UNIT 1: ELECTRONIC INSTRUMENTATION - SHORT NOTES

I. CATHODE RAY OSCILLOSCOPE (CRO) & CRT FUNDAMENTALS

A. CRT Construction & Electrostatic Deflection

• Construction: Consists of an electron gun (cathode, control grid, focusing & accelerating anodes), deflection system (electrostatic plates), phosphor screen (fluorescent screen), and a glass envelope (evacuated).

• Electrostatic Deflection: Deflection of the electron beam by applying a voltage across a pair of parallel plates. The force on an electron is $$\displaystyle F = eE = e \frac{V_d}{d} $$, where $$\displaystyle V_d $$ is deflecting voltage and $d$ is plate separation.

• Deflection Sensitivity (S): Deflection on screen (mm or cm) per unit deflecting voltage.

$$S = \frac{D}{V_d} \quad \text{(mm/V)}$$

• Deflection Factor (D_f): Reciprocal of sensitivity. Deflecting voltage required for unit deflection.

$$D_f = \frac{1}{S} = \frac{V_d}{D} \quad \text{(V/mm)}$$

• Post-Deflection Acceleration (PDA): An additional high-voltage anode placed after the deflection plates.

  • Purpose: Increases the velocity of the electron beam after deflection.

  • Effect: Reduces the spot size (less time for beam spreading) and increases brightness. It does not affect the deflection sensitivity $S$ (since deflection occurs before acceleration), but the deflection factor $$\displaystyle D_f $$ becomes larger.

[!TIP] Common Pitfall: Students often confuse the effect of PDA on sensitivity. Remember: Deflection happens before PDA, so sensitivity (D/V_d) remains unchanged. Only spot size and brightness improve.

B. CRO Block Diagram & General Purpose CRO

• Functional Blocks:

  1. Vertical Amplifier: Amplifies the input signal.

  2. Horizontal Amplifier (Time Base): Generates a linearly increasing sweep voltage.

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

  4. Power Supply: Provides high voltage for CRT and low voltages for circuits.

  5. CRT: The display device.

• Applications: Voltage/time measurement, frequency measurement (Lissajous), phase difference measurement, waveform observation, testing digital circuits.

C. Types of Oscilloscopes

• Sampling Oscilloscope:

  • Principle: For very high frequencies (> 100 MHz). Takes samples of the input waveform over many cycles and reconstructs the waveform (Equivalent-time sampling).

  • Types: Real-time (for repetitive signals), Equivalent-time (for very high freq).

  • Precaution: Cannot display non-repetitive or one-shot signals.

• Wobbly Scope (Sweep Generator): A CRO where the horizontal sweep is externally controlled by a low-frequency signal. Used to display frequency response (magnitude vs. frequency) of a circuit on the screen (X: freq, Y: output amplitude).

D. Time Base Circuits & Sweep Generation

• Sawtooth Wave Generation: Circuits like Miller (bootstrap) integrator and Phantastron are used.

• Sweep Synchronization (Triggering): The process of starting the sweep voltage at a specific point on the input waveform.

  • Effect on Accuracy: Proper synchronization (triggering) ensures a stationary, stable waveform display. Without it, the waveform drifts or rolls, making measurement impossible.

E. Graticules & Lissajous Patterns

• Graticules: Grids on the CRT face for measurement. Types: Internal (etched on glass), External (removable plastic), Illuminated (edge-lit).

• Lissajous Patterns: Formed when two sinusoidal signals are applied to X and Y plates.

  • Frequency Ratio: $$\displaystyle \frac{f_x}{f_y} = \frac{N_x}{N_y} $$, where $$\displaystyle N_x $$ = number of horizontal tangencies, $$\displaystyle N_y $$ = number of vertical tangencies.

  • Phase Measurement: Shape of the ellipse gives phase difference $\phi$ (for equal frequencies).

$$\sin \phi = \frac{B}{A} \quad \text{or} \quad \cos \phi = \frac{C}{A}$$

    (Where A = major axis, B = minor axis, C = intercept on Y-axis).

F. X-Y Recorders & Display Comparisons

• Analog XY Recorder: Uses two servo-controlled D'Arsonval movements (one for X, one for Y) to plot Y vs X. Slow, mechanical wear.

• Digital XY Recorder: Uses A/D converters and digital plotter/display. Faster, no wear, can store/print.

