\displaystyle t_s \approx \frac{4}{\zeta\omega_n} $$. Need $\zeta$: $$\displaystyle 2\zeta \times 7.07 = 10 \Rightarrow \zeta \approx 0.707 $$. Then $$\displaystyle t_s \approx \frac{4}{0.707 \times 7.07} \approx 0.8 \text{ s} $$.

II. TRANSDUCERS AND SENSORS (BY MEASURAND)

A. Displacement Measurement

• Potentiometer:

  • Principle: Variable resistance. Wiper movement on resistive element changes output voltage $$\displaystyle V_{out} = V_{in} \frac{R_2}{R_1+R_2} $$.

  • Construction: resistive element (cermet, wire-wound), wiper, shaft.

  • Limitations: Wear, limited resolution (wire-wound), finite force required, limited bandwidth.

• Linear/Rotary Variable Differential Transformer (LVDT/RVDT):

  • Principle: Mutual induction. Primary coil excited with AC. Two secondary coils connected in series opposition. Core displacement induces voltage difference.

  • Construction: Cylindrical core, primary winding, two secondary windings, housing.

  • Output: $$\displaystyle V_{out} \propto $$ displacement $x$. Phase indicates direction.

  • Advantages: Infinite resolution, no physical contact (wear-free), high reliability, linear over wide range, single-axis sensitivity.

  • Provides precise measurement due to differential output rejecting common-mode noise.

[!DIAGRAM] LVDT:

DiagramSEARCH: LVDT cross-section diagram

B. Pressure Measurement

• Fluid Column Devices:

  • Manometer (U-tube): $$\displaystyle P_1 - P_2 = \rho g h $$. Simple, no calibration needed, but slow, large size.

  • Inclined Manometer: Amplifies deflection $h$ for low pressures ($$\displaystyle h = L \sin\theta $$).

  • Barometer: Measures atmospheric pressure (mercury or aneroid).

• Elastic Elements:

  • Bellows: Sealed, corrugated cylinder. Pressure causes expansion. Used for moderate pressures, high displacement.

  • Diaphragm: Flat or corrugated disk. Deflection $\propto$ pressure difference. Used for higher pressures, smaller displacement.

• Dead Weight Tester: (See I.C). Primary standard for calibrating pressure gauges. Piston area must be precisely known.

• Electronic Types:

  • Piezoelectric: (See II.F) For dynamic pressure.

  • Ionisation Transducer: For vacuum. Measures ion current from gas ionisation, which depends on pressure.

C. Temperature Measurement

Contact Methods:

• Thermocouple (TC):

  • Seebeck Effect: Two dissimilar metals joined at two junctions produce net EMF if junctions at different T.

  • Requirement: Two metals (A & B). Output $$\displaystyle E_{AB}(T_1, T_2) $$ depends on difference $$\displaystyle T_1 - T_2 $$.

  • Cold-Junction Compensation: Reference junction at known T (often 0°C ice bath or electronic compensation).

  • Types: J, K, T, E, S, R, B (material pairs). Selection based on T range, atmosphere, sensitivity.

• Resistance Temperature Detector (RTD):

  • Principle: $$\displaystyle R_T = R_0 [1 + \alpha (T - T_0)] $$ (linear approx). $\alpha$ = temperature coefficient.

  • Material: Platinum (Pt100, Pt1000) most common (stable, repeatable, linear).

  • Characteristics: High accuracy, stability, but slower response than TC, more expensive.

• Thermistor:

  • Principle: Semiconductor. Resistance changes exponentially with T.

  • Types: NTC (Negative TC, $\downarrow R$ with $\uparrow T$, most common), PTC (Positive TC, $\uparrow R$ sharply at Curie point).

  • Characteristics: High sensitivity, non-linear, limited range (NTC: -50 to 150°C), self-heating error.

• Bimetallic Thermometer:

  • Principle: Differential thermal expansion. Two strips with different $\alpha$ bonded. Bends with T change.

  • Construction: Helical or cantilever. Mechanical linkage to pointer.

  • Applications: HVAC, thermostats. Simple, no power, but low accuracy, mechanical wear.

• Liquid-in-Glass (e.g., Mercury/Alcohol):

  • Principle: Thermal expansion of liquid in capillary.

  • Advantages: Simple, no power, direct reading.

