UNIT 2: ELECTRONIC MEASUREMENTS AND INSTRUMENTATION

1.0 FUNDAMENTALS OF MEASUREMENT

1.1 Static Characteristics of Instruments

• Accuracy: Closeness of the measured value to the true value. It is a qualitative term.

• Precision: Closeness of agreement between a set of measurements. It has repeatability and reproducibility aspects.

• Resolution: Smallest change in input quantity that produces a detectable change in output.

[!TIP] Exam Focus: Accuracy vs. Precision is a very common 7-mark question. Use the target diagram in your answer.

• Static Error (e): Difference between measured value ($$\displaystyle A_m $$) and true value ($$\displaystyle A_t $$).

$$e = A_m - A_t$$

• Static Correction (C): Negative of static error. $$\displaystyle C = -e = A_t - A_m $$. It is added to the measured value to get the true value.

• Relative Error ($\delta$): Ratio of absolute error to the true value.

$$\delta = \frac{e}{A_t}$$

• Percentage Relative Error ($\%\delta$): $$\displaystyle \%\delta = \delta \times 100\% $$

1.2 Errors in Measurement

• Gross Errors: Human mistakes (reading, recording). Minimized by careful reading and repetition.

• Systematic Errors: Consistent, predictable. Sub-classified as:

  • Instrumental: Defect in instrument (e.g., friction, calibration).

  • Environmental: Due to external conditions (temp, humidity, EMI).

  • Observational: Parallax, bias.

  • Systematic errors can be minimized by calibration.

• Random Errors: Unpredictable fluctuations (noise). Reduced by statistical analysis (mean, standard deviation).

[!TIP] Calibration is the process of comparing an instrument with a standard to determine its error and apply correction. It is fundamental to reducing systematic errors.

1.3 Dynamic Characteristics

Describe how an instrument responds to time-varying inputs.

• Damping: Opposition to oscillation. Types:

  • Under-damped: Oscillates before settling. Slightly under-damped is often preferred as it settles faster than over-damped and avoids endless oscillation of critically damped.

  • Critically Damped: Fastest approach to final value without oscillation.

  • Over-damped: Slow, sluggish response.

  [!TIP] "Preference for Slightly Under-damped" is a repeatedly asked concept. Explain with a step response curve.

• Response to Standard Inputs:

  • Step Input: Tests damping and time constant ($\tau$).

  • Ramp Input: Tests tracking ability (e.g., speedometer).

  • Sinusoidal Input: Tests frequency response (bandwidth).

2.0 ANALOG VOLTMETERS & SPECIAL INSTRUMENTS

2.1 AC Voltmeters

• Average Responding: Measures average of rectified AC. Scale calibrated in RMS for sine wave ($$\displaystyle V_{rms} = 1.11 \times V_{avg} $$). Inaccurate for non-sinusoidal waves.

• True RMS Responding: Uses thermal or electronic techniques to measure heating effect. Accurate for any waveform.

• Peak Responding: Measures peak value. Scale calibrated in RMS for sine wave ($$\displaystyle V_{rms} = V_p/\sqrt{2} $$). Sensitive to peaks/spikes.

2.2 Chopper Type DC Voltmeter (Frequently Asked)

Principle: Converts DC input into AC using a chopper (modulator), amplifies the AC with a high-gain AC amplifier, then demodulates back to DC. Eliminates DC drift and low-frequency noise. Block Diagram & Working:

1. Chopper (Modulator): Mechanical (vibrating reed) or electronic (FET switch). Converts DC $$\displaystyle V_{in} $$ to a square wave AC signal centered around zero.

2. AC Amplifier: High-gain, high-input impedance amplifier. Amplifies the AC signal. No DC offset drift as it's an AC amplifier.

3. Demodulator (Detector): Rectifies and filters the amplified AC back to a DC proportional to $$\displaystyle V_{in} $$.

4. Output Meter: PMMC meter calibrated in DC volts.

[!TIP] Key advantage: High input impedance and elimination of 1/f noise & drift. Essential for measuring small DC signals.

2.3 Electrostatic Instruments

• Principle: Force between charged plates. Used as electrostatic voltmeters.

• Construction: Fixed plate (guard ring) and movable plate. Movement is by a spring.