• Electrophoretic Image Display: Used in medical imaging (ECG, EEG). Paper moves under a row of electrodes; ion migration creates visible image.

• Liquid Vapor Display (LVD): Uses heated filaments to vaporize a colored dye onto paper. Obsolete, replaced by thermal printers.

II. AC BRIDGE CIRCUITS FOR PARAMETER MEASUREMENT

A. General Bridge Theory & Errors

• Balance Condition (General): For a four-arm bridge (Z₁, Z₂, Z₃, Z₄), balance occurs when:

$$Z_1 Z_4 = Z_2 Z_3$$

In terms of impedances (R ± jX) or admittances (G ± jB).

• Sources of Errors & Reduction:

  • Contact Resistance & Thermal EMF: Use four-terminal connections, AC excitation, copper bars.

  • Stray Capacitance: Shielding, guarding, using Wagner earth.

  • Frequency: Use appropriate bridge for frequency, or operate at standard frequency (1 kHz).

  • Temperature: Temperature-controlled environment, temperature compensation.

• Detectors: Vibration Galvanometer (for power freq), Headphones (audio freq), CRO (wide freq), Tuned Amplifier (high sensitivity at specific freq).

B. Specific Bridges

[!TIP] Exam Focus: Be prepared to derive balance equations for Maxwell, Schering, and Wien bridges. Also, know which bridge is used for loss tangent (Schering) vs high Q inductance (Hay's).

C. Bridge Applications & Problem Solving

• Loss Factor & Q-Factor Relation: For a capacitor, $$\displaystyle \tan \delta = \frac{1}{Q} $$. For an inductor, $$\displaystyle Q = \frac{\omega L}{R} $$.

• Example (Maxwell Bridge): Given $$\displaystyle C_1=0.01\mu F $$, $$\displaystyle R_1=470k\Omega $$, $$\displaystyle R_2=5.1k\Omega $$, $$\displaystyle R_3=100k\Omega $$:

$$L_x = R_2 R_3 C_1 = (5.1 \times 10^3)(100 \times 10^3)(0.01 \times 10^{-6}) = 5.1 \ \text{H}$$

$$R_x = \frac{R_2 R_3}{R_1} = \frac{5.1 \times 100}{470} \approx 1.085 \ \text{k}\Omega$$

\boxed{Z_x = 1.085 \ \text{k}\Omega + j 2\pi f \cdot 5.1 \ \text{H}}

III. TRANSDUCERS & SENSORS

A. Fundamental Concepts

• Transducer: A device that converts a physical quantity (input) into another (usually electrical) output.

• Primary Transducer: Senses the input and produces a mechanical output (e.g., Bourdon tube in pressure gauge).

• Secondary Transducer: Converts the mechanical output into an electrical signal (e.g., LVDT attached to Bourdon tube).

• Digital Multiplexing: Connecting multiple sensors to a single ADC/data logger via a multiplexer (MUX). Improves efficiency by reducing wiring cost, signal conditioning channels, and data acquisition hardware in industrial systems.

B. Resistive Transducers

• Strain Gauge:

  • Principle: Piezoresistive effect – resistance changes with strain ($$\displaystyle \Delta R / R = GF \cdot \varepsilon $$).

  • Gauge Factor (GF):

$$GF = \frac{\Delta R / R}{\varepsilon} = 1 + 2\nu + \frac{\Delta \rho / \rho}{\varepsilon}$$

    For metals: $GF \approx 2$ (mainly from geometry term $1+2\nu$). For semiconductors: $GF \approx 50-150$ (dominant $\Delta \rho/\rho$ term).

*   **Types:** Metal foil (most common), wire-wound, semiconductor (higher sensitivity, more temperature-sensitive).

*   **Temperature Compensation:** Use **dummy gauge** in adjacent arm of bridge, or use **three-wire/ four-wire** connection to eliminate lead resistance.

• RTD (Resistance Temperature Detector): Pure metals (Pt, Ni, Cu). Positive TCR. PT100 (100Ω at 0°C). Linear over narrow range, stable, used for -200°C to 850°C.

• Thermistor: Semiconductors (metal oxides). NTC (most common, resistance ↓ with temp), PTC (resistance ↑ sharply at Curie point). Highly non-linear, high sensitivity, used for -50°C to 150°C.

