UNIT 5: Electronic Measurements and Instrumentation

I. Fundamental Concepts in Measurement

Accuracy, Precision, and Resolution

• Accuracy: Closeness of the measured value to the true value or standard. It is a measure of systematic error.

• Precision: Closeness of agreement among a series of measurements made on the same quantity. It indicates repeatability or reproducibility and is a measure of random error.

• Resolution: The smallest change in the input quantity that produces a detectable change in the output. For digital instruments, it is the value of the least significant digit (LSD).

[!TIP] Exam Focus: A precise instrument can be inaccurate (consistent but wrong). A high-resolution instrument is not necessarily accurate. Accuracy requires both calibration and precision.

Static Characteristics

• Static Error ($$\displaystyle e_s $$): The difference between the measured value ($$\displaystyle A_m $$) and the true value ($$\displaystyle A_t $$).

$$e_s = A_m - A_t$$

• Static Correction ($$\displaystyle C_s $$): The correction to be applied to the instrument reading to obtain the true value. It is the negative of static error.

$$C_s = -e_s = A_t - A_m$$

• Relative Error ($$\displaystyle \epsilon_r $$): Error expressed as a fraction of the true value.

$$\epsilon_r = \frac{e_s}{A_t}$$

• Percentage Relative Error ($$\displaystyle \% \epsilon_r $$): Relative error in percentage.

$$\% \epsilon_r = \frac{e_s}{A_t} \times 100\%$$

Calibration

The process of determining the experimental transfer function of an instrument by comparing it with a standard of known accuracy. It establishes the relationship between the input and output. Regular calibration is essential to maintain accuracy and traceability.

Types of Errors

1. Gross Errors: Human mistakes (reading, recording).

2. Systematic Errors: Consistent, predictable errors (instrumental, environmental, observational). Can be calibrated out.

3. Random Errors: Unpredictable fluctuations (noise). Reduced by statistical averaging.

Dynamic Characteristics & Damping

Describes the response of an instrument to a time-varying input. For a second-order damped system, the response is governed by:

$$\frac{d^2y}{dt^2} + 2\zeta\omega_n\frac{dy}{dt} + \omega_n^2 y = K \omega_n^2 x(t)$$

where $\zeta$ is the damping factor, $$\displaystyle \omega_n $$ is the natural frequency.

[!TIP] Why Slightly Underdamped? A damping factor $\zeta \approx 0.6 - 0.7$ (slightly underdamped) provides a fast response without excessive overshoot or oscillation. Critically damped ($$\displaystyle \zeta=1 $$) is slower; overdamped ($$\displaystyle \zeta>1 $$) is very sluggish.

Digital vs. Analog Instruments

II. Voltage Measurement Instruments

A. AC Voltmeters

Types: Rectifier Type (average responding, RMS calibrated), True RMS Voltmeters, Thermal Voltmeters, Sampling Voltmeters.

Detailed Working: Rectifier Type AC Voltmeter

• Principle: Converts AC to DC using a rectifier, then measures the average value of the rectified signal. The scale is calibrated in RMS assuming a sinusoidal input.

• Circuit: AC input → Rectifier (half-wave or full-wave) → DC Ammeter (or DC voltmeter with series resistor).

• For Sinusoidal Input: $$\displaystyle V_{rms} = \frac{V_{avg}}{\text{Form Factor}} $$. For sine wave, Form Factor = 1.11.

$$V_{rms} = 1.11 \times V_{avg}$$

• Limitation: Inaccurate for non-sinusoidal waveforms because the form factor changes.

B. DC Voltmeters

1. Chopper Type DC Voltmeter (Detailed Operation)

• Purpose: Measures very low DC voltages (µV range) with high input impedance and low drift.

• Principle: Uses a chopper amplifier (modulator + amplifier + demodulator) to convert DC signal into AC, amplify it with high gain and stability, then convert back to DC.

• Block Diagram:

  DC Input → Chopper (Modulator) → AC Amplifier → Demodulator → Filter → DC Output/Meter

• Advantage: Eliminates drift and 1/f noise of DC amplifiers, provides high gain.

2. Electrostatic Voltmeter

• Construction: Two sets of parallel plates: Fixed (stator) and Movable (rotor). Often uses guard rings to ensure uniform electric field.

• Principle: Based on electrostatic force of attraction between oppositely charged plates.

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

where $C$ is capacitance, $x$ is displacement, $V$ is applied voltage.

