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Annex 1 (BE01) — Lesson BE.01.07 • TLO: 28XX-MAIN-1002 • ELOs: 28XX-MAIN-1002z, 1002ac, 1002ag, 1002ai, 1002ao, 1002av

Parallel Resistive Circuits

A parallel circuit is any circuit in which there is more than one path for current flow. Each independent current path is called a branch, and every point where current splits or recombines across multiple paths is called a node.

1. Topology & The Four Laws of Parallel Circuits

Rule 1 — Current ($I$) is ADDITIVE in Parallel (Kirchhoff’s Current Law)

• Node Definition: A node (junction) is any electrical point in a circuit where there is more than one path for current flow.
• Kirchhoff’s Current Law (KCL):
  1. The algebraic sum of all currents entering and leaving a node must equal zero:
     $$I_{\text{entering}} - I_{\text{leaving}} = 0 \implies I_{\text{in}} = I_{\text{out}}$$
  2. Total circuit current ($I_T$) supplied by the source equals the sum of all individual branch currents:
     $$I_T = I_{R1} + I_{R2} + I_{R3} + \dots + I_{RN}$$
• Calculating & Measuring Branch Current:
  • Calculation: Apply Ohm’s Law to each branch independently: $I_{RN} = \frac{E_{RN}}{R_N}$.
  • Measurement: Break the specific branch path you wish to measure (or break the main supply line to measure $I_T$) and insert the ammeter in series (“Make a break – fill the break”).

Rule 2 — Voltage ($E$ or $V$) is COMMON in Parallel

Because all parallel branