3.1 Fundamental Electrical Laws, Ohm's Law & Power Calculations

Fundamental Electrical Laws, Ohm's Law & Power Calculations

Every physiological monitoring system, therapeutic life-support device, and diagnostic laboratory instrument in the clinical environment relies on the predictable motion and control of subatomic electric charge. For the Certified Biomedical Equipment Technician (CBET), a rigorous understanding of basic electrical theory is not merely academic—it is the direct foundation for diagnosing power supply faults, isolating component degradation, calculating signal attenuation, preventing thermal equipment damage, and safeguarding patients against lethal microshock and macroshock hazards.

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1. Atomic Structure, Electric Charge & Charge Carriers

All matter is composed of atoms containing a central nucleus of positively charged protons ($q_p = +1.602 \times 10^{-19}\text{ C}$) and electrically neutral neutrons, surrounded by orbiting shells of negatively charged electrons ($q_e = -1.602 \times 10^{-19}\text{ C}$). The chemical and electrical properties of any element are governed by its outermost orbital shell, termed the valence shell.

+-----------------------------------------------------------------------------+
|                        ENERGY BAND THEORY OF SOLIDS                         |
|                                                                             |
|   [CONDUCTOR]                 [SEMICONDUCTOR]              [INSULATOR]      |
|   (e.g., Copper, Gold)        (e.g., Silicon, Ge)          (e.g., PVC, PTFE)|
|                                                                             |
|   +-----------------+         +-----------------+          +---------------+|
|   | Conduction Band |         | Conduction Band |          |Conduction Band||
|   |=================|         +-----------------+          +---------------+|
|   | Overlapping     |                 ^                            ^        |
|   | Valence Band    |          Bandgap| Eg ~ 1.1 eV         Bandgap|Eg > 5eV|
|   +-----------------+                 v                            v        |
|                               +-----------------+          +---------------+|
|                               |  Valence Band   |          | Valence Band  ||
|                               +-----------------+          +---------------+|
+-----------------------------------------------------------------------------+

Classification of Electrical Materials:

1. Conductors (e.g., Copper, Silver, Gold, Aluminum): Atoms possess 1 to 3 valence electrons that are weakly bound to the parent nucleus. The valence band and conduction band overlap (forbidden energy gap $E_g = 0\text{ eV}$), allowing valence electrons to transition into the conduction band at room temperature with negligible energy input. Gold is extensively utilized on clinical ECG snap connectors, defibrillator relay contacts, and internal PCB edge connectors due to its high conductivity combined with outstanding resistance to atmospheric oxidation and chemical corrosion.
2. Semiconductors (e.g., Silicon, Germanium): Atoms possess 4 valence electrons forming a stable crystalline covalent lattice. At absolute zero ($0\text{ K}$), semiconductors act as pure insulators; at room temperature ($300\text{ K}$), thermal energy excites a small fraction of electrons across a narrow energy bandgap ($E_g \approx 1.12\text{ eV}$ for Silicon; $E_g \approx 0.67\text{ eV}$ for Germanium) into the conduction band, leaving behind mobile "holes". Doping with trivalent or pentavalent impurities creates p-type and n-type semiconductor materials used in diodes, transistors, and operational amplifiers.
3. Insulators (e.g., Polyvinyl Chloride [PVC], Polytetrafluoroethylene [PTFE / Teflon], Ceramic, Glass, Rubber): Atoms possess tightly bound valence shells (typically 5 to 8 valence electrons) and a wide forbidden energy bandgap ($E_g > 5\text{ to }10\text{ eV}$). Extremely high electric fields (dielectric breakdown strength) are required to liberate electrons. Insulators provide galvanic isolation and enclosure shielding to protect clinical operators and patients from mains potentials.