• Force-Capacitance Relationship: For a parallel plate capacitor with guard rings, the force is:

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

where $$\displaystyle \varepsilon_0 $$ = permittivity of free space, $A$ = plate area, $d$ = plate separation.

• Characteristics: Very high input impedance (no current drawn), used for high voltage (kV range) measurement. Non-linear scale.

[!EXAMPLE] Numerical: Given $$\displaystyle V=10 $$ kV, $$\displaystyle F=5 \times 10^{-3} $$ N, $d$ changes by 1 mm, plate diameter 100 mm ($$\displaystyle A = \pi (0.05)^2 $$). Find $\Delta C$.

Solution: From $$\displaystyle F = \frac{1}{2} \frac{dC}{dx} V^2 \Rightarrow dC = \frac{2F dx}{V^2} $$. Substitute values to find $\Delta C$.

III. Cathode Ray Oscilloscope (CRO)

A. Basic Structure and Operation

• Block Diagram:

  Vertical Amplifier → Delay Line → Vertical Deflection Plates

  Horizontal Amplifier (Time Base) → Horizontal Deflection Plates

  Trigger Circuit → Time Base

  CRT → Graticule → Display

• Internal Structure of CRT:

  1. Electron Gun: Cathode (electron emitter), Control Grid (intensity focus), Focusing System (electrostatic/electromagnetic), Accelerating Anode.

  2. Deflection System: Electrostatic (plates) for CROs; Electromagnetic (yokes) for TVs.

  3. Fluorescent Screen: Converts electron energy to light (phosphor).

• Electrostatic Focusing: Uses a focus electrode (part of anode) to create an electrostatic lens that converges the electron beam to a fine spot on the screen.

• Graticules: A grid of lines on the CRT face (usually 1 cm x 1 cm) used for visual measurement of amplitude and time.

B. Types of CROs

• Dual Beam CRO: Uses two separate electron guns and two sets of deflection plates. Can display two signals simultaneously with independent control. Faster, more expensive.

• Dual Trace CRO: Uses one electron gun and a fast electronic switch to alternate between two input channels. Displays signals alternately on the same beam. Cheaper, but cannot show very fast, unrelated signals simultaneously.

• Sampling Oscilloscope: Used for very high frequencies (>1 GHz). Takes samples of the input waveform at precise intervals, stores them, and reconstructs the waveform. Stroboscopic principle.

• Special Purpose CROs: Storage CROs (DSO), Digital CROs, Vector CROs (for Lissajous), PC-based CROs.

C. Digital Storage Oscilloscope (DSO)

• Working Principle:

  1. Analog Front End: Input signal → Attenuator/Amplifier → Anti-aliasing filter → Sample & Hold (S/H) circuit.

  2. ADC: S/H output → Analog-to-Digital Converter (Flash or SAR type) → Digital data.

  3. Memory & Control: Digital data stored in memory (FIFO). Microprocessor controls acquisition, display, and measurement.

  4. DAC & Display: Stored digital data → Digital-to-Analog Converter → Analog signal for CRT/LCD display. Also used for on-screen menus and automated measurements.

• Storage & Retrieval: Waveform data is stored in digital memory. Can be frozen, processed (math operations), and saved to disk/USB.

D. CRO Probes

• Importance: Isolate the CRO from the circuit under test, minimize loading, provide safety, and attenuate high voltages.

• Types: 1:1 (direct), 10:1 (attenuating), Active (for high impedance, high frequency).

• 10:1 Probe Circuit & Operation:

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

  • Operation: Forms a voltage divider with $$\displaystyle C_{in} $$. $$\displaystyle R_s $$ provides 10:1 attenuation. $$\displaystyle C_c $$ is adjusted (compensated) so that $$\displaystyle R_s C_c = R_{in} C_{in} $$, ensuring frequency-independent attenuation.

IV. Bridge Circuits for Impedance Measurement

A. General Bridge Concepts

• Bridge: Four-arm circuit (AB, BC, CD, DA) with a source across one diagonal (AC) and a detector (null meter) across the other (BD).

• Balance Condition: $$\displaystyle Z_1 Z_4 = Z_2 Z_3 $$ (product of opposite arms equal). At balance, detector current = 0.

• Bridge Sensitivity ($S$): Defined as the deflection of detector per unit change in unknown parameter.

$$S = \frac{\theta}{\Delta Z_x}$$

**Condition for Maximum Sensitivity:** The **impedance of the detector** should be matched to the **Thevenin impedance** of the bridge network at the balance point. Also, source voltage should be as high as possible (within limits).

