Electronic Measurements and Instrumentation (EC-302) - Unit 3 Short Notes

UNIT 3: Electronic Measurements and Instrumentation – Short Notes

1.0 Fundamental Concepts in Measurement Systems

Static and Dynamic Characteristics

• Static Characteristics: Describe instrument performance under steady-state conditions.

  • Accuracy: Closeness of measured value to true value.

  • Precision: Degree of reproducibility or repeatability of measurements.

  • Resolution: Smallest detectable change in input.

  • Sensitivity: Ratio of output change to input change ($$\displaystyle S = \frac{\Delta \text{output}}{\Delta \text{input}} $$).

  • Linearity: Maximum deviation of actual calibration curve from ideal straight line.

  • Drift: Slow change in output over time with constant input.

• Dynamic Characteristics: Response to time-varying inputs.

  • Speed of Response: How quickly instrument responds to input changes.

  • Fidelity: Ability to follow rapid input changes without distortion.

  • Lag: Delay in response.

  • Overshoot: Temporary exceedance of steady-state value.

  • Damping: Mechanism to suppress oscillations. Slightly underdamped systems are preferred as they reach steady-state quickly without excessive oscillations.

[!TIP] Common Pitfall: Do not confuse Accuracy (correctness) with Precision (repeatability). An instrument can be precise but inaccurate.

Types of Errors

Calibration

• Importance: Ensures instrument accuracy by comparing against a standard. Establishes relationship between input and output. Periodic calibration compensates for drift, wear, and aging.

• Procedure: Apply known standards, record output, adjust instrument or generate calibration curve/table.

Accuracy, Precision, and Resolution Interrelationship

• High resolution does not guarantee high accuracy or precision.

• Precision requires good repeatability (low random errors).

• Accuracy requires both precision and minimal systematic errors (bias).

• Example: A digital multimeter with 6½ digits (high resolution) may still be inaccurate if not calibrated.

Damping in Instruments

• Overdamped: Slow response, no oscillation. Unacceptable for fast measurements.

• Underdamped: Oscillatory response, takes time to settle.

• Slightly Underdamped: Optimal compromise—quick response with minimal overshoot and rapid settling. Critical for readable pointer movement.

Differential Amplifier using Two FETs

• Circuit: Two matched FETs (Q1, Q2) with common source resistor $$\displaystyle R_S $$ and drain resistors $$\displaystyle R_{D1}, R_{D_{D2}} $$. Inputs applied to gates, output from drains.

• Operation: Common-mode rejection. Differential input $$\displaystyle v_{id} = v_{i1} - v_{i2} $$ causes current change in $$\displaystyle R_S $$, producing differential output $$\displaystyle v_{od} = v_{o1} - v_{o2} $$.

• Output Voltage Derivation:

  For small signals, transconductance $$\displaystyle g_m $$:

$$ i_{d1} = g_m v_{gs1}, \quad i_{d2} = g_m v_{gs2} $$

Differential output:

$$ v_{od} = (i_{d1} - i_{d2}) R_D = g_m (v_{gs1} - v_{gs2}) R_D $$

With source degeneration, differential gain:

$$ A_d = \frac{v_{od}}{v_{id}} \approx \frac{g_m R_D}{1 + g_m R_S} \quad (\text{for large } g_m R_S) $$

Comparison: Digital vs. Analog Instruments

2.0 Analog Measuring Instruments

2.1 Voltmeters

DC Voltmeters

Chopper Type DC Voltmeter

• Working Principle: Converts DC input to AC using a chopper (modulator), amplifies AC with high-gain AC amplifier, then demodulates back to DC. Eliminates drift and offset issues of DC amplifiers.

• Circuit: Input DC → Chopper (mechanical/electronic switch) → AC amplifier → Demodulator → Output meter.

• Advantage: High stability, low drift, suitable for microvolt measurements.

Electrostatic Voltmeter

• Construction: Two plates (fixed and movable) with guard rings. Attraction force due to electric field causes deflection.

• Force Calculation:

  For parallel plates with area $A$, separation $d$, voltage $V$:

$$ F = \frac{1}{2} \frac{dC}{dx} V^2 $$

For linear variable capacitance $$\displaystyle C = \frac{\varepsilon_0 A}{d-x} $$ (x = deflection):

$$ F = \frac{\varepsilon_0 A V^2}{2(d-x)^2} \approx \frac{\varepsilon_0 A V^2}{2d^2} \quad (\text{for small } x) $$

Example Problem: Given $$\displaystyle V=10 $$ kV, $$\displaystyle F=5 \times 10^{-3} $$ N, plate diameter $$\displaystyle D=100 $$ mm ($$\displaystyle A = \pi D^2/4 $$), find $\Delta C$ for $$\displaystyle \Delta x = 1 $$ mm.

Solution: Use $$\displaystyle F = \frac{1}{2} \frac{\Delta C}{\Delta x} V^2 \Rightarrow \Delta C = \frac{2F \Delta x}{V^2} $$.

AC Voltmeters

• Types:

  • Average Responding with Rectifier: Measures average value of rectified AC, calibrated to read RMS for sine wave ($$\displaystyle V_{rms} = 1.11 \times V_{avg} $$).

  • Peak Responding: Captures peak value, calibrated for sine ($$\displaystyle V_{rms} = V_p/\sqrt{2} $$).

  • True RMS: Uses thermal converters or electronic squaring/root-mean-square circuits; accurate for any waveform.

• Working of Average Responding:

  1. AC input → Precision rectifier (full-wave) → DC proportional to average.

  2. DC amplified and drives a moving-coil meter calibrated in RMS for sine waves.

2.2 Cathode Ray Oscilloscopes (CRO)

Block Diagram

[Input] → [Vertical Amplifier] → [Delay Line] → [Vertical Deflection Plates]

                              ↓

[Trigger Circuit] ← [Sweep Generator] → [Horizontal Amplifier] → [Horizontal Deflection Plates]

                              ↓

                         [Power Supply]

• Signal Flow: Vertical channel amplifies input for Y-deflection. Horizontal sweep (time base) provides X-deflection. Trigger synchronizes sweep with input.

Cathode Ray Tube (CRT)

• Internal Structure:

  • Electron Gun: Cathode (heated emitter), control grid (intensity control), focusing anode (electrostatic lens), accelerating anode.

  • Deflection System: Electrostatic plates (vertical/horizontal) for beam deflection.

  • Fluorescent Screen: Emits light when struck by electrons (phosphor coating).

  • Glass Envelope: Evacuated, with conductive coating (aquadag) for voltage stabilization and light absorption.

• Functions: Electron generation, beam formation, focusing, deflection, and display.

Electrostatic Focusing and Deflection

• Focusing: Variable voltage on focusing anode creates electrostatic lens that converges electron beam to a fine spot on screen.

• Deflection: Voltage on deflection plates creates transverse electric field, deflecting electron beam. Deflection sensitivity: $$\displaystyle S = \frac{D}{V_d} \text{ (mm/V)} $$, where $D$ = post-deflection anode voltage.

Probes

• Purpose: Connect CRO to circuit without loading, provide attenuation, protect input.

• Types: 1:1 (direct), 10:1 (attenuating), active (with buffer).

• 10:1 Probe Circuit:

  • Series resistor $$\displaystyle R_s $$ (9 MΩ) and shunt capacitor $$\displaystyle C_s $$ (compensation capacitor) with cable capacitance $$\displaystyle C_c $$.

  • Attenuation: $$\displaystyle V_{in} : V_{out} = 10:1 $$.

  • Compensation: Adjust $$\displaystyle C_s $$ so $$\displaystyle R_s C_s = R_{in} C_{in} $$ (where $$\displaystyle R_{in} $$ = oscilloscope input resistance, $$\displaystyle C_{in} $$ = input capacitance) for frequency-independent attenuation.

Graticules

• Purpose: Grid overlay on screen for measuring amplitude and time.

• Types: Internal (etched on glass), external (transparent plastic). Standard: 1 cm × 1 cm grid with 0.2 cm subdivisions.

Dual Trace vs. Dual Beam CRO

Special Purpose CROs

• Digital Storage Oscilloscope (DSO): Samples input, stores in memory, displays digitally. Enables waveform capture, processing, and persistence.

• Sampling Oscilloscope: Samples high-frequency signal at lower rate, reconstructs waveform. Used for signals beyond bandwidth limit (e.g., GHz).

• Dual Beam Storage CRO: Combines dual beam with storage capability.

3.0 Impedance Measurement Techniques

3.1 Bridge Circuits

Schering Bridge

• Circuit:

  
      A
  
     / \
  
  R1   C1
  
     \ /
  
      B---[R2]---C
  
     / \
  
  R3   C3 (unknown)
  
     \ /
  
      D
  
  

  • Arms: AB: $$\displaystyle R_1 $$, $$\displaystyle C_1 $$ (standard); BC: $$\displaystyle R_2 $$ (standard); CD: $$\displaystyle R_3 $$, $$\displaystyle C_3 $$ (unknown $$\displaystyle C_x $$, $$\displaystyle R_x $$); DA: $$\displaystyle C_4 $$ (standard capacitor).

• Balance Condition:

  At balance, $$\displaystyle Z_1 Z_3 = Z_2 Z_4 $$:

$$ (R_1 - j/\omega C_1) (R_3 + 1/j\omega C_3) = R_2 \cdot (-j/\omega C_4) $$

Separating real and imaginary:

$$ R_3 = \frac{R_1 R_2 C_4}{C_1} \quad \text{and} \quad C_3 = \frac{C_1 C_4}{C_1 + C_4 (1 + \omega^2 R_1^2 C_1^2)} \approx C_4 \frac{R_1}{R_2} \quad (\text{if } \omega R_1 C_1 \ll 1) $$

• Operation: Adjust $$\displaystyle R_1 $$ and $$\displaystyle C_1 $$ until detector (headphones/oscilloscope) shows null.

• Applications: Testing capacitors, measuring dielectric properties, insulation resistance.

Hay's Bridge

• Circuit:

  
      A
  
     / \
  
  R1   L1 (known, with series R1)
  
     \ /
  
      B---[R2]---C
  
     / \
  
  R3   C3 (unknown: R, C)
  
     \ /
  
      D
  
  

  • Arms: AB: $$\displaystyle R_1 $$, $$\displaystyle L_1 $$ (known inductor with series $$\displaystyle R_1 $$); BC: $$\displaystyle R_2 $$ (standard); CD: $$\displaystyle R_3 $$, $$\displaystyle C_3 $$ (unknown $$\displaystyle R_x $$, $$\displaystyle C_x $$); DA: $$\displaystyle C_4 $$ (standard).

• Balance Condition:

$$ (R_1 + j\omega L_1) (R_3 - j/\omega C_3) = R_2 \cdot (-j/\omega C_4) $$

Equating real and imaginary:

$$ R_3 = \frac{R_1 R_2 C_4}{C_3} \quad \text{and} \quad L_1 = R_2 R_3 C_4 $$

Solving for unknown $$\displaystyle R_x $$, $$\displaystyle C_x $$:

$$ R_x = \frac{R_2 R_1}{\omega^2 L_1 C_4} \quad \text{and} \quad C_x = \frac{R_1 C_4}{\omega^2 L_1} $$

• Operation: Adjust $$\displaystyle R_3 $$ and $$\displaystyle C_3 $$ for null.

• Applications: Measuring inductance of coils with high Q ($Q \gg 1$). Advantage over Maxwell bridge: simpler expressions.

• Problem Example (from Jun 2024):

  Given: AB = 600 Ω + 0.18 H, BC = DA = 1200 Ω, CD = R_x + C_x, $$\displaystyle f=50 $$ Hz, $$\displaystyle V=3 $$ V.

