BE Computer Engineering (Pokhara University) Instrumentation (PU, ELE 172) Question Paper 2079 Nepal

(a) Generalized Configuration of a Measurement System [8]

A measurement system is conventionally divided into five functional stages. The signal (measurand) flows from left to right:

 Measurand --> [Primary    ] --> [Variable    ] --> [Variable    ] --> [Data        ] --> [Data         ]
              [Sensing    ]     [Conversion  ]     [Manipulation]     [Transmission ]     [Presentation ]
              [Element    ]     [Element     ]     [Element      ]     [Element      ]     [Element       ]

1. Primary Sensing Element The element that first receives energy from the measured medium and produces an output dependent on the measurand. It is the transducer/sensor in direct contact with the quantity being measured. Example: a thermocouple junction sensing temperature, a Bourdon tube sensing pressure.

2. Variable Conversion Element Converts the output of the primary sensing element into a more suitable variable (usually electrical) while preserving the information. Example: an LVDT converting mechanical displacement of a Bourdon tube into a voltage.

3. Variable Manipulation Element Manipulates (changes the magnitude of) the signal without changing its physical nature — i.e. amplification, attenuation, linearization or filtering. Example: an amplifier raising a low-level mV signal to a usable level.

4. Data Transmission Element Carries the signal from one location to another, e.g. from the field to the control room. Example: cables, telemetry links, 4–20 mA current loop.

5. Data Presentation Element Conveys the processed information to a human observer or to a recording/control device in a readable form. Example: a moving-pointer scale, digital display, chart recorder or CRT/monitor.

(b) Accuracy, Precision, Resolution, Sensitivity [6]

• Accuracy: the closeness of a measured value to the true (accepted) value of the quantity. Usually expressed as a percentage of full-scale deflection.
• Precision (repeatability): the closeness of agreement among a set of repeated measurements of the same quantity under the same conditions. It describes reproducibility, not correctness.
• Resolution: the smallest change in the measured quantity that the instrument can detect/indicate.
• Sensitivity: the ratio of the change in output to the change in input, $S = \dfrac{\Delta\,\text{output}}{\Delta\,\text{input}}$ (slope of the calibration curve).

Accuracy vs Precision (target-diagram description):

Think of a dartboard: precision is how tightly grouped the darts are; accuracy is how close that group sits to the bull's-eye. An instrument can be precise (repeatable) yet inaccurate if it carries a systematic bias.

(a) Linear Variable Differential Transformer (LVDT) [8]

Construction: An LVDT has one primary winding ($P$) and two identical secondary windings ($S_1$, $S_2$) wound on a cylindrical former. The secondaries are connected in series opposition (phase opposition). A movable soft-iron/ferromagnetic core slides axially inside the former and is attached to the object whose displacement is measured.

        S1        P        S2
      |||||||  |||||||||  |||||||
      ====== [ core ----> ] ======
   secondary  primary    secondary  (S1 and S2 in series opposition)

Principle of operation: The primary is excited with an AC voltage. The alternating flux links both secondaries. Because they are connected in opposition, the net output is $E_{out}=E_{S1}-E_{S2}$.

• When the core is central (null position), equal flux links both secondaries, $E_{S1}=E_{S2}$, so $E_{out}=0$.
• When the core moves toward $S_1$, $E_{S1}>E_{S2}$, giving a net output; moving toward $S_2$ gives output of opposite phase.

Thus the magnitude of $E_{out}$ is proportional to the displacement and the phase (relative to the primary) indicates direction.

Output characteristic: A V-shaped curve of $|E_{out}|$ versus displacement, passing through (ideally) zero at the null point, with a linear region on either side. With phase taken into account, the output is a straight line passing through the origin (positive on one side, negative on the other). A small residual voltage usually remains at null.

Advantages (any two): frictionless operation / infinite mechanical resolution; rugged, high sensitivity and good linearity over its range. Limitations (any two): sensitive to stray/external magnetic fields; needs AC excitation and demodulation; output affected by temperature and vibration.