• Force Equation: For parallel plates with area $A$, spacing $d$, voltage $V$:

$$F = \frac{1}{2} \frac{dC}{dd} V^2 = \frac{1}{2} \frac{\varepsilon_0 A}{d^2} V^2$$

Deflection $$\displaystyle \propto V^2 $$. Scale is **non-linear (square law)**.

• Permissible Errors in Ammeters/Voltmeters: Refer to the maximum allowable error specified by the manufacturer, often given as a percentage of full-scale deflection (FSD) or reading.

2.4 Signal Generators

• Sweep Frequency Generator (Frequently Asked)

  Purpose: Generates a sinusoidal output whose frequency varies smoothly (sweeps) over a specified range.

  Block Diagram:

  [Waveform Generator (e.g., VCO)] -> [Sweep Control Circuit (Ramp Generator)] -> [Output Amplifier]

  Working: A low-frequency ramp voltage from the sweep control circuit is applied to the Voltage Controlled Oscillator (VCO). The VCO's output frequency is a function of its control voltage. Thus, as the ramp increases, the output frequency sweeps linearly (or logarithmically) upward, then resets. Used for frequency response analysis of filters, amplifiers.

  DiagramSEARCH: sweep frequency generator block diagram

• Pulse Wave Generator

  Purpose: Generates standard pulse waveforms (rectangular).

  Working: Uses a multivibrator (astable or monostable) circuit. Key controls: Pulse repetition frequency (PRF), pulse width, amplitude. Often uses a step recovery diode for very fast rise times.

3.0 CATHODE RAY OSCILLOSCOPE (CRO) & VARIANTS

3.1 Basic CRO

• Block Diagram & Functional Description:

  1. Vertical Amplifier: Amplifies the input signal under test.

  2. Delay Line: Provides small delay to allow sweep circuit to start.

  3. Trigger Circuit: Synchronizes sweep with input signal for stable display.

  4. Time Base (Sweep) Generator: Generates a linear ramp voltage for horizontal deflection (time axis).

  5. Horizontal Amplifier: Amplifies sweep voltage.

  6. Power Supply: Provides high voltages for CRT.

  DiagramSEARCH: CRO block diagram

• CRT Construction & Internal Structure:

  • Electron Gun: Cathode (electron emitter), Control Grid (intensity control), Focusing System (electrostatic or magnetic), Accelerating Anode.

  • Deflection System: Electrostatic (most common). Two pairs of plates: Vertical (Y) and Horizontal (X).

  • Screen: Fluorescent material (e.g., P31) that glows when hit by electrons.

  • Glass Envelope: Evacuated to high vacuum.

  DiagramSEARCH: CRT internal structure diagram

• Graticules: Grid lines on the screen (usually 1cm x 1cm) for measuring voltage (vertical) and time (horizontal).

3.2 CRO Probes

• Importance: Isolate CRO input capacitance from circuit, prevent loading, provide attenuation, safety.

• Types: 1:1 (direct), 10:1 (most common), 100:1, active, current probes.

• 10:1 Probe (Circuit & Operation):

  • Circuit: Series resistor ($$\displaystyle R_s $$, typically 9 MΩ) and a compensating capacitor ($$\displaystyle C_c $$) in parallel with the CRO's input capacitance ($$\displaystyle C_{in} $$).

  • Operation: Forms an RC attenuator. At the probe tip, $$\displaystyle R_s $$ and $$\displaystyle C_{in} $$ form a voltage divider. $$\displaystyle C_c $$ is adjusted (compensated) so that the time constants $$\displaystyle R_s C_{in} $$ and $$\displaystyle R_s C_c $$ are equal. This ensures frequency-independent attenuation (10:1) over a wide bandwidth. Prevents signal distortion due to CRO capacitance.

  DiagramCANVAS: 10:1 oscilloscope probe circuit showing 9MΩ resistor, trimmer capacitor, and CRO input capacitance

3.3 Types of CROs

• Dual Beam vs Dual Trace (Frequently Asked)

  | Feature | Dual Beam CRO | Dual Trace CRO | | :--- | :--- | :--- | | CRT | Two separate electron guns & deflection systems | Single electron gun, fast electronic switch | | Simultaneity | True simultaneous display of two signals | Alternate/Chopped display (time-shared) | | Bandwidth | Higher (no switching) | Lower (limited by switching speed) | | Cost/Complexity | Higher | Lower | | Use | High-speed, unrelated signals | General-purpose, related signals |

  DiagramSEARCH: dual beam vs dual trace CRO block diagram comparison

• Sampling Oscilloscope (Frequently Asked)

  Purpose: Measure very high-frequency signals (GHz) beyond the bandwidth of a real-time CRO.