  Balance:

$$ R_x = \frac{R_2 R_1}{\omega^2 L_1} = \frac{1200 \times 600}{(2\pi \times 50)^2 \times 0.18} \approx 203.7 \ \Omega $$

$$ C_x = \frac{R_1}{\omega^2 L_1} = \frac{600}{(2\pi \times 50)^2 \times 0.18} \approx 0.678 \ \mu\text{F} $$

Wien Bridge

• Circuit:

  
      A
  
     / \
  
  R1   C1
  
     \ /
  
      B---[R2]---C
  
     / \
  
  R3   C3 (unknown: R, C)
  
     \ /
  
      D
  
  

  • Arms: AB: $$\displaystyle R_1 $$, $$\displaystyle C_1 $$; BC: $$\displaystyle R_2 $$; CD: $$\displaystyle R_3 $$, $$\displaystyle C_3 $$; DA: $$\displaystyle R_4 $$, $$\displaystyle C_4 $$ (often $$\displaystyle R_1=R_2=R $$, $$\displaystyle C_1=C_2=C $$).

• Balance Condition:

$$ Z_1 Z_3 = Z_2 Z_4 \Rightarrow (R_1 + 1/j\omega C_1)(R_3 + 1/j\omega C_3) = R_2 (R_4 + 1/j\omega C_4) $$

For $$\displaystyle R_1=R_2=R $$, $$\displaystyle C_1=C_2=C $$, $$\displaystyle R_3=R_x $$, $$\displaystyle C_3=C_x $$:

$$ \omega^2 = \frac{1}{R_x C_x R C} \quad \text{and} \quad R_x = R \frac{C}{C_x} $$

• Use for Frequency Measurement:

  If $$\displaystyle R_x $$, $$\displaystyle C_x $$ known (unknown frequency $f$):

$$ f = \frac{1}{2\pi \sqrt{R_x C_x R C}} $$

Adjust $R$ or $C$ until balance; read frequency from calibrated dial.

• Applications: Audio frequency oscillator calibration, frequency measurement in audio range (20 Hz–20 kHz).

Maxwell Bridge

• Circuit: Similar to Hay's but with known capacitor in series with resistor in one arm.

• Limitation for High Q Coils: Balance equations become:

$$ L_x = R_2 R_3 C_1 \quad \text{and} \quad R_x = \frac{R_2 R_3}{R_1} $$

For high Q coils ($$\displaystyle R_x $$ small), $$\displaystyle R_1 $$ must be very small (difficult to achieve precisely). Also, $$\displaystyle C_1 $$ must be lossless (hard at high frequencies). Hay's bridge avoids this by using known inductor.

Bridge Sensitivity

• Definition: Rate of change of detector output (e.g., galvanometer deflection) per unit change in unknown impedance.

• Condition for Maximum Sensitivity:

  Detector resistance $$\displaystyle R_g $$ should match the Thevenin resistance $$\displaystyle R_{th} $$ seen by the detector at balance:

$$ R_g = R_{th} = \frac{R_1 R_3}{R_1 + R_3} \quad (\text{for simple resistive bridges}) $$

This maximizes power transfer and deflection.

3.2 Q-Meter

Principle of Impedance Measurement

• Q-Meter: Based on series resonance circuit. Coil (unknown) with series resistance $R$ and inductance $L$ is connected to a variable capacitor $C$ and a source. At resonance ($$\displaystyle X_L = X_C $$), voltage across capacitor $$\displaystyle V_C $$ is $Q$ times the applied voltage $V$: $$\displaystyle Q = V_C / V $$.

• Indirect Measurement: $Q$, $f$, $C$ known → calculate $L$, $R$.

Connection Methods

1. Series Connection:

• Unknown impedance $$\displaystyle Z_x = R_x + jX_x $$ in series with Q-meter coil.

• At resonance: $$\displaystyle X_C = X_L + X_x $$.

• Expressions:

$$ Q = \frac{V_C}{V} = \frac{1}{\omega C (R + R_x)} \quad \Rightarrow \quad R_x = \frac{1}{\omega C Q} - R $$

$$ X_x = \frac{1}{\omega C} - \omega L \quad (\text{from } X_C = X_L + X_x) $$

2. Parallel Connection (more common):

• Unknown impedance connected in parallel with Q-meter capacitor $C$.

