Units, Significant Figures & Introductory Circuit Quantities – Comprehensive Lecture Notes

Introduction & Lecture Objectives

• Course focus: foundational electrical quantities and variables used in Electrical & Computer Engineering (ECE).

• Lecture goals:

  • Identify and define charge, current, voltage, power, energy.

  • Present SI-units and the system of prefixes (powers of 10) that will be used constantly in the course.

  • Review significant figures, accuracy, precision, rounding rules, and their application when reporting answers.

  • Develop competence in converting units, working with powers of ten, and choosing correct notation (scientific vs. engineering).

  • Discuss mathematical relationships linking $q,i,v,p,w$ and their sign conventions (positive vs. negative power, passive sign convention).

  • Connect these ideas to conservation of energy (Tellegen’s Theorem) in electric circuits.

  • Introduce element taxonomy: active vs. passive, independent vs. dependent sources.

  • Build a conceptual bridge between physical metaphors (water analogy) and actual circuit schematics.

International System of Units (SI) & Standards of Measurement

• Engineers rely on measurable quantities expressed in a common, international language: SI (Système International), adopted in 1960.

• Seven principal SI units (plus charge as a derived unit):

  • Length $\to$ meter (m)

  • Mass $\to$ kilogram (kg)

  • Time $\to$ second (s)

  • Electric current $\to$ ampere (A)

  • Thermodynamic temperature $\to$ kelvin (K)

  • Luminous intensity $\to$ candela (cd)

  • Charge $\to$ coulomb (C)

• Energy unit: joule (J)

  • 1\,\text{J}=2.78\times10^{-7}\,\text{kW·h}=2.39\times10^{-4}\,\text{kcal}

  • Because 1\,\text{kW·h}=1000\;\text{W}\times3600\;\text{s}=3.6\,\text{MJ}.

• Comparison of systems (English, CGS, MKS, SI) highlights conversion pitfalls (e.g.
  $1\,\text{lb}=4.45\,\text{N}$, 1\,\text{ft·lb}=1.356\,\text{J}).

• Standards:

  • Meter defined via speed of light ($c=299\,792\,458\ \text{m/s}$).

  • Kilogram historically defined as mass of platinum-iridium cylinder (now redefined via Planck constant—though original lecture uses classic definition).

  • Second defined by $9\,192\,631\,770$ periods of Cs-133 radiation.

Prefixes & Powers of Ten

• SI advantage: decimal prefixes to scale base units.

  • Examples: $600\,\text{km}=600\,000\,\text{m}=600\,000\,000\,\text{mm}$.

• Prefix table (multiples of 3 emphasized for engineering notation):

  • $\text{G}=10^9$, $\text{M}=10^6$, $\text{k}=10^3$, $\text{m}=10^{-3}$, $\mu=10^{-6}$, $\text{n}=10^{-9}$, $\text{p}=10^{-12}$, etc.

• Power-of-ten algebra:

  • $10^n\times10^m=10^{n+m}$

  • $10^n/10^m=10^{n-m}$

  • $(10^n)^m=10^{n\,m}$

  • Addition/subtraction: $A\times10^n\pm B\times10^n=(A\pm B)\times10^n$ (exponents identical).

Scientific vs. Engineering Notation

• Scientific notation: mantissa $\in[1,10)$, exponent any integer.

  • Example: $\frac13=3.33\times10^{-1}$, $2300=2.30\times10^{3}$.

• Engineering notation: exponent restricted to multiples of 3; mantissa $\in[1,1000)$.

  • $\frac13=333.33\times10^{-3}$, $2300=2.30\times10^{3}$ remain identical for powers of 3.

• Metric prefixes map naturally to engineering notation.

Significant Figures • Accuracy • Precision • Rounding

• Two number types:

  • Exact (counting integers, definitions)—no measurement uncertainty.

  • Approximate (measurements)—subject to uncertainty.

• Significant digit: any digit known to be correct.

• Accuracy: closeness to true value.

• Precision: repeatability (scatter) of measurements.

• Rules:

  • Retain one uncertain digit when reporting results.

  • Rounding examples: $3.186\to3.19\to3.2$ depending on required sig-figs.

  • Example error propagation: For $R=V/I$, with $V=(100\pm5)\,\text{V}$ and $I=(10\pm0.2)\,\text{A}$, percent error:
    $\frac{\Delta R}{R}=\sqrt{\left(\frac{\Delta V}{V}\right)^2+\left(\frac{\Delta I}{I}\right)^2}\approx7\%$ (lecture used linear addition $7\%$).

• Reporting guideline checklist before substituting numerical values:

  • Ensure each quantity expressed with correct units & magnitude.

  • Convert to consistent unit system.

  • Confirm numerical reasonableness.

Unit-Conversion Example

• Velocity example: Given $d=4000\,\text{m}$, $t=1\,\text{min}$, want km/h.

  • Correct: v=\frac{4000\times10^{-3}\,\text{km}}{1/60\,\text{h}}=240\,\text{km·h}^{-1}.

  • Incorrect: substituting without unit consideration yields 4000\,\text{km·h}^{-1}.

Practical Energy-Cost Example

• Toaster: 1.2\,\text{kW·h} per heating cycle lasts 4 min.

  • Energy per cycle =\frac{1.2}{60}\,\text{kW} \times4\,\text{min}=0.08\,\text{kW·h}.

  • Monthly (30 days) cost at 0.5\,\text{TL/kW·h}: $0.08\times30\times0.5=1.2\,\text{TL}$.

• CPU clock: $2.6\,\text{GHz}=2.6\times10^{3}\,\text{MHz}=2.6\times10^{6}\,\text{kHz}=2.6\times10^{9}\,\text{Hz}$.

Water Analogy for Electricity

• Water head $\rightarrow$ Voltage (potential difference): high $\rightarrow$ low level.

• Water flow rate $\rightarrow$ Electric current ($i$).

• Height reference (sea level) $\rightarrow$ Ground node (0 V).

Charge ($q$)

• Fundamental property causing electromagnetic interaction.

• Electron charge magnitude $e=1.6\times10^{-19}\,\text{C}$.

• Steady-state notation: $Q$; time-varying notation: $q(t)$.

Current ($i$)

• Definition: $i=\dfrac{dq}{dt}$; units ampere (A) = coulomb/second.

• Net charge over interval $[t<em>1,t</em>2]$: $Q=\int<em>{t</em>1}^{t_2} i\,dt$.

• Conventional direction: from positive to negative potential (opposite to electron flow).

• Examples 04–06 illustrate differentiating q(t) to obtain i(t) and integrating i(t) to find total charge.

• DC ⇒ current constant; AC ⇒ sinusoidal current with zero average per period.

Voltage ($v$)

• Electromotive force (emf) causing charge movement: $v=\dfrac{dw}{dq}$ (J/C = volt).

• Always defined between two points; ground chosen as common 0 V reference.

Power ($p$) & Energy ($w$)

• Instantaneous power: $p=\dfrac{dw}{dt}=v\,i$; unit watt (W).

• Energy transferred between $t<em>1$ and $t</em>2$: $w=\int<em>{t</em>1}^{t<em>2} p\,dt=\int</em>{t<em>1}^{t</em>2} v\,i\,dt$.

• Positive $p$: element absorbs/consumes energy.

• Negative $p$: element delivers/generates energy.

Passive Sign Convention (PSC)

• If current enters the positive-labeled terminal of an element, $p=+v\,i$ (absorption).

• If current leaves the positive terminal, $p=-v\,i$ (generation).

• Diagrammatic reminder: arrows for $i$, $+/-$ polarity orientation.

Conservation of Energy (Tellegen’s Theorem)

• For a closed circuit: $\sum p(t)=0$ at every instant.

• Example 07: With components dissipating 3 W (C#2) and 5 W (C#3) and source #1 supplying 2 W, remaining component #4 must supply $6\,\text{W}$ to satisfy $-2+3+5-6=0$.

Classification of Circuit Elements

• Active elements: supply energy.

  • Independent sources: fixed-valued or time-varying voltage/current sources regardless of circuit conditions.

  • Dependent (controlled) sources: value determined by some other voltage or current in the circuit.

  • Types: CCCS, VCCS, VCVS, CCVS.

• Passive elements: cannot generate net energy; dissipate or temporarily store.

  • Resistor ($R$): dissipation; $v=iR$.

  • Capacitor ($C$): stores electric field energy.

  • Inductor ($L$): stores magnetic field energy.

Independent Source Behavior & Example 02

• Independent voltage source enforces prescribed $v$; current determined by external circuit.

• Independent current source enforces prescribed $i$; voltage adjusts accordingly.

• Example: 1 V dropped across unknown element in series with 3 A current source ⇒ current source generates $p=-3\,\text{A}\times1\,\text{V}=-3\,\text{W}$ (negative, i.e., delivers).