  Principle: Equivalent-time sampling. Takes one sample from each successive cycle of the input signal, with a small incremental delay. After many cycles, these samples are stitched together to reconstruct the waveform.

  Circuit & Operation:

  1. Input signal is fed to a sample-and-hold (S/H) circuit.

  2. A sampling clock (derived from the input trigger) controls the S/H. The sampling instant is advanced by a tiny amount ($\Delta t$) on each cycle.

  3. The sampled values are stored and displayed on a slow X-Y plotter or digital storage.

  Limitation: Cannot capture non-repetitive or single-shot events.

  DiagramSEARCH: sampling oscilloscope block diagram equivalent time sampling

• Digital Storage Oscilloscope (DSO) (Frequently Asked)

  Block Diagram & Working:

  1. Vertical System: Analog input -> Attenuator/Amplifier -> Anti-aliasing Filter -> Sample-and-Hold (S/H).

  2. Analog-to-Digital Converter (ADC): Converts sampled voltage to digital word (e.g., 8-bit, 12-bit). Key component.

  3. Memory (FIFO): Stores digital samples.

  4. Time Base & Trigger Control: Manages sampling rate and display start point.

  5. Microprocessor/Controller: Processes data (measurements, averaging, FFT), controls display.

  6. Display: LCD/LED screen. Waveform is digitally reconstructed from memory.

  Advantages: Storage, processing (measurements, math), hard copy output, high bandwidth (with sampling), no drift.

  DiagramSEARCH: DSO block diagram

• Special Purpose CROs: Vector CRO (for Lissajous), Storage CRO (analog storage tube), Logic Analyzer interface.

4.0 AC BRIDGES & IMPEDANCE MEASUREMENT

4.1 General Bridge Theory

• Bridge Balance Condition: For a 4-arm bridge (ABCD), balance occurs when:

$$Z_1 Z_3 = Z_2 Z_4$$

Or in terms of admittances: $$\displaystyle Y_1 Y_3 = Y_2 Y_4 $$.

At balance, **detector current = 0**.

• Bridge Sensitivity: Deflection of detector (galvanometer) per unit change in unknown parameter.

  Condition for Maximum Sensitivity: The bridge should be initially balanced and the detector resistance should be matched to the Thevenin resistance of the bridge network at the detector terminals. Maximum power transfer occurs when $$\displaystyle R_g = R_{th} $$.

4.2 Specific Bridges

• Schering Bridge (Frequently Asked)

  Use: Measure unknown capacitance (C_x) and its dissipation factor (D.F. or tan δ).

  Circuit:

  • Arm AB: Known capacitor $$\displaystyle C_1 $$ (standard) in parallel with resistor $$\displaystyle R_1 $$.

  • Arm BC: Unknown capacitor $$\displaystyle C_x $$ in parallel with its loss resistor $$\displaystyle R_x $$ (represents D.F.).

  • Arm DA & CD: Pure resistors $$\displaystyle R_2 $$, $$\displaystyle R_3 $$.

  DiagramSEARCH: Schering bridge circuit diagram

  Balance Equations:

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

$$\left( R_1 - j \frac{1}{\omega C_1} \right) \left( R_3 \right) = \left( R_x - j \frac{1}{\omega C_x} \right) \left( R_2 \right)$$

Separating real & imaginary parts:

$$R_3 = \frac{R_x R_2}{R_1} \quad \text{and} \quad \frac{1}{\omega C_1} = \frac{R_2}{\omega C_x R_1}$$

**Final Expressions**:

$$C_x = C_1 \frac{R_1}{R_2}$$

$$\tan \delta_x = \omega C_x R_x = \omega C_1 R_1$$

**Applications**: Testing of capacitors, cables, insulators.

• Hay's Bridge (Frequently Asked)

  Use: Measure high-Q coils (inductance $$\displaystyle L_x $$ and resistance $$\displaystyle R_x $$).

