Measurement, Instrumentation and Control

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This section covers static and dynamic measurement errors, and the AC bridges (Maxwell, Schering, Wien) commonly used to measure inductance, capacitance and frequency.

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Static & Dynamic Errors Error Type Description Static error Occurs while measuring a constant or slowly-varying quantity, due to the instrument's own imperfections Dynamic error Additional error that arises because the instrument cannot respond fast enough to a rapidly time- varying quantity Gross error Caused by human mistakes — misreading, incorrect recording, wrong instrument choice Systematic error Consistent, repeatable error from instrumental, environmental, or observational sources Random error Unpredictable variation from unknown causes; can never be fully eliminated, only reduced statistically (e.g., by averaging) AC Bridges Bridge Primarily Used to Measure Maxwell’s bridge Unknown inductance (medium-Q coils), by comparison against a standard capacitance Schering bridge Unknown capacitance and dissipation factor — widely used for insulation/cable testing Wien bridge Frequency; also used as the frequency-selective feedback network in RC oscillators General AC bridge balance condition: the products of impedances in opposite arms are equal, Z₁Z₄ = Z₂Z₃ (the AC analogue of the Wheatstone bridge balance condition).

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This section covers transducer classification and selection, common resistive/capacitive/inductive transducers and sensors, temperature and optical transducers, linearization and signal conditioning, and digital signal conditioning (DAC/ADC types).

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Transducer Basics & Classification A transducer converts one form of energy/physical quantity into another — typically a physical/mechanical quantity into a proportional electrical signal.

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Active (self-generating) transducer: generates its own output signal, needs NO external power (e.g., thermocouple, piezoelectric crystal, photovoltaic cell).

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Passive transducer: requires an external excitation/power source to produce an output (e.g., resistive, capacitive, inductive transducers).

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Selection factors: range, accuracy, sensitivity, linearity, response time, cost, and environmental compatibility.

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Resistive, Capacitive & Inductive Transducers Type Principle Example Resistive Resistance changes with the measurand Strain gauge, RTD, thermistor, potentiometer Capacitive Capacitance changes (plate area / distance / dielectric) Capacitive level or pressure sensor Inductive Inductance / mutual inductance changes LVDT (Linear Variable Differential Transformer) LVDT: a 3-coil transformer (1 primary, 2 secondary) whose differential output voltage is proportional to core displacement — used for precise linear position measurement.

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Sensors for Physical Quantities Sensor Typically Measures Position / proximity sensors Presence or distance of an object (inductive, capacitive, optical) Motion sensors Velocity, acceleration, or movement Pressure sensors Fluid or gas pressure Level sensors Liquid or solid level in a container Flow sensors Rate of fluid flow Strain gauges Mechanical strain/deformation, via a resistance change Load cells Force or weight (usually strain-gauge based) Temperature & Optical Transducers Transducer Principle RTD Resistance rises with temperature (positive temp. coefficient; typically platinum) Thermistor Resistance changes sharply with temperature (usually a negative temp. coefficient, NTC) Thermocouple Seebeck effect — a junction of two dissimilar metals generates a voltage ∝ temperature difference Thermopile Several thermocouples in series, for higher output/sensitivity Piezoelectric Generates a voltage when mechanically stressed (self-generating) Photoconductive cell (LDR) Resistance DECREASES as light intensity increases Photodiode Reverse-biased PN junction; current varies with incident light Photovoltaic cell Converts light directly into voltage (solar cell) — self-generating Phototransistor Light-controlled transistor; amplifies a photodiode-like current Linearization & Signal Conditioning Linearization corrects a transducer’s nonlinear output (e.g., via bridge circuits or op-amp compensation) so it becomes proportional to the measurand.

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Signal conditioning covers amplification, filtering, and conversion of a raw transducer signal into a standard, usable form.

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Impedance/power matching (often via buffer amplifiers) preserves signal integrity and maximizes power transfer between a sensor and the next stage.

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Digital Signal Conditioning — DAC DAC Type Description Binary-weighted resistor DAC Uses resistors in binary-weighted ratios (R, 2R, 4R, …) summed through an op-amp R/2R ladder DAC Uses only two resistor values (R and 2R) in a ladder network — easier to fabricate precisely DAC performance parameters: resolution (smallest output step, = V_ref/2ⁿ), accuracy, settling time, linearity, monotonicity.