B. Specific Bridges

1. Maxwell Inductance-Capacitance Bridge

• Circuit: $$\displaystyle Z_1 = R_1 $$ (known), $$\displaystyle Z_2 = R_2 $$ (known), $$\displaystyle Z_3 = R_3 + j\omega L_3 $$ (standard inductor), $$\displaystyle Z_4 = R_4 + j\omega L_4 $$ (unknown $$\displaystyle L_x = L_4 $$, $$\displaystyle R_x = R_4 $$).

• Balance Equations:

$$R_4 = \frac{R_1 R_3}{R_2}$$

$$L_4 = R_1 R_3 C_1$$

• Limitation for High Q Coils: The standard capacitor $$\displaystyle C_1 $$ must be very small and loss-free for high $Q$ ($$\displaystyle Q = \omega L_x / R_x $$). This is difficult to achieve practically. Also, the bridge requires a variable standard capacitor in parallel with a known resistor.

2. Hay's Bridge (Improved Maxwell Bridge)

• Circuit: $$\displaystyle Z_1 = R_1 $$ (known), $$\displaystyle Z_2 = R_2 $$ (known), $$\displaystyle Z_3 = R_3 + j\omega L_3 $$ (standard inductor), $$\displaystyle Z_4 = R_4 + \frac{1}{j\omega C_4} $$ (unknown $$\displaystyle L_x = L_4 $$, $$\displaystyle R_x = R_4 $$).

• Balance Equations:

$$R_4 = \frac{\omega^2 R_1 R_2 R_3 C_4}{1 + \omega^2 R_3^2 C_4^2}$$

$$L_4 = \frac{R_1 R_2 C_4}{1 + \omega^2 R_3^2 C_4^2}$$

For **high Q coils** ($$\displaystyle \omega R_3 C_4 \ll 1 $$), simplifies to:

$$R_4 \approx \omega^2 R_1 R_2 R_3 C_4$$

$$L_4 \approx R_1 R_2 C_4$$

• Advantage over Maxwell: Uses a standard capacitor in series with a resistor. Easier to obtain high-quality, larger-value capacitors. Suitable for medium to high Q coils.

[!EXAMPLE] Numerical (Hay's Bridge): Given: $$\displaystyle R_1=600\Omega $$, $$\displaystyle L_3=0.18H $$, $$\displaystyle R_2=R_1=1200\Omega $$, $$\displaystyle f=50Hz $$, $$\displaystyle V=3V $$. Find $$\displaystyle R_4 $$ and $$\displaystyle C_4 $$.

Solution: First derive balance conditions. $$\displaystyle \omega = 2\pi f = 100\pi $$. Use simplified equations for high Q (verify $$\displaystyle \omega R_3 C_4 \ll 1 $$). Solve for $$\displaystyle C_4 $$ from $$\displaystyle L_4 $$ equation, then $$\displaystyle R_4 $$.

3. Schering Bridge

• Circuit: $$\displaystyle Z_1 = R_1 $$ (known), $$\displaystyle Z_2 = R_2 $$ (known), $$\displaystyle Z_3 = R_3 + \frac{1}{j\omega C_3} $$ (standard capacitor with parallel loss $$\displaystyle R_3 $$), $$\displaystyle Z_4 = R_4 + \frac{1}{j\omega C_4} $$ (unknown $$\displaystyle C_x = C_4 $$, with parallel loss $$\displaystyle R_4 $$).

• Balance Equations:

$$C_4 = \frac{R_1 C_3}{R_2}$$

$$R_4 = \frac{R_2}{R_1} R_3$$

• Derivation: From $$\displaystyle Z_1 Z_4 = Z_2 Z_3 $$, separate real and imaginary parts.

• Applications: Measurement of capacitors, insulation resistance (loss tangent), cable testing, quality factor of capacitors.

4. Wien Bridge (for Frequency Measurement)

• Circuit for Audio Frequency: Used in oscillators and frequency measurement. Bridge arms: $$\displaystyle Z_1 = R_1 $$, $$\displaystyle Z_2 = R_2 + C_2 $$ (series RC), $$\displaystyle Z_3 = R_3 $$, $$\displaystyle Z_4 = R_4 + C_4 $$ (series RC). Often $$\displaystyle R_1=R_3=R $$, $$\displaystyle R_2=R_4=R' $$, $$\displaystyle C_2=C_4=C' $$.

• Frequency Equation: At balance,

$$\omega = \frac{1}{RC'} \quad \text{or} \quad f = \frac{1}{2\pi RC'}$$

and $$\displaystyle \frac{R'}{R} = 2 $$ for symmetrical case.