Units of Charge ($Q$):

The fundamental unit of electric charge is the Coulomb (C). One Coulomb represents the total electrical charge carried by approximately $6.2415 \times 10^{18}$ electrons:

$Q = I \cdot t = n \cdot e$

Where:

• $Q = \text{Electric charge in Coulombs (C)}$
• $I = \text{Electric current in Amperes (A)}$
• $t = \text{Time in seconds (s)}$
• $n = \text{Number of discrete charge carriers}$
• $e = 1.60217663 \times 10^{-19}\text{ C (elementary electronic charge)}$

Electron Flow vs. Conventional Current Flow

In electrical engineering and biomedical instrumentation schematics, two opposing conventions describe current direction:

• Conventional Current Flow: Posited by Benjamin Franklin, conventional current defines electric current as flowing from the positive (+) terminal to the negative (-) terminal of an external circuit. All standard circuit analysis conventions (Kirchhoff's Laws, diode symbols, transistor schematic arrows, operational amplifier polarities) are built on conventional current flow.
• Electron Flow: Describes the physical reality of charge carrier transport in metallic conductors. Negatively charged free electrons are repelled by the negative terminal and attracted toward the positive (+) terminal.

[!NOTE] Throughout the CBET examination and all standard biomedical service schematics, circuit analysis formulas, node equations, and diagnostic polarities strictly assume conventional current flow unless electron beam physics (such as legacy CRT deflection or X-ray tube anode target bombardment) is explicitly stated.

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2. Potential Difference (Voltage), Current & Conductance

+-----------------------------------------------------------------------------+
|                   ELECTROMOTIVE FORCE & CURRENT MECHANICS                   |
|                                                                             |
|   [HIGH POTENTIAL] (+)                                  (-) [LOW POTENTIAL] |
|            |                                                     |          |
|            +=======> Conventional Current Flow (I) =======>======+          |
|            |                                                     |          |
|            +-------< Physical Electron Drift (e-) <--------------+          |
|                                                                             |
|   POTENTIAL DIFFERENCE (V): Work required to move 1 Coulomb of charge       |
|   CURRENT (I): Rate of charge transport across a cross-section per second   |
|   CONDUCTANCE (G): Ease with which current flows (G = 1 / R)                |
+-----------------------------------------------------------------------------+

Potential Difference (Voltage, $V$ or $E$)

Voltage—termed Electromotive Force (EMF) or potential difference—is the electrical pressure or work required to move a unit charge between two distinct points against an electrostatic field. One Volt (V) is defined as the potential difference between two points when one Joule (J) of energy is expended in moving one Coulomb (C) of charge between those points:

$V = \frac{W}{Q} = \frac{\text{Energy in Joules}}{\text{Charge in Coulombs}}$

Clinical voltage orders of magnitude encountered by BMETs span more than nine orders of magnitude:

• Bioelectric Signals: Electroencephalogram (EEG) $\approx 10\text{ to }100,\mu\text{V}$; Electrocardiogram (ECG) $\approx 0.5\text{ to }4.0\text{ mV}$; Electromyogram (EMG) $\approx 1.0\text{ to }10.0\text{ mV}$.
• Digital Logic & Microcontroller Rails: $+1.8\text{ V}{\text{DC}}, +3.3\text{ V}{\text{DC}}, +5.0\text{ V}_{\text{DC}}$.
• Analog Operational Amplifier Rails: $\pm 12.0\text{ V}{\text{DC}}, \pm 15.0\text{ V}{\text{DC}}$.
• Rechargeable Battery Packs: $+3.7\text{ V}{\text{DC}}$ (single Li-ion cell) to $+14.4\text{ V}{\text{DC}}$ (4S Li-ion pack in transport monitors).
• Mains AC Power Distribution: $120\text{ V}{\text{RMS}}$ (standard hospital receptacle, $60\text{ Hz}$) or $208/240/480\text{ V}{\text{RMS}}$ (X-ray generators, CT scanners, steam autoclaves).
• Defibrillator Discharge: $+1,200\text{ to }+5,000\text{ V}_{\text{DC}}$ stored across high-voltage discharge capacitors.
• Electrosurgical Generators (ESU): Peak RF voltages of $1,000\text{ to }10,000\text{ V}_{\text{pk-pk}}$ during unmodulated coagulation modes.

Electric Current (Amperes, $I$)

Current is the rate at which charge passes through a specified cross-sectional boundary in an electric circuit. One Ampere (A) corresponds to the movement of one Coulomb of charge per second:

$I = \frac{dQ}{dt} = \frac{\text{Coulombs}}{\text{Seconds}}$

Conductance (Siemens, $G$)

Conductance represents the ease with which electrical current passes through a material or component. It is the mathematical reciprocal of resistance ($R$):

$G = \frac{1}{R} = \frac{I}{V}$

The SI unit of conductance is the Siemens (S) (historically referred to as the mho, $\mho$). In parallel circuit analysis, converting branch resistances to conductances allows total equivalent conductance to be calculated by simple addition ($G_{\text{total}} = G_1 + G_2 + \dots + G_n$).