  Circuit:

  • Arm AB: Unknown inductor $$\displaystyle L_x $$ in series with $$\displaystyle R_x $$.

  • Arm BC: Standard capacitor $C$.

  • Arm DA & CD: Non-inductive resistors $$\displaystyle R_1 $$, $$\displaystyle R_2 $$.

  DiagramSEARCH: Hay's bridge circuit diagram

  Balance Equations:

$$Z_1 Z_3 = Z_2 Z_4 \implies (R_x + j\omega L_x) R_2 = R_1 \left( \frac{1}{j\omega C} \right)$$

Separating:

$$R_2 R_x = -\frac{R_1}{\omega C} \quad \text{(Impossible as LHS positive, RHS negative)}$$

**Correction**: Hay's bridge uses a **standard capacitor in series with a resistor** in one arm (typically arm BC). Let arm BC be $$\displaystyle R_3 $$ in series with $C$. Then:

$$(R_x + j\omega L_x) R_2 = R_1 \left( R_3 - j\frac{1}{\omega C} \right)$$

Real: $$\displaystyle R_x R_2 = R_1 R_3 \implies R_x = \frac{R_1 R_3}{R_2} $$

Imag: $$\displaystyle \omega L_x R_2 = -\frac{R_1}{\omega C} \implies L_x = \frac{R_1}{R_2} \cdot \frac{1}{\omega^2 C} $$

**Final Expressions**:

$$L_x = \frac{R_1 R_3}{\omega^2 C R_2}$$

$$Q = \frac{\omega L_x}{R_x} = \frac{1}{\omega^2 C R_3}$$

**Advantage over Maxwell**: Better for **high Q** coils ($$\displaystyle Q > 10 $$).

• Wien Bridge (Frequently Asked)

  Use: Frequency measurement in audio range (20 Hz - 20 kHz) and as a notch filter.

  Circuit: Arms AB & BC form a series RC and parallel RC network respectively. Arms DA & CD are equal resistors ($$\displaystyle R_1 = R_2 = R $$).

  DiagramSEARCH: Wien bridge circuit diagram

  Balance Condition:

  Let $$\displaystyle Z_1 = R + \frac{1}{j\omega C} $$, $$\displaystyle Z_2 = R \parallel \frac{1}{j\omega C} = \frac{R}{1 + j\omega RC} $$

  Balance: $$\displaystyle Z_1 Z_3 = Z_2 Z_4 $$, with $$\displaystyle Z_3 = Z_4 = R $$.

$$R \left( R + \frac{1}{j\omega C} \right) = R \cdot \frac{R}{1 + j\omega RC}$$

Simplifying:

$$R^2 + \frac{R}{j\omega C} = \frac{R^2}{1 + j\omega RC}$$

Cross-multiply & equate real/imag parts leads to:

$$\omega^2 = \frac{1}{R^2 C^2} \quad \text{and} \quad R_3 = R_4$$

**Frequency of Balance**:

$$\boxed{f = \frac{1}{2\pi RC}}$$

**Operation**: At balance, bridge is balanced **only at one frequency** for given R, C. By varying R or C and noting balance, frequency can be measured.

• Maxwell's Inductance-Capacitance Bridge

  Use: Measure medium-Q coils.

  Limitation for High Q Coils: Requires a standard variable capacitor which becomes impractically small and difficult to adjust for high-Q coils. Also, the balance condition involves a series combination of R and C, making it less convenient than Hay's bridge for high Q.

4.3 Q-Meter

• Principle: Based on series resonance in an RLC circuit. At resonance, $$\displaystyle X_L = X_C $$, circuit impedance is minimum (pure R), current is maximum. Q-factor is given by $$\displaystyle Q = \frac{\omega L}{R} = \frac{1}{\omega C R} $$.

• Parallel-Connection Method (Frequently Asked)

  Setup: Unknown impedance $$\displaystyle Z_x $$ is connected in parallel with the Q-meter's known coil (with $L$, $R$) and standard capacitor $C$.

  Expressions:

  Let the Q-meter reading (without $$\displaystyle Z_x $$) be $$\displaystyle Q_0 = \frac{\omega L}{R} $$.

