INSTRUMENTATION & CONTROL (ME-402) - Unit 4 Short Notes

1.0 FOUNDATIONS OF MEASUREMENT SCIENCE

1.1 Elements of a Generalized Measurement System

A measurement system typically consists of four functional stages:

1. Primary Sensing Element / Transducer: Converts the measured physical quantity (measurand) into a usable electrical/mechanical signal (e.g., thermocouple, strain gauge).

2. Signal Conditioning / Transmitter: Modifies the primary signal into a suitable form (amplification, filtering, linearization, conversion to standard signal like 4-20 mA).

3. Signal Processing / Data Handling: Further manipulates the signal (computation, correction, digitization, communication).

4. Display / Recording / Output: Presents the measurement result to the user (digital display, chart recorder, computer interface).

[!TIP] Exam Focus: Be prepared to map a specific instrument (e.g., pressure gauge, temperature controller) to these four elements.

1.2 Measurement Errors & Uncertainties

Definitions:

• Error: The difference between the measured value and the true value.

  • Systematic Error: Consistent, repeatable deviation (calibration error, zero error, environmental effect). Correctable.

  • Random Error: Unpredictable fluctuations (noise, observer variation). Reducible by averaging.

  • Gross Error: Mistake (reading error, recording error). Eliminated by care.

• Uncertainty: A quantitative estimate of the doubt about the measurement result. Expressed as:

  • Absolute Uncertainty (±Δx): Same units as measurement.

  • Relative/Percentage Uncertainty: $$\displaystyle \frac{\Delta x}{x} \times 100\% $$.

Sources of Errors:

• Instrumental: Imperfect calibration, hysteresis, wear.

• Environmental: Temperature, pressure, humidity, vibration changes.

• Observational: Parallax, estimation error.

• Methodological: Inherent limitations of the measurement method.

Key Calculation: Accuracy Specification

• % of Full Scale Deflection (FSD): $$\displaystyle \text{Error} = \pm (\% \text{ of FSD}) $$. Absolute error is constant across range.

  • Example (DEC 2024): Gauge range = 1000 kN/m², accuracy ±1% FSD → Max error = ±10 kN/m². At 100 kN/m², reading = $100 \pm 10$ kN/m².

• % of True Value (Reading): $$\displaystyle \text{Error} = \pm (\% \text{ of reading}) $$. Absolute error varies with reading.

  • Example (DEC 2024): Same gauge, ±1% of true value → At 100 kN/m², max error = ±1 kN/m². Reading = $100 \pm 1$ kN/m².

Mitigation: Regular calibration, averaging readings, environmental control, proper training.

[!TIP] Common Pitfall: Confusing %FSD (worse at low readings) with %True Value (constant relative error). Always check how instrument accuracy is specified.

1.3 Calibration & Standards

• Need: To establish traceability to national/international standards, ensure accuracy, and determine system equation (output vs. input).

• Standards Hierarchy:

  • Primary Standard: Highest accuracy, maintained by national labs (e.g., NPL India).

  • Secondary Standard: Calibrated against primary, used in industry labs.

  • Working Standard: Used routinely for calibrating instruments.

• Procedure:

  1. Apply known, precise inputs (from a standard) across the instrument's range.

  2. Record corresponding outputs.

  3. Plot calibration curve (output vs. input).

  4. Determine system equation (often linear: $$\displaystyle y = mx + c $$) and correction factors.

• Dead Weight Tester (Pressure - JUN 2025):

  • Principle: $$\displaystyle P = \frac{F}{A} = \frac{mg}{A} $$. Known masses (m) on a precision piston (area A) generate exact pressure.

  • Construction: Piston-cylinder assembly (low friction, high finish), fluid medium, weight set.

  • Role: Primary standard for pressure calibration.

1.4 Instrument Classification & Selection

• By Function: Indicating (dial), Recording (chart), Integrating (totalizer).

• By Output: Analog (continuous), Digital (discrete), Smart (digital with communication).

