INSTRUMENTATION & CONTROL (ME-402) - Unit 2 Short Notes

1.0 FUNDAMENTALS OF MEASUREMENT SYSTEMS

1.1 Elements of a Generalized Measurement System

A measurement system typically consists of four functional elements:

1. Primary Sensor/Transducer: Converts the measured physical quantity (input) into a corresponding output signal (often electrical or mechanical).

2. Signal Conditioner: Modifies the sensor output into a suitable form (e.g., amplification, filtering, linearization).

3. Signal Processor: Further processes the conditioned signal (e.g., analog-to-digital conversion, computation, storage).

4. Display/Recorder: Presents the measurement result to the user (e.g., analog pointer, digital readout, chart recorder).

1.2 Classification of Instruments

• Based on Function:

  • Indicating: Provides immediate visual readout (e.g., pressure gauge, thermometer).

  • Recording: Plots the variation of the measured variable against time (e.g., chart recorder).

  • Integrating: Totals or integrates the measured variable over time (e.g., totalizing gas meter).

• Based on Operating Principle:

  • Mechanical: (e.g., Bourdon tube, bimetallic thermometer).

  • Electrical: (e.g., potentiometer, strain gauge).

  • Electronic: (e.g., digital multimeter, piezoelectric transducer with amplifier).

1.3 Role of Standards and Calibration

• Standards (International, National, Primary, Secondary) provide universally accepted reference values.

• Calibration is the process of comparing an instrument's output against a known standard under specified conditions.

• Purpose: To establish the relationship between the instrument's indication and the true value, thereby ensuring accuracy, traceability, and reliability of measurements. Regular calibration is essential for quality control and compliance.

1.4 Calibration Procedures and Standards

• Reference/Primary Standard: Highest accuracy, maintained by national labs (e.g., NIST). Used to calibrate transfer standards.

• Transfer Standard: High-accuracy instrument used to calibrate working standards in a laboratory.

• Working Standard: Used for routine calibration of instruments in the field or production line.

• Procedure: Typically involves applying known input values (from a standard) across the instrument's range and recording the output. A calibration curve (plot of true value vs. instrument output) is generated to determine correction factors.

2.0 MEASUREMENT ERRORS, UNCERTAINTIES & STATIC PERFORMANCE CHARACTERISTICS (High-Frequency)

2.1 Errors in Measurement

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

• Types:

  • Systematic Error: Consistent, reproducible deviation. Causes: Instrumental (miscalibration), Environmental (temperature drift), Observational (parallax). Correctable.

  • Random Error (Precision Error): Unpredictable fluctuations. Causes: Noise, inherent variability. Reducible by averaging.

  • Gross Error: Mistake (e.g., misreading, recording error). Should be eliminated.

2.2 Uncertainty in Measurement

• Definition: A quantitative estimate of the doubt about the measurement result. Expressed as a range ($X \pm U$).

• Absolute Uncertainty ($U$): Margin of error in the same units as the measurement (e.g., $\pm 0.02$ g).

• Relative Uncertainty ($$\displaystyle U_r $$): Fraction or percentage of the measurement. $$\displaystyle U_r = \frac{U}{X} \times 100\% $$.

• From Accuracy Specs:

  • % of Full Scale Deflection (FSD): $$\displaystyle U = \frac{\%}{100} \times \text{FSD} $$. Uncertainty is constant across range.

  • % of True Value (Reading): $$\displaystyle U = \frac{\%}{100} \times \text{Measured Value} $$. Uncertainty scales with reading.

• Propagation of Uncertainties: For a calculated result $$\displaystyle R = f(A, B, ...) $$, the combined uncertainty $$\displaystyle U_R $$ is found using partial derivatives (root-sum-square method for independent errors).

$$U_R = \sqrt{\left( \frac{\partial R}{\partial A} U_A \right)^2 + \left( \frac{\partial R}{\partial B} U_B \right)^2 + ...}$$

• Mitigation: Regular calibration, environmental control, proper technique, using instruments with higher precision, statistical analysis.

2.3 Static Performance Characteristics (Core Topic)

Defined for steady-state (constant) inputs.

• Accuracy: Closeness of agreement between measured value and true value. Global specification.

• Precision: Closeness of agreement between a series of measurements (repeatability). Does not imply accuracy.

• Repeatability: Degree of agreement among repeated measurements under identical conditions (same operator, instrument, short time).

• Reproducibility: Degree of agreement among measurements under changed conditions (different operator, time, lab).

• Sensitivity: Ratio of change in output signal to change in input signal. $$\displaystyle S = \frac{\Delta Output}{\Delta Input} $$. Slope of calibration curve.

• Scale Factor: Same as sensitivity, often used for digital instruments.

• Resolution: Smallest detectable change in input.

• Threshold: Minimum input that produces a detectable output change.

• Hysteresis: Difference in output for the same input depending on direction of approach (increasing vs. decreasing). Caused by friction, magnetic effects.

• Linearity: Maximum deviation of actual calibration curve from a specified straight line (e.g., least-squares fit).

• Drift: Slow change in output over time with constant input.

  • Zero Drift: Change in zero output.

  • Sensitivity Drift: Change in sensitivity/scale factor.

• Determination: Obtained experimentally via calibration over the full range. Specifications are listed in the instrument's datasheet.

• Instrument Selection: Based on required accuracy, range, environment, and cost. A high-precision instrument may be unnecessary for a rough measurement.

[!TIP]

Exam Focus: Be prepared to calculate uncertainty from %FSD vs. %Reading (a very common question). Distinguish clearly between Accuracy, Precision, Repeatability, and Reproducibility. Know how hysteresis and linearity are determined from a calibration curve.

3.0 DYNAMIC PERFORMANCE CHARACTERISTICS & SIGNAL ANALYSIS (High-Frequency)

3.1 Dynamic vs. Static

• Static: Response to a steady (DC) input. Characteristics from §2.3 apply.

• Dynamic: Response to a time-varying (AC) input. Characterized by how quickly and accurately the system follows changes.

3.2 Dynamic Characteristics (Time Domain)

• Speed of Response: General term for how fast the instrument responds.

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

• Settling Time ($$\displaystyle t_s $$): Time for response to stay within a specified band (e.g., ±2%) of the final value.

• Measuring Lag (Time Lag): Delay in response due to system inertia. For a first-order system, it's related to the time constant $\tau$.

• Dead Time ($\theta$): Period after a step change where output shows no response.

• Fidelity: Degree to which the output waveform accurately reproduces the input waveform shape.

• Overshoot & Damping Ratio ($\zeta$): For underdamped second-order systems. Overshoot is the maximum peak above the final value. $\zeta$ controls the amount of overshoot and oscillation.

3.3 System Order & Standard Responses

• First-Order System: Governed by $$\displaystyle \tau \frac{dy}{dt} + y = Kx(t) $$.

  • Time Constant ($\tau$): Time to reach 63.2% of final value for a step input. Measure of speed.

  • Step Response: $$\displaystyle y(t) = K(1 - e^{-t/\tau}) $$.

  • Ramp Response: Exhibits a tracking error (steady-state error) of $\tau K$.

  • Examples: Mercury-in-glass thermometer, RC circuit.

• Second-Order System: Governed by $$\displaystyle \frac{d^2y}{dt^2} + 2\zeta\omega_n \frac{dy}{dt} + \omega_n^2 y = K\omega_n^2 x(t) $$.

  • Natural Frequency ($$\displaystyle \omega_n $$): Frequency of undamped oscillation.

  • Damping Ratio ($\zeta$): Determines response type:

    • $$\displaystyle \zeta > 1 $$: Overdamped (slow, no oscillation).

    • $$\displaystyle \zeta = 1 $$: Critically damped (fastest without oscillation).

    • $$\displaystyle 0 < \zeta < 1 $$: Underdamped (oscillatory).