  When $$\displaystyle Z_x $$ is connected in parallel, the new Q becomes $Q$.

  For inductive $$\displaystyle Z_x = R_x + j\omega L_x $$:

$$R_x = \frac{1}{Q - Q_0} \cdot \frac{1}{\omega C}$$

$$L_x = \frac{1}{\omega^2 C} \left( \frac{Q}{Q - Q_0} - 1 \right)$$

For **capacitive** $$\displaystyle Z_x = -j/(\omega C_x) $$:

$$C_x = C \left( \frac{Q_0}{Q} - 1 \right)$$

> **Note**: Derivation involves calculating the equivalent parallel resistance and reactance of the combined circuit at resonance.

5.0 TRANSDUCERS

5.1 Introduction & Classification

• Transducer: Device that converts a physical quantity (non-electrical) into an electrical signal (or vice versa).

• Classification:

  1. Based on Energy: Active (self-generating, e.g., piezo, thermocouple) vs. Passive (require external power, e.g., potentiometer, LVDT).

  2. Based on Principle: Resistive, Inductive, Capacitive, Piezoelectric, Photoelectric, etc.

  3. Based on Output: Analog vs. Digital.

  4. Based on Application: Primary (directly senses) vs. Secondary (converts primary's output).

5.2 Resistive Transducers

• Potentiometer: Simple, used for displacement/position measurement. Wiper voltage $$\displaystyle V_{out} = V_{in} \frac{R_2}{R_1+R_2} $$. Limited by contact resistance & wear.

• Thermistor (Frequently Asked)

  • Importance: Highly sensitive temperature sensor (resistance changes ~3-6%/°C). Used in temperature measurement, compensation, inrush current limiting.

  • Advantages:

    • High sensitivity (much higher than RTD).

    • Small size, fast response.

    • Low cost.

    • Can measure small temperature changes.

  • Disadvantages: Non-linear resistance-temperature curve, needs linearization. Limited temperature range (typically -50°C to 150°C). Self-heating error.

• Strain Gauge: Resistance changes with strain (metallic foil or semiconductor). Uses Wheatstone bridge.

5.3 Inductive Transducers

• Linear Variable Differential Transformer (LVDT) (Very Frequently Asked)

  Construction: One primary winding (P) on a central bobbin, two identical secondary windings (S1, S2) on either side, connected in series opposition. Movable ferromagnetic core.

  DiagramSEARCH: LVDT construction diagram

  Principle & Operation:

  • AC voltage $$\displaystyle V_p $$ applied to primary.

  • Induces voltages $$\displaystyle V_{s1} $$, $$\displaystyle V_{s2} $$ in secondaries.

  • Core at Null Position (Center): Flux linkage equal, $$\displaystyle V_{s1} = V_{s2} $$, net output $$\displaystyle V_o = 0 $$.

  • Core Displaced Right: Flux in S1 > S2, $$\displaystyle V_o = V_{s1} - V_{s2} > 0 $$ (in-phase with $$\displaystyle V_p $$).

  • Core Displaced Left: Flux in S2 > S1, $$\displaystyle V_o = V_{s1} - V_{s2} < 0 $$ (out-of-phase with $$\displaystyle V_p $$ by 180°).

  • Output magnitude $\propto$ displacement magnitude. Phase indicates direction.

  Advantages: Infinite resolution, frictionless, high reliability, linear over wide range (around null), differential output rejects common-mode noise.

  Disadvantages: Requires AC excitation & demodulation, bulky, sensitive to stray magnetic fields.

5.4 Capacitive Transducers

• Principle: Capacitance $$\displaystyle C = \frac{\varepsilon A}{d} $$ changes with area (A), distance (d), or dielectric constant ($\varepsilon$).

• Types:

  • Parallel Plate: Change in d (displacement) or A (angular displacement). $C \propto 1/d$ (non-linear) or $\propto A$ (linear).

  • Differential Capacitor: Two capacitors, opposite changes. Improves linearity & sensitivity (used in precision micrometers).

  • Rotary: Change in overlapping area.

5.5 Piezo-electric Transducers (Frequently Asked)

• Working Principle: Certain crystals (Quartz, Rochelle salt, PZT) generate electric charge on their surfaces when subjected to mechanical stress (direct effect). Conversely, they deform when voltage is applied (converse effect).