(b) Platinum Resistance Thermometer (PT-100) [6]

Resistance variation with temperature:

$$R_t = R_0\,(1 + \alpha t)$$

Given $R_0 = 100\ \Omega$, $\alpha = 0.00385\ /^{\circ}\text{C}$, $t = 150\ ^{\circ}\text{C}$:

$$R_{150} = 100\,(1 + 0.00385 \times 150) = 100\,(1 + 0.5775) = 100 \times 1.5775$$ $$\boxed{R_{150} = 157.75\ \Omega}$$

Three-wire compensation: When the RTD is remote from the bridge, the two lead wires carrying current have resistance that adds directly to $R_t$, causing error that also drifts with ambient temperature. In a three-wire connection, one lead is placed in the RTD arm and a second, identical lead is placed in the adjacent bridge arm, while the third wire is the sense/galvanometer connection carrying negligible current. Because the two leads have equal resistance and are in opposite arms, their effects cancel in the bridge balance equation, so the indicated value reflects only the RTD resistance, independent of lead length and lead temperature.

(a) Digital Data Acquisition System (DAS) [7]

[Transducers] -> [Signal      ] -> [Multiplexer] -> [Sample &] -> [ADC] -> [Computer/   ] -> [Display/
                 [Conditioning]      (MUX)            Hold (S/H)            Controller    ]    Storage]
                                        ^________________________________________|
                                              channel-select & control

• Transducers + signal conditioning: sensors convert physical quantities to electrical signals; conditioning amplifies, filters and level-shifts them.
• Multiplexer (MUX): an electronic switch that sequentially connects many input channels to a single ADC, sharing the costly converter among channels (time-division multiplexing).
• Sample-and-Hold (S/H): samples the multiplexed analog voltage at an instant and holds it constant on a capacitor during conversion, preventing the input from changing while the ADC works.
• ADC (Analog-to-Digital Converter): converts the held analog voltage into an $n$-bit digital word.
• Controller / computer: generates timing and channel-address signals, controls MUX and S/H, reads the ADC output, and stores, processes, displays or transmits the data.

(b) Nyquist Sampling Theorem [5]

Statement: A band-limited signal containing no frequency components above $f_m$ can be completely reconstructed from its samples provided the sampling frequency satisfies

$$f_s \ge 2 f_m.$$

For $f_m = 4\ \text{kHz}$:

$$f_{s(\min)} = 2 \times 4\ \text{kHz} = \boxed{8\ \text{kHz}}.$$

Aliasing: If $f_s < 2 f_m$, high-frequency components fold back and masquerade as lower frequencies in the sampled data — the original and the alias become indistinguishable, corrupting the reconstruction.

Anti-aliasing filter: a low-pass analog filter placed before the sampler. It removes (attenuates) all frequency components above $f_s/2$ so the input is genuinely band-limited, ensuring the Nyquist criterion is met and aliasing cannot occur.

(a) Three-Op-Amp Instrumentation Amplifier [7]

 V1 o--+--[A1]--+
        |       R3
        Rg              [A2 ] differential
        |       R3      stage
 V2 o--+--[A2]--+ ----> Vout
   (input buffers A1,A2 with gain resistor Rg ; output diff-amp A3)

Stage 1 (input buffers): Two non-inverting op-amps $A_1$ and $A_2$ buffer $V_1$ and $V_2$, giving very high input impedance. They are cross-coupled through the gain resistor $R_g$ and two equal resistors $R_1$. The differential gain of this stage is

$$\frac{V_{o1}-V_{o2}}{V_1-V_2} = 1 + \frac{2R_1}{R_g}.$$

The common-mode signal passes with unity gain (no current flows through $R_g$ for a common input).

Stage 2 (difference amplifier): $A_3$ with matched resistors $R_2, R_3$ subtracts the two buffered signals with gain $R_3/R_2$.

Overall gain:

$$\boxed{A_V = \frac{V_{out}}{V_1 - V_2} = \left(1 + \frac{2R_1}{R_g}\right)\frac{R_3}{R_2}}$$

Gain is set by a single resistor $R_g$.