    • $$\displaystyle \zeta = 0 $$: Undamped (continuous oscillation).

  • Step Response: Characterized by overshoot, settling time, rise time.

  • Examples: Spring-mass-damper, LCR circuit, moving-coil galvanometer.

3.4 Signal Analysis for Dynamic Systems

• Periodic Signals: Repeat exactly after a fixed period $T$. Frequency $$\displaystyle f = 1/T $$.

• Harmonic Signals: A special type of periodic signal that is a pure sine or cosine wave at a single frequency.

• Non-Harmonic Periodic Signals: Periodic but not pure sine waves (e.g., square, triangular). Can be decomposed into a sum of harmonic signals (Fourier Series).

• Sources of Non-Harmonic Signals: Digital switching, engine firing pulses, non-sinusoidal mechanical vibrations.

• Random Signals: Non-deterministic, described statistically (e.g., thermal noise, turbulence). Characterized by probability density function and power spectral density. Instrument response to random signals is often analyzed in the frequency domain.

3.5 Frequency Domain Analysis

• Fourier Transform (FT): Mathematical tool to transform a time-domain signal $x(t)$ into its frequency-domain representation $X(f)$.

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

*   **Significance:** Decomposes any signal into its constituent frequencies (spectrum).

*   **Linear Property:** FT of a sum is the sum of FTs. $$\displaystyle FT\{a x_1(t) + b x_2(t)\} = a X_1(f) + b X_2(f) $$.

• Frequency Response: The steady-state response of a system to a sinusoidal input of frequency $f$. Described by magnitude ratio $$\displaystyle M(f) = |Y(f)/X(f)| $$ and phase shift $\phi(f)$.

• Bode Plots: Graphical representation of frequency response (log magnitude vs. log frequency, phase vs. log frequency). Used to assess stability and bandwidth.

[!TIP]

Exam Focus: Derive/identify step response equations for 1st/2nd order systems. Calculate $\tau$, $$\displaystyle \omega_n $$, $\zeta$ from given transfer functions. Understand the physical meaning of these parameters. Know the FT's linear property and its use in spectral analysis.

4.0 DISPLACEMENT, POSITION & ANGULAR MEASUREMENT SENSORS

4.1 Potentiometers (Resistive Type)

• Principle: Voltage divider. A uniform resistance element with a sliding contact (wiper). Output voltage $$\displaystyle V_o = V_{in} \frac{R_2}{R_1+R_2} $$ is proportional to displacement.

• Construction: Conductive plastic or wire-wound element, wiper, terminals.

• Conversion: Linear or rotational mechanical motion → variable resistance → variable voltage.

• Limitations:

  • Loading: Measurement circuit draws current, causing error (use high-impedance buffer).

  • Wear & Finite Resolution: Especially for wire-wound types (stepwise output).

  • Inertia: Moving parts limit high-frequency response.

4.2 Variable Differential Transformers (VDTs)

• Linear Variable Differential Transformer (LVDT):

  • Principle: Mutual inductance. Primary coil excited with AC. Two secondary coils connected in series opposition. A movable ferromagnetic core changes the coupling.

  • Output: Differential voltage $$\displaystyle V_{out} = V_{sec1} - V_{sec2} $$. Magnitude ∝ displacement, sign indicates direction.

  • Characteristics: Infinite resolution (analog), contactless, robust, linear over limited range (~±1 cm). Requires AC excitation and demodulation for DC output.

• Rotary Variable Differential Transformer (RVDT):

  • Same principle as LVDT but with a rotary core. Measures angular displacement (typically ±30° to ±60°).

4.3 Other Displacement Sensors (Brief)

• Capacitive: Change in capacitance with plate separation/overlap. High sensitivity, used for small displacements.

• Inductive (Proximity): Eddy current loss changes with target distance. Non-contact.

• Optical (Encoders):

  • Incremental: Outputs pulses proportional to displacement. Requires reference point.

  • Absolute: Unique digital code for each position.