  Charge Generated: $$\displaystyle Q = d \cdot F $$, where $d$ is piezo coefficient, $F$ is force.

  Voltage Generated: $$\displaystyle V = \frac{Q}{C} = \frac{d \cdot F}{C} $$, where C is crystal capacitance.

• Modes of Operation:

  • Longitudinal (Thickness): Stress & electric field along same axis. $$\displaystyle d_{33} $$ mode.

  • Transverse (Width): Stress along one axis, field along another. $$\displaystyle d_{31} $$ mode.

  • Shear: Shear stress produces charge. $$\displaystyle d_{15} $$ mode.

• Binders & Twisters: Terms used in ultrasonic transducers.

  • Binder: A matching layer to impedance match the high-impedance piezo crystal to the low-impedance medium (e.g., tissue, water). Thickness = λ/4.

  • Twister: A backing material (damping) to absorb backward vibrations, shorten pulse duration, improve resolution.

5.6 Optical Transducers (Frequently Asked)

• Photoemissive (Photoemission): Light ejects electrons from a cathode (photocell, photomultiplier tube). Requires high vacuum. Current output.

• Photoconductive: Light decreases resistance of a semiconductor (LDR, photodiode in photoconductive mode). Resistance change.

• Photovoltaic: Light generates voltage across a PN junction (solar cell, photodiode in photovoltaic mode). Voltage output.

• Comparison:

  | Type | Output | Speed | Sensitivity | Common Device | | :--- | :--- | :--- | :--- | :--- | | Photoemissive | Current | Very Fast | Very High | PMT | | Photoconductive | Resistance | Fast | Medium | LDR, Photodiode | | Photovoltaic | Voltage | Fast | Medium | Solar Cell, Photodiode |

• Photodiode (Detailed Working):

  • Construction: PN junction with a window for light to enter. Operated in reverse bias (photoconductive mode) or zero bias (photovoltaic).

  • Working: Photons with energy > bandgap generate electron-hole pairs in the depletion region. These are swept by the electric field, creating a photocurrent proportional to light intensity.

  • Modes:

    • Photovoltaic: No bias. Generates voltage. Used in solar cells, light meters.

    • Photoconductive: Reverse biased. Faster response, higher gain. Used in optical communication, fiber optics.

• Phototransistor: Photodiode base current controls transistor current. Higher gain (β), slower than photodiode.

• Light Emitting Diode (LED) (Frequently Asked)

  • Construction: PN junction, heavily doped, forward biased. Encased in transparent epoxy.

  • Working: Recombination of electrons & holes across the junction releases energy as photons (light). Color depends on semiconductor bandgap (e.g., GaAs for IR, GaAsP for red/green, GaN for blue/white).

  • Advantages: Low voltage, low current, fast switching, long life, small size.

  • Disadvantages: Monochromatic (unless phosphor coated), viewing angle limited, temperature sensitive.

5.7 Other Transducers (Short Notes)

• Thermocouple: Two dissimilar metals joined. Seebeck effect: temperature difference generates voltage. Measures temperature, wide range.

• Accelerometer: Measures acceleration. Types: piezoelectric, capacitive, LVDT. Used in vibration, shock, inertial navigation.

• Bolometer: Measures RF/microwave power. Absorbs radiation, heats a thermistor/thermocouple, resistance/voltage changes. High sensitivity.

6.0 DIGITAL INSTRUMENTS: DVM, ADC & DAC

6.1 Digital Voltmeter (DVM)

• Significance of ½ Digit (Frequently Asked):

  • A "digit" can display 0-9.

  • ½ Digit can only display 0 or 1 (most significant digit).

  • Example: 3½ digit DVM has 3 full digits (0-9) and 1 half-digit (0-1). Max reading = 1999 counts.

  • Significance: Defines the maximum count and over-range capability. The half-digit indicates the instrument can show an over-range condition (e.g., "1" in 3½ digit when input > 199.9 mV on a 200 mV range).

• 3½ Digit & 5½ Digit DVM:

  • 3½ Digit: Max display 1999. Common in multimeters. Resolution ~1 mV on 2V range.

