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Annex 1 (BE01) — Lesson BE.01.06 • TLO: 28XX-MAIN-1002 • ELOs: 28XX-MAIN-1002m through 28XX-MAIN-1002y (13 ELOs)

Series Resistive Circuits

A series circuit is any electrical circuit in which components are connected end-to-end so that there is only one path for current flow. Every calculation, voltage divider design, and fault diagnosis in a series resistive circuit stems from four fundamental parameter laws.

1. The Four Fundamental Laws of Series Circuits

Rule 1 — Current ($I$) is COMMON in Series

Because there is only one conductive path from the negative terminal of the source, through each resistor, and back to the positive terminal, the exact same current flows through every component:

Rule 2 — Voltage ($E$ or $V$) is ADDITIVE in Series (Kirchhoff’s Voltage Law)

• Kirchhoff’s Voltage Law (KVL):
  1. The algebraic sum of all voltages (rises and drops) around any closed circuit loop is equal to zero:
     $$V_A - V_{R1} - V_{R2} - \dots - V_{RN} = 0$$
  2. Equivalently: The sum of the voltage drops across all series components equals the sum of the applied voltage rises:
     $$V_A = V_{R1} + V_{R2} + V_{R3} + \dots + V_{RN}$$
• Voltage Rise ($+$): Energy added to the circuit by the power source ($V_A$ or $V_{\text{in}}$).
• Voltage Drop ($-$): Potential expended in pushing current through a power-dissipating resistance ($V_{Rx} = I_T \times R_x$). Because $I_T$ is common, the voltage drop across any series resistor is directly proportional to its resistance.

Rule 3 — Resistance ($R$) is ADDITIVE in Series

Substituting Ohm’s Law ($V = I \times R$) into KVL ($V_A = V_1 + V_2 + \dots + V_N$) and canceling the common current $I_T$ proves that total series resistance is the sum of all individual resistances:

Rule 4 — Power ($P$) is ALWAYS ADDITIVE (Watt’s / Joule’s Law)

• Definition: Power ($P$) is the rate at which work is done (energy expended) over a period of time, measured in Watts ($\text{W}$). Resistors dissipate electrical power in the form of heat.
• Watt’s / Joule’s Law Formulas:
  $$P = I \times E \qquad\qquad I = \frac{P}{E} \qquad\qquad E = \frac{P}{I}$$
• Ohm’s Law Substitutions for Power:
  • When current ($I$) is unknown, substitute $I = \frac{E}{R}$ into $P = I \times E$:
    $$P = \left(\frac{E}{R}\right) \times E \implies \mathbf{P = \frac{E^2}{R}}$$
  • When voltage ($E$) is unknown, substitute $E = I \times R$ into $P = I \times E$:
    $$P = I \times (I \times R) \implies \mathbf{P = I^2 \times R}$$
• Total Power Rule: Whether a circuit is series, parallel, or series-parallel, total power is ALWAYS additive:
  $$P_T = P_{R1} + P_{R2} + P_{R3} + \dots + P_{RN} = I_T \times V_A$$

Summary Table of Series Resistive Circuit Properties

2. Polarity & Subscript Reference Notation ($V_{AB}$ vs. $V_A$)

Every DC voltage measurement possesses both amplitude (magnitude) and polarity ($+$ or $-$) depending on test probe placement.

1. Double-Subscript Notation ($V_{AB}$)

• Meaning: $V_{AB}$ represents the voltage at Point $A$ with respect to Point $B$.
• Multimeter Lead Placement Rule:
  • Red Probe (Measurement Lead): Placed on the first subscript ($A$).
  • Black Probe (Reference Lead): Placed on the second subscript ($B$).
• Reversing the Probes ($V_{BA}$ vs. $V_{AB}$): Reversing the red and black leads across the exact same two points flips the algebraic sign ($+$ becomes $-$, or $-$ becomes $+$) while the numerical magnitude remains identical ($V_{BA} = -V_{AB}$).

2. Single-Subscript Notation ($V_A$)

• Meaning: When only a single subscript is written ($V_A$, $V_B$), the second reference point is implicitly circuit ground ($0,\text{V}$).
• Multimeter Lead Placement Rule: Place the Red Probe on Point $A$ and the Black Probe on Ground.