5.0 PRESSURE MEASUREMENT (High-Frequency)

5.1 Fluid Column Manometers

• Principle: Hydrostatic pressure $$\displaystyle P = \rho g h $$. Height difference $h$ of fluid column balances pressure.

• U-tube Manometer:

  • Construction: U-shaped tube, one end connected to pressure source, other open to atmosphere (or reference).

  • Working: $$\displaystyle P_{measured} - P_{ref} = \rho g h $$. $h$ is read directly.

  • Applications: Simple, accurate for low pressures, calibration standard.

• Inclined Manometer: One leg inclined. Amplifies $h$ for better resolution of small pressure differences.

• Micromanometer: Uses a very light fluid (e.g., alcohol) and a magnified scale for high sensitivity.

5.2 Mechanical Pressure Gauges

• Bourdon Tube:

  • Principle: Flattened, curved tube (C-type, helical) tends to straighten under internal pressure.

  • Construction: Tube, link, gear train, pointer, dial.

  • Applications: Widely used for local indication (steam, hydraulic systems).

• Bellows Gauge:

  • Principle: Corrugated bellows expands axially with pressure.

  • Construction: Bellows, mechanical linkage to pointer.

  • Applications: Low to medium pressures, good for gases.

• Diaphragm Gauge:

  • Principle: Flexible diaphragm deflects under pressure difference.

  • Construction: Diaphragm capsule, often with one side sealed (gauge pressure) or both sides connected (differential pressure).

  • Applications: Low pressures, differential pressure measurement.

5.3 Elastic Element Pressure Transducers

• Strain Gauge based: Pressure deflects a diaphragm or Bourdon tube. Strain gauges bonded to the stressed element measure strain → pressure. (See Unit 6).

• Capacitive Pressure Sensor:

  • Principle: Pressure deflects a diaphragm, changing capacitance $$\displaystyle C = \frac{\epsilon A}{d} $$ between diaphragm and fixed plate.

  • Construction: Sealed capacitor, diaphragm as one plate.

  • Output: Capacitance change → measured via AC bridge or oscillator circuit.

5.4 Piezoelectric Pressure Transducers

• Principle: Piezoelectric effect (e.g., Quartz, PZT). Certain crystals generate an electric charge $Q$ on their surface when mechanically stressed (compressed).

  • $$\displaystyle Q = d \cdot F $$, where $d$ is the piezoelectric constant, $F$ is force.

  • For pressure: $$\displaystyle Q = d_p \cdot P \cdot A $$, where $A$ is crystal area.

• Construction: Crystal stack, housing, electrodes. Often includes a charge amplifier.

• Advantages: Very high frequency response, small size, high rigidity.

• Limitations: AC output only (charge dissipates), sensitive to vibration and temperature, cannot measure static pressure (charge leaks away).

5.5 Calibration of Pressure Instruments

• Dead Weight Tester (Pressure Calibrator):

  • Principle: Applies a known, precise pressure by balancing the force of calibrated weights on a precision piston-cylinder.

  • Construction: Cylinder (highly polished, close tolerance), piston, weights, handle to generate pressure, gauge under test.

  • Reference Pressure: $$\displaystyle P = \frac{F}{A} = \frac{mg}{A} $$, where $m$ is total mass of weights, $g$ is gravity, $A$ is piston cross-sectional area.

  • Procedure: Place weights, pump fluid to lift piston, ensure it floats freely (no drag). Pressure in system equals weight force divided by piston area. Highly accurate primary standard.

6.0 STRAIN, FORCE & LOAD MEASUREMENT (High-Frequency)

6.1 Electrical Resistance Strain Gauges

• Principle: Piezoresistive effect. Strain $\epsilon$ changes the resistance $R$ of a conductor/semiconductor.

$$ \frac{\Delta R}{R} = GF \cdot \epsilon $$

• Gauge Factor (GF): Sensitivity factor. For metallic foil gauges, $GF \approx 2$ (mostly from geometric change). For semiconductors, $GF$ is much larger (~100) but temperature-sensitive.

