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Annex 1 (BE01) — Lesson BE.01.03 • TLOs: 28XX-MAIN-1001, 28XX-MAIN-1002 • ELOs: 1001d–e, 1002a–g

DC Theory

Direct Current (DC) theory provides the mathematical notation, atomic physics, component characteristics, color-code standards, and Fluke 77/BN measurement procedures required for every circuit analyzed in the Basic Electronics Course.

1. Scientific & Engineering Notation

Electronic quantities span twenty-four orders of magnitude—from picofarads ( $10^{-12}$ ) to terahertz ( $10^{12}$ ). Standardizing large and small numbers prevents decimal-place errors during circuit analysis.

Scientific vs. Engineering Notation

Notation System	Coefficient Requirement	Exponent Requirement	Example ( $135{,}000{,}000$ )	Example ( $0.00781$ )
Scientific Notation	$1 \le \text{Coefficient} < 10$	Any integer power of $10$ ( $10^n$ )	$1.35 \times 10^8$	$7.81 \times 10^{-3}$
Engineering Notation	$1 \le \text{Coefficient} < 1{,}000$	Multiples of $3$ only ( $10^{\pm 3}, 10^{\pm 6}, 10^{\pm 9}, 10^{\pm 12}$ )	$135 \times 10^6 = \mathbf{135,\text{M}}$	$7.81 \times 10^{-3} = \mathbf{7.81,\text{m}}$

The `LARS` Decimal Conversion Rule

When shifting the decimal point to convert a number into Scientific or Engineering Notation, apply the LARS mnemonic:

Left Add: Moving the decimal point to the LEFT makes the power-of-ten exponent more positive (add to the exponent).
Right Subtract: Moving the decimal point to the RIGHT makes the power-of-ten exponent more negative (subtract from the exponent).

Complete Engineering Notation Metric Prefix Chart

Prefix Name	Symbol	Word Meaning	Power of Ten	Numerical Value
Tera	$\text{T}$	Trillion times	$10^{12}$	$1{,}000{,}000{,}000{,}000$
Giga	$\text{G}$	Billion times	$10^{9}$	$1{,}000{,}000{,}000$
Mega	$\text{M}$	Million times	$10^{6}$	$1{,}000{,}000$
Kilo	$\text{k}$	Thousand times	$10^{3}$	$1{,}000$
Unity (Base Unit)	—	One	$10^{0}$	$1$
Milli	$\text{m}$	Thousandth of	$10^{-3}$	$0.001$
Micro	$\mu$	Millionth of	$10^{-6}$	$0.000001$
Nano	$\text{n}$	Billionth of	$10^{-9}$	$0.000000001$
Pico	$\text{p}$	Trillionth of	$10^{-12}$	$0.000000000001$

Place-Value Alignment Map:
  1 , 001 , 001 , 001 , 000 . 001   001   001   001
  T    G     M     k   [Unit]  m     μ     n     p
(10¹²)(10⁹) (10⁶) (10³) (10⁰) (10⁻³)(10⁻⁶)(10⁻⁹)(10⁻¹²)

Case-Sensitivity Warning (M vs. m): Uppercase $\text{M}$ (Mega, $10^6$ ) and lowercase $\text{m}$ (milli, $10^{-3}$ ) differ by nine orders of magnitude ( $10^9$ ). Confusing $5,\text{mA}$ ( $0.005,\text{A}$ ) with $5,\text{MA}$ ( $5{,}000{,}000,\text{A}$ ) is a frequent exam distractor.

BEC Rounding Standard

Round the FINAL answer only: Never round intermediate calculation steps; carry full calculator precision until the final result.
Two Decimal Places: Round all final numerical answers to two significant digits past the decimal point (e.g., $6.8181,\text{mA} \rightarrow 6.82,\text{mA}$ ) unless otherwise specified.
Standard $5+$ Rule: If the third digit past the decimal is $5$ or greater, round the second digit up by $1$ ; if $4$ or less, leave it unchanged.