• Key Specifications for Selection (from Datasheet):

  • Range & Span: Operating limits (Span = Max - Min).

  • Accuracy: % of reading or %FSD.

  • Resolution: Smallest detectable change.

  • Precision/Repeatability: Closeness of repeated readings.

  • Linearity: Max deviation from straight line.

  • Hysteresis: Difference between up-scale and down-scale readings for same input.

  • Deadband: Range of input change with no output change.

  • Drift: Change in output over time for constant input (zero/sensitivity drift).

2.0 STATIC PERFORMANCE CHARACTERISTICS (No time variation)

Definitions & Significance:

[!TIP] Exam Link: These characteristics directly contribute to total measurement uncertainty. A question may ask: "How do hysteresis and nonlinearity impact accuracy?"

3.0 DYNAMIC PERFORMANCE CHARACTERISTICS & SIGNAL ANALYSIS

3.1 Dynamic Response Fundamentals

Needed when input varies with time. Instruments have inertia/lag, so output doesn't instantly track input. Characterized by time domain (response vs. time) and frequency domain (response vs. frequency).

3.2 System Modeling & Differential Equations

• Lumped Parameter Models: Assume physical properties concentrated at a point.

• First-Order System (e.g., thermometer, RC circuit):

$$ \tau \frac{dy}{dt} + y = Kx $$

Where:

*   $y$ = output, $x$ = input

*   $K$ = static sensitivity (steady-state gain)

*   $\tau$ = **time constant** (time to reach 63.2% of final value for step input).

• Second-Order System (e.g., spring-mass-damper, RLC circuit):

$$ \frac{d^2y}{dt^2} + 2\zeta\omega_n\frac{dy}{dt} + \omega_n^2 y = K\omega_n^2 x $$

Where:

*   $$\displaystyle \omega_n $$ = **natural frequency** (rad/s)

*   $\zeta$ = **damping ratio** (dimensionless)

*   Response type: Underdamped ($$\displaystyle \zeta<1 $$), Critically damped ($$\displaystyle \zeta=1 $$), Overdamped ($$\displaystyle \zeta>1 $$).

3.3 Time Domain Response

• Standard Test Inputs: Step (sudden change), Ramp (linear increase), Impulse (very short pulse), Parabolic.

• First-Order to Step Input ($$\displaystyle x(t)=X_0u(t) $$):

$$ y(t) = KX_0 (1 - e^{-t/\tau}) $$

*   Reaches **98%** of final value at $t \approx 4\tau$, **99.3%** at $$\displaystyle t=5\tau $$.

• Second-Order to Step Input ($$\displaystyle \zeta<1 $$):

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

  • Peak Time ($$\displaystyle t_p $$): Time to first maximum overshoot.

  • Maximum Overshoot ($$\displaystyle M_p $$): $$\displaystyle M_p = e^{\frac{-\zeta\pi}{\sqrt{1-\zeta^2}}} \times 100\% $$.

  • Settling Time ($$\displaystyle t_s $$): Time to stay within a band (e.g., ±2% → $$\displaystyle t_s \approx \frac{4}{\zeta\omega_n} $$).

  • Damped Natural Frequency ($$\displaystyle \omega_d $$): $$\displaystyle \omega_d = \omega_n\sqrt{1-\zeta^2} $$.

3.4 Frequency Domain Analysis

• Fourier Transform (FT): Decomposes a time-domain signal $x(t)$ into its frequency components $X(f)$.

$$ X(f) = \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt $$

*   **Significance (JUN 2025, NOV 2023):** Analyzes how a system responds to sinusoidal inputs of different frequencies. Essential for understanding **bandwidth** and **filtering**.

*   **Property (NOV 2023):** **Linearity:** FT of a sum is sum of FTs: $$\displaystyle \mathcal{F}\{a x_1(t) + b x_2(t)\} = a X_1(f) + b X_2(f) $$.

• Frequency Response: System's steady-state output to sinusoidal input $$\displaystyle x(t)=X_0\sin(\omega t) $$. Output: $$\displaystyle y(t)=Y_0\sin(\omega t + \phi) $$.