• Derivation:

  At resonance, net reactance infinite → susceptances cancel:

$$ B_C + B_x = 0 \quad \Rightarrow \quad \omega C - \frac{1}{\omega L_x} = 0 \quad \text{for coil} $$

But with parallel $$\displaystyle Z_x = R_x \parallel jX_x $$, total admittance $$\displaystyle Y = j\omega C + \frac{1}{R_x} - j\frac{1}{X_x} $$.

Resonance: Imaginary part zero:

$$ \omega C = \frac{1}{X_x} \quad \Rightarrow \quad X_x = \frac{1}{\omega C} $$

Real part: $$\displaystyle G_x = 1/R_x = \omega C / Q $$ (since $$\displaystyle Q = \omega C / G_x $$ for parallel circuit).

Thus:

$$ R_x = \frac{Q}{\omega C} \quad \text{and} \quad X_x = \frac{1}{\omega C} $$

For coil with series $$\displaystyle R_s $$, $L$: $$\displaystyle R_x = \frac{Q^2 R_s}{\omega^2 L^2} \approx \frac{R_s}{Q^2} $$ (if $Q \gg 1$).

4.0 Transducers

4.1 Classification of Transducers

4.2 Resistive Transducers

Thermistor:

• Principle: Resistance of semiconductor material changes with temperature (NTC: negative temperature coefficient; PTC: positive).

• Importance: High sensitivity ($\sim -5\%/\degree C$), small size, fast response.

• Advantages: High sensitivity, low cost, suitable for temperature compensation.

• Applications: Temperature measurement, inrush current limiting, temperature compensation in circuits.

4.3 Inductive Transducers

Linear Variable Differential Transformer (LVDT):

• Construction: Primary winding, two secondary windings (series/parallel opposition), movable ferromagnetic core.

• Working Principle: AC excitation on primary. Core displacement changes mutual inductance, inducing voltages in secondaries. Output $$\displaystyle V_{out} = V_{s1} - V_{s2} $$.

• Displacement Detection:

  • Magnitude: $$\displaystyle |V_{out}| \propto $$ core displacement from null position.

  • Direction: Phase of $$\displaystyle V_{out} $$ relative to excitation indicates direction (0° or 180°).

• Advantages: Infinite resolution, no physical contact, robust, linear over wide range.

• Applications: Displacement, position, vibration measurement.

4.4 Capacitive Transducers

• Principle: Capacitance changes with plate area, separation, or dielectric constant.

• Types:

  • Parallel Plate: $$\displaystyle C = \frac{\varepsilon A}{d} $$ → displacement changes $d$ or $A$.

  • Differential: Two capacitors in series, opposite changes with displacement → better linearity.

• Applications: Displacement, pressure, humidity measurement.

4.5 Piezoelectric Transducers

• Working Principle: Piezoelectric effect (e.g., quartz, PZT): Mechanical stress → electric charge; vice versa.

• Modes of Operation:

  • Benders: Cantilevered piezoelectric strip; deflection perpendicular to stress.

  • Twisters: Disk or tube with electrodes on opposite faces; twist under shear stress.

• Applications: Accelerometers, pressure sensors, ultrasonic generators, microphones.

• Limitation: Cannot measure static signals (charge leaks away).

4.6 Thermoelectric Transducers

Thermocouple:

• Principle: Seebeck effect: Two dissimilar metals joined → temperature difference generates EMF.

• Construction: Two wires (e.g., Chromel-Alumel) welded at measuring junction, connected to cold junction (reference).

• Output: Small voltage ($\sim 40 \ \mu V/\degree C$). Requires cold-junction compensation and amplification.

• Applications: Wide-range temperature measurement (-200°C to +2000°C), industrial processes.

4.7 Optical Transducers

Photoemissive:

• Principle: Light ejects electrons from cathode (photoelectric effect). Example: Photomultiplier tube (PMT).