2. Atomic Theory & Fundamental Electrical Quantities

Hierarchy of Matter & The Planetary Atom

Matter: Anything that occupies space, has mass (weight), and requires energy to move; composed of molecules.
Molecule: The smallest particle a compound can be reduced to while retaining the chemical properties of that compound; composed of elements.
Atom: The smallest particle of an element that retains the characteristics of that element.
- Nucleus (Center): Contains the two heaviest subatomic particles—Protons (positively charged, $+$ ) and Neutrons (electrically neutral, no charge).
- Electrons (Orbiting Shells): Lightweight, negatively charged ( $-$ ) particles orbiting the nucleus in discrete concentric shells.

Valence Electrons & Material Conductivity

The outermost electron shell is the valence ring (shell), and electrons occupying it are valence electrons:

The further an electron orbits from the nucleus, the less energy is required to free it from orbit.
Because applied energy distributes equally among all valence electrons in the outer ring, the force required to liberate a valence electron is directly proportional to the number of valence electrons.

Material Classification	Valence Electrons	Electrical Behavior
Conductors (e.g., Copper, Silver, Gold)	$\le 3$ ( $1$ to $3$ )	Valence electrons are loosely bound and easily freed with minimal applied force, supporting ready current flow.
Intrinsic Semiconductors (e.g., Silicon, Germanium)	$4$	Neither a good conductor nor a good insulator in pure crystal form (analyzed in detail in `BE03`).
Insulators (e.g., Glass, Rubber, Mica, Plastic)	$\ge 5$ ( $5$ to $8$ )	Applied energy is divided across many valence electrons, tightly binding them and opposing electron flow.

Coulomb’s Law & Electrical Charge ( $Q$ )

Coulomb’s Law of Charged Bodies: Charged bodies attract or repel each other with a force that is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
- Unlike charges attract ( $+$ and $-$ ); like charges repel ( $-$ and $-$ , or $+$ and $+$ ).
- Separation of charges establishes a difference of potential capable of performing electrical work.
The Coulomb ( $\text{C}$ ): The fundamental unit of electrical charge ( $Q$ ): $1\,\text{Coulomb (C)} = 6.25 \times 10^{18}\,\text{electrons}$

Summary of Fundamental Electronic Quantities

Quantity	Symbol(s)	Unit of Measure (Abbrev.)	Physical Definition & Benchmark Relationship
Charge	$Q$	Coulomb ( $\text{C}$ )	Combined electrostatic charge of $6.25 \times 10^{18}$ electrons.
Voltage / EMF	$E$ or $V$	Volt ( $\text{V}$ )	Difference of potential (Electromotive Force) that drives charge flow. $1,\text{V}$ is the potential required for $1,\text{Joule}$ of energy to move $1,\text{Coulomb}$ of charge ( $1,\text{V} = \frac{1,\text{J}}{1,\text{C}}$ ).
Current	$I$	Ampere ( $\text{A}$ )	Directed flow of charge through a circuit ( $I$ = intensity). $1,\text{A}$ equals $1,\text{Coulomb}$ ( $6.25 \times 10^{18}$ electrons) passing a given point in $1,\text{second}$ ( $1,\text{A} = \frac{1,\text{C}}{1,\text{s}}$ ).
Resistance	$R$	Ohm ( $\Omega$ )	Opposition to current flow, converting electrical energy into heat. $1,\Omega$ limits current to $1,\text{A}$ when $1,\text{V}$ of potential is applied.

BEC Electron Flow Convention: Throughout BEC, current is analyzed using Electron Flow: negatively charged electrons are repelled from the negative ( $-$ ) terminal of the source and flow through the external circuit toward the positive ( $+$ ) terminal.

3. Circuit Requirements, Ground Symbols & Protection

Three Requirements of a Complete Electrical Circuit

Every functional circuit requires three elements:

Source: Supplies the electromotive force (potential difference) required to move electrons (e.g., battery or DC power supply).
Conductors: Provide a low-resistance physical path for electron flow from the source, through the load, and back to the source.
Load: The component or device that consumes electrical energy to perform work (e.g., resistor, lamp, motor, relay).