Why high CMRR is desirable: Bridge transducer outputs are small differential voltages (mV) riding on a large common-mode voltage (excitation, supply noise, pickup). A high

$$\text{CMRR}=20\log_{10}\frac{A_d}{A_{cm}}\ \text{dB}$$

means the amplifier strongly rejects the common-mode part while amplifying only the wanted differential signal, preventing the large common-mode component from swamping or distorting the measurement.

(b) Balanced vs Unbalanced (Deflection) Wheatstone Bridge [3]

• Balanced (null) mode: the variable arm is adjusted until the detector reads zero ($R_1 R_4 = R_2 R_3$). The unknown is found from the known arm values. It is independent of supply voltage and gives high accuracy — preferred for precise static/calibration measurements.
• Unbalanced (deflection) mode: the bridge is left in a fixed state and the output (off-balance) voltage is measured; it is proportional to the change in the transducer resistance. Preferred for dynamic/continuous measurement (e.g. strain-gauge transducers) where rapid, automatic readout is needed.

Classification of Measurement Errors

• Gross errors: human mistakes — wrong reading, incorrect recording, parallax, wrong scale. Example: misreading 25.0 V as 35.0 V.
• Systematic errors: consistent, repeatable errors from instrument imperfection, environment or loading. Example: a meter with a zero-offset that reads 0.5 V high on every reading.
• Random errors: unpredictable, scattered errors from unknown fluctuating causes; reduced by averaging many readings. Example: small variations due to noise/thermal effects giving slightly different readings each time.

Statistical Calculations

Readings (V): 50.1, 49.8, 50.0, 50.2, 49.9, $n = 5$.

Arithmetic mean:

$$\bar{x} = \frac{50.1+49.8+50.0+50.2+49.9}{5} = \frac{250.0}{5} = 50.0\ \text{V}$$

Deviations $d_i = x_i - \bar{x}$: $+0.1,\,-0.2,\,0.0,\,+0.2,\,-0.1$.

$$\sum d_i^2 = 0.01+0.04+0+0.04+0.01 = 0.10$$

Standard deviation (using $n-1$, sample):

$$\sigma = \sqrt{\frac{\sum d_i^2}{n-1}} = \sqrt{\frac{0.10}{4}} = \sqrt{0.025} = 0.158\ \text{V}$$

(If population $n$ is used: $\sigma = \sqrt{0.10/5}=0.1414$ V.)

Probable error of one reading:

$$r = 0.6745\,\sigma = 0.6745 \times 0.158 \approx \boxed{0.107\ \text{V}}$$

Results: mean = 50.0 V, $\sigma \approx 0.158$ V, probable error $\approx 0.107$ V.

Transducer

A transducer is a device that converts one form of energy (a physical quantity such as pressure, temperature, displacement) into another, usually into an electrical signal proportional to the measured quantity.

Active vs Passive Transducers

(any two examples of each are sufficient)

Piezoelectric Transducer

Principle: Certain crystals (quartz, barium titanate, PZT) exhibit the piezoelectric effect — when mechanical stress (force/pressure) is applied across a defined crystallographic axis, electric charges of opposite polarity appear on opposite faces, producing a voltage proportional to the applied force.

   Force F
      |
      v
  +++++++++  <- electrode (top face, +Q)
  [ crystal ]  --> Vout across faces
  ---------  <- electrode (bottom face, -Q)

The output voltage $V = \dfrac{Q}{C} = \dfrac{d\,F}{C}$, where $d$ is the piezoelectric charge constant and $C$ the capacitance between electrodes.

It is an active transducer and responds only to changing (dynamic) inputs.

Application (any one): measurement of dynamic pressure / vibration / acceleration (accelerometers), force/load measurement, and ultrasonic transmitters.

Dual-Slope Integrating DVM

Working principle: Conversion occurs in two timed phases using an integrator (op-amp + capacitor), a comparator, a control logic and a counter.