C. Capacitive Transducers

• Principle: $$\displaystyle C = \frac{\varepsilon A}{d} $$. Change in Area (A), Distance (d), or Dielectric (ε).

• Applications: Displacement (parallel plate, change in d), pressure (diaphragm), humidity (change in ε).

D. Inductive Transducers

• LVDT (Linear Variable Differential Transformer):

  • Construction: One primary winding, two identical secondary windings (series/parallel opposing), movable ferromagnetic core.

  • Working: AC excitation in primary. Core displacement changes mutual inductance, producing differential output voltage $$\displaystyle e_{out} = e_{s1} - e_{s2} $$.

    • Null Position: Core centered → $$\displaystyle e_{s1} = e_{s2} $$ → $$\displaystyle e_{out}=0 $$.

    • Displacement: Core moves → $$\displaystyle e_{out} \neq 0 $$. Amplitude ∝ displacement, phase indicates direction (±90° or ±180° shift).

  • Characteristics: Excellent linearity (~±1% over ±1 inch), infinite resolution, no physical contact (wear-free).

  • Advantages: High reliability, long life, frictionless.

  • Limitations: Large size for long travel, needs AC excitation & demodulation.

• RVDT (Rotary LVDT): For angular displacement.

E. Piezoelectric Transducers

• Piezoelectric Effect: Certain crystals (Quartz, Rochelle salt, PZT) generate charge on surfaces when mechanically stressed (direct effect). Conversely, they strain when voltage applied (converse effect).

• Modes of Operation:

  • Thickness/Extensional: Force applied parallel to polar axis (charge on faces).

  • Transverse: Force applied perpendicular to polar axis (charge on sides).

  • Shear: Shear stress applied.

• Equivalent Circuit: Voltage source $$\displaystyle V_p = \frac{F}{C_e \cdot S} $$ in series with capacitor $$\displaystyle C_e $$ (crystal capacitance), or charge source $$\displaystyle Q_p = d \cdot F $$ in parallel with $$\displaystyle C_e $$.

• Applications: Force, pressure, acceleration (seismic mass), ultrasonic generation/detection.

• Calculation (Example): Given crystal dimensions, charge sensitivity $d$ (C/N), strain $\varepsilon$:

$$F = Y \cdot A \cdot \varepsilon \quad (Y = \text{Young's modulus})$$

$$Q = d \cdot F$$

$$V = \frac{Q}{C} = \frac{Q}{\varepsilon_r \varepsilon_0 A / t} \quad (t = thickness)$$

F. Magnetic Transducers

• Hall Effect Transducer:

  • Hall Voltage: When current $I$ flows through a conductor/semiconductor in a perpendicular magnetic field $B$, a transverse voltage $$\displaystyle V_H $$ develops.

$$V_H = \frac{R_H I B}{t} = \frac{IB}{n e t}$$

    Where $$\displaystyle R_H = 1/(ne) $$ is Hall coefficient, $t$ is thickness, $n$ is carrier density.

*   **Geometrical Correction Factor (k):** For rectangular samples, $$\displaystyle V_H = k \frac{IB}{nt} $$. $k$ depends on aspect ratio (width/length).

*   **Applications:** Magnetic field measurement, current sensing (clamp meters), position/speed sensing (magnetic encoder).

G. Thermal Transducers

• Thermocouple:

  • Seebeck Effect: When two dissimilar metals are joined at two junctions at different temperatures, an EMF is generated proportional to the temperature difference.

  • Materials: Reference junction (constant, e.g., Cu-Constantan), measuring junction (chosen for high Seebeck coefficient, e.g., Chromel-Alumel - Type K).

  • Cold Junction Compensation (CJC): Since output depends on difference between measuring and reference junctions, the reference junction must be kept at known temperature (ice bath at 0°C) or its temperature measured and compensated electronically.

• Thermopile: Series/parallel connection of multiple thermocouples for higher output.

• Radiation Thermometer (Pyrometer): Measures temperature from emitted infrared radiation (non-contact).

H. Photoelectric Transducers

I. Other Special Transducers

• Digital Tachometer:

  • Optical Encoder: Rotating disk with slots; photodiode/phototransistor detects pulses. Frequency $$\displaystyle f = N \cdot RPM / 60 $$.

  • Magnetic Encoder: Uses Hall effect or variable reluctance.