• Operation: Vary standard capacitor $C'$ or frequency of source until null. Read frequency from calibrated $C'$ or known $R,R'$.

C. Q-Meter

• Principle: Based on series resonance in an $L-C$ circuit. At resonance, $$\displaystyle X_L = X_C $$, impedance is minimum ($$\displaystyle = R $$), current is maximum, and voltage across $C$ or $L$ is $Q$ times the source voltage.

$$Q = \frac{V_C}{V_{in}} = \frac{1}{\omega_0 C R}$$

• Parallel-Connection Method (for measuring $R$, $X$, $Q$ of an unknown impedance $$\displaystyle Z_x $$):

  1. Connect unknown in parallel with the resonant circuit.

  2. Tune to resonance (max current, min voltage across $C$).

  3. Let $$\displaystyle C_0 $$ = capacitance at resonance without $$\displaystyle Z_x $$.

  4. With $$\displaystyle Z_x $$ connected, tune to new resonance at $$\displaystyle C = C_0 + \Delta C $$.

• Expressions:

  • Reactance: $$\displaystyle X_x \approx \frac{1}{\omega_0 \Delta C} $$ (if $$\displaystyle R_x $$ large)

  • Resistance: $$\displaystyle R_x \approx \frac{1}{\omega_0^2 C_0^2 \Delta C \cdot R} $$ (where $R$ is known circuit resistance)

  • Q-Factor: $$\displaystyle Q_x \approx \omega_0 C_0 R_x $$

V. Transducers

A. Classification

1. By Operating Principle: Resistive, Inductive, Capacitive, Piezoelectric, Optical, Electrochemical, etc.

2. By Output: Analog/Digital, Active (generate output, e.g., thermocouple) / Passive (need external power, e.g., strain gauge).

3. By Input: Displacement, Force, Pressure, Temperature, Light, etc.

B. Resistive Transducers

• Thermistor: Thermally sensitive resistor (NTC: resistance ↓ with temp ↑; PTC: resistance ↑ with temp ↑).

• Importance: High sensitivity, small size, fast response (for bead type).

• Advantages: High sensitivity, low cost, simple circuitry.

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

C. Inductive Transducers

• Linear Variable Differential Transformer (LVDT):

  • Construction: Primary winding (center-tapped), two secondary windings (series-opposite), movable soft iron core.

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

  • Displacement Detection:

    • Magnitude: $$\displaystyle |V_{out}| \propto $$ displacement $x$.

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

  • Advantages: Infinite resolution, frictionless, robust, linear over wide range.

D. Capacitive Transducers

• Basic Principle: Capacitance $$\displaystyle C = \frac{\varepsilon A}{d} $$. Change in area (A), distance (d), or dielectric constant ($\varepsilon$) converts physical quantity to capacitance change.

• Types: Parallel plate (linear for area change, non-linear for $d$ change), cylindrical.

• Applications: Displacement, pressure, humidity measurement.

E. Piezoelectric Transducers

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

• Modes of Operation:

  • Benders: Crystal bonded to a metal strip. Stress causes bending. Used for pressure, acceleration.

  • Twisters: Crystal in a shear mode. Used for torque measurement.

  • Sketches: Show deformation direction relative to applied force.

F. Optical Transducers

• Photoemissive: Light ejects electrons from a photocathode (vacuum tube, e.g., phototube). Current $\propto$ light intensity.

• Photoconductive: Light reduces resistance of a semiconductor (e.g., photoresistor (LDR), photodiode in photoconductive mode).

• Photovoltaic: Light generates voltage/current at a PN junction without bias (e.g., solar cell, photodiode in photovoltaic mode).

• Photodiode (Detailed):

  • Operation: Reverse-biased PN junction. Incident light generates electron-hole pairs in depletion region → photocurrent ($$\displaystyle I_{ph} $$) proportional to light intensity.

  • Modes: Photovoltaic (zero bias, high resistance), Photoconductive (reverse bias, faster response, linear).

• Phototransistor: Similar to photodiode but with current gain. Base current is replaced by photocurrent. Higher sensitivity, slower.

G. Display Transducers

• Light Emitting Diode (LED):

  • Working: Injection electroluminescence. Forward-biased PN junction. Electrons recombine with holes, releasing energy as light (color depends on bandgap).

  • Details: Requires current limiting resistor. Advantages: Low voltage, fast switching, long life, small size. Disadvantages: Requires DC, limited viewing angle.