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3. Resistance, Resistivity & Temperature Coefficients

Resistance ($R$) is the opposition that a material offers to the movement of electric charge carriers, converting electrical energy into thermal energy (Joule heating). Resistance is measured in Ohms ($\Omega$).

+-----------------------------------------------------------------------------+
|                        PHYSICAL RESISTANCE MECHANICS                        |
|                                                                             |
|                    |<---------------- L ---------------->|                  |
|                    +-------------------------------------+  ---             |
|             Current|                                     |   |              |
|             =====> |     Resistivity (rho) Material      |   A (Area)       |
|                    |                                     |   |              |
|                    +-------------------------------------+  ---             |
|                                                                             |
|                    Formula: R = rho * (L / A)                               |
|                    - Length (L) doubles       ---> Resistance DOUBLES       |
|                    - Area (A) doubles         ---> Resistance HALVES        |
|                    - Temperature (T) increases ---> RT = R0 * (1 + alpha*dT)|
+-----------------------------------------------------------------------------+

Physical Laws Governing Resistance

The resistance of any uniform physical conductor is determined by its material resistivity, length, and cross-sectional area:

$R = \rho \cdot \frac{L}{A}$

Where:

• $R = \text{Resistance in Ohms (}\Omega\text{)}$
• $\rho = \text{Specific resistivity of the material in Ohm-meters (}\Omega\cdot\text{m)}$
• $L = \text{Conductor length in meters (m)}$
• $A = \text{Conductor cross-sectional area in square meters (m}^2\text{)}$

American Wire Gauge (AWG) Rules of Thumb for BMETs:

In biomedical facilities and hospital power distribution cords (NFPA 99 / UL 60601-1 standards), conductor cross-sectional area is specified in American Wire Gauge (AWG):

1. The 3-Gauge Rule: Decreasing the wire gauge by 3 numbers (e.g., from 18 AWG to 15 AWG) doubles the cross-sectional area ($A$) and cuts the resistance per unit length in half.
2. The 10-Gauge Rule: Decreasing the wire gauge by 10 numbers (e.g., from 20 AWG to 10 AWG) increases the cross-sectional area by a factor of approximately 10, reducing resistance per unit length by a factor of 10.
3. Hospital-Grade Power Cords: Portable medical devices with high current draws (e.g., mobile X-ray units, blood warmers, autoclaves) require 14 AWG or 12 AWG conductors to minimize line voltage drops and ground-conductor impedance ($R_{\text{ground}} < 0.10,\Omega$ or $100\text{ m}\Omega$ per NFPA 99 standards).

Temperature Coefficient of Resistance ($\alpha$)

The resistivity of all conductive and semiconductive materials changes as temperature fluctuates. The relationship is expressed as:

$R_T = R_0 \left[1 + \alpha_0 (T - T_0)\right] = R_0 (1 + \alpha_0 \Delta T)$

Where:

• $R_T = \text{Resistance at temperature } T$
• $R_0 = \text{Nominal resistance at reference temperature } T_0\text{ (standardly } 0^\circ\text{C or } 20^\circ\text{C)}$
• $\alpha_0 = \text{Temperature coefficient of resistance (}\Omega/\Omega/^\circ\text{C or } ^\circ\text{C}^{-1}\text{)}$
• $\Delta T = (T - T_0) = \text{Temperature change in } ^\circ\text{C}$

Material Behavior Profiles:

