8.3 Measurement Systems

8.3 Measurement Systems

1. Standards and Calibration

1. Standards Hierarchy:

   • Primary Standards:

     • Highest accuracy at national labs (NIST, NPL).

     • Maintain fundamental units (kg, m, s, A, etc.).

     • Used to calibrate secondary standards.

   • Secondary Standards:

     • Reference standards in industrial/calibration labs.

     • Traceable to primary standards.

     • Used to calibrate working standards.

   • Working Standards:

     • Used daily for calibrating test/measurement equipment.

     • Highest accuracy instruments in regular use.

   • International Standards:

     • SI System (Système International d'Unités).

     • Ensures global consistency.

2. Calibration Process:

   • Comparison against higher-accuracy standard.

   • Establishment of correction factors.

   • Documentation of measurement uncertainty.

   • Issuance of calibration certificate.

   • Calibration Interval: Based on usage, stability, criticality.

3. Traceability:

   • Unbroken chain of comparisons to national standards.

   • Required for quality systems (ISO 9001, ISO/IEC 17025).

   • Ensures measurement validity and comparability.

4. Key Standards:

   • Length: Gauge blocks, laser interferometers.

   • Mass: Standard weights, balances.

   • Temperature: Fixed points (triple point of water).

   • Electrical: Josephson junction (voltage), Quantum Hall (resistance).

   • Pressure: Dead weight testers.

2. Static and Dynamic Characteristics

1. Static Characteristics (for constant or slowly varying inputs):

   • Accuracy: Closeness to true value.

     • Often expressed as: $Accuracy = \frac{|True\ Value - Measured\ Value|}{True\ Value} \times 100\%$

   • Precision: Repeatability of measurements.

   • Resolution: Smallest detectable change in input.

   • Sensitivity: Output change per input change.

     • $S = \frac{\Delta Output}{\Delta Input}$

   • Linearity: Maximum deviation from best-fit straight line.

     • Usually expressed as % of full scale.

   • Hysteresis: Different outputs for same input depending on direction.

   • Dead Band: Range of input where no output change occurs.

   • Threshold: Minimum input needed to produce output.

   • Drift: Slow change in output with constant input.

   • Repeatability: Ability to reproduce readings under same conditions.

   • Reproducibility: Ability to reproduce readings under different conditions.

2. Dynamic Characteristics (for time-varying inputs):

   • Response Time: Time to reach specified percentage (e.g., 90%, 95%) of final value.

   • Time Constant ($\tau$):

     • For first-order systems: time to reach 63.2% of final value.

     • Affects speed of response.

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

   • Settling Time ($t_s$): Time to reach and stay within specified tolerance band.

   • Bandwidth: Range of frequencies where response is within specified limits (usually -3dB point).

   • Natural Frequency ($\omega_n$): Frequency at which system oscillates without damping.

   • Damping Ratio ($\zeta$): Measure of oscillation decay.

   • Dynamic Error: Difference between measured and true time-varying value.

3. Error Sources:

   • Static Errors: Affected by calibration, environment, instrument condition.

   • Dynamic Errors: Caused by system inertia, capacitance, inductance.

   • Systematic vs Random: As defined in previous sections.

3. First and Second Order Systems

1. First Order Systems:

   • Characterized by single energy storage element.

   • Mathematical Model: $\tau \frac{dy}{dt} + y = Kx(t)$ Where: $\tau$ = time constant, $K$ = static sensitivity, $y$ = output, $x$ = input.

   • Step Response: $y(t) = Kx_0(1 - e^{-t/\tau})$

   • Frequency Response: $G(j\omega) = \frac{K}{1 + j\omega\tau}$ Magnitude: $|G| = \frac{K}{\sqrt{1 + (\omega\tau)^2}}$ Phase: $\phi = -\tan^{-1}(\omega\tau)$

   • Examples: RC circuit, thermal system (thermometer), simple pressure sensor.

2. Second Order Systems:

   • Characterized by two energy storage elements.

   • Mathematical Model: $\frac{d^2y}{dt^2} + 2\zeta\omega_n \frac{dy}{dt} + \omega_n^2 y = K\omega_n^2 x(t)$ Where: $\zeta$ = damping ratio, $\omega_n$ = natural frequency.

   • Damping Classification:

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

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

     • Underdamped (0 < $\zeta < 1$): Oscillations with exponential decay.

     • Undamped ($\zeta = 0$): Continuous oscillations.

   • Step Response Characteristics:

     • Peak Time ($t_p$): $t_p = \frac{\pi}{\omega_d}$ where $\omega_d = \omega_n\sqrt{1-\zeta^2}$

     • Percent Overshoot ($OS\%$): $OS\% = 100 \times e^{-\pi\zeta/\sqrt{1-\zeta^2}}$

     • Settling Time (2% criterion): $t_s \approx \frac{4}{\zeta\omega_n}$

   • Frequency Response: $G(j\omega) = \frac{K\omega_n^2}{(j\omega)^2 + 2\zeta\omega_n(j\omega) + \omega_n^2}$ Resonant frequency: $\omega_r = \omega_n\sqrt{1-2\zeta^2}$ (for $\zeta < 0.707$)

   • Examples: Mass-spring-damper, LRC circuit, accelerometers, pressure transducers with cavities.

3. System Comparison:

   • First Order: Simpler, predictable, monotonic response.

   • Second Order: Can exhibit oscillations, resonant behavior.

   • Selection: Based on required speed, damping, and accuracy for application.

4. Measurement System Design Considerations:

   • Match system order to measurement requirements.

   • Ensure adequate bandwidth for signal frequencies.

   • Consider damping to avoid oscillations.

   • Account for time delays in feedback systems.