  • Magnitude Ratio: $$\displaystyle M(\omega) = \frac{Y_0}{X_0} $$.

  • Phase Angle: $\phi(\omega)$.

  • Plotted as Bode plots (log-log magnitude, log-linear phase).

• Bandwidth: Frequency range where $M(\omega)$ is within -3 dB (or 70.7%) of low-frequency gain.

3.5 Input Signal Classification

• Periodic Signals: Repeat after period $T$ (e.g., sine, square). Can be harmonic (single frequency sine) or non-harmonic periodic (complex waveform, requires Fourier series).

• Aperiodic/Random Signals: Non-repeating, described statistically (mean, RMS, probability density). Response analysis uses statistical methods (JUN 2024).

3.6 Dynamic Performance Specs

• Fidelity: Ability to reproduce input waveform shape accurately. High for systems with wide bandwidth.

• Speed of Response: How quickly system reaches steady-state (e.g., time constant $\tau$, settling time $$\displaystyle t_s $$).

• Measuring Lag / Time Lag (NOV 2023): Delay between input change and observable output change. Inversely related to speed.

• Bandwidth: Higher bandwidth → faster response, better fidelity for rapid inputs.

4.0 SENSORS & TRANSDUCERS BY MEASURAND

4.1 Displacement, Position & Angular Measurements

• Potentiometer (JUN 2025):

  • Principle: Variable resistance voltage divider. Displacement moves wiper on resistive element.

  • Construction: Wire-wound (resolution limited by wire turns), carbon film (continuous).

  • Conversion: Linear/rotary displacement → Voltage/current.

  • Limitations: Loading effect (needs high impedance buffer), mechanical wear, limited frequency response, finite resolution.

  • DiagramSEARCH: "potentiometer construction working principle"

• LVDT & RVDT (JUN 2025 - 14m):

  • LVDT (Linear):

    • Construction: One primary winding, two identical secondary windings (series opposing), movable ferromagnetic core.

    • Principle: AC excitation on primary. Core position induces voltages in secondaries. Null position (core centered) → equal & opposite secondary voltages → net output zero. Displacement from null → unbalanced output. Amplitude proportional to displacement, phase indicates direction.

    • Output: AC voltage (needs demodulation for DC output). Infinite resolution (theoretical), no electrical contact.

    • Advantages: Frictionless, long life, high reliability, linear over small range.

    • DiagramSEARCH: "LVDT construction and working principle diagram"

  • RVDT (Rotary): Same principle for angular displacement.

4.2 Pressure Measurements

• Fluid Column Manometers (DEC 2024, JUN 2023, NOV 2023):

  • Principle: Hydrostatic balance. $$\displaystyle P = \rho g h $$ (for simple U-tube). Differential pressure $$\displaystyle \Delta P = (\rho_m - \rho_f)gh $$.

  • Types:

    • U-tube: Simple, measures gauge/differential pressure.

    • Inclined: Amplifies reading (longer scale), for low pressures.

    • Well-type: One leg large well, for high pressures.

    • Differential: Two pressures applied to two legs.

  • DiagramSEARCH: "U-tube manometer inclined well-type differential manometer diagrams"

• Bourdon Tube (C-type, Helical, Spiral):

  • Principle: Flattened tube tends to straighten under pressure. Motion converted to pointer via gear.

  • Construction: Curved tube, closed end (pressure port), free end (mechanical linkage).

  • Applications: Wide range pressure gauges.

• Bellows Gauge (JUN 2023):

  • Principle: Pressure causes elastic bellows to expand. Expansion moves pointer.

  • Construction: Corrugated bellows, mechanical linkage.

  • Applications: Medium pressure, low pressure, differential pressure.

• Diaphragm Gauge (NOV 2023):

  • Principle: Pressure deflects diaphragm (flat or corrugated). Deflection measured capacitively, piezoresistively, or mechanically.