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Sources of DAC error: resistor mismatch/tolerance, offset error, gain error, nonlinearity, reference-voltage drift.

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Digital Signal Conditioning — ADC ADC Type Description Single-ramp (single-slope) Compares the input to a linearly rising ramp voltage; a counter runs until the ramp matches the input Dual-slope Integrates the input for a fixed time, then de-integrates at a fixed rate — high accuracy and noise rejection, but slow Successive approximation (SAR) Performs a binary search using an internal DAC — fast, widely used, moderate resolution Flash (parallel) Uses a bank of 2ⁿ−1 comparators for direct parallel conversion — the fastest type, but expensive at high resolution

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This section covers the essential requirements of indicating instruments, the PMMC movement, common measuring instruments (voltmeter, ammeter, wattmeter, energy meter, etc.), and special-purpose meters.

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Essential Requirements of an Indicating Instrument System Function Deflecting system Produces a deflecting torque/force proportional to the quantity being measured Controlling system Opposes the deflecting torque, bringing the pointer to rest at a position corresponding to the true value (spring control or gravity control) Damping system Prevents pointer oscillation and brings it to rest quickly (air-friction, fluid-friction, or eddy-current damping) PMMC & Basic Instrument Types A PMMC (Permanent Magnet Moving Coil) instrument works on the motor principle: a current-carrying coil in a magnetic field experiences a torque; it responds only to the AVERAGE (DC) value and is the basis of many DC ammeters/voltmeters — highly accurate and sensitive.

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Instrument Working Principle Voltmeter Meter movement with a high series (multiplier) resistance; connected in PARALLEL with the circuit; ideally has very high (infinite) internal resistance Ammeter Meter movement with a low parallel (shunt) resistance; connected in SERIES with the circuit; ideally has very low (zero) internal resistance Ohmmeter Combines an internal battery with a meter movement; deflection depends on the unknown resistance in the circuit Clamp- meter Uses the current-transformer (CT) principle — clamps around a conductor to measure current WITHOUT breaking the circuit Multimeter Combines voltmeter, ammeter, and ohmmeter functions via a range/function selector switch Wattmeter Electrodynamometer type: a fixed current coil (in series with the load) and a moving voltage coil (in parallel) — deflection ∝ power (VIcosφ) Energy meter Induction type (e.g., household kWh meter): a rotating aluminium disc whose speed ∝ power; cumulative rotations record energy Special-Purpose Meters Meter Purpose Time-of-day meter Records energy consumption against time-of-use tariff periods Power factor meter Directly indicates the power factor (cosφ) of a circuit Frequency meter Measures the supply frequency Phase meter Measures the phase-angle difference between two AC signals Instrument transformer (CT/PT) Scales down high currents (CT) or voltages (PT) to safe, standard levels for metering/protection Megger Measures very high resistance, mainly insulation resistance Tachometer Measures rotational speed (RPM) of a shaft or rotating machine

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This section covers control system classification (open-loop vs closed-loop), transfer functions, mathematical modeling of mechanical, electrical, liquid-level and thermal systems, and the force-voltage / force-current analogies.

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Control Systems — Definition & Classification Type Feature Open-loop system Output has NO effect on the control action (no feedback); simple and cheap, but inaccurate and sensitive to disturbances Closed-loop system Output is fed back and compared with the input; more accurate and self-correcting, but more complex and can face stability issues Mathematical Modeling of Physical Systems Transfer function:

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G(s) = Y(s)/X(s), the ratio of the Laplace transform of the output to that of the input, assuming zero initial conditions.

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Mechanical systems: modeled with mass (M), spring (K), and damper (B) elements — directly analogous to electrical R, L, C.

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Electrical circuits: modeled using KVL/KCL to derive differential equations, then transformed via the Laplace transform.

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Liquid level systems: modeled using fluid resistance and (tank) capacitance — analogous to an RC electrical circuit.

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Thermal systems: modeled using thermal resistance and thermal capacitance — also analogous to an RC electrical circuit.