  • Stroboscopic: Flashing light at known frequency; when flash rate matches rotation, image appears stationary.

• Temperature Transducers by Range:

  • -200°C to 0°C: Platinum RTD, Thermocouple (Type T, E).

  • 0°C to 500°C: Platinum RTD (PT100), Thermocouple (Type J, K, T).

  • 500°C to 1500°C: Thermocouples (Type K, S, R, B).

  • >1500°C: Optical pyrometers.

IV. SIGNAL GENERATORS & WAVE ANALYZERS

A. Function Generators (AF & RF)

• Block Diagram: Function Generator (produces sine, square, triangle) → Attenuator → Output Amplifier.

• Sine Wave Generation: RC Wien Bridge Oscillator (uses frequency-selective positive feedback). Frequency $$\displaystyle f = \frac{1}{2\pi RC} $$.

• Square/Triangle Generation: Comparator/Schmitt Trigger (square from sine) → Integrator (triangle from square).

• Frequency Control: VCO (Voltage-Controlled Oscillator). External voltage changes the capacitance (varactor) or resistance in the oscillator, thus changing frequency.

• Beat Frequency Oscillator (BFO): Two close-frequency oscillators (one fixed, one variable). Their outputs are mixed, producing beat frequency $$\displaystyle f_{beat} = |f_1 - f_2| $$. Used for audio-frequency generation (1-20 kHz) with good stability.

B. Sweep & Fixed-Frequency Generators

• Fixed-Frequency: Outputs a single, precise frequency (e.g., crystal oscillator).

• Sweep-Frequency (Wobbler): Output frequency varies continuously and repetitively over a range (e.g., 10 Hz - 100 kHz). Used to drive the horizontal input of an oscilloscope in X-Y mode to plot frequency response (Bode plot) of a DUT.

C. Wave Analyzers

• Frequency Selective (Filter) Type:

  • Principle: Series of tuned filters (LC or RC) to select a narrow frequency band.

  • Limitations: Poor selectivity at high frequencies, limited sensitivity.

• Heterodyne (Superheterodyne) Type:

  • Principle: 1. Mix input signal with local oscillator (LO) → produces sum & difference frequencies. 2. IF amplifier (high-Q, fixed center frequency, e.g., 455 kHz) selects one (usually difference). 3. Detector extracts amplitude.

  • Advantages: High sensitivity (due to high-gain IF amp) and high selectivity (due to narrow IF filter). Used for RF signal analysis.

• Spectrum Analyzer:

  • Swept-Tuned: Most common. LO sweeps, mixer produces IF, IF filter selects, detector displays amplitude vs frequency.

  • FFT (Fast Fourier Transform): Digitizes time-domain signal, computes FFT to get frequency spectrum. Faster for transient signals.

[!TIP] Key Difference: Heterodyne wave analyzer uses frequency conversion (mixing) to a fixed IF for better performance at RF. Filter-type directly filters the input, limited to audio frequencies.

V. DIGITAL INSTRUMENTS & METERS

A. Digital Voltmeter (DVM) Types

• Ramp Type (Integrating - Dual-Slope):

  • Principle: 1. Integrate input voltage $$\displaystyle V_x $$ for fixed time $$\displaystyle T_1 $$ → capacitor voltage ∝ $$\displaystyle V_x $$. 2. De-integrate with reference voltage $$\displaystyle -V_{ref} $$ until capacitor returns to zero → time $$\displaystyle T_2 $$ ∝ $$\displaystyle V_x $$. Count $$\displaystyle T_2 $$.

  • Output: $$\displaystyle V_x = V_{ref} \cdot (T_2 / T_1) $$.

  • Advantages: Excellent noise rejection (integration averages), high accuracy, no need for precision components.

  • Disadvantages: Slow (conversion time ~ T₁+T₂).

• Successive Approximation Type:

  • Principle: Uses a Successive Approximation Register (SAR) and a DAC. SAR tries bits from MSB to LSB, DAC generates trial voltage, comparator decides bit value.

  • Speed: Fast (conversion in n clock cycles, e.g., 1 µs).

  • Accuracy: Good, but depends on DAC linearity.

• DVM Specifications (3½ Digit):

  • Resolution: Smallest change detectable. For a 10V range: $$\displaystyle 1 \text{ count} = \frac{10V}{1999} \approx 5 \text{mV} $$. Resolution = 10 mV / 1999 ≈ 0.5 mV.