  • Applications: Low pressure, differential pressure, vacuum.

• Piezoelectric Transducer (JUN 2022, JUN 2023):

  • Principle: Certain crystals (Quartz, PZT) generate surface charge when strained (direct piezoelectric effect). $$\displaystyle Q = d \cdot F $$, $$\displaystyle V = g \cdot t \cdot P $$.

  • Characteristics: High frequency response, AC output only (charge leaks away), very high output impedance (needs charge amplifier). For dynamic pressure only.

  • DiagramSEARCH: "piezoelectric pressure transducer diagram"

• Dead Weight Tester (JUN 2025): (See 1.3)

4.3 Temperature Measurements

• Thermometric Expansion:

  • Glass Thermometer (JUN 2023): Liquid (Hg, alcohol) in calibrated glass capillary. Simple, direct reading. Limitations: Fragile, limited range (Hg: -38 to 350°C), no remote output.

  • Bimetallic Thermometer (DEC 2024, JUN 2023, JUN 2024):

    • Principle: Two metals with different $\alpha$ bonded. Temperature change causes bending.

    • Construction: Strip (helix/spiral for amplification). Bending moves pointer.

    • Curvature Formula (NOV 2023): $$\displaystyle \frac{1}{r} = \frac{6(\alpha_1 - \alpha_2)\Delta T}{t(1+m)^2} $$ (approx), where $t$=total thickness, $m$=modulus ratio.

    • Applications: Industrial thermostats, dial thermometers.

• Electrical Resistance Thermometry:

  • RTD (JUN 2024, JUN 2022): $$\displaystyle R_T = R_0 [1 + \alpha T + \beta T^2 + ...] $$. PT100 ($$\displaystyle R_0=100\Omega $$ at 0°C). Materials: Platinum (best stability/linearity), Nickel, Copper.

    • Advantages: High accuracy, stability, wide range (-200 to 850°C).

    • Limitations: Self-heating, requires excitation current, lead wire resistance error (use 3/4-wire compensation).

  • Thermistor (DEC 2024 x2, JUN 2023, JUN 2022): $$\displaystyle R_T = R_0 e^{\beta (\frac{1}{T} - \frac{1}{T_0})} $$ (NTC). High sensitivity (large $$\displaystyle \frac{dR}{dT} $$), non-linear, limited range (-50 to 300°C). PTC used as inrush limiter/sensor.

    • Comparison (JUN 2022): RTD = linear, stable, wide range, expensive. Thermistor = non-linear, high sensitivity, cheap, limited range.

• Thermoelectric Thermometry (Thermocouple - JUN 2023):

  • Principle: Seebeck Effect. Two dissimilar metals joined at two junctions generate EMF proportional to temperature difference.

  • Key Points: Requires reference (cold) junction at known temperature (compensation needed). Types: J (Fe-CuNi), K (NiCr-NiAl - most common), T (Cu-CuNi), E (NiCr-CuNi), S/R/B (Pt-Rh/Pt - high temp).

  • Advantages: Wide range (-200 to 2300°C), rugged, no excitation.

  • Limitations: Low output (mV), requires cold-junction compensation, susceptible to noise.

• Radiation Pyrometry (JUN 2022):

  • Optical Pyrometer: Compares brightness of hot object to calibrated filament. Disappearing filament type. For high temperatures (>700°C). No contact.

  • Infrared Pyrometer: Measures IR radiation energy. Wider range, faster.

4.4 Strain, Load & Force Measurements

• Resistance Strain Gauge (Core Topic):

  • Principle: Piezoresistive Effect. Strain $\epsilon$ changes resistance $R$.

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

  • Derivation (JUN 2023, JUN 2022): $$\displaystyle R = \frac{\rho l}{A} $$. $$\displaystyle \frac{\Delta R}{R} = \frac{\Delta \rho}{\rho} + \frac{\Delta l}{l} - \frac{\Delta A}{A} $$. For Poisson's ratio $\nu$, $$\displaystyle \frac{\Delta A}{A} = -2\nu \epsilon $$. $$\displaystyle \frac{\Delta \rho}{\rho} \propto \epsilon $$. Hence $$\displaystyle GF = 1 + 2\nu + \frac{\frac{\Delta \rho}{\rho}}{\epsilon} \approx 2 $$ for metals.