• Calculation: Given $R$, $\Delta R$, $\epsilon$, solve for $GF$. Or, given load $F$, area $A$, Young's modulus $E$: $$\displaystyle \epsilon = \frac{\sigma}{E} = \frac{F/A}{E} $$.

6.2 Strain Gauge Configurations (Wheatstone Bridge)

• Balanced Condition: $$\displaystyle \frac{R_1}{R_2} = \frac{R_3}{R_4} $$ → $$\displaystyle V_o = 0 $$.

• Quarter Bridge: One active gauge. $$\displaystyle V_o \approx \frac{V_{ex}}{4} \cdot GF \cdot \epsilon $$. Temperature compensation requires a dummy gauge in another arm.

• Half Bridge: Two active gauges (e.g., tension/compression on opposite sides of a beam). Output doubles, provides temperature compensation.

• Full Bridge: Four active gauges. Maximum output ($$\displaystyle V_o \approx V_{ex} \cdot GF \cdot \epsilon $$), excellent temperature compensation.

• Bridge Output: For small $\Delta R/R$, $$\displaystyle V_o \approx \frac{V_{ex}}{4} \cdot GF \cdot \epsilon \cdot n $$, where $n$ is number of active tension/compression gauges in the correct configuration.

6.3 Mounting Procedure (Paper Backing)

1. Clean surface with solvent.

2. Apply adhesive (e.g., cyanoacrylate) to gauge backing or surface.

3. Position gauge using tweezers, remove paper backing.

4. Press gently, ensure no air bubbles.

5. Cure adhesive (time/temperature per spec).

6. Remove gauge tabs, lead wires.

6.4 Application for Bending, Compression, Tensile Strains

• Bending (Beam): Top in compression, bottom in tension. Place gauges on opposite surfaces (half/full bridge) to double signal and cancel temperature effects.

• Tensile/Compressive: Single gauge or full bridge on all sides of a tensile member.

6.5 Load Cells

• Principle: Convert force/load into strain → electrical signal (via strain gauges in bridge).

• Types:

  • Compression: Load applied to compress a column.

  • Tension: Load applied to stretch a member.

  • Universal (Shear Beam): Common design; load causes shear strain in a machined beam section.

• Construction: Typically a full-bridge strain gauge arrangement on a precision-machined elastic element (e.g., diaphragm, shear beam). Output mV/V proportional to load.

7.0 TEMPERATURE MEASUREMENT (High-Frequency)

7.1 Thermometric Liquids & Expansion

• Glass Thermometers (Liquid-in-Glass):

  • Construction: Bulb (reservoir), capillary stem, liquid (mercury, alcohol), scale.

  • Working: Liquid expands/contracts with temperature, meniscus position indicates temperature.

  • Advantages: Simple, no power needed, good for calibration.

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

• Bimetallic Thermometers:

  • Principle: Differential thermal expansion. Two bonded metals with different coefficients of expansion (CTE). Bends with temperature change.

  • Construction: Bimetallic strip (helix or spiral for amplification). One end fixed, other connected to pointer.

  • Applications: Thermostats, dial thermometers (room temp, industrial).

7.2 Electrical Resistance Thermometry

• Resistance Temperature Detectors (RTDs):

  • Principle: Positive Temperature Coefficient (PTC) of pure metals (Pt, Ni, Cu). $$\displaystyle R_T = R_0 [1 + \alpha (T - T_0) + \beta (T - T_0)^2 + ...] $$. For Pt, $\alpha \approx 0.00385 /°C$.

  • Construction: Wire-wound (coil on ceramic core) or thin-film (deposited on substrate).

  • Advantages: Excellent stability, accuracy, linearity (over limited range), repeatability.

  • Limitations: Lower sensitivity than thermistors, cost (Pt), self-heating.

• Thermistors:

  • Principle: Negative Temperature Coefficient (NTC) of semiconductor ceramics. $$\displaystyle R = R_0 e^{B(\frac{1}{T} - \frac{1}{T_0})} $$, where $B$ is material constant.