• Liquid Crystal Display (LCD):

  • Working: Light modulation using liquid crystals. Crystals align under electric field, changing polarization of light. Requires external light (reflective/transflective) or backlight.

  • Advantages: Very low power, flat, no radiation.

  • Disadvantages: Slow response, limited viewing angle, requires drive circuitry, temperature sensitive.

H. Special Transducers (Short Notes)

• Accelerometer: Measures acceleration. Common type: Piezoelectric accelerometer (mass on crystal; force = mass × acceleration).

• Bolometer: Measures radiation power (IR, microwave). Absorbs radiation → temperature rise → resistance change (e.g., thermistor or superconducting bolometer).

• Thermocouple: Based on Seebeck effect. Two dissimilar metals joined → temperature difference → thermo-emf proportional to temperature difference. Measures temperature.

VI. Digital Measurement Systems

A. Digital Voltmeter (DVM)

Digits Concept

• n½ Digits: A display with n full digits (0-9) and one half-digit (0 or 1 only).

• 3½ Digits: Can display from 0000 to 1999. Maximum count = 1999.

• 5½ Digits: Can display from 00000 to 19999. Maximum count = 19999.

• Significance of ½ Digit: The most significant digit (MSD) can only be 0 or 1. It indicates the over-range capability (e.g., 1.999V on a 2V range for a 3½ DVM). Prevents overload indication.

Resolution and Sensitivity

• Resolution: Smallest change in input voltage that causes a change in the LSB of the output display.

$$\text{Resolution} = \frac{\text{Full Scale Range (FSR)}}{\text{Maximum Count}}$$

For a 3½ DVM on 10V range: Resolution = $10V / 1999 \approx 5mV$.

• Sensitivity: Minimum change in input that produces a detectable change in output. Often same as resolution for DVMs.

B. Digital-to-Analog Converters (DAC)

1. Weighted Resistor DAC

• Circuit (4-bit):

  Digital Inputs (b3 b2 b1 b0) → Switches (S3...S0) → Weighted Resistors (R, 2R, 4R, 8R) → Summing Amplifier (Op-Amp) → Analog Output

• Transfer Characteristic Derivation:

  Output voltage $$\displaystyle V_o = -\frac{R_f}{R} V_{ref} \left( \frac{b_3}{2} + \frac{b_2}{4} + \frac{b_1}{8} + \frac{b_0}{16} \right) $$ for R-2R ladder in inverting summing config.

  For weighted resistor (all resistors to virtual ground):

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

(Assuming $$\displaystyle R_f = R $$ and $$\displaystyle V_{ref} $$ positive for 1's).

• Step Size (LSB): $$\displaystyle \Delta V = \frac{V_{ref}}{2^n} $$ for unipolar positive output.

2. R-2R Ladder DAC

• Circuit (3-bit): See diagram. Uses only two resistor values (R and 2R). Each bit position has an R-2R network.

• Operation: Each bit switch connects either to $$\displaystyle V_{ref} $$ (for 1) or ground (for 0). The ladder presents constant Thevenin resistance ($2R$) to the op-amp input, preventing interaction between bits.

• Advantage: Easier to fabricate with high accuracy (only two resistor values needed).

• Numerical: For 4-bit, $$\displaystyle V_{ref}=5V $$, Input 0111 (binary 7) and 1111 (15).

  • Step size $$\displaystyle \Delta V = 5V / 16 = 0.3125V $$.

  • Output for 0111 = $$\displaystyle 7 \times 0.3125V = 2.1875V $$.

  • Output for 1111 = $$\displaystyle 15 \times 0.3125V = 4.6875V $$.

C. Analog-to-Digital Converters (ADC)

Types and Comparison

1. Successive Approximation ADC (SAR)

• Circuit: Successive Approximation Register (SAR), DAC, Comparator, Control Logic.

• Operation:

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

  2. Comparator checks: if $$\displaystyle V_{in} > V_{DAC} $$, MSB stays 1; else, reset to 0.

  3. Set next bit to 1, compare... repeat for all bits.

  4. After n cycles, digital output is the approximation.

• Advantage: Fixed conversion time ($n+1$ clock cycles), good speed-accuracy trade-off.

2. Dual Slope ADC (Integrating Type)

• Circuit: Integrator (op-amp with capacitor), Comparator, Control Logic, Counter, Clock.

