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Key formulas
| Name | Formula | Variables / Notes |
|---|---|---|
| Wheatstone Bridge Balance Condition | P/Q = R/S or P x S = Q x R | P, Q, R, S = four bridge arm resistances; when balanced, galvanometer current is zero; unknown resistance S = R x P / Q |
| Percentage Error | % Error = ((Measured - True) / True) x 100 | Measured = instrument reading; True = actual value; positive % error is an overestimate; negative is an underestimate |
| Sensitivity of a Measuring System | S = delta_output / delta_input | delta_output = change in output quantity (e.g., voltage in mV); delta_input = change in input measurand (e.g., temperature in degC); units depend on transducer type |
| LVDT Output Voltage | V_out = K x x | K = sensitivity constant (mV/mm or V/mm); x = core displacement from null position (mm); V_out is zero at null, polarity indicates direction |
| Resolution of ADC | Resolution = V_FS / (2^N - 1) | V_FS = full-scale voltage range; N = number of bits; a 12-bit ADC with 5 V range gives resolution = 5 / 4095 = 1.22 mV per step |
| Q factor of a Resonant Circuit (Measurement) | Q = f_r / BW = f_r / (f_2 - f_1) | f_r = resonant frequency; BW = -3 dB bandwidth; f_1 and f_2 are lower and upper half-power frequencies; higher Q means sharper resonance |
Key concepts
Types of Errors in Measurement
Gross errors arise from human mistakes (misreading, wrong settings). Systematic errors are repeatable and include instrument errors (calibration), environmental errors (temperature, humidity), and observational errors (parallax). Random errors are non-repeatable and follow a statistical distribution; they are reduced by averaging repeated measurements.
Static Characteristics of Instruments
Accuracy: closeness to true value. Precision: repeatability of readings. Sensitivity: output change per unit input change. Resolution: smallest detectable change. Linearity: how closely output follows a straight line. Hysteresis: difference in output for same input approached from opposite directions.
Wheatstone Bridge Applications
Used to measure unknown resistance, strain (using strain gauges in one or more arms), temperature (using RTD or thermistor). A deflection bridge produces output proportional to imbalance. A null bridge requires adjustment until galvanometer reads zero, giving higher accuracy.
LVDT (Linear Variable Differential Transformer)
LVDT has one primary and two secondary coils. At null position, secondary voltages are equal and output is zero. Displacement moves the ferromagnetic core, increasing flux linkage in one secondary and decreasing it in the other. Output voltage magnitude gives displacement magnitude; phase relative to primary gives direction.
CRO Measurements
Time period T = (time/div setting) x (number of divisions per cycle). Frequency f = 1/T. Voltage amplitude = (V/div setting) x (number of divisions peak to peak) / 2. Phase difference between two signals: phi = (time shift between signals / time period) x 360 degrees.
Signal Conditioning
Amplification increases signal level for ADC input range. Filtering removes noise (low-pass for slow sensors, band-pass for AC sensors). Impedance matching prevents loading error. Isolation (optical or transformer) protects measurement circuits from high voltages. Bridge circuits convert resistance changes to voltage.
Tables
Common Transducers and Their Principles
| Transducer | Measurand | Principle |
|---|---|---|
| Thermocouple | Temperature | Seebeck effect (EMF) |
| RTD (Pt100) | Temperature | Resistance change |
| Strain Gauge | Force / Strain | Piezoresistive effect |
| LVDT | Displacement | Mutual inductance change |
| Piezoelectric | Pressure / Force | Charge generation |
| Hall Effect Sensor | Magnetic field / Current | Lorentz force |
Bridge Circuit Types
| Bridge | Measures | Null Condition |
|---|---|---|
| Wheatstone | Resistance | P x S = Q x R |
| Maxwell | Inductance | L_x = R2 x R3 x C1 |
| Schering | Capacitance | C_x = C1 x R3 / R2 |
| Hay | High-Q inductance | L_x = R2R3C1/(1+1/Q^2) |
Quick facts
- A thermocouple Type K (Chromel-Alumel) has a sensitivity of approximately 41 uV per degC at room temperature.
- Pt100 RTD has a resistance of exactly 100 ohms at 0 degC and approximately 138.5 ohms at 100 degC.
- A strain gauge gauge factor (GF) is typically 2.0 for metallic gauges and 100 to 150 for semiconductor gauges.
- The loading effect of a voltmeter reduces the measured voltage; error is minimised when voltmeter resistance is much greater than circuit resistance.
- A 4-wire (Kelvin) connection eliminates lead resistance error in precision resistance measurement.
- Parallax error in analog meters is eliminated by aligning the pointer with its reflection in the mirror behind the scale.
- CRO bandwidth determines the highest frequency it can accurately display; for a 100 MHz CRO, accuracy is guaranteed up to about 33 MHz (one-third rule).
- A digital multimeter with 4.5 digit display reads up to 19999 counts, giving resolution of 0.005% of full scale.
Exam shortcuts
- Wheatstone bridge unknown: S = (Q/P) x R. Identify which arm is unknown and which is the variable (ratio) arm. If P = 100 ohm, Q = 1000 ohm, R = 48.5 ohm, then S = 10 x 48.5 = 485 ohm.
- For error propagation in a product Z = A x B: percentage error in Z = percentage error in A + percentage error in B. For a quotient Z = A/B: same rule applies. This avoids lengthy partial derivative calculations.
- Sensitivity and accuracy are independent: a highly sensitive instrument can be inaccurate if not calibrated. A precise instrument is repeatable but not necessarily accurate. Know these definitions clearly for theory questions.
- CRO phase measurement: count the x-axis divisions between corresponding points on two waveforms, divide by total divisions per period, multiply by 360. If shift is 2 div and period is 8 div, phase = (2/8) x 360 = 90 degrees.
- For LVDT output, if the problem states sensitivity as 2 mV/mm and asks for displacement when output is 14 mV: x = V_out / K = 14 / 2 = 7 mm. Always check units match.