Ground Reference Symbols ( $0,\text{V}$ Reference Point)

Ground establishes the $0,\text{V}$ reference potential for circuit measurements and safely dissipates fault currents:

Earth Ground: Physical connection to copper rods driven into the earth (e.g., $120,\text{VAC}$ facility power distribution).
Chassis Ground: Connection to the metal frame or housing of equipment (e.g., tactical vehicles, man-portable radios).
Common Ground (Signal/Circuit Common): Common return plane on printed circuit boards and schematic diagrams.

Open Circuit vs. Short Circuit Characteristics

Circuit Condition	Physical Cause	Resistance ( $R$ )	Current Flow ( $I$ )	Voltage Across Fault ( $V$ )
Open Circuit	Broken conductor, lifted lead, blown fuse, or burned-out resistor (incomplete path)	Maximum ( $\infty$ / `OL`)	Minimum ( $0,\text{A}$ )	Maximum ( $\approx V_A$ ) — Full applied voltage appears across the open break
Short Circuit	Accidental low-resistance bridge bypassing the normal load	Minimum ( $\approx 0,\Omega$ )	Maximum (Excessive current)	Minimum ( $\approx 0,\text{V}$ ) — Virtually zero voltage dropped across the short

Overcurrent Protection: Fuses vs. Circuit Breakers

Fuses: Connected in series with the power source to protect against excessive current (shorts/overloads) by melting an internal fusible link and creating an open circuit.
- Current Rating: Maximum continuous current before the fuse element melts and opens.
- Voltage Rating: Maximum voltage the open fuse can withstand across its terminals without arcing.
- Fast-Blow Fuse: Opens immediately when rated current is exceeded; protects sensitive semiconductors against surge current.
- Slow-Blow (Time-Delay) Fuse: Both current- and time-sensitive; tolerates brief, harmless inrush surges for a predetermined period before opening on a sustained overload.
Circuit Breakers: Perform the same series overcurrent protection function as a fuse, but can be mechanically reset after tripping rather than requiring replacement.

4. Resistors: Compositions & Variable Mechanics

A resistor introduces a precise opposition to current flow ( $R$ , in Ohms) while dissipating electrical energy as heat.

Physical Size Rule: The physical size of a resistor determines its power (wattage/heat) dissipation rating—NOT its ohmic resistance value. A physically larger resistor has greater surface area to dissipate heat and can safely handle higher current/power without burning up.

Four Fixed Resistor Compositions

Composition Type	Physical Construction	Resistive Range	Power & Operational Characteristics
Carbon Composition	Finely ground carbon/graphite blended with an insulating binder	$1,\Omega$ to $20,\text{M}\Omega$ (Widest resistance range)	Low power ( $< 2,\text{W}$ ); resistance value can drift when exposed to high voltages, currents, or humidity.
Wire-Wound	Precision resistance wire wound around a ceramic core (longer wire = higher $R$ )	$< 1,\Omega$ to $3,\text{k}\Omega$ (Low resistance range)	High current & power ( $\ge 1{,}000,\text{W}$ ); highly accurate and thermally stable; most expensive.
Metal-Film	Thin metallic film sprayed onto a ceramic rod and laser-cut in a spiral	$5,\Omega$ to $10,\text{M}\Omega$ (Mid resistance range)	Low temperature coefficient — resistance value remains highly stable across extreme temperature shifts.
Surface Mount (SMD / “Chip”)	Micro-miniature ceramic chip ( $0.125"\times 0.063"\times 0.028"$ ) soldered directly to PCB pads	$< 1,\Omega$ to $1,\text{M}\Omega$ (Small resistance range)	Very low power ( $\frac{1}{8},\text{W}$ to $\frac{1}{4},\text{W}$ ) due to minimal physical surface area.