  • 5½ Digit: Max display 19999. Higher precision lab instruments. Resolution ~10 µV on 2V range.

• Resolution & Sensitivity (Frequently Asked):

  • Resolution: Smallest change in input that causes a change in the LSB of the output. For an n-bit ADC over range V, resolution = $$\displaystyle V/(2^n - 1) $$ or $$\displaystyle V/2^n $$ (approx).

  • Sensitivity: Minimum input signal that produces a full-scale output. Often same as resolution for a given range.

  • Example: 3½ digit DVM on 10V range: 4-digit display (max 1999). Resolution = $10V / 1999 \approx 5mV$. Sensitivity = 5mV.

• Types of DVMs (Based on Conversion):

  1. Integrating Type (Dual-slope): High noise rejection, good accuracy.

  2. Successive Approximation Type: Fast, common in general-purpose DVMs.

  3. Flash (Parallel) Type: Fastest, used in high-speed scopes.

  4. Voltage-to-Frequency Type: Good for noisy environments, long-distance transmission.

6.2 Analog-to-Digital Converters (ADC)

• Resolution (n-bit ADC): Number of discrete levels = $$\displaystyle 2^n $$. Smallest step (LSB size) = $$\displaystyle V_{FS} / (2^n - 1) $$, where $$\displaystyle V_{FS} $$ is full-scale voltage range.

• Successive Approximation ADC (Very Frequently Asked)

  Circuit: SAR register, DAC (usually R-2R ladder), comparator, control logic.

  Operation:

  1. Start: SAR sets MSB to 1, others 0. DAC output = $$\displaystyle V_{FS}/2 $$.

  2. Compare: Comparator compares $$\displaystyle V_{in} $$ with $$\displaystyle V_{DAC} $$.

     • If $$\displaystyle V_{in} > V_{DAC} $$: MSB remains 1.

     • If $$\displaystyle V_{in} < V_{DAC} $$: MSB cleared to 0.

  3. Next Bit: SAR sets next bit to 1, repeats comparison.

  4. Continue for all n bits. n clock cycles needed.

  Speed: Moderate (µs range). Advantage: Good speed-accuracy trade-off.

  DiagramSEARCH: successive approximation ADC block diagram

• Dual-Slope (Integrating) ADC (Frequently Asked)

  Circuit: Integrator, comparator, control logic, counter, reference voltage ($$\displaystyle V_{ref} $$).

  Operation:

  1. Integrate Phase (Fixed Time $$\displaystyle T_1 $$): $$\displaystyle V_{in} $$ (positive or negative) is integrated. Output slope $$\displaystyle \propto V_{in} $$. Final integrator voltage $$\displaystyle V_1 \propto V_{in} $$.

  2. De-integrate Phase (Variable Time $$\displaystyle T_2 $$): Switch to $$\displaystyle -V_{ref} $$ (opposite polarity). Integrator output ramps down at fixed slope. Counter counts clock pulses until output returns to 0. $$\displaystyle T_2 \propto V_1 \propto V_{in} $$.

  Digital Output: Count in $$\displaystyle T_2 $$ is proportional to $$\displaystyle V_{in} $$.

  Advantages: Excellent noise rejection (averages input over $$\displaystyle T_1 $$), high accuracy, no need for precise DAC. Disadvantage: Slow (max speed ~100 Hz).

  DiagramSEARCH: dual slope ADC block diagram

• Clock Pulses Comparison:

  | ADC Type | Clock Pulses Needed (for n-bit) | Speed | Notes | | :--- | :--- | :--- | :--- | | Flash | 1 cycle | Fastest (ns) | $$\displaystyle 2^n - 1 $$ comparators, expensive | | SAR | n cycles | Moderate (µs) | Most common in DVMs | | Dual Slope | Max = $$\displaystyle 2^n $$ | Slow (ms) | Noise immune, accurate | | Counter Type | Up to $$\displaystyle 2^n $$ | Slow | Simple, but inefficient |

6.3 Digital-to-Analog Converters (DAC)

• Weighted Resistor DAC (Frequently Asked)

  Circuit: n-bit binary input controls n switches. Each switch connects a resistor ($$\displaystyle R, 2R, 4R... 2^{n-1}R $$) to either $$\displaystyle V_{ref} $$ or ground. Summing amplifier adds currents.