  • Construction: Metallic foil (most common), wire-wound, semiconductor (high GF, high temp. sensitivity).

  • Mounting (DEC 2024): Surface prep (smooth, clean), adhesive bonding (epoxy), paper backing for handling, orientation along principal strain axis.

  • Wheatstone Bridge (JUN 2024): Converts small $\Delta R$ to measurable voltage $$\displaystyle V_o $$.

    • Quarter Bridge (1 active gauge): $$\displaystyle V_o \approx \frac{V_{ex}}{4} \cdot \frac{\Delta R}{R} $$

    • Half Bridge (2 active, opposite arms): $$\displaystyle V_o \approx \frac{V_{ex}}{2} \cdot \frac{\Delta R}{R} $$

    • Full Bridge (4 active): $$\displaystyle V_o \approx V_{ex} \cdot \frac{\Delta R}{R} $$ (max sensitivity, temperature compensation if arranged properly).

    • Temperature Compensation: Use dummy gauge (identical, unstrained) in adjacent arm.

  • DiagramSEARCH: "Wheatstone bridge strain gauge quarter half full bridge configurations"

• Load Cell (NOV 2023): Force sensor using strain gauges in a bridge. Types: Compression/tension, beam, canister. Applications: weighing scales, industrial process.

• Piezoelectric Force Sensor: For dynamic force only (like pressure).

4.5 Velocity & Rotational Speed Measurements

• Stroboscope (JUN 2025, NOV 2023):

  • Principle: Freezes/reverses apparent motion when flash rate $$\displaystyle f_s $$ equals object speed $n$ (rpm) or sub-multiple.

$$ n = 60 \times f_s \quad (\text{if } f_s = n) \quad \text{or} \quad n = \frac{60 \times f_s}{k} \quad (\text{if } f_s = k \cdot n) $$

*   **Operation:** Adjust flash rate until object appears stationary/slow.

*   **Applications:** Measure rotational speed, observe vibrating/rotating machinery.

• Electromagnetic Tachometers:

  • DC Tachogenerator: Voltage $V \propto \omega$. Requires contact/rotating contact.

  • AC Tachogenerator (Induction): Output frequency $f \propto \omega$. No brushes.

  • Proximity/Eddy Current: Non-contact. Measures speed via passing ferromagnetic target.

• Mechanical Tachometers (JUN 2022): Centrifugal (flyball), vibrating reed. Disadvantages: Wear, friction, low accuracy, contact needed.

5.0 CONTROL SYSTEMS FUNDAMENTALS

5.1 Basic Definitions

• Control System: Interconnected components to achieve desired response.

• Plant: The part being controlled (e.g., boiler, motor).

• Input (Command/Reference): Desired value ($r(t)$).

• Output (Controlled Variable): Actual value ($c(t)$).

• Disturbance: Unwanted input affecting output ($d(t)$).

• Manipulated Variable: Input to plant adjusted by controller ($u(t)$).

• Controller: Compares input & output, generates control action.

5.2 Open-Loop vs. Closed-Loop (Feedback)

• Open-Loop (JUN 2025, DEC 2024, JUN 2023, JUN 2022, NOV 2023):

  • Block Diagram: Input → Controller → Plant → Output. No feedback path.

  • Operation: Output does not influence control action.

  • Examples: Washing machine timer, traffic light (fixed timing), toaster.

  • Advantages: Simple, stable, low cost.

  • Limitations: No correction for disturbances, sensitive to parameter changes (calibration critical).

• Closed-Loop (Feedback - JUN 2025, NOV 2023):

  • Block Diagram (Standard):

    
    r(t) → [Σ] → Controller → Plant → c(t)
    
            ↑           ↓
    
            ←──── Sensor ←──
    
    

    Error $$\displaystyle e(t) = r(t) - c(t) $$.