• Construction: Photocathode → dynodes (cascade amplification) → anode.

• Applications: Low-light detection, scintillation counters.

Photoconductive:

• Principle: Light decreases resistance of semiconductor.

  • Photodiode: Reverse-biased pn junction. Current increases with light intensity. Fast response.

  • Phototransistor: Photodiode base current → transistor amplification. Higher sensitivity, slower.

• Applications: Light meters, optical switches, fiber-optic communication.

Photovoltaic:

• Principle: Light generates voltage across pn junction (solar cell). No external bias needed.

• Applications: Solar panels, light sensors.

4.8 Other Transducers

Accelerometer:

• Principle: Measures acceleration via force on mass (Newton's law $$\displaystyle F=ma $$).

• Types: Piezoelectric (charge output), capacitive (change in capacitance), servo (force-balanced).

• Applications: Vibration analysis, inertial navigation, structural monitoring.

Bolometer:

• Principle: Absorbs RF/microwave power → temperature rise → resistance change (e.g., thermistor or barretter).

• Use in RF Measurement: Measures RF power by converting to heat. High accuracy, wide bandwidth.

• Applications: RF power meters, microwave detection.

5.0 Signal Generators

Sweep Frequency Generator

• Block Diagram:

  
  [Oscillator] → [Voltage-Controlled Oscillator (VCO)] → [Amplifier] → [Output]
  
                ↑
  
          [Sweep Control (ramp generator)]
  
  

• Working Principle: A ramp voltage (from sweep generator) controls the frequency of a VCO, causing continuous, automatic frequency variation over a specified range (sweep). Used with a spectrum analyzer or for frequency response testing.

Pulse Wave Generator

• Block Diagram:

  
  [Clock] → [Frequency Divider] → [Monostable Multivibrator] → [Amplifier] → [Output]
  
  

• Operation: Clock triggers a monostable (one-shot) producing pulses of width $$\displaystyle t_p $$ determined by RC timing. Frequency set by clock rate. Adjustable pulse width and repetition rate.

• Applications: Digital circuit testing, timing analysis, pulse modulation.

6.0 Digital Techniques in Measurements

6.1 Digital Voltmeters (DVM)

Digits Notation

• 3½ Digits: Can display 0 to 1999 (three full digits 0–9, and a half digit 0 or 1). Range: ±1.999 V (for 2 V range).

• 5½ Digits: Can display 0 to 19999 (five full digits, half digit 0/1). Range: ±1.9999 V (for 2 V range).

• Significance of ½ Digit: Leftmost digit can only be 0 or 1, indicating polarity and limiting maximum reading to just under 2. Provides overrange indication (e.g., "1" on 2 V range means >1.999 V).

Resolution and Sensitivity

• Resolution: Smallest change in input voltage that produces a detectable output change. For n-bit ADC with range $$\displaystyle V_{FS} $$:

$$ \text{Resolution} = \frac{V_{FS}}{2^n - 1} \approx \frac{V_{FS}}{2^n} $$

• Sensitivity: Minimum input voltage that causes a one-count change in output. Same as resolution for ideal ADC.

• Example: 10-bit DVM, 10 V range: Resolution = $$\displaystyle 10/(2^{10}-1) \approx 9.77 $$ mV.

6.2 Analog-to-Digital Converters (ADC)

Types and Comparison

Resolution and Step Size

• Resolution (n-bit): Number of discrete levels = $$\displaystyle 2^n $$.

• Step Size (LSB): Voltage corresponding to 1 LSB:

$$ \text{LSB} = \frac{V_{ref}}{2^n} \quad (\text{for unipolar}) $$

• Percentage Error: $$\displaystyle \% \text{ error} = \frac{\text{measured} - \text{true}}{\text{full-scale}} \times 100\% $$.