  • Types: NTC (resistance ↓ with T), PTC (resistance ↑ sharply at Curie point).

  • Characteristics: High sensitivity (large $\Delta R/\Delta T$), non-linear, limited range (NTC: -50°C to 150°C).

  • Applications: Temperature compensation, inrush current limiting (PTC), precision thermometry (NTC with linearization).

7.3 Thermocouples

• Principle: Seebeck Effect. When two dissimilar metals (A and B) are joined at two junctions at different temperatures ($$\displaystyle T_j $$, $$\displaystyle T_{ref} $$), an EMF (voltage) is generated proportional to the temperature difference.

  • $$\displaystyle E_{AB}(T_j, T_{ref}) = \int_{T_{ref}}^{T_j} (S_A - S_B) dT $$, where $S$ is Seebeck coefficient.

• Why Two Metals? A single metal cannot generate a net thermoelectric voltage with a temperature gradient; the effect relies on the difference between two materials.

• Junctions:

  • Measuring/Hot Junction: At unknown temperature $$\displaystyle T_j $$.

  • Reference/Cold Junction: At known temperature $$\displaystyle T_{ref} $$ (usually 0°C or ambient). Cold Junction Compensation (CJC) is essential to correct for $$\displaystyle T_{ref} \neq 0°C $$.

• Common Types (ISA/IEC):

  • Type J (Fe-CuNi): -210°C to 760°C, general purpose, sensitive.

  • Type K (NiCr-NiAl): -270°C to 1260°C, wide range, oxidation resistant.

  • Type T (Cu-CuNi): -270°C to 400°C, stable, good for cryogenics.

  • Type E (NiCr-CuNi): -270°C to 1000°C, highest sensitivity.

• Output: Small mV signal (e.g., ~41 µV/°C for Type K). Requires amplification and CJC.

8.0 FLOW & VELOCITY MEASUREMENT

8.1 Differential Pressure Flowmeters

• Principle: Bernoulli's equation: $$\displaystyle P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2 $$ (for horizontal, inviscid flow). A constriction (orifice, venturi) increases velocity ($$\displaystyle v_2 > v_1 $$) and decreases pressure ($$\displaystyle P_2 < P_1 $$). Flow rate $Q$ is proportional to $\sqrt{\Delta P}$.

• Orifice Meter:

  • Construction: Flat plate with concentric hole (orifice plate) inserted in pipe.

  • Working: $\Delta P$ measured upstream and downstream (tapping). $$\displaystyle Q = C_d A_o \sqrt{\frac{2 \Delta P}{\rho (1 - \beta^4)}} $$, where $$\displaystyle C_d $$ = discharge coefficient, $$\displaystyle A_o $$ = orifice area, $$\displaystyle \beta = d/D $$.

  • Advantages: Simple, cheap, no moving parts.

  • Limitations: High permanent pressure loss, $$\displaystyle C_d $$ varies with $\beta$ and Reynolds number, sensitive to upstream disturbances.

• Venturi Meter:

  • Construction: Converging cone → throat → diverging cone.

  • Working: Same principle. $$\displaystyle C_d \approx 0.98 $$ (very high).

  • Advantages: Very low permanent pressure loss, high accuracy, handles solids/liquids with solids.

  • Limitations: Expensive, large size.

8.2 Velocity Measurement

• Stroboscope:

  • Principle: Alias/Frozen Motion. A flashing light at frequency $$\displaystyle f_s $$ illuminates a moving object. If $$\displaystyle f_s = n \cdot f_o $$ ($n$ = integer, $$\displaystyle f_o $$ = object frequency), the object appears stationary. If $$\displaystyle f_s \approx (n \pm \delta) f_o $$, it appears to move slowly.

  • Use: Measure rotational speed ($$\displaystyle f_o = f_s / n $$) or linear speed (with markings).