Fixed vs. Variable Resistor Schematic Symbols

Every Variable Resistor schematic symbol has three mandatory parts:

Resistive Element: The zigzag body between fixed end terminals $A$ and $B$ .
The Arrow (Wiper Arm): The movable sliding contact ( $C$ ) that divides the resistive element.
Positional Reference (CW or CCW): Indicates which direction the wiper contact moves when the control shaft is rotated Clockwise (CW) or Counter-Clockwise (CCW).

Wiper Movement Rules (Given fixed terminals $A$ [left] and $B$ [right], wiper $C$ in middle, and CW pointing toward $B$ ):

Rotating Shaft CW (wiper moves toward $B$ ): Resistance $R_{AC}$ increases, resistance $R_{BC}$ decreases, and end-to-end resistance $R_{AB}$ remains constant.
Rotating Shaft CCW (wiper moves toward $A$ ): Resistance $R_{AC}$ decreases, resistance $R_{BC}$ increases, and end-to-end resistance $R_{AB}$ remains constant.

5. EIA 4-Band Resistor Color Code

Axial carbon-composition and film resistors use four colored bands read from the band closest to the lead edge (Band 1) toward the tolerance band (Band 4).

Band 1: 1st Significant Digit (Never Black).
Band 2: 2nd Significant Digit.
Band 3: Multiplier (Power of $10$ / number of zeroes added to the first two digits; use Gold $\times 0.1$ or Silver $\times 0.01$ for resistors under $10,\Omega$ ).
Band 4: Tolerance Percentage ( $\pm%$ ). If only 3 bands are present (No 4th Band), tolerance is $\pm 20%$ .

Master 4-Band Resistor Color Code Table

Color	Band 1 (1st Digit)	Band 2 (2nd Digit)	Band 3 (Multiplier)	Band 4 (Tolerance)
Black	— (2nd band only)	$0$	$\times 1$ ( $10^0$ )	—
Brown	$1$	$1$	$\times 10$ ( $10^1$ )	—
Red	$2$	$2$	$\times 100$ ( $10^2$ )	—
Orange	$3$	$3$	$\times 1{,}000$ ( $10^3$ )	—
Yellow	$4$	$4$	$\times 10{,}000$ ( $10^4$ )	—
Green	$5$	$5$	$\times 100{,}000$ ( $10^5$ )	—
Blue	$6$	$6$	$\times 1{,}000{,}000$ ( $10^6$ )	—
Violet (Purple)	$7$	$7$	$\times 10{,}000{,}000$ ( $10^7$ )	—
Gray	$8$	$8$	$\times 100{,}000{,}000$ ( $10^8$ )	—
White	$9$	$9$	$\times 1{,}000{,}000{,}000$ ( $10^9$ )	—
Gold	—	—	$\times 0.1$ ( $10^{-1}$ )	$\pm 5%$
Silver	—	—	$\times 0.01$ ( $10^{-2}$ )	$\pm 10%$
None (3 Bands)	—	—	—	$\pm 20%$

Tolerance Range Calculation Walkthrough

Given a resistor banded Yellow – Violet – Red – Silver:

Nominal Value: Yellow ( $4$ ), Violet ( $7$ ), Red ( $\times 100$ ) $\implies 47 \times 100 = \mathbf{4{,}700,\Omega\ (4.7,\text{k}\Omega)}$ .
Tolerance Value: Silver ( $\pm 10%$ ) $\implies 4{,}700,\Omega \times 0.10 = \mathbf{\pm 470,\Omega}$ .
Minimum Acceptable Value: $4{,}700,\Omega - 470,\Omega = \mathbf{4{,}230,\Omega\ (4.23,\text{k}\Omega)}$ .
Maximum Acceptable Value: $4{,}700,\Omega + 470,\Omega = \mathbf{5{,}170,\Omega\ (5.17,\text{k}\Omega)}$ .

6. Ohm’s Law & Proportionality

Formulated by George Simon Ohm, Ohm’s Law states:

“The amount of current in a circuit is directly proportional to the electromotive force (voltage) applied to that circuit, and inversely proportional to the opposition (resistance) to which that force is applied.”