  • Operation: Output measured, compared to input, error drives controller.

  • Examples (DEC 2024, NOV 2023): Boiler water level control (float/level sensor → controller → feed valve). Temperature control (thermostat), position control (servo).

  • Advantages: High accuracy, rejects disturbances, reduces sensitivity to parameter variations.

  • Limitations: More complex, potential instability (oscillations), higher cost, need sensor maintenance.

  [!TIP] Exam Focus: Be ready to draw block diagrams for boiler water level control (DEC 2024, NOV 2023) and temperature control.

5.3 Block Diagram Representation (JUN 2025, NOV 2023)

• Basic Elements: Arrow (signal), Summing point (Σ), Take-off point, Block (transfer function $G(s)$).

• Interconnections:

  • Series: $$\displaystyle G_1 G_2 $$

  • Parallel: $$\displaystyle G_1 + G_2 $$

  • Feedback: Forward path $G(s)$, Feedback path $H(s)$. Closed-loop TF: $$\displaystyle \frac{C(s)}{R(s)} = \frac{G(s)}{1 + G(s)H(s)} $$ (negative feedback).

• Construction: Identify input, output, sensor, controller, plant, disturbance path. Draw blocks and connect with arrows/summing points.

5.4 Control System Classification

• By Control Action:

  • On-Off (Two-position): Controller output is either max or min (e.g., thermostat). Simple, causes hunting.

  • Proportional (P): $$\displaystyle u(t) = K_P e(t) $$. Steady-state error usually exists.

  • Integral (I): $$\displaystyle u(t) = K_I \int e(t) dt $$. Eliminates steady-state error.

  • Derivative (D): $$\displaystyle u(t) = K_D \frac{de(t)}{dt} $$. Anticipates error, improves stability.

  • PID: Combination $$\displaystyle u(t) = K_P e + K_I \int e dt + K_D \frac{de}{dt} $$.

• By I/O: SISO (Single-Input Single-Output), MIMO (Multi-Input Multi-Output).

• By Nature: Linear/Non-linear, Time-invariant/Time-varying, Continuous/Discrete (sampled).

5.5 Specific Control System Examples

• Servomechanism (JUN 2022): Feedback system where output is mechanical position/velocity. Uses sensors (pot, encoder), amplifier, motor. Example: antenna positioning, CNC machine axis.

• Temperature Control System (JUN 2022): Heater (plant), thermocouple/RTD (sensor), controller (PID), setpoint. May have disturbance (ambient change, load change).

• Position Control System (JUN 2022): Similar to servomechanism. Motor drives load, position sensor (LVDT, encoder) provides feedback.

• Boiler Water Level Control (DEC 2024, NOV 2023):

  • Objective: Maintain constant water level in boiler drum.

  • Elements: Level sensor (float, differential pressure), controller (PID), manipulated variable = feedwater valve opening.

  • Disturbances: Steam demand changes, feedwater pressure changes.

  • DiagramSEARCH: "boiler drum water level control system block diagram"

6.0 MATHEMATICAL MODELING OF SYSTEMS

6.1 General Approach

Apply physical laws (Newton's, Kirchhoff's) to system boundaries. Define inputs, outputs, state variables. Assume linearity & time-invariance for transfer function.

6.2 Modeling of Electrical Systems (JUN 2024)

• Component Equations:

  • Resistor: $$\displaystyle v_R = i R $$

  • Capacitor: $$\displaystyle i_C = C \frac{dv_C}{dt} $$ or $$\displaystyle v_C = \frac{1}{C} \int i_C dt $$

  • Inductor: $$\displaystyle v_L = L \frac{di_L}{dt} $$ or $$\displaystyle i_L = \frac{1}{L} \int v_L dt $$

• Network Analysis: Apply KVL (sum voltages = 0) and KCL (sum currents = 0) to write differential equations.