  • Limitations: Fragile, limited range (mercury: -39 to 357°C), parallax error, slow response.

Non-Contact:

• Optical Pyrometer: Measures radiation intensity from hot body. Compares with calibrated lamp brightness. Used for >600°C. No contact needed.

Humidity:

• Sling Psychrometer: Measures dry-bulb (ambient T) and wet-bulb T (evaporative cooling). Psychrometric chart gives relative humidity.

D. Strain and Force Measurement

• Resistance Strain Gauge:

  • Principle: Strain $\epsilon$ changes resistance $R$: $$\displaystyle \frac{\Delta R}{R} = GF \times \epsilon $$.

  • Gauge Factor (GF): $$\displaystyle GF = \frac{\Delta R / R}{\epsilon} $$. For metallic gauges, $GF \approx 2$.

  • Bonding: Gauge glued to specimen with adhesive (mica paper backing common). Transfer strain.

• Wheatstone Bridge:

  • Quarter Bridge: One active gauge. Output $$\displaystyle V_o \approx \frac{V_{ex}}{4} \frac{\Delta R}{R} $$.

  • Half Bridge: Two active gauges (tension/compression). Output doubles.

  • Full Bridge: Four active gauges. Max output, temperature compensation.

  • Output: $$\displaystyle V_o = V_{ex} \left( \frac{R_3}{R_3+R_4} - \frac{R_2}{R_1+R_2} \right) $$. Balanced when $$\displaystyle R_1/R_2 = R_3/R_4 $$.

• Load Cell: Force transducer using strain gauges in bridge. Types: compression, tension, shear.

  • Strain Gauge Load Cell: Multiple gauges on diaphragm/beam. Output mV/V proportional to load.

  • Applications: Weighing scales, industrial process control.

[!CALCULATION] Gauge Factor (JUN 2023/2022):

Given: $$\displaystyle R = 150 \Omega $$, $$\displaystyle A = 0.5 \times 10^{-4} \text{ m}^2 $$, $$\displaystyle E = 200 \text{ GN/m}^2 $$, $$\displaystyle F = 60 \text{ kN} $$, $$\displaystyle \Delta R = 1.5 \Omega $$.

Stress $$\displaystyle \sigma = F/A = 60,000 / (0.5 \times 10^{-4}) = 1.2 \times 10^9 \text{ N/m}^2 = 1200 \text{ MN/m}^2 $$.

Strain $$\displaystyle \epsilon = \sigma / E = (1.2 \times 10^9) / (200 \times 10^9) = 0.006 $$.

$$\displaystyle GF = \frac{\Delta R / R}{\epsilon} = \frac{1.5 / 150}{0.006} = \frac{0.01}{0.006} \approx \boxed{1.67} $$.

E. Velocity and Flow Measurement

• Velocity:

  • Stroboscope: Produces high-frequency light pulses. If flash rate matches object frequency, object appears stationary. Used for rotating/reciprocating machinery speed measurement.

  • Electromagnetic Sensor: Based on Faraday's Law. Conductor (fluid or rotating toothed wheel) moving in magnetic field $B$ induces $$\displaystyle e = B l v $$. Measures linear or rotational velocity.

• Flow:

  • Venturimeter: Converging section, throat, diverging section. Pressure drop $$\displaystyle \Delta P \propto v^2 $$ (Bernoulli). Low permanent loss, used for high flow rates.

  • Orifice Meter: Thin plate with hole. Simpler, cheaper than venturi, but higher permanent pressure loss. $$\displaystyle \Delta P \propto v^2 $$. Discharge coefficient $$\displaystyle C_d $$ needed.

F. Piezoelectric Transducers (General)

• Principle: Piezoelectric Effect. Certain crystals (quartz) & ceramics (PZT) generate charge $Q$ when stressed ($$\displaystyle Q = d \cdot F $$, where $d$ = charge sensitivity) or deform when voltage applied.

• Materials: Quartz (stable, low $d$), Barium Titanate, PZT (high $d$, but temperature sensitive).

• Applications: Pressure, force, acceleration (seismic mass), ultrasound (both transmit/receive).

• Calibration: For accelerometers, use back-to-back comparison with reference accelerometer on shaker table, or laser interferometry.

G. Photo electric Transducers

• Principle: Photoelectric effect. Light incident on cathode emits electrons.