  Operation: Output voltage $$\displaystyle V_o = -\frac{V_{ref}}{R} \left( \frac{b_1}{2} + \frac{b_2}{4} + ... + \frac{b_n}{2^n} \right) R_{f} $$ (if $$\displaystyle R_f = R $$).

  Transfer Characteristic (3-bit example): Step size (LSB) = $$\displaystyle V_{ref}/8 $$. Output jumps by LSB for each increment in binary input. Non-ideal: Switch resistance mismatch causes non-linearity.

  DiagramSEARCH: weighted resistor DAC circuit

• R-2R Ladder DAC (Very Frequently Asked)

  Principle: Uses only two resistor values (R and 2R). Each bit position has identical structure, simplifying fabrication.

  Circuit (3-bit):

  • Ladder network of R and 2R resistors.

  • Each bit node connected to $$\displaystyle V_{ref} $$ (if bit=1) or ground (if bit=0) via a switch.

  • Output taken from the "virtual ground" node of an op-amp summing junction.

  DiagramSEARCH: R-2R ladder DAC 3-bit circuit

  Operation: The ladder presents Thevenin equivalent resistance = R to ground at each node. Each bit contributes a current $$\displaystyle I_b = V_{ref}/(2R) $$ if high. Output $$\displaystyle V_o = -R_f \cdot \sum I_b $$.

  For 3-bit (b2 b1 b0): $$\displaystyle V_o = -\frac{V_{ref}}{8} \left( 4b_2 + 2b_1 + b_0 \right) $$ (if $$\displaystyle R_f = R $$).

  Step Size (LSB): $$\displaystyle V_{LSB} = V_{ref} / 2^n $$.

  Example Calculation (Dec 2024 Q): 4-bit R-2R, $$\displaystyle V_{ref}=5V $$.

  • Input 0111 (7): $$\displaystyle V_o = -\frac{5}{16} \times 7 = -2.1875 V $$

  • Input 1111 (15): $$\displaystyle V_o = -\frac{5}{16} \times 15 = -4.6875 V $$

  Advantage: Easy to fabricate, high accuracy, good speed.

7.0 DISPLAY DEVICES

7.1 Light Emitting Diode (LED)

• Construction: PN junction, forward biased. Epoxy lens shapes light.

• Working: Electron-hole recombination releases photons (electroluminescence).

• Advantages: Low voltage/current, fast, long life, small, rugged, high brightness.

• Disadvantages: Monochromatic (unless phosphor), viewing angle, temperature sensitive, requires current limiting resistor.

7.2 Liquid Crystal Display (LCD) (Frequently Asked)

• Construction & Working Principle:

  1. Structure: Two glass plates with transparent ITO electrodes. Spacer maintains gap (~10µm). Liquid crystal (LC) material fills gap. Polarizers on front & back.

  2. LC Properties: Rod-shaped molecules, twist naturally (90° in Twisted Nematic - TN). Anisotropic refractive index.

  3. Operation (TN type):

     • No Voltage: LC molecules twist light's polarization by 90°. Light passes through front polarizer -> LC -> rear polarizer (aligned perpendicularly to front) -> ON (bright).

     • Voltage Applied: Electric field aligns LC molecules vertically. No twist. Light's polarization not rotated. Blocked by rear polarizer -> OFF (dark).

  4. Color LCD: Uses color filter (RGB sub-pixels) and white backlight.

  DiagramSEARCH: twisted nematic LCD operation with and without voltage

• Advantages: Very low power (bias only), no self-emission (good for dark environments), cheap, wide viewing angle (in modern IPS).

• Disadvantages: Slow response time (ms), limited temperature range, requires backlight (for transmissive), viewing angle dependent (TN).

• Comparison with LED:

  | Feature | LED | LCD | | :--- | :--- | :--- | | Emission | Self-emissive | Transmissive/Reflective (needs light) | | Power | Higher per pixel | Very low (bias only) | | Viewing Angle | Wide | Limited (TN), Wide (IPS) | | Response Time | Very Fast (ns) | Slow (ms) | | Brightness | High | Medium (depends on backlight) | | Use | Indicators, displays (low res) | Monitors, TVs, instrument panels |

END OF UNIT 2 NOTES