• Electromagnetic Flowmeters:

  • Principle: Faraday's Law of Induction. A conductor (fluid) moving with velocity $v$ perpendicular to a magnetic field $B$ induces an EMF $$\displaystyle e = B l v $$, where $l$ is electrode spacing.

  • Construction: Non-magnetic pipe, magnetic coils (create $B$), electrodes (pick up $e$).

  • Advantages: No obstruction to flow, bidirectional, measures volumetric flow directly, good for slurries.

  • Limitations: Fluid must be electrically conductive (e.g., water, acids), requires lining for corrosive fluids.

• Mechanical Tachometers:

  • Centrifugal (Flyball): Rotating weights fly out against spring, pointer indicates speed. Disadvantage: Inertia limits high-speed response, wear.

  • Vibro-tachometer: Vibrating reed tuned to specific frequency. Disadvantage: Only for specific fixed speeds.

9.0 CONTROL SYSTEMS FUNDAMENTALS (High-Frequency)

9.1 Basic Control System Concepts

• Control System: An interconnection of components to achieve a desired response by controlling the Controlled Variable.

• Manipulated Variable: The input to the process/plant that is adjusted by the controller.

• Set Point (SP): The desired value of the controlled variable.

• Error Signal (e): $$\displaystyle e = SP - PV $$ (Measured value). Controller acts to minimize $e$.

9.2 Open-Loop vs. Closed-Loop (Feedback) Control

• Open-Loop:

  • Block Diagram: Controller → Plant/Process → Output. No measurement feedback.

  • Examples: Washing machine timer, traffic light sequence, toaster.

  • Advantages: Simple, cheap, stable (no oscillations).

  • Limitations: No correction for disturbances or changes in plant characteristics. Accuracy depends entirely on calibration.

• Closed-Loop (Feedback):

  • Block Diagram: SP → (+) → Controller → Plant → Output → Sensor → (-). Output measured and fed back.

  • Examples: Boiler water level, cruise control, oven temperature.

  • Advantages: High accuracy, rejects disturbances, less sensitive to component variations.

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

• Comparison Table:

9.3 Block Diagrams for Closed-Loop Systems

• Basic Components:

  • Controller (C): Generates control signal from error (e.g., PID).

  • Plant/Process (P/G): The system being controlled (e.g., motor, tank).

  • Sensor/Measurement (H): Measures output (PV).

  • Comparator (∑): Computes error $$\displaystyle e = r - y $$.

  • Disturbance (d): Unwanted input affecting the output.

• Signal Flow: Reference input $R(s)$ → Error $E(s)$ → Controller output $U(s)$ → Plant output $Y(s)$ → Feedback $$\displaystyle B(s) = H(s)Y(s) $$ → back to comparator.

• Transfer Function: $$\displaystyle Y(s) = \frac{C(s)G(s)}{1 + C(s)G(s)H(s)} R(s) + \frac{G(s)}{1 + C(s)G(s)H(s)} D(s) $$.

• Reduction Techniques: Series ($$\displaystyle C_1C_2 $$), Parallel ($$\displaystyle C_1 + C_2 $$), Feedback ($$\displaystyle \frac{C}{1 \pm CH} $$).

10.0 MATHEMATICAL MODELLING OF SYSTEMS (High-Frequency)

10.1 Need for Modelling

To predict system behavior, design controllers, simulate performance, and analyze stability before building physical systems. Provides a mathematical framework (differential equations, transfer functions).

10.2 Modelling of Electrical Systems

• Kirchhoff's Laws:

  • KVL: Sum of voltages around a closed loop = 0.

  • KCL: Sum of currents at a node = 0.