• Positive Temperature Coefficient (PTC): Pure metallic conductors (Copper $\alpha \approx +0.00393/^\circ\text{C}$, Platinum $\alpha \approx +0.00385/^\circ\text{C}$) have a positive temperature coefficient. As temperature rises, increased lattice vibrations impede electron drift velocity, causing resistance to increase linearly.
• Resistance Temperature Detectors (RTD): Platinum RTDs (such as the international standard PT100: $R_0 = 100.00,\Omega$ at $0^\circ\text{C}$, $R_{100} = 138.50,\Omega$ at $100^\circ\text{C}$; and PT1000: $1000.0,\Omega$ at $0^\circ\text{C}$) are standardly employed in patient blood warmers, sterilization autoclaves, and clinical laboratory incubators due to their exceptional linearity, long-term repeatability, and drift-free calibration.
• Negative Temperature Coefficient (NTC): Metal-oxide ceramic semiconductors (thermistors) exhibit an exponential negative temperature coefficient where resistance drops drastically as temperature increases. NTC thermistors are standard in neonatal skin temperature probes, Foley catheter core temperature probes, and esophageal probes (e.g., YSI 400 series: $2,252,\Omega$ at $25^\circ\text{C}$; YSI 700 series dual thermistor probes).

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4. Ohm's Law & Electrical Power Calculations

+-----------------------------------------------------------------------------+
|                        THE OHM'S LAW & POWER WHEEL                          |
|                                                                             |
|                         P / I            V^2 / R                            |
|                             \          /                                    |
|                      I * R ---\      /--- I^2 * R                           |
|                                [VOLTAGE]                                    |
|                                    V                                        |
|                                    |                                        |
|            [CURRENT] I ----------( + )---------- R [RESISTANCE]             |
|                                    |                                        |
|                                [ POWER ]                                    |
|                                    P                                        |
|                      V / R ---/      \--- V / I                             |
|                             /          \                                    |
|                        P / V             V^2 / P                            |
+-----------------------------------------------------------------------------+

Fundamental Mathematical Formulations:

$\text{Ohm's Law: } V = I \cdot R \quad \iff \quad I = \frac{V}{R} \quad \iff \quad R = \frac{V}{I}$

$\text{Joule's Power Law: } P = V \cdot I = I^2 \cdot R = \frac{V^2}{R}$

Where:

• $V = \text{Potential difference in Volts (V)}$
• $I = \text{Current in Amperes (A)}$
• $R = \text{Resistance in Ohms (}\Omega\text{)}$
• $P = \text{Electrical power in Watts (W, where } 1\text{ W} = 1\text{ Joule/second)}$

Electrical Energy ($E$ or $W$)

Energy represents the total work performed over a finite period of time:

$E = P \cdot t$

• In the SI system: Energy is expressed in Joules (J) or Watt-seconds (W·s), where $1\text{ J} = 1\text{ W} \times 1\text{ s}$. Defibrillator discharge energy is universally calibrated in Joules ($10\text{ to }360\text{ J}$).
• In commercial power distribution and battery metrics: Energy is expressed in Watt-hours (Wh) or Kilowatt-hours (kWh) ($1\text{ Wh} = 3,600\text{ J}$; $1\text{ kWh} = 3.6 \times 10^6\text{ J}$).

Efficiency ($\eta$) and Power Dissipation Calculations

No real-world electronic subsystem is $100%$ efficient. Internal conversion losses are dissipated as waste heat, which must be extracted from medical device enclosures via convection, heatsinks, or cooling fans:

$\eta = \left(\frac{P_{\text{out}}}{P_{\text{in}}}\right) \times 100\%$

$P_{\text{loss}} = P_{\text{in}} - P_{\text{out}} = P_{\text{in}} (1 - \eta)$

[!IMPORTANT] Clinical Case Study: Linear Power Supply Regulator Thermal Dissipation An older clinical telemetry central station uses an LM7805 three-terminal linear voltage regulator to step down a $+12.0\text{ V}{\text{DC}}$ unregulated rail to a clean $+5.0\text{ V}{\text{DC}}$ logic rail powering microcontroller and ADC ICs drawing a continuous $I_{\text{load}} = 1.20\text{ A}$.