Worked Polarity Walkthrough (BE.01.06 Reference Loop)

Consider a series loop with a $+4,\text{V}$ source between Point $A$ (top positive) and Point $D$ (ground), top resistor $R_1$ dropping $1,\text{V}$ between $A$ and $B$, and right resistor $R_2$ dropping $3,\text{V}$ between $B$ and $C$ (ground):

• $V_A$ (Red at $A$, Black at Ground $D$): $+\mathbf{4,\text{V}}$ (full source rise above ground).
• $V_{AB}$ (Red at $A$, Black at $B$): $+\mathbf{1,\text{V}}$ (Point $A$ is $1,\text{V}$ more positive than Point $B$).
• $V_{BA}$ (Red at $B$, Black at $A$): $-\mathbf{1,\text{V}}$ (Point $B$ is $1,\text{V}$ more negative than Point $A$).
• $V_{BC}$ (Red at $B$, Black at $C$): $+\mathbf{3,\text{V}}$ (Point $B$ is $3,\text{V}$ more positive than ground Point $C$).

Shifting the Reference Lead Between Resistors (BE.01.06 Lab Exercise): In a 3-resistor series string ($R_1$ at top, $R_2$ in middle, $R_3$ at bottom connected to negative source rail), if you anchor the Black (reference) lead at the junction between $R_2$ and $R_3$:

• Probing above the reference (top of $R_1$, or junction of $R_1/R_2$) measures Positive ($+$) because those points are closer to the positive terminal of the source.
• Probing below the reference (bottom of $R_3$ at the negative supply rail) measures Negative ($-$) because that point is more negative than the $R_2/R_3$ reference junction.

3. Unloaded Series Voltage Dividers

A voltage divider is a series string of resistors engineered to divide a single source voltage ($V_A$) into one or more smaller output potentials.

• Direct Proportionality: Because current $I_T$ is identical through every series resistor, each resistor drops a fraction of $V_A$ equal to its fraction of total resistance $R_T$:
  $$V_{Rx} = V_A \times \left(\frac{R_x}{R_T}\right)$$
• Benchmark Example: Given $V_A = 12,\text{V}$ across $R_1 = 2,\text{k}\Omega$ and $R_2 = 4,\text{k}\Omega$ ($R_T = 6,\text{k}\Omega$):
  • $R_2$ accounts for two-thirds of total resistance ($\frac{4,\text{k}\Omega}{6,\text{k}\Omega} = \frac{2}{3}$), so $R_2$ drops two-thirds of the source voltage: $V_{R2} = \mathbf{8,\text{V}}$.
  • $R_1$ accounts for one-third of total resistance ($\frac{2,\text{k}\Omega}{6,\text{k}\Omega} = \frac{1}{3}$), so $R_1$ drops one-third of the source voltage: $V_{R1} = \mathbf{4,\text{V}}$.

4. Worked Circuit Calculations & Parameter Trend Analysis

Example A: Solving from Applied Voltage & Resistances

Given: Series circuit with $V_A = 15,\text{V}$, $R_1 = 12,\text{k}\Omega$, $R_2 = 2,\text{k}\Omega$, $R_3 = 1,\text{k}\Omega$.

1. Total Resistance ($R_T$):
   $$R_T = R_1 + R_2 + R_3 = 12\,\text{k}\Omega + 2\,\text{k}\Omega + 1\,\text{k}\Omega = \mathbf{15\,\text{k}\Omega}$$
2. Total & Individual Currents ($I_T$):
   $$I_T = \frac{V_A}{R_T} = \frac{15\,\text{V}}{15\,\text{k}\Omega} = \mathbf{1\,\text{mA}} \implies I_{R1} = I_{R2} = I_{R3} = \mathbf{1\,\text{mA}}$$
3. Individual Voltage Drops ($V_{Rx} = I_T \times R_x$):
   • $V_{R1} = 1,\text{mA} \times 12,\text{k}\Omega = \mathbf{12,\text{V}}$
   • $V_{R2} = 1,\text{mA} \times 2,\text{k}\Omega = \mathbf{2,\text{V}}$
   • $V_{R3} = 1,\text{mA} \times 1,\text{k}\Omega = \mathbf{1,\text{V}}$ (KVL Check: $12,\text{V} + 2,\text{V} + 1,\text{V} = 15,\text{V} = V_A$)
4. Individual & Total Power ($P_{Rx} = I_T \times V_{Rx}$):
   • $P_{R1} = 1,\text{mA} \times 12,\text{V} = \mathbf{12,\text{mW}}$
   • $P_{R2} = 1,\text{mA} \times 2,\text{V} = \mathbf{2,\text{mW}}$
   • $P_{R3} = 1,\text{mA} \times 1,\text{V} = \mathbf{1,\text{mW}}$
   • $P_T = 12,\text{mW} + 2,\text{mW} + 1,\text{mW} = \mathbf{15,\text{mW}}$