  • Display Examples:

    • 11.52V on 10V range → Overrange (display shows "1" or "OL").

    • 0.5234V on 1V range → 0.5234 V (full 4 digits used).

    • 0.5234V on 10V range → 0.523 V (only 3 decimal places, leading zero suppressed).

B. Digital Frequency Meter

• Block Diagram: Input Conditioner (amplifier, Schmitt trigger) → Gate (controlled by time base) → Counter (counts input pulses) → Latch & Display.

• Working: Gate opens for a precise time interval $T$ (e.g., 1 second from crystal clock). Number of input cycles $N$ counted. Frequency $$\displaystyle f = N / T $$.

C. Digital pH Meter

• Principle: Glass electrode develops a voltage proportional to pH (Nernst equation: $$\displaystyle E = E_0 - \frac{2.303 RT}{F} pH $$). This high-impedance signal (MΩ) is fed to a high-input-impedance amplifier (FET input op-amp) and then to an ADC/display.

• Block Diagram: pH Electrode → High-Z Amplifier → ADC → Digital Display. Includes temperature compensation circuit.

D. Data Logger vs Data Acquisition System (DAS)

VI. DISPLAY DEVICES & RECORDERS

A. Display Devices

• LED (Light Emitting Diode):

  • Principle: Electroluminescence – recombination of electrons/holes in semiconductor (GaAs, GaP) emits light.

  • Advantages: High brightness, fast response time (ns), wide viewing angle.

  • Disadvantages: Higher power consumption, cost per digit higher than LCD.

• LCD (Liquid Crystal Display - Twisted Nematic):

  • Theory: Liquid crystals between polarizers. Twisted (90°) in OFF state, blocks light. Untwisted by applied voltage, allows light through.

  • Advantages: Very low power (microwatts), flat, cheap.

  • Disadvantages: Slow response time (ms), poor viewing angle, needs backlight (for transmissive type).

• Comparison:

  • Choose LED for high brightness, outdoor, fast updates.

  • Choose LCD for battery-powered, low-cost, static displays (calculators, meters).

B. Recorders

• X-Y Recorder:

  • Working: Two independent servo systems (each with amplifier, motor, position feedback) control the X and Y positions of a pen on paper. Input voltages control the setpoints.

  • Applications: Plotting characteristics (I-V, transfer function), Lissajous patterns, process variables.

• Analog vs Digital Recorders:

  • Analog (Chart Recorder): Pen on paper, mechanical/thermal. Continuous trace, no storage, wear & tear.

  • Digital: Data stored in memory, can be printed/plotted later. No mechanical wear, can store large amounts.

VII. INTERFACING & COMMUNICATION IN INSTRUMENTATION

A. Standard Instrumentation Buses & Interfaces

B. Grounding & Safety

• Wagener's Earthing Device:

  • Purpose: To safely discharge the high voltage (HV) anode of a CRT (CRO) when the instrument is switched off or during maintenance.

  • Principle: A bleeder resistor (high value, e.g., 1 MΩ) is permanently connected between the HV terminal and earth. Additionally, a manual discharge probe (with high-value resistor) is provided for immediate safe discharge.

VIII. SPECIALIZED TOPICS (SHORT NOTES)

A. Q-Meter

• Circuit: Series resonant circuit: Unknown coil (L_x, R_x) in series with a standard capacitor (C_s) and a low-loss coil (L_s) with very low resistance. A variac controls the voltage across the circuit. A voltmeter (V) across C_s measures voltage.

• Working: At resonance ($$\displaystyle \omega L_x = 1/(\omega C_s) $$), impedance is minimum ($$\displaystyle R_x + R_s $$), current maximum, voltage across C_s is $Q$ times the input voltage.

$$Q = \frac{V}{V_{in}} \quad \text{(at resonance)}$$

$$L_x = \frac{1}{\omega^2 C_s} \quad \text{(if } R_s \ll R_x\text{)}$$

• Application: Direct reading of Q-factor and inductance of coils at a specific frequency.