1. Output Power: $P_{\text{out}} = V_{\text{out}} \times I_{\text{load}} = 5.0\text{ V} \times 1.20\text{ A} = 6.00\text{ W}$
2. Input Power: $P_{\text{in}} = V_{\text{in}} \times I_{\text{load}} = 12.0\text{ V} \times 1.20\text{ A} = 14.40\text{ W}$
3. Power Dissipated as Pure Heat: $P_{\text{loss}} = (V_{\text{in}} - V_{\text{out}}) \times I_{\text{load}} = (12.0\text{ V} - 5.0\text{ V}) \times 1.20\text{ A} = 7.0\text{ V} \times 1.20\text{ A} = 8.40\text{ W}$
4. Electrical Efficiency: $\eta = (6.00\text{ W} / 14.40\text{ W}) \times 100% = 41.67%$

Troubleshooting Takeaway: Dissipating $8.40\text{ W}$ across a TO-220 package without an aluminum heatsink will raise junction temperature above $150^\circ\text{C}$ within seconds, triggering thermal shutdown or permanent failure. Modern medical devices replace linear regulators with high-efficiency ($>90%$) buck switching converters to minimize thermal build-up.

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5. Resistor Color Coding, Component Types & Power Ratings

+-----------------------------------------------------------------------------+
|                     RESISTOR COLOR CODE IDENTIFICATION                      |
|                                                                             |
|   4-BAND RESISTOR:  [Band 1][Band 2]  x  [Multiplier]   [Tolerance]         |
|                     (1st Dig)(2nd Dig)   (10^n)         (+/- %)             |
|                                                                             |
|   5-BAND RESISTOR:  [Band 1][Band 2][Band 3] x [Multiplier] [Tolerance]    |
|   (Precision 1%)    (1st Dig)(2nd Dig)(3rd Dig) (10^n)       (+/- %)        |
|                                                                             |
|   Mnemonic: Bad Boys Race Our Young Girls But Violet Generally Wins         |
+-----------------------------------------------------------------------------+

EIA Color Code Reference Table:

Decoding Examples:

1. 4-Band Example: Resistor bands are Yellow – Violet – Orange – Gold
   • Digit 1 = Yellow (4)
   • Digit 2 = Violet (7)
   • Multiplier = Orange ($10^3 = 1,000$)
   • Tolerance = Gold ($\pm 5%$)
   • Nominal Value = $47 \times 1,000,\Omega = 47,000,\Omega = 47\text{ k}\Omega \pm 5%$ (Acceptable range: $44.65\text{ k}\Omega\text{ to }49.35\text{ k}\Omega$).
2. 5-Band Precision Example: Resistor bands are Brown – Black – Black – Red – Brown
   • Digit 1 = Brown (1)
   • Digit 2 = Black (0)
   • Digit 3 = Black (0)
   • Multiplier = Red ($10^2 = 100$)
   • Tolerance = Brown ($\pm 1%$)
   • Nominal Value = $100 \times 100,\Omega = 10,000,\Omega = 10.0\text{ k}\Omega \pm 1%$ (Acceptable range: $9.90\text{ k}\Omega\text{ to }10.10\text{ k}\Omega$).

Resistor Types and Clinical Applications:

• Metal Film Resistors: Feature a thin metallic oxide film deposited on a ceramic substrate. Highly stable, low temperature coefficient ($25\text{ to }50\text{ ppm/K}$), tight tolerances ($0.1%\text{ to }1%$), and extremely low current noise. Standard choice for patient-isolation preamplifiers, ECG Wilson Central Terminal networks, and active filter stages.
• Carbon Composition: Made from powdered carbon and resin binder. Inexpensive and withstands high energy transient surges without catastrophic open circuits, but exhibits poor tolerance ($\pm 5%\text{ to }\pm 20%$), high moisture absorption drift, and significant thermal noise ($1/f$ flicker noise). Largely obsolete in modern medical devices except for specialized high-voltage surge dampers.
• Wirewound Resistors: High-power resistive wire (e.g., Nichrome) wound around a ceramic core. Capable of dissipating high power ($5\text{ W to }>100\text{ W}$) with tight precision. Crucial limitation: Coil structure exhibits significant parasitic series inductance, making standard wirewound resistors unsuitable for high-frequency RF electrosurgical generator circuits (ESU). Used in power supply dummy loads, motor brake circuits, and autoclave heater controls.
• Surface Mount Device (SMD) Chips: 3-digit marking (e.g., "223" = $22 \times 10^3 = 22\text{ k}\Omega$), 4-digit precision marking (e.g., "4702" = $470 \times 10^2 = 47.0\text{ k}\Omega$), or "R" as a decimal point (e.g., "0R05" = $0.05,\Omega$ current-sense shunt resistor).
• Variable Resistors:
  • Potentiometer (Pot): 3-terminal device configured as an adjustable voltage divider (e.g., manual volume control, touch-screen optical encoder backup, ventilator exhalation valve position sensor).
  • Rheostat: 2-terminal device configured in series to vary current directly.