Example B: Qualitative Parameter Trend Analysis (↑, ↓, SAME)

Scenario: In a 3-resistor series circuit ($R_1, R_2, R_3$), $V_A$, $R_1$, and $R_2$ remain the SAME, while $R_3$ INCREASES ($\uparrow$). How does every circuit parameter respond?

• $R_T \uparrow$ (Increases): Because resistance is additive ($R_T = R_1 + R_2 + R_3\uparrow$).
• $I_T, I_{R1}, I_{R2}, I_{R3} \downarrow$ (Decrease): By Ohm’s Law ($I_T = \frac{V_A}{R_T\uparrow}$), higher total resistance reduces current throughout the entire series loop.
• $V_{R1} \downarrow$ and $V_{R2} \downarrow$ (Decrease): Because $R_1$ and $R_2$ are unchanged but the current flowing through them decreased ($V_{R1} = I_T\downarrow \times R_1$).
• $V_{R3} \uparrow$ (Increases): By KVL ($V_{R3} = V_A - V_{R1}\downarrow - V_{R2}\downarrow$), since $V_A$ is constant and $R_1, R_2$ drop less voltage, $R_3$ now drops a larger share of $V_A$.
• $P_T \downarrow$ (Decreases): Since $P_T = V_A \times I_T\downarrow$.

5. Bench & Troubleshooting Application (BE.01.06 Lab)

In the BE.01.06 breadboard lab, students build a 3-resistor series circuit across the $5,\text{VDC}$ supply using:

• $R_1 = 1,\text{k}\Omega$ (Brown – Black – Red – Gold)
• $R_2 = 2,\text{k}\Omega$ (Red – Black – Red – Gold)
• $R_3 = 4.7,\text{k}\Omega$ (Yellow – Violet – Red – Gold)
• Nominal Baseline: $R_T = 7.7,\text{k}\Omega$, $I_T = 649.35,\mu\text{A}$, $V_{R1} = 649.35,\text{mV}$, $V_{R2} = 1.30,\text{V}$, $V_{R3} = 3.05,\text{V}$.

Series Fault Diagnostic Signatures (Open vs. Short)

Critical Diagnostic Distinction ($0,\text{V}$ Across a Component in Series): Measuring $0,\text{V}$ across a resistor in a series circuit can happen in two completely different faults:

1. Another component is OPEN: Circuit current is $0,\text{A}$, so every good resistor reads $0,\text{V}$ while the open component reads full source voltage ($V_A$).
2. The component itself is SHORTED: Current is higher than normal, the shorted component reads $0,\text{V}$ (and $0,\Omega$), and all remaining good resistors read higher-than-normal voltage drops.

Lab Handout Errata Notes (BE.01.06 Student Lab):

1. On Page 10 (Step C4), the recall prompt mentions $9,\text{VDC}$, whereas the breadboard regulator and Step E6 use $5,\text{VDC}$ ($\frac{5,\text{V}}{7.7,\text{k}\Omega} = 649.35,\mu\text{A}$).
2. On Page 23 (Troubleshooting Procedure 2 Table — $R_2$ Shorted), the observed $R_T$ cell is printed as 5.7Ω due to a missing k prefix; the actual total resistance with $R_2$ shorted is $R_1 + R_3 = 1,\text{k}\Omega + 4.7,\text{k}\Omega = \mathbf{5.7,\text{k}\Omega}$.

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