  • The red passive element absorbs $+3\,\text{W}$.

  • If red element were an independent voltage source (e.g., a battery being charged), then that battery absorbs power (charging).

Dependent Source Example

• Given $v<em>2=3\,\text{V}$, VCVS with gain 5 ⇒ $v</em>L=5v_2=15\,\text{V}$ across load terminals.

Basic Circuit Components & Symbols

• Resistor, capacitor, inductor symbols & commercial ranges (from µΩ to TΩ for resistors; µF capacitors shown).

• Measurement devices: Ammeter (in-series, measures $i$), Voltmeter (in-parallel, measures $v$).

• Wire: ideal conductor (R≈0); all nodes along same wire share same potential.

• Switches: mechanical or semiconductor; open = break path; closed = complete path.

• Protective devices: fuse, circuit breaker, ground fault interrupter—open circuit when abnormal currents detected.

• Ground variants: earth, chassis, digital, analog symbols.

Wires & AWG

• Current-carrying ability depends on cross-sectional area.

• AWG (American Wire Gauge) — larger gauge number ⇒ thinner wire ⇒ lower ampacity.

  • Example data: AWG 12 ⇒ diameter $\approx2.05\,\text{mm}$, area $3.31\,\text{mm}^2$; AWG 26 ⇒ diameter $0.40\,\text{mm}$, area $0.13\,\text{mm}^2$.

General Circuit Rules (Ideal)

• All nodes on same conductor share voltage.

• Source values are enforced regardless of attached elements.

• Ground node fixed at $0\,\text{V}$.

• Current flows high $\to$ low potential, provided a conductive path exists.

Ohm’s Law & Resistance Fundamentals

• Linear relationship $v=iR$ (Ohm’s Law).

• Resistance factors for wires:

  • Material resistivity $\rho$ (Ω·m)—copper, aluminum, etc.

  • Length $\ell$: $R\propto\ell$.

  • Cross-sectional area $A$: $R\propto1/A$.

  • Temperature (positive coefficient for metals).

• Conductance $G=1/R$; unit siemens (S).

  • Open circuit: $R\to\infty\Rightarrow i=0$.

  • Short circuit: $R=0\Rightarrow v=0$.

Resistor Colour Code & Tolerances

• 4-band system:

  • Band 1: first digit, Band 2: second digit, Band 3: multiplier $10^{\text{band}}$, Band 4: tolerance.

• Example: Brown-Red-Orange-Gold ⇒ $12\times10^{3}\,\Omega=12\,\text{k}Ω$, $\pm5\%$.

• Example question: For $50\,Ω\pm10\%$ resistor with $i=2\,\text{mA}$ ⇒ $v= iR$ ranges $0.09\,\text{V}$ to $0.11\,\text{V}$.

Example 08: Source & Resistor Power

• Circuit: $i<em>s = 2.5\,\text{mA}$ DC current entering negative node of $v</em>s = 10\,\text{V}$.

• Source power $p<em>s = v</em>s ( -i_s ) = -10\times 2.5\text{mW} = -25\,\text{mW}$ (generates).

• Resistor $R=4\,\text{k}Ω$ absorbs $p_R=10\times2.5=25\,\text{mW}$; PSC satisfied.

Sign Conventions Recap

• Always label $+/-$ across element; choose arrow for $i$.

  • Resistor must align $i$ from $+\to-$ to keep absorption $p=+v\,i$.

• Negative voltage/current illustrated: reversing polarity or direction is equivalent to negating magnitude.

Key Terms Summary (from Ch. 1 slide)

• Engineering notation: 1–3-digit mantissa with exponent multiple of 3.

• Exponent: power to which base-10 is raised.

• Metric prefix: alphabetic symbol representing power of ten (e.g.
  $\text{M},\text{k},\mu$).

• Power of ten: representation $10^n$.

• Accuracy: closeness to true value.

• Precision: repeatability of measurements.

• Significant digit: trusted digit in measurement.

• Scientific notation: mantissa $\in[1,10)$ times $10^n$.

Ethical, Practical & Real-World Implications

• Correct unit usage prevents catastrophic design errors (Mars Climate Orbiter lost due to metric/English mix-up).

• Proper fuse selection & wiring gauge ensure safety, adhering to electrical codes.

• Accurate power calculations support sustainability (energy billing, device efficiency, battery charging).

• Understanding passive sign convention critical when integrating renewable sources (solar panels, batteries) with loads.