• Component Equations (Time Domain):

  • 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 $$\displaystyle i_L = \frac{1}{L} \int v_L dt $$

• Deriving Transfer Function:

  1. Write differential equation using KVL/KCL and component laws.

  2. Apply Laplace Transform (zero initial conditions).

  3. Solve for output/input ratio: $$\displaystyle G(s) = \frac{Y(s)}{X(s)} $$.

• Example (Series RLC):

  $$\displaystyle V_{in} = Ri + L\frac{di}{dt} + \frac{1}{C}\int i dt $$

  Laplace: $$\displaystyle V_{in}(s) = [R + Ls + \frac{1}{Cs}] I(s) $$

  Transfer Function (impedance): $$\displaystyle G(s) = \frac{I(s)}{V_{in}(s)} = \frac{1}{Ls^2 + Rs + \frac{1}{C}} $$

10.3 Modelling of Mechanical Systems

• Translational Systems (Newton's Law): $$\displaystyle \sum F = M \ddot{x} $$.

  • Mass (M): Stores kinetic energy. Force: $$\displaystyle F_M = M \ddot{x} $$.

  • Spring (K): Stores potential energy. Force: $$\displaystyle F_K = K x $$ (linear spring).

  • Damper/Viscous Friction (B): Dissipates energy. Force: $$\displaystyle F_B = B \dot{x} $$.

• Rotational Systems (Torque Balance): $$\displaystyle \sum T = J \ddot{\theta} $$.

  • Moment of Inertia (J): $$\displaystyle T_J = J \ddot{\theta} $$.

  • Torsional Spring (K_t): $$\displaystyle T_{K_t} = K_t \theta $$.

  • Viscous Damper (B_t): $$\displaystyle T_{B_t} = B_t \dot{\theta} $$.

• Deriving Transfer Function: Write differential equation (force/torque balance), Laplace transform, solve for output/input.

10.4 Transfer Function Representation

• Definition: Ratio of Laplace transform of output to input, assuming zero initial conditions. $$\displaystyle G(s) = \frac{Y(s)}{X(s)} $$.

• Properties: Linear, time-invariant systems. Encodes system dynamics via poles (roots of denominator) and zeros (roots of numerator).

• Example (Given Servo Motor):

  $$\displaystyle H(s) = \frac{50}{s(s+10)} $$

  • Standard Form: $$\displaystyle H(s) = \frac{K}{\tau s (s/\omega_n + 1)} $$? Not exactly second-order canonical form.

  • Poles: At $$\displaystyle s=0 $$ (integrator) and $$\displaystyle s=-10 $$.

  • Type: Second-order system with an integrator (Type 1).

  • Time Constant: From pole at -10, $$\displaystyle \tau = 1/10 = 0.1 $$ sec.

  • Settling Time (2% criterion): Dominant pole at -10. $$\displaystyle t_s \approx \frac{4}{\zeta \omega_n} $$? For first-order dominant pole, $$\displaystyle t_s \approx 4\tau = 4 \times 0.1 = 0.4 $$ sec. (More precisely, for real pole, $$\displaystyle t_s = \frac{4}{|p|} $$).

11.0 ADVANCED SENSORS & TRANSDUCERS (From Past Questions)

11.1 Ionisation Transducers

• Principle: Ionizing radiation (alpha, beta, gamma) ionizes a gas in a chamber. The resulting ion pairs create a current proportional to radiation intensity.

• Example: Ionization Chamber. Used for radiation detection (nuclear plants, medical).

11.2 Optical Pyrometer

• Principle: Non-contact temperature measurement by comparing the brightness of the target object (at high T) with a calibrated internal lamp. Based on Planck's law and Wien's displacement law. Measures brightness temperature.

• Use: Furnaces, molten metals, high-temperature processes (>600°C).

11.3 Photo-electric Transducers

• Photovoltaic Mode: Light generates a voltage/current directly (e.g., solar cell, photodiode in open circuit).

• Photoconductive Mode: Light decreases the resistance of a semiconductor (e.g., LDR, photodiode in reverse bias). Used in light meters, burglar alarms.

11.4 Sling Psychrometer

• Principle: Uses evaporation cooling. Two thermometers: dry-bulb (ambient air temperature) and wet-bulb (covered with wet wick, lower due to evaporation).

• Use: Rotate (sling) to ensure air flow over wet bulb. Read both temperatures, use psychrometric chart to determine relative humidity and dew point.