• Operation:

  1. Integrate Phase (fixed time $$\displaystyle T_1 $$): Input $$\displaystyle V_{in} $$ (positive or negative) integrated. Output slope $$\displaystyle \propto V_{in} $$. At end, integrator output = $$\displaystyle -k V_{in} T_1 $$.

  2. Deintegrate Phase (variable time $$\displaystyle T_2 $$): Switch to $$\displaystyle -V_{ref} $$. Integrator output ramps toward zero at fixed slope $$\displaystyle \propto V_{ref} $$. Counter stops when output crosses zero.

  3. Digital Output: Count in $$\displaystyle T_2 \propto V_{in} $$. $$\displaystyle V_{in} = V_{ref} \times (T_2 / T_1) $$.

• Advantages: Excellent noise rejection (especially 50/60 Hz), high accuracy, low cost. Disadvantage: Slow.

3. Resolution of ADC

• Definition: Smallest change in input voltage that can be detected, equal to 1 LSB.

$$\text{Resolution} = \frac{\text{FSR}}{2^n}$$

where $n$ = number of bits.

• Calculation Example: 5-bit ADC, range 0-10V.

  • Resolution = $$\displaystyle 10V / 2^5 = 10V / 32 = 0.3125V $$.

  • Range for MSB (bit 4): $5.0V - 10.0V$ (since $$\displaystyle 2^4 = 16 $$ LSBs = $$\displaystyle 16 \times 0.3125 = 5V $$).

  • % Error (Quantization Error): Maximum error = ±½ LSB = ±0.15625V. % Error = $$\displaystyle (0.15625 / 10) \times 100\% = 1.5625\% $$.

VII. Signal Generators

A. Sweep Frequency Generator

• Working Principle: Generates a sinusoidal output whose frequency varies smoothly (sweeps) over a specified range (typically audio or RF). Used for frequency response analysis of filters, amplifiers.

• Block Diagram:

  LF Oscillator (e.g., Wein bridge) → Voltage-Controlled Oscillator (VCO) or Switched Capacitor Bank → Output Amplifier → Attenuator

  Sweep Control (ramp generator) → VCO control voltage

• Operation: A low-frequency ramp (sweep control voltage) varies the capacitance or inductance in the VCO, causing its output frequency to sweep linearly (or logarithmically) with time.

B. Pulse Wave Generator

• Working with Diagram:

  • Block Diagram: Trigger Source → Multivibrator (Astable) → Shaper/Amplifier → Output

  • Operation: A monostable or astable multivibrator (using transistors, 555 timer, or logic gates) generates a square/pulse wave. A shaper circuit (differentiator + clipper) can convert square wave to narrow pulses. Pulse width and repetition rate are controlled by RC components.

VIII. Additional Topics and Circuits

A. Differential Amplifier using Two FETs

• Construction: Two N-channel JFETs (or MOSFETs) with common source resistor ($$\displaystyle R_S $$) and gate bias (often from a current mirror or voltage divider). Drain loads are often current mirrors.

• Output Voltage Derivation:

  For small signals, differential gain $$\displaystyle A_d = g_m R_D $$ (if $$\displaystyle R_D $$ is drain load, ignoring $$\displaystyle R_S $$ degeneration). With source degeneration $$\displaystyle R_S $$:

$$A_d = \frac{g_m R_D}{1 + g_m R_S}$$

Common-mode gain $$\displaystyle A_{cm} \approx \frac{R_D}{2R_{SS}} $$ (if $$\displaystyle R_{SS} $$ is large resistance from source to ground).

**CMRR** = $$\displaystyle |A_d / A_{cm}| $$.

B. Binary Adder

• Role in DAC/ADC:

  • In Flash ADC: The outputs of comparators are binary-weighted (thermometer code). A binary encoder (priority encoder) converts this to binary. A binary adder is part of the encoder logic.

  • In Weighted Resistor DAC: The digital inputs are applied to switches. Conceptually, the summing amplifier performs a weighted sum of binary digits, analogous to a binary-weighted addition.

C. Permissible Errors in Ammeters and Voltmeters

• Explanation: The maximum error an instrument can have under specified conditions (temperature, humidity, etc.) without being considered defective. Defined by standards (e.g., IEC 60051, ANSI C39.1).

• Expression: Often given as a percentage of full-scale deflection (FSD) or percentage of reading.

$$\text{Permissible Error} = \pm (a\% \text{ of FSD} + b\% \text{ of reading})$$

• Importance: Specifies the accuracy class of the instrument (e.g., 0.5 class means error ≤ 0.5% of FSD). Guides selection for measurement tasks.