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Force-Voltage & Force-Current Analogies Mechanical Quantity F-V Analogy F-I Analogy Force (F) Voltage (V) Current (I) Velocity (v) Current (I) Voltage (V) Mass (M) Inductance (L) Capacitance (C) Damping (B) Resistance (R) Conductance (1/R) Spring stiffness (K) Elastance (1/C) Inverse inductance (1/L) Gears, Sensors & Electromechanical Systems Gears/transformers: a gear (or transformer turns) ratio relates torque/speed (or voltage/current) between two coupled sides — modeled like an ideal transformer.

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Sensors, encoders & generators are typically modeled as a simple proportional (gain) transfer function relating a physical input to an electrical output.

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Electromechanical/mixed systems (e.g., a DC motor) combine electrical and mechanical sub-models, coupled via the back-EMF and torque constants.

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This section covers block diagram reduction rules, signal flow graphs (SFGs) and Mason’s gain formula, and the time-domain response of first- and second-order systems to standard test signals.

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Block Diagram Reduction Rules Configuration Combined Transfer Function Blocks in series (cascade) Multiply:

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G₁·G₂ Blocks in parallel Add (or subtract):

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G₁ ± G₂ Feedback loop (forward G, feedback H) G/(1±GH) — negative feedback:

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G/(1+GH); positive feedback:

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G/(1−GH) Moving a summing/take-off point Requires inserting a compensating block to preserve the overall signal relationships Signal Flow Graphs (SFGs) A signal flow graph represents a system of linear equations as nodes (variables) connected by directed, weighted branches (gains) — an alternative to the block diagram.

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Construction rule: a node’s value equals the sum of all incoming branch signals (each branch carries a gain in the direction of signal flow).

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Block diagrams and signal flow graphs are mathematically equivalent — each can be converted to the other.

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Mason’s Gain Formula computes the overall transfer function directly from an SFG’s forward paths and loop gains, without step-by-step reduction.

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Response to Standard Test Signals Standard test signals: impulse, step, ramp, parabolic — used to evaluate a system’s transient and steady- state behaviour.

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First-order system response to a unit step: rises exponentially toward the final value with time constant τ, with NO overshoot.

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Second-order system response to a unit step is governed by damping ratio ζ and natural frequency ωₙ: underdamped/oscillatory (ζ<1), critically damped (ζ=1), or overdamped (ζ>1) — directly analogous to the RLC transient behaviour of Chapter 1.

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This section covers the definition of stability, the Routh-Hurwitz criterion, root locus, and frequency-domain stability analysis using Bode plots, polar plots, and the Nyquist criterion.

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Concept of Stability A system is BIBO stable if its output remains bounded for every bounded input, and returns to equilibrium once a disturbance is removed.

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Stability is determined by the location of the closed-loop poles: poles in the LEFT half of the s-plane give a stable system (as introduced in Section 1.5).

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Routh-Hurwitz Criterion A tabular method that determines the number of characteristic-equation roots in the RIGHT-half s-plane WITHOUT solving for the roots directly.

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Necessary condition: all coefficients of the characteristic equation must be present and of the same sign.

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Sufficient condition: no sign changes down the first column of the Routh array.

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Root Locus The root locus plots the paths of the closed-loop poles as a system parameter (usually gain K) is varied from 0 to ∞.

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It is used to study how relative stability and transient response change with gain; standard rules give the number of branches, asymptotes, and breakaway points.

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Frequency-Domain Analysis Plot Description Bode plot Two graphs — magnitude (dB) and phase (degrees) — vs. frequency on a log scale; simplifies analysis since each pole/zero contributes a standard shape Polar plot A single plot of magnitude and phase as frequency varies from 0 to ∞, in the complex plane Bode stability criterion: the system is stable if the gain-crossover frequency occurs BEFORE the phase- crossover frequency (equivalently, both gain margin and phase margin are positive).

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Nyquist stability criterion: determines closed-loop stability from the open-loop frequency response by counting encirclements of the point (−1, 0) in the complex plane.

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Gain Margin (GM): the additional gain (in dB) that can be tolerated before instability.

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Phase Margin (PM): the additional phase lag that can be tolerated before instability.

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Both must be positive for a stable system.