6.3 Digital-to-Analog Converters (DAC)

Weighted Resistor DAC

• Circuit (4-bit):

  
  V_ref
  
    |
  
    R
  
    |
  
    ├───[2R]───[4R]───[8R]───[16R]───→ Output (summing junction)
  
    |      |      |      |
  
  b3     b2     b1     b0   (binary inputs, 0/1)
  
  

• Operation: Each bit controls a switch connecting a weighted resistor ($$\displaystyle 2^k R $$) to summing junction. Output:

$$ V_o = -\frac{V_{ref}}{R} \left( \frac{b_0}{2^0} + \frac{b_1}{2^1} + \cdots + \frac{b_{n-1}}{2^{n-1}} \right) \quad (\text{for inverting summing amp}) $$

• Transfer Characteristic (3-bit):

  Digital input $$\displaystyle D = b_2 b_1 b_0 $$ → Analog output:

$$ V_o = -\frac{V_{ref}}{8} (4b_2 + 2b_1 + b_0) $$

Step size = $$\displaystyle V_{ref}/8 $$.

R-2R Ladder DAC

• Circuit (4-bit):

  
  V_ref
  
    |
  
    R
  
    |
  
    ├───┬───┬───┬───→ Output
  
    |   |   |   |
  
   R  2R  R  2R  ...
  
    |   |   |   |
  
  b3  b2  b1  b0  (to ground or V_ref via switches)
  
  

• Operation: Each bit sees equivalent resistance of $2R$ looking into the ladder. Output voltage:

$$ V_o = V_{ref} \left( \frac{b_0}{2} + \frac{b_1}{4} + \frac{b_2}{8} + \frac{b_3}{16} \right) \quad (\text{non-inverting}) $$

• Example (from Dec 2024): 4-bit, $$\displaystyle V_{ref}=5 $$ V.

  • Input 0111: $$\displaystyle V_o = 5 \times (0/2 + 1/4 + 1/8 + 1/16) = 5 \times (0.25+0.125+0.0625) = 2.1875 $$ V.

  • Input 1111: $$\displaystyle V_o = 5 \times (0.5+0.25+0.125+0.0625) = 4.6875 $$ V.

• Advantage: Only two resistor values (R, 2R), better matching, scalable.

6.4 Digital Circuits in Measurement Systems

Binary Adder:

• Purpose: Adds binary numbers. Used in DACs (to sum weighted currents/voltages), digital signal processing, and error correction.

• Types:

  • Half Adder: Adds two bits, outputs sum and carry. $$\displaystyle S = A \oplus B $$, $$\displaystyle C = A \cdot B $$.

  • Full Adder: Adds three bits (A, B, carry-in). $$\displaystyle S = A \oplus B \oplus C_{in} $$, $$\displaystyle C_{out} = AB + BC_{in} + AC_{in} $$.

• Role in Measurement: In weighted resistor DAC, the summing amplifier effectively adds weighted voltages. In digital systems, adders compute averages, sums, or digital filtering.

7.0 Display Devices

Light Emitting Diode (LED)

• Construction: pn junction (GaAs, GaP) with lens encapsulation.

• Working: Recombination of electrons and holes across pn junction emits light (electroluminescence). Color depends on semiconductor bandgap.

• Advantages: Low voltage, long life, fast switching, high brightness, small size.

• Disadvantages: Higher power consumption than LCD, requires current limiting, ambient light visibility issues.

Liquid Crystal Display (LCD)

• Working Principle (Twisted Nematic):

  1. Liquid crystals naturally form helical twist (90°).

  2. Polarizers on front/back at 90°.

  3. Without voltage, light passes through both polarizers (twist rotates polarization).

  4. With voltage, crystals align → no rotation → light blocked by second polarizer → dark pixel.

• Advantages: Very low power (battery-operated), no glare, wide viewing angle (modern).

• Disadvantages: Slow response (ms), temperature sensitive, requires backlight (transmissive type), limited brightness.

[!TIP] Exam Focus: Be prepared to derive bridge balance conditions (Schering, Hay's, Wien) and Q-meter expressions. For CRO, know block diagram, CRT structure, and dual trace vs dual beam differences. For digital converters, understand conversion logic, clock pulses, and step size calculations.