B. Total Harmonic Distortion (THD)

• Definition: Measure of harmonic distortion in a signal. Ratio of the RMS value of all harmonic components to the RMS value of the fundamental component.

$$THD = \frac{\sqrt{V_2^2 + V_3^2 + V_4^2 + \dots}}{V_1} \times 100\%$$

Where $$\displaystyle V_1 $$ is fundamental RMS, $$\displaystyle V_2, V_3... $$ are harmonic RMS voltages.

• Measurement: Using a wave analyzer (heterodyne type) to isolate and measure the fundamental and each harmonic individually, then computing THD.

C. Multi-Input Sampling Oscilloscope

• Principle: Extension of sampling oscilloscope. Uses multiple sampling channels (e.g., 4 or 8) to capture different points of a repetitive waveform simultaneously (in one cycle) or across multiple cycles.

• Advantage: Can reconstruct high-speed waveforms with multiple channels (e.g., digital buses) where a single-channel sampler would miss timing relationships.

D. Heterodyne Wave Analyzer (Detailed)

• Operation: 1. Input signal → Attenuator. 2. Mixed with tunable local oscillator (LO) in ** mixer**. 3. IF amplifier (high-Q, fixed center freq, e.g., 455 kHz) selects difference frequency ($$\displaystyle f_{signal} - f_{LO} $$). 4. Detector (diode) → Low-pass filter → Meter/Display.

• Tuning: Vary $$\displaystyle f_{LO} $$ so that $$\displaystyle f_{signal} - f_{LO} = f_{IF} $$ (constant). Meter reading s to amplitude of $$\displaystyle f_{signal} $$.

• Sensitivity & Selectivity: Much higher than filter-type because of high-gain, narrow-band IF amplifier.

E. Sweep Generator (Wobbly Scope)

• Detailed Note: A sweep-frequency generator whose output frequency is linearly swept (wobbled) by a low-frequency sawtooth (e.g., 50 Hz). This output drives the horizontal (X) input of an oscilloscope. The vertical (Y) input is the output of the DUT (Device Under Test) being tested.

• Application: Displays the frequency response (gain vs. frequency) of an amplifier, filter, or any network directly on the oscilloscope screen (X-axis = frequency, Y-axis = amplitude).

F. Thermocouple (Detailed)

• Seebeck Effect: In a circuit of two dissimilar metals (A and B), if the two junctions are at temperatures $$\displaystyle T_1 $$ and $$\displaystyle T_2 $$, the net EMF is:

$$E_{AB}(T_1, T_2) = \int_{T_1}^{T_2} (S_A - S_B) dT$$

Where $$\displaystyle S_A, S_B $$ are Seebeck coefficients (material-dependent).

• Materials: Reference junction often uses Constantan (copper-nickel alloy) paired with various metals (Iron, Copper, Chromel). Measuring junction uses Chromel-Alumel (Type K) for wide range and durability.

• Cold Junction Compensation (CJC): Since tables give $E$ vs. $T$ for $$\displaystyle T_{ref}=0°C $$, if $$\displaystyle T_{ref} \neq 0°C $$, an equal and opposite EMF must be added. Done electronically by measuring $$\displaystyle T_{ref} $$ with an RTD/thermistor and adding a compensating voltage.

G. Hall Effect Transducer (Detailed)

• Hall Voltage Derivation:

  Consider a thin slab of material (thickness $t$, width $w$, length $l$) with current $I$ along length ($x$), magnetic field $B$ perpendicular (z-direction).

  Lorentz force on charge carriers: $$\displaystyle F_L = q(v_d \times B) $$, deflects them to sides, creating transverse electric field $$\displaystyle E_H $$.

  At equilibrium: $$\displaystyle q E_H = q v_d B $$ → $$\displaystyle E_H = v_d B $$.

  Current density $$\displaystyle J = n e v_d = I / (w t) $$ → $$\displaystyle v_d = I / (n e w t) $$.

  Therefore, $$\displaystyle V_H = E_H \cdot w = \frac{I B}{n e t} = \frac{R_H I B}{t} $$.

• Geometrical Correction Factor (k): For a rectangular sample, the exact Hall voltage is $$\displaystyle V_H = k \frac{IB}{n e t} $$, where $k$ depends on the aspect ratio ($w/l$). For $$\displaystyle w >> l $$, $k \approx 1$. For $$\displaystyle w = l $$, $k \approx 0.8$. This factor must be accounted for in precise measurements.

END OF UNIT 1 NOTES