MODULE 1 — Measurement Fundamentals
1.1 Accuracy, Precision & Error Types
Errors
Abs: $$ \delta A = A_{measured} - A_{true} $$
Rel: $$ \epsilon_r = \frac{\delta A}{A_{true}} \times 100\%
$$
Accuracy & Precision
$$ \text{Acc} = 1 - \left|\frac{A_{m} -
A_{t}}{A_{t}}\right| $$
$$ P = 1 - \frac{|x_n - \bar{x}|}{x_{max} - x_{min}} $$
Resolution
$$ \text{Res} = \frac{\text{FS}}{2^n} $$
$$ S = \frac{\Delta\theta}{\Delta Q} $$
1.2 Random Errors & Statistics
$$ \sigma = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}} $$
$$ \text{Probable Error} = 0.6745\sigma $$
Error Propagation
$$ \delta(A \pm B) = \delta A + \delta B $$
$$ \frac{\delta(AB)}{AB} = \frac{\delta A}{A} + \frac{\delta
B}{B} $$
1.3 Calibration & Characteristics
$$ \text{Hysteresis} = \frac{O_{up} - O_{down}}{O_{FS}}
\times 100\% $$
$$ \text{Dead Zone} = \text{Smallest input } (\Delta I)
\text{ causing output} $$
Dynamic Response (2nd Order)
$$ \tau = \frac{1}{\omega_n \sqrt{1-\zeta^2}} $$
$$ \text{Rise Time} = \frac{1.8}{\omega_n} $$
1.4 Methods of Measurement
Direct Method
$$ \text{Unknown quantity directly compared against a standard} $$
$$ \text{Inaccurate — Examples: Ruler, Ammeter reading} $$
Indirect Method
$$ \text{Unknown quantity computed from measured related quantities} $$
$$ P = V \times I \quad \text{(measure V, I separately)} $$
$$ \text{Accurate, More Sensitive, Preferred} $$
1.5 Classification of Instruments
(i) Absolute vs Secondary
$$ \text{Absolute: value from constants (no calibration)} $$
$$ \text{Secondary: must be calibrated} $$
(ii) Deflection vs Null
$$ \text{Deflection: fast, less accurate} $$
$$ \text{Null: slow, high accuracy \& sensitivity} $$
(iii) Recording vs Integrating
$$ W = \int_0^T P\,dt = \int_0^T VI\cos\phi\,dt \;\text{(kWh)} $$
1.6 Static Characteristics of Instruments
1. Accuracy
$$ \%\text{Acc} = \left(1 - \frac{|A_m - A_t|}{A_t}\right) \times 100 $$
2. Precision
$$ \text{Closeness of measurements to each other} $$
$$ \text{Precise} \neq \text{Accurate} $$
3. Sensitivity
$$ S = \frac{\Delta V_o}{\Delta V_i} = \frac{1}{I_{FSD}} \;\Omega/\text{V} $$
4. Repeatability
$$ \text{Short-term consistency (same observer)} $$
5. Dead Time / Zone
$$ t_{dead} = \text{delay before response} $$
$$ \Delta x_{dead} = \text{input with no output} $$
6. Linearity
$$ y = kx \quad \text{(ideal)} $$
$$ e_{lin} = \frac{|y_{act} - y_{ideal}|}{y_{FS}} \times 100\% $$
7. Resolution
$$ R = \frac{\text{FSD}}{n} $$
8. Drift
$$ \text{Zero drift | Sensitivity drift | Span drift} $$
1.7 Errors in Measurement & Analysis
Static Error
$$ \delta A = A_m - A_t $$
$$ |\delta A| = |A_m - A_t| \;\text{(Absolute)} $$
$$ \epsilon_r = \frac{\delta A}{A_t} \;\text{(Relative)} $$
$$ \%\text{Error} = \frac{A_m - A_t}{A_t} \times 100 $$
% Accuracy
$$ \%\text{Acc} = (1 - |\epsilon_r|) \times 100 $$
$$ \%\text{Error}_{class} = \frac{\text{Limiting error}}{\text{FSD}} \times 100 $$
Error Types
$$ \text{Gross: Human blunders} $$
$$ \text{Systematic: Pattern (+ve or -ve)} $$
$$ \text{Random: Gaussian distributed} $$
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1.8 Limiting Error & Static Correction
Relative Limiting Error
$$ \epsilon_r = \frac{A_m - A_t}{A_t} \times 100\% $$
Static Correction
$$ \boxed{\delta C = A_t - A_m = -\delta A} $$
$$ A_t = A_m + \delta C $$
Non-Linearity Error
$$ \%NL = \frac{\text{Max deviation from ideal line}}{\text{Desired Value}} \times 100 $$
1.9 Combination of Errors
Sum / Difference
$$ \delta X = \pm(\delta X_1 + \delta X_2 + \cdots) $$
Product / Quotient
$$ \frac{\delta x}{x} = \pm\left(\frac{\delta x_1}{x_1} + \frac{\delta x_2}{x_2}\right) $$
Composite (Power)
$$ x = x_1^a \cdot x_2^b \Rightarrow \epsilon_x = a\epsilon_1 + b\epsilon_2 $$
$$ P = \frac{V^2}{R} \Rightarrow \frac{\delta P}{P} = 2\frac{\delta V}{V} + \frac{\delta R}{R} $$
1.10 Arithmetic Mean & Deviation Analysis
Mean & Deviation
$$ \bar{X} = \frac{\sum X_i}{n} $$
$$ d_i = X_i - \bar{X} \quad;\quad \bar{D} = \frac{\sum|d_i|}{n} $$
Std Dev & Variance
$$ \sigma = \sqrt{\frac{\sum d_i^2}{n-1}} \;\text{(n<20)} $$
$$ V = \sigma^2 $$
Probable & Std Error
$$ r = 0.6745\,\sigma $$
$$ S_{\bar{X}} = \frac{\sigma}{\sqrt{n}} $$
Confidence Intervals
$$ \pm\sigma \rightarrow 68.27\% $$
$$ \pm 2\sigma \rightarrow 95.45\% $$
$$ \pm 3\sigma \rightarrow 99.73\% $$
1.11 Error Types & Analog Instruments
Gross Errors
$$ \text{Human mistakes — cannot treat mathematically} $$
$$ \text{Parallax, wrong reading, wrong setting} $$
Systematic Errors
$$ \text{Instrumental | Environmental | Observational} $$
$$ \delta R = R_0 \alpha \Delta T $$
Random Errors
$$ \text{Follow Gaussian distribution} $$
$$ \text{Error reduces as } \frac{1}{\sqrt{n}} $$
Analog Instrument Types
$$ \text{Indicating | Recording | Integrating | Null} $$
$$ W = \int_0^T VI\cos\phi\,dt \;\text{(energy meter)} $$