• Transfer Function (TF): $$\displaystyle G(s) = \frac{L\{output(t)\}}{L\{input(t)\}} $$ assuming zero initial conditions. TF is ratio of polynomials in Laplace variable $s$.

• Analogies:

  • Force-Voltage (Mobility) Analogy: Force ↔ Voltage, Velocity ↔ Current, Mass ↔ Capacitor, Spring ↔ Inductor, Damper ↔ Resistor.

  • Force-Current (Impedance) Analogy: Force ↔ Current, Velocity ↔ Voltage, Mass ↔ Inductor, Spring ↔ Capacitor, Damper ↔ Resistor.

6.3 Modeling of Mechanical Systems (JUN 2024)

• Translational:

  • Mass ($m$): $$\displaystyle F = m \frac{d^2x}{dt^2} $$

  • Spring ($k$): $$\displaystyle F = k x $$

  • Damper ($b$): $$\displaystyle F = b \frac{dx}{dt} $$

• Rotational:

  • Inertia ($J$): $$\displaystyle T = J \frac{d^2\theta}{dt^2} $$

  • Torsional spring ($$\displaystyle k_t $$): $$\displaystyle T = k_t \theta $$

  • Rotational damper ($$\displaystyle b_r $$): $$\displaystyle T = b_r \frac{d\theta}{dt} $$

• Write equations using Newton's 2nd law (sum forces/torques = mass/inertia × acceleration).

6.4 Transfer Function Examples

• First-Order (RC circuit): $$\displaystyle V_o(s)/V_i(s) = \frac{1}{RCs + 1} $$, $$\displaystyle \tau = RC $$.

• Second-Order (Spring-mass-damper): $$\displaystyle X(s)/F(s) = \frac{1}{ms^2 + bs + k} $$. Standard form: $$\displaystyle \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} $$.

7.0 SPECIALIZED & SHORT NOTE TOPICS

• Flow Measurement:

  • Venturimeter (DEC 2024, NOV 2023): Converging section, throat, diverging section. Based on Bernoulli & continuity. $$\displaystyle Q = C_d A_t \sqrt{\frac{2(P_1 - P_2)}{\rho(1 - (A_t/A_1)^2)}} $$. Low permanent pressure loss, expensive.

  • Orifice Meter (DEC 2024, NOV 2023): Simple orifice plate. Higher pressure loss, prone to wear, cheaper. Same principle as venturi.

  • DiagramSEARCH: "venturimeter and orifice meter flow measurement diagram"

• Bimetallic Thermometer (DEC 2024, JUN 2023): (See 4.3.1)

• Stroboscope (JUN 2025, NOV 2023): (See 4.5.1)

• Load Cell (NOV 2023): (See 4.4.2)

• Manometer (NOV 2023): (See 4.2.1)

• Ionisation Transducer (NOV 2023): Measures pressure in vacuum systems. Gas ionized by current, ion current ∝ pressure.

• Photo-electric Transducer (JUN 2022): Light intensity → electrical signal (photocell, photodiode, phototransistor).

• Sling Psychrometer (JUN 2022): Measures dry & wet bulb temperatures. Psychrometric chart gives relative humidity from $$\displaystyle T_{dry} $$, $$\displaystyle T_{wet} $$, and atmospheric pressure.

• Accelerometer Calibration (JUN 2022): Shake on vibration table of known frequency/amplitude, compare output.

• Impulse Function (JUN 2023): $\delta(t)$: Infinite amplitude, zero width, area=1. $$\displaystyle \int_{-\infty}^{\infty} \delta(t) dt = 1 $$. $$\displaystyle f(t)\delta(t-a) = f(a)\delta(t-a) $$. Used for system identification (output = TF * input).

[!TIP] Final Exam Strategy: For "short note" questions, structure as: 1. Principle/Definition, 2. Construction/Key Components, 3. Working/Operation, 4. Applications, 5. Advantages/Limitations. Use diagrams where possible. For calculation problems (error, GF, bimetallic), show formula, substitute, box answer.