• Types: Photocell (vacuum), phototube (gas), photomultiplier (high gain).

• Applications: Position detection (interrupt beam), speed measurement (chopper), light meters, counters.

III. DYNAMIC ANALYSIS AND SIGNAL PROCESSING

A. Signal Types in Dynamic Analysis

• Periodic Inputs: Repeats after period $T$. Harmonic signal: $$\displaystyle x(t) = A \sin(\omega t + \phi) $$. Single frequency $\omega$.

• Non-Harmonic Periodic: Complex periodic waveforms (square, triangular). Composed of harmonics (Fourier series).

• Random Signals: No deterministic pattern (e.g., thermal noise, road vibration). Characterized by statistical properties (mean, RMS, power spectral density). Instruments must have sufficient bandwidth.

• Impulse Function (Dirac Delta): $\delta(t)$: infinite amplitude, zero width, unit area. Impulse Response $h(t)$: Output when $\delta(t)$ is input. Contains all system dynamics.

B. Fourier Transform and Frequency Domain Analysis

• Definition: $$\displaystyle X(f) = \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt $$. Transforms time-domain $x(t)$ to frequency-domain $X(f)$.

• Significance: Decomposes any signal into its constituent frequencies and their amplitudes/phases. Reveals frequency content.

• Linear Property: If $$\displaystyle x_1(t) \leftrightarrow X_1(f) $$ and $$\displaystyle x_2(t) \leftrightarrow X_2(f) $$, then $$\displaystyle a x_1(t) + b x_2(t) \leftrightarrow a X_1(f) + b X_2(f) $$.

  • Proof: Direct from integral definition: $$\displaystyle \mathcal{F}\{a x_1(t) + b x_2(t)\} = \int [a x_1(t) + b x_2(t)] e^{-j2\pi ft} dt = a \int x_1(t) e^{-j2\pi ft} dt + b \int x_2(t) e^{-j2\pi ft} dt = a X_1(f) + b X_2(f) $$.

C. Mathematical Modeling of Systems

• Electrical Systems:

  • Resistor (R): $$\displaystyle v = iR $$ (Ohm's Law).

  • Capacitor (C): $$\displaystyle i = C \frac{dv}{dt} $$ or $$\displaystyle v = \frac{1}{C} \int i dt $$.

  • Inductor (L): $$\displaystyle v = L \frac{di}{dt} $$.

  • RLC Circuit: $$\displaystyle L\frac{di}{dt} + Ri + \frac{1}{C}\int i dt = v_{in}(t) $$. Differential equation.

• Mechanical Systems:

  • Translational: Mass ($M$), Damper ($B$, force $$\displaystyle = Bv $$), Spring ($K$, force $$\displaystyle = Kx $$). Newton's 2nd Law: $$\displaystyle M\ddot{x} + B\dot{x} + Kx = F(t) $$.

  • Rotational: Moment of inertia ($J$), Viscous damper ($B$, torque $$\displaystyle = B\omega $$), Torsional spring ($K$, torque $$\displaystyle = K\theta $$). $$\displaystyle J\ddot{\theta} + B\dot{\theta} + K\theta = T(t) $$.

• Generalised System & Block Diagrams:

  • Block: Transfer function $$\displaystyle G(s) = \frac{Output(s)}{Input(s)} $$.

  • Summing Point: Adds/subtracts signals.

  • Take-off Point: Copies signal to multiple branches.

  • Example (Mechanical): Force $F(s)$ → [$$\displaystyle \frac{1}{Ms^2 + Bs + K} $$] → Displacement $X(s)$.

D. System Response and Order

• First-Order System: $$\displaystyle G(s) = \frac{K}{\tau s + 1} $$.

  • Examples: Thermometer (thermal mass), RC circuit.

  • Step Response: $$\displaystyle y(t) = K(1 - e^{-t/\tau}) $$. $\tau$ = time to reach 63.2% of final value.

• Second-Order System: $$\displaystyle G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} $$.

  • Examples: Spring-mass-damper, RLC circuit, servo motor.

  • Parameters: $$\displaystyle \omega_n $$ = natural frequency (rad/s), $\zeta$ = damping ratio.

  • Step Response:

    • $$\displaystyle \zeta < 1 $$ (Underdamped): Overshoot, oscillations. $$\displaystyle \%OS = e^{-\frac{\zeta\pi}{\sqrt{1-\zeta^2}}} \times 100\% $$.

