Mittausjärjestelmien dynamiikka ja epävarmuus Summary & Study Notes

Luento 7_pruju_v2026.pdf 📘

• What this source covers

  • Explains the difference between measurements that do not change during measurement and measurements of time-varying quantities.
  • Introduces system dynamics for measurement systems: amplitude (gain) and phase behavior as functions of frequency, and how to analyze them.

• Start from the smallest building blocks

  • A measurement records a quantity (what we want to know) using a measurement system that produces an output related to that quantity.
  • Two basic measurement classes:
    • Static measurement: the measured quantity stays essentially constant during the measurement (example: length with a ruler).
    • Dynamic measurement: the measured quantity changes over time and we care about its instantaneous or time-dependent value (example: EKG, vibration, AC voltage).

• Why dynamics matter (intuition)

  • If the measured quantity changes too quickly and the measurement system is too "slow", the output will not follow the true changes; we call this a dynamic limitation.
  • If different frequency components of a signal are amplified or attenuated differently, the signal shape will distort (because non-sinusoidal signals are sums of sinusoids).

Basic frequency ideas (tiny steps)

• Frequency f: how many cycles per second a repeating signal has, measured in Hz.
• Angular frequency $\omega$: another frequency unit often used in analysis, $\omega = 2\pi f$ (rad/s).
• A sinusoid at frequency f can be written as a sine or cosine with angular frequency $\omega$.

What we measure to describe dynamics

• Amplitude response (gain): how the amplitude of the output compares to the amplitude of the input at each frequency; numerically output amplitude / input amplitude.
  • Key term: amplitude response.
• Phase response: the time (or angular) shift between input and output peaks, as a function of frequency.
  • Key term: phase response.

Example system: flow into tank → tank level (physical intuition)

• Input: oscillating inflow (volume per time).
• Output: tank surface level.
• Low-frequency input changes: tank level follows the inflow closely (large output amplitude).
• High-frequency input changes: tank level hardly moves (output amplitude attenuated) because the tank integrates/averages fast fluctuations.
• Also: at higher input frequencies the output peaks lag the input peaks (phase shift grows).

Electrical example: RC low-pass filter (step-by-step)

• Circuit: resistor R in series with capacitor C; input across series, output measured across capacitor.
• Physical idea: capacitor resists rapid voltage changes, so high-frequency content is blocked (attenuated).
• Transfer function (Laplace-domain ratio of output to input):
  • Define Laplace variable $s$. For the RC low-pass: $H(s)=\dfrac{1}{1+sRC}$.
  • For steady-state sinusoidal analysis set $s=j\omega$ and get frequency response $H(j\omega)=\dfrac{1}{1+j\omega RC}$.
• Amplitude response (magnitude):
  • $|H(j\omega)| = \dfrac{1}{\sqrt{1+(\omega RC)^2}}$.
• Phase response (angle):
  • $\phi(\omega)=\arg(H(j\omega)) = -\arctan(\omega RC)$.
• Cutoff (corner) frequency $f_c$ (the frequency where amplitude drops to $1/\sqrt{2}$ of low-frequency value):
  • $f_c = \dfrac{1}{2\pi RC}$.
  • Key term: cutoff frequency (also called corner frequency).

Numerical example (from the lecture numbers)

• Given R = 100 kΩ (1.0e5 Ω), C = 100 nF (1.0e-7 F) → $RC = 1.0\times10^{-2}$ s.
• Compute cutoff: $f_c = 1/(2\pi RC) \approx 1/(2\pi\cdot0.01) \approx 15.915\ \text{Hz}$ (matches lecture example).
• Compute amplitude ratio at f = 100 Hz:
  1. Convert to angular frequency: $\omega = 2\pi f = 2\pi\cdot100 \approx 628.32\ \text{rad/s}$.
  2. Compute $\omega RC = 628.32\cdot0.01 = 6.2832$.
  3. Amplitude ratio $|H|=1/\sqrt{1+(6.2832)^2} \approx 1/\sqrt{1+39.478} \approx 1/\sqrt{40.478} \approx 0.157$ (≈0.16 in lecture table).
• Phase at 100 Hz: $\phi = -\arctan(6.2832) \approx -80.96^\circ$ (matches lecture table).

Representations and plotting

• Two common plots show the system response vs frequency: amplitude (gain) plot and phase plot.
• Practical plotting conventions:
  1. Frequency axis is often logarithmic to spread decades evenly.
  2. Amplitude often expressed in decibels (dB):
     • For amplitude quantities (voltage, current): $G_{dB}=20\log_{10}(A_{out}/A_{in})\ \text{dB}$.
     • 0 dB means output amplitude equals input amplitude (ratio = 1).