I = \frac{E}{R} \qquad\qquad E = I \times R \qquad\qquad R = \frac{E}{I}

Direct & Inverse Proportionality Rules

Voltage & Current ( $E \propto I$ at constant $R$ ): Directly proportional. If applied voltage ( $E$ ) doubles while resistance ( $R$ ) stays constant, circuit current ( $I$ ) doubles.
Resistance & Current ( $I \propto \frac{1}{R}$ at constant $E$ ): Inversely proportional. If circuit resistance ( $R$ ) doubles while applied voltage ( $E$ ) stays constant, circuit current ( $I$ ) is cut in half.
Equal Scaling: If both applied voltage ( $E$ ) and circuit resistance ( $R$ ) are doubled simultaneously, circuit current ( $I$ ) remains unchanged.

7. Bench & Troubleshooting Application: Fluke 77/BN & Breadboard

Solderless Breadboard Internal Continuity

Power Rails (Vertical Columns along Left/Right Edges): Electrically continuous vertically from top to bottom. Used to distribute the positive ( $+$ ) and negative ( $-$ ) supply rails.
Terminal Strips (Horizontal Rows 1–30, Columns A–E and F–J): Electrically continuous horizontally in groups of 5 pins (A-B-C-D-E are connected together; F-G-H-I-J are connected together, separated by the center ravine).
Common Wiring Mistake: Plugging both leads of a resistor into the same horizontal 5-pin row (e.g., CP 7A and CP 7E) shorts the resistor out through the underlying metal clip. Always place resistor leads in two different numbered rows (e.g., CP 7E and CP 21E).

Fluke 77/BN Digital Multimeter Setup & Measurement Rules

Auto-Scale (Yellow Center Button): Keep the Fluke 77/BN in AUTO range mode. If a small encircled square appears on the LCD, the meter is locked in manual range—press the yellow center button to return to Auto-Scale.
Port Connections:
- COM (Black Jack): The black test lead remains plugged into COM for all measurements.
- V / Ω (Right Red Jack): Used for all DC/AC Voltage, Resistance ( $\Omega$ ), and Diode tests.
- 300mA (Center Fused Red Jack): Used for all BEC current measurements. BEC Rule: Never measure a current exceeding $300,\text{mA}$ in BEC.

Parameter	Circuit Power State	Meter Connection to Circuit	Red Probe Jack	Critical Safety / Accuracy Rule
DC Voltage ( $V_{\text{DC}}$ )	ENERGIZED (ON)	In Parallel (Across) the component	`V / Ω`	Place Black probe on reference/negative side, Red probe on positive side.
Resistance ( $\Omega$ )	DE-ENERGIZED (OFF)	Across the isolated component	`V / Ω`	Always remove power jumper first. Measuring $\Omega$ on a live circuit gives false readings and risks meter damage.
DC Current ( $A_{\text{DC}}$ )	ENERGIZED (ON)	In Series (“Make a Break – Fill the Break”)	`300mA`	Physically lift one circuit lead to create an open gap, bridge the gap with the meter probes, and move the red lead back to `V/Ω` immediately afterward.

Curriculum & Lab Handout Clarification (BE.01.03 Student Lab $9,\text{V}$ vs. $5,\text{V}$ Discrepancy): In the BE.01.03 Student Lab handout, a $9,\text{V}$ battery snaps onto a clip-on MB102 breadboard power regulator module that steps the rail voltage down to a regulated $5,\text{VDC}$ (yielding $I_T = \frac{5,\text{V}}{1,\text{k}\Omega} = 5,\text{mA}$ , as shown in the lab schematics). However, paragraphs 1.g, 1.n, and 1.s in the text contain legacy unedited prompts referencing $9,\text{V}$ and $9,\text{mA}$ . Always use your actual measured rail voltage ( $\approx 5,\text{VDC}$ on the regulated breadboard supply) when verifying $I = \frac{V}{R}$ at the bench.

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