Phase 1 — fixed-time integration of the unknown: The unknown input $V_{in}$ is applied to the integrator for a fixed time $T_1$ (a fixed count $N_1$ of clock pulses). The integrator output ramps with a slope proportional to $V_{in}$, reaching

$$V_p = -\frac{V_{in}}{RC}\,T_1.$$

Phase 2 — fixed-slope de-integration with reference: A known reference voltage $V_{ref}$ of opposite polarity is now applied, and the integrator ramps back toward zero at a fixed slope. The counter counts clock pulses (time $T_2$) until the comparator detects zero crossing.

Since the up-ramp and down-ramp charge are equal:

$$\frac{V_{in}}{RC}T_1 = \frac{V_{ref}}{RC}T_2 \;\Rightarrow\; V_{in} = V_{ref}\,\frac{T_2}{T_1} = V_{ref}\,\frac{N_2}{N_1}.$$

The count $N_2$ is therefore directly proportional to $V_{in}$ and is displayed.

Waveform (described): A triangular ramp — a rising linear ramp during $T_1$ (slope $\propto V_{in}$) up to a peak, followed by a falling linear ramp of fixed slope during $T_2$ back to zero. A larger $V_{in}$ gives a higher peak and longer $T_2$.

Advantages over single-slope (ramp) type (any two):

1. The result depends only on the ratio $T_2/T_1$, so it is independent of $R$, $C$ and exact clock frequency (they cancel), giving high accuracy and stability.
2. Integration over a fixed time averages out (rejects) noise, especially line-frequency hum when $T_1$ is an integer multiple of the mains period.

PMMC Instrument — Construction and Working

Construction: A light rectangular coil of many turns of fine copper wire is wound on an aluminium former and pivoted on jewelled bearings in the air gap between the poles of a permanent magnet and a soft-iron cylindrical core. Hairsprings (also serving as current leads) provide the controlling torque and a pointer moves over a scale. Eddy currents in the aluminium former provide damping.

      N |  [coil on former]  | S
        |   pointer ----> scale
  permanent magnet + soft-iron core

Working: When current $I$ passes through the coil placed in the radial magnetic field $B$, a deflecting torque is produced:

$$T_d = N B A I = G I \quad (\text{proportional to } I).$$

The spring's controlling torque $T_c = K\theta$ balances it, so at equilibrium $\theta \propto I$ — giving a uniform (linear) scale. PMMC responds only to DC (average value).

Conversion of the Galvanometer

Let full-scale deflection current be $I_g$ and coil resistance $R_m$.

(i) Ammeter — shunt: Connect a low resistance $R_{sh}$ in parallel with the movement to bypass the excess current. To read full scale $I$:

$$I_g R_m = (I - I_g) R_{sh} \;\Rightarrow\; \boxed{R_{sh} = \frac{I_g R_m}{I - I_g}}$$

(ii) Voltmeter — series multiplier: Connect a high resistance $R_s$ in series with the movement to drop the excess voltage. To read full scale $V$:

$$V = I_g (R_m + R_s) \;\Rightarrow\; \boxed{R_s = \frac{V}{I_g} - R_m}$$

(a) LED vs LCD [4]

(b) Seven-Segment Display [2]

A seven-segment display has seven bar-shaped segments labelled a, b, c, d, e, f, g arranged in a figure-8 (plus an optional decimal point). Each decimal digit 0–9 is formed by turning ON the appropriate subset of segments:

• 0 -> a,b,c,d,e,f
• 1 -> b,c
• 7 -> a,b,c, etc.

A BCD-to-seven-segment decoder/driver (e.g. 7447 for common-anode, 7448 for common-cathode) takes the 4-bit BCD code and energizes the correct segments to display the digit.

Successive Approximation ADC (SAR)

 Vin --->|+\
         |  comparator |--> [Successive Approximation Register (SAR)] --> n-bit digital output
      +--|-/                         |  (control + bit logic)
      |                             v
      +-------------------------[ DAC ]

Working: The conversion uses a binary search controlled by the Successive Approximation Register (SAR) and an internal DAC:

1. The SAR first sets the MSB = 1; the DAC produces the corresponding analog voltage.
2. The comparator compares this DAC output with $V_{in}$. If $V_{DAC} \le V_{in}$, the bit is kept; otherwise it is reset to 0.
3. The process repeats from MSB to LSB, one bit per clock pulse.