From time domain to frequency domain (why Laplace helps)

• Differential equations in time are often hard to solve directly.
• Laplace transform converts time-domain derivatives to algebraic multiplication by $s$ (if initial conditions are zero), turning differential equations into algebraic equations.
  • Laplace transform definition: $F(s)=\int_0^{\infty} f(t) e^{-st},dt$.
• Convolution in time (output = impulse response * input) becomes multiplication in Laplace domain: $Y(s)=H(s)X(s)$, where $H(s)$ is the transfer function (Laplace transform of impulse response).
  • Key term: transfer function = $H(s)=Y(s)/X(s)$ under zero initial conditions.

System order, step responses and resonance (short)

• System order: determined by highest derivative in time-domain model (e.g., 1st-order for RC, 2nd-order for mass-spring-damper or RLC).
• 1st-order systems respond monotonically to steps (exponential approach).
• 2nd-order and higher can oscillate (underdamped) or overshoot, producing ringing in step responses.

Practical measurement notes from the lecture

• When measuring a time-varying signal, ensure the measurement system's passband (bandwidth) includes the signal frequencies of interest.
• Low-pass filtering can be beneficial to remove high-frequency noise before A/D conversion (anti-aliasing).
• Always compare the phenomenon frequency content with system cutoff frequency: if the phenomenon lies outside the passband, the measurement will be attenuated or distorted.

Quick glossary (terms to memorize)

• static measurement: measurement when the measurand does not change during the measurement.
• dynamic measurement: measurement of time-varying quantities or instantaneous values.
• amplitude response: output amplitude relative to input amplitude vs frequency.
• phase response: phase shift between input and output vs frequency.
• transfer function: Laplace-domain ratio $H(s)=Y(s)/X(s)$ describing linear system behavior.

If you want worked practice from this lecture

• I can:
  1. reproduce the whole frequency table (f, Uin, Uout, ratio, phase) and show calculations.
  2. show step-response derivation for a first-order RC and for a second-order system (mass-spring-damper) with step-by-step math.

Luento 6_pruju_2026.pdf 📙

• What this source covers (from provided excerpt)

  • Starts with how independent uncertainty components combine (sum in quadrature) and the standard uncertainty of the mean.
  • Lists many contributors to measurement uncertainty: calibration, data collection, signal conditioning, processing, fitting, unknown systematics, correlated variables, method uncertainty, constants, instrument interaction, transformation of point measurements, environment.

• Tiny building blocks: what is "uncertainty"?

  • Every measurement has an uncertainty: a statement of how much the measured value might deviate from the true value.
  • Two broad ways to estimate uncertainty:
    • Type A: statistical evaluation from repeated measurements (uses sample statistics).
    • Type B: other evaluations (calibration certificates, manufacturer specs, previous data, judgement).

Combining independent uncertainties (very small steps)

• If you have several independent uncertainty components $u_1, u_2, \dots, u_n$, the combined standard uncertainty is
  • $u_c = \sqrt{u_1^2 + u_2^2 + \dots + u_n^2}$.
  • This is called summing in quadrature.
  • Key term: combined standard uncertainty.

Standard uncertainty of the mean (Type A example)

• From repeated measurements compute sample standard deviation $s$.
• Standard uncertainty of the sample mean (standard error): $u_{\bar{x}} = s/\sqrt{n}$ where $n$ is the sample size.
• Example:
  1. Suppose $s=0.1$ and $n=25$.
  2. Then $u_{\bar{x}} = 0.1/\sqrt{25} = 0.1/5 = 0.02$.

Propagation of uncertainty through a function (small steps)

• If the quantity of interest $y$ depends on measured variables $x_1, x_2, \dots, x_m$ by $y=f(x_1,\dots,x_m)$:
  • For independent input uncertainties, approximate combined uncertainty by:
    $u_y \approx \sqrt{\sum_{i=1}^m \left(\dfrac{\partial f}{\partial x_i} u_{x_i}\right)^2 }$.
  • If inputs are correlated, include covariance terms:
    $u_y^2 = \sum_i \left(\dfrac{\partial f}{\partial x_i} u_{x_i}\right)^2 + 2\sum_{i<j} \dfrac{\partial f}{\partial x_i}\dfrac{\partial f}{\partial x_j} \operatorname{cov}(x_i,x_j)$.
  • Key term: propagation of uncertainty (law of propagation).

Expanded uncertainty and coverage factor

• Often we want an interval that covers the true value with a chosen confidence (coverage probability).
• Multiply combined standard uncertainty by a coverage factor $k$ (often $k=2$ approximates ~95% for normal distributions):
  • Expanded uncertainty $U = k u_c$.
  • Key term: expanded uncertainty.

Practical sources of uncertainty (lecture list explained)

• Calibration uncertainty: how well the instrument is calibrated to a standard.
• Data collection: digitization noise, sampling, resolution.
• Signal conditioning: amplification, filtering, offset errors.
• Data processing: rounding, fitting model error.
• Unknown systematic errors: biases not accounted for in uncertainty budget.
• Correlated variables: e.g., two readings affected by the same temperature drift.
• Method uncertainty: repeatability of the measurement method itself.
• Environmental effects: temperature, humidity, electromagnetic interference.