    • $$\displaystyle \zeta = 1 $$ (Critically damped): Fastest without overshoot.

    • $$\displaystyle \zeta > 1 $$ (Overdamped): Slow, no overshoot.

  • Response to Harmonic Input: Magnification factor $$\displaystyle M = \frac{1}{\sqrt{(1-(\omega/\omega_n)^2)^2 + (2\zeta\omega/\omega_n)^2}} $$. Resonance near $$\displaystyle \omega_r \approx \omega_n $$ for low $\zeta$.

IV. CONTROL SYSTEMS

A. Basic Concepts and Classifications

• Control System: Arrangement to cause a quantity (controlled variable) to conform to a desired value (reference input).

• Open-Loop: No feedback. Output does not affect control action.

  • Example: Washing machine timer, toaster.

  • Advantages: Simple, cheap, stable.

  • Limitations: Inaccurate with disturbances, no correction for errors.

• Closed-Loop (Feedback): Output measured and fed back to compare with reference. Error signal drives actuator.

  • Example: Boiler water level, room temperature (thermostat), position control (servo).

  • Advantages: High accuracy, rejects disturbances, reduced sensitivity to parameter changes.

  • Limitations: More complex, can be unstable (oscillations), higher cost.

[!EXAMPLE] Boiler Water Level Control (JUN 2023/2022): Reference level → Comparator → Controller (e.g., PID) → Valve actuator → Steam flow. Feedback: Level sensor (float, differential pressure) measures actual level and feeds back to comparator.

B. Block Diagrams and System Representation

• Basic Components:

  • Block: System/component with transfer function.

  • Summing Point: $\oplus$ or $\ominus$ (adds/subtracts inputs).

  • Take-off Point: Branches signal.

• Closed-Loop Interconnection:

  
  R(s) → [Σ] → (+) → [G(s)] → [H(s)] → C(s)
  
          ↑        ↓
  
          [H(s)] ←──┘
  
  

  Error $$\displaystyle E(s) = R(s) - H(s)C(s) $$. Output $$\displaystyle C(s) = \frac{G(s)}{1+G(s)H(s)} R(s) $$. $1+G(s)H(s)$ = Characteristic equation.

• Construction: Identify input (R), output (C), feedback path (H), forward path (G). Draw summing point for error.

C. Dynamic Response in Control Systems

• Transient Response: Response to a change (usually step input) before reaching steady-state.

  • Rise Time ($$\displaystyle t_r $$): Time to go from 10% to 90% (or 0% to 100%) of final value.

  • Settling Time ($$\displaystyle t_s $$): Time to enter and remain within ±2% (or 5%) band.

  • Overshoot ($$\displaystyle M_p $$): Max peak - steady-state value. $$\displaystyle \%OS = \frac{\text{Peak} - \text{Steady-state}}{\text{Steady-state}} \times 100\% $$.

  • Steady-State Error ($$\displaystyle e_{ss} $$): Difference between desired and actual output as $t \to \infty$. Depends on system type (number of integrators) and input type (step, ramp, parabola). Briefly mentioned in outline.

D. Applications and Case Studies

• Automatic Boiler Water Level Control:

  • Problem: Maintain water level in steam boiler despite steam demand changes.

  • Block Diagram: Reference level → Comparator → Controller (P or PI) → Feedwater valve. Feedback: Level transmitter (differential pressure across water column).

  • Working: If level drops (steam draw-off), error signal opens valve to increase feedwater.

• Temperature Control System:

  • Elements: Thermostat/RTD (sensor), Controller (ON/OFF or PID), Heater/cooler (actuator).

  • ON/OFF: Simple, causes oscillations. PID: Provides smooth, accurate control.

• Position Control System (Servomechanism):

  • Goal: Control position/angle of load (e.g., radar antenna, CNC machine).

  • Elements: Command input (potentiometer), Error detector, Amplifier, Servo motor, Feedback tachometer/encoder.

  • Block Diagram: Command → [Σ] → Amp → Motor → Load. Feedback from Load (position &/or velocity) → Σ.

[!TIP] JUN 2025 & NOV 2023 ask for block diagrams. Always label signals (R, E, C) and blocks (G, H). For boiler control, explicitly show level transmitter in feedback path.