After $n$ clock cycles the SAR holds the digital equivalent of $V_{in}$. It needs a fixed $n$ clock periods regardless of input — fast and widely used.

Resolution and Quantization Error

For an $n$-bit ADC with full-scale range $V_{FS}$:

$$\text{Resolution} = \frac{V_{FS}}{2^{n}-1} \;\; (\text{or } V_{FS}/2^{n} \text{ for step size}), \qquad \text{Max quantization error} = \pm\tfrac{1}{2}\,\text{LSB}.$$

8-bit, 0–5 V:

$$\text{Resolution} = \frac{5\ \text{V}}{2^{8}-1} = \frac{5}{255} = 0.0196\ \text{V} \approx \boxed{19.6\ \text{mV}}$$

(Using $2^8 = 256$ steps: $5/256 = 19.5$ mV.) Quantization error $= \pm 9.8$ mV $(\pm\tfrac12$ LSB$)$.

Strain Gauge Principle

A strain gauge is a passive resistive transducer whose electrical resistance changes when it is mechanically strained. When a metal wire/foil of length $L$, area $A$ and resistivity $\rho$ is stretched, $L$ increases and $A$ decreases, so its resistance ($R = \rho L/A$) increases. Bonding the gauge to a member lets it sense the surface strain.

Gauge Factor

The gauge factor (GF) relates fractional resistance change to strain $\varepsilon$:

$$\text{GF} = \frac{\Delta R / R}{\Delta L / L} = \frac{\Delta R / R}{\varepsilon}$$

For common metal gauges GF $\approx 2$.

Numerical

Given GF $= 2.0$, $R = 120\ \Omega$, strain $\varepsilon = 500\ \mu\varepsilon = 500\times10^{-6}$.

$$\Delta R = \text{GF}\times R \times \varepsilon = 2.0 \times 120 \times 500\times10^{-6}$$ $$\Delta R = 2.0 \times 120 \times 0.0005 = \boxed{0.12\ \Omega}$$

The gauge resistance increases by 0.12 Ω (to 120.12 Ω).

Cathode Ray Oscilloscope (CRO)

[Vertical input]->[Vert. amp]->+---------------------+
                               |   Cathode Ray Tube  |
[Trigger]->[Time-base/sweep]-->|  (CRT) electron gun |->[screen]
            (sawtooth) -> Horiz. amp -> X-plates       Y-plates
[Power supply for EHT & heaters]

Working: The electron gun (heater, cathode, grid, focusing & accelerating anodes) produces and focuses a narrow electron beam that strikes the phosphor screen and glows. Two pairs of deflecting plates steer the beam:

• Vertical (Y) plates receive the amplified input signal.
• Horizontal (X) plates receive a sawtooth (time-base) voltage that sweeps the spot left-to-right at a known rate, while the trigger circuit synchronizes the sweep so the waveform appears stationary. The result is a plot of the input voltage versus time on the screen.

Frequency and Phase Measurement using Lissajous Figures

Feed the unknown signal to the Y-plates and a known-frequency signal to the X-plates (time base off). The pattern formed is a Lissajous figure.

Frequency: Count the points where the pattern is tangent to a horizontal line and a vertical line:

$$\frac{f_y}{f_x} = \frac{\text{number of horizontal tangencies}}{\text{number of vertical tangencies}}.$$

From the known $f_x$ the unknown $f_y$ is found. (A 1:1 ratio with equal frequencies gives a circle/ellipse/line.)

Phase difference: For two equal-frequency sinusoids the figure is an ellipse. With

$$\sin\phi = \frac{y_1}{y_2} = \frac{x_1}{x_2},$$

where $y_2$ is the maximum vertical intercept and $y_1$ the y-intercept where the ellipse crosses the vertical axis, the phase difference $\phi$ between the two signals is obtained. A straight diagonal line means $0^\circ$ or $180^\circ$; a circle means $90^\circ$.