Resistor Power Ratings & Safety Derating

Resistors are manufactured in standard physical power ratings: $1/8\text{ W } (0.125\text{ W}), 1/4\text{ W } (0.25\text{ W}), 1/2\text{ W } (0.50\text{ W}), 1\text{ W}, 2\text{ W}, 5\text{ W}, 10\text{ W}+$. Physical package size correlates directly with power rating because larger surface areas dissipate heat to ambient air more effectively.

[!WARNING] The 50% BMET Engineering Derating Rule: In medical equipment maintenance and component-level board repair, always select a replacement resistor rated for at least twice (200%) the calculated maximum steady-state power dissipation ($P_{\text{rated}} \ge 2 \times P_{\text{actual}}$). Inside enclosed patient monitors where ambient internal temperatures reach $40^\circ\text{C to }50^\circ\text{C}$, operating a resistor at $100%$ of its rated power accelerates thermal substrate degradation, shifts resistance out of tolerance, and scorches the PCB.

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6. Practical Medical Troubleshooting Calculations

Case 1: LED Status Indicator Current-Limiting Resistor Sizing

A frontline defibrillator front panel requires an amber "Battery Charging" LED indicator operating from a $+12.0\text{ V}_{\text{DC}}$ power bus. The LED datasheet specifies a forward voltage drop $V_f = 2.10\text{ V}$ at a forward operating current $I_f = 15.0\text{ mA}$ ($0.015\text{ A}$).

1. Calculate Voltage Drop Across Resistor ($V_R$): $V_R = V_{\text{supply}} - V_f = 12.0\text{ V} - 2.10\text{ V} = 9.90\text{ V}$
2. Calculate Required Resistance ($R$): $R = \frac{V_R}{I_f} = \frac{9.90\text{ V}}{0.015\text{ A}} = 660\,\Omega$ (Select closest standard EIA 5% value: $680,\Omega$)
3. Calculate Power Dissipation in the Resistor ($P_R$): $P_R = I_f^2 \cdot R = (0.015\text{ A})^2 \times 680\,\Omega = 0.000225 \times 680 = 0.153\text{ W } (153\text{ mW})$
4. Select Appropriate Resistor Power Rating: Using the $50%$ derating rule ($P_{\text{rated}} \ge 2 \times 0.153\text{ W} = 0.306\text{ W}$), a $1/2\text{ W } (0.50\text{ W})$ metal film resistor is selected. A standard $1/8\text{ W}$ or $1/4\text{ W}$ resistor would run excessively hot and fail prematurely.

Case 2: Transport Monitor Battery Run-Time Calculation

A portable transport vital signs monitor is powered by an internal 4-cell Lithium-Ion battery pack with a nominal rating of $14.4\text{ V}_{\text{DC}}$ and $4,800\text{ mAh}$ ($4.80\text{ Ah}$). With continuous multi-parameter monitoring (ECG, SpO2, NIBP every 15 min, and active Wi-Fi telemetry), the monitor draws an average electrical power of $18.5\text{ W}$.

1. Calculate Total Stored Energy Capacity: $E_{\text{battery}} = V_{\text{nominal}} \times \text{Capacity (Ah)} = 14.4\text{ V} \times 4.80\text{ Ah} = 69.12\text{ Watt-hours (Wh)}$
2. Account for Internal DC-DC Converter Efficiency (\eta = 88%): $E_{\text{usable}} = 69.12\text{ Wh} \times 0.88 = 60.825\text{ Wh}$
3. Calculate Continuous Operating Run-Time: $t = \frac{E_{\text{usable}}}{P_{\text{draw}}} = \frac{60.825\text{ Wh}}{18.5\text{ W}} = 3.288\text{ hours } \approx 3\text{ hours and } 17\text{ minutes}$

If the measured battery run-time during preventive maintenance testing falls below $80%$ of the factory specification ($< 2.63\text{ hours}$), the battery pack has reached end-of-life due to internal chemical impedance growth and must be replaced.