Small checklist for making an uncertainty budget

1. List all input quantities that affect the result.
2. For each, decide Type A (statistical) or Type B (other) and assign standard uncertainty $u_i$.
3. If inputs are correlated, estimate covariances or correlation coefficients.
4. Use propagation (partial derivatives) to get combined uncertainty $u_c$.
5. If desired, choose coverage factor $k$ to get expanded uncertainty $U=k u_c$.

Short example (propagation)

• Suppose you measure voltage $V$ and resistance $R$ to compute power $P = V^2/R$.
• Given standard uncertainties $u_V$ and $u_R$ (assumed independent):
  1. Compute partial derivatives: $\partial P/\partial V = 2V/R$, $\partial P/\partial R = -V^2/R^2$.
  2. Combined standard uncertainty:
     • $u_P = \sqrt{\left(\dfrac{2V}{R} u_V\right)^2 + \left(\dfrac{V^2}{R^2} u_R\right)^2 }$.
  3. If you want ~95% coverage, take $U \approx 2 u_P$.

If you want more from this lecture

• I can expand with:
  • More worked examples (calibration uncertainty, combination of Type A and B).
  • Monte Carlo uncertainty propagation explanation and example.
  • Guidance on estimating Type B components from datasheets and calibration certificates.

Luento 5_pruju_2026.pdf 📗

• What I can tell from the submission

  • The file was listed but no content was supplied in your message, so I could not extract lecture text.
  • I will not invent specific lecture content; instead I provide a compact study scaffold and request the file if you want full notes.

• If the lecture is missing, here is a small first-principles scaffold you can use immediately

  • Expectation: Lecture 5 typically (in this course order) might cover sensors, transducers, calibration or signal conditioning—but confirm by uploading the file.

• Study scaffold (what to collect from the file and how to convert to notes)

  1. Identify 2–3 main learning goals stated by the lecturer.
  2. For each goal, list definitions first (no jargon before explanation).
  3. Break formulas into one-line interpretations (what variable means physically).
  4. Extract 2–3 worked examples and rewrite them step-by-step.

• Quick generic checklist for common Lecture-5 topics (sensors & conditioning) that helps you study even before upload

  • For each sensor type note: what it measures, physical principle, typical output (voltage, resistance), dynamic behavior (time constant), typical errors.
  • For signal conditioning: list amplification, filtering (LP/HP), offset removal, and ADC anti-aliasing—draw block diagram mentally.
  • For calibration: record procedure, reference standards, and expression for correction (e.g., calibration curve and uncertainty).

• Next step

  • Upload Luento 5_pruju_2026.pdf content and I will produce full first-principles notes like for Lectures 6 and 7, with formulas, examples, and highlighted key terms.

Luento 4_pruju_2026.pdf 📕

• What I can tell from the submission

  • File listed but content not included in your message, so I cannot extract specific lecture text.
  • I provide a short study-template you can use and will create full notes if you upload the file text.

• Quick study-template to convert lecture contents into first-principles notes

  1. Start with 2–3 bullets that say exactly what the lecture aims to teach.
  2. Define all physical quantities used (units and intuitive meaning).
  3. For each equation, add a one-line plain-language explanation of what it expresses.
  4. Produce one worked numerical example per major equation.

• Typical Lecture 4 topics (common to measurement courses) — checklist to verify against your file

  • Instrument calibration and traceability steps.
  • Error types: random vs systematic, and how to detect them.
  • Basic time-domain response examples (step response) and time constants.

• Next step

  • Upload Luento 4_pruju_2026.pdf or paste its text and I will turn it into a full section with highlighted key terms, worked examples, and a compact glossary.

Luento 3_pruju2026.pdf 📔

• What I can tell from the submission

  • The file name was listed but no lecture content was provided in your message.
  • I include a short primer and an action request so I can produce full lecture notes when you supply the file.

• Quick primer (first principles) likely matching early-lecture content

  • Measurements begin with a measurand: the physical quantity to be measured (e.g., temperature, voltage).
  • Sensors convert the measurand into a readable signal (voltage, current, resistance).
  • Every sensor has characteristics: sensitivity, linearity, range, resolution, bandwidth, and noise.

• Tiny building-block checklist to extract from Lecture 3 when available

  1. Identify the measurand and its typical range.
  2. Identify the sensor/transducer used and the conversion principle.
  3. List the key sensor parameters and how they affect measurement accuracy and dynamics.

• Next step

  • Paste or upload the content of Luento 3_pruju2026.pdf and I will produce detailed, first-principles study notes with worked examples, highlighted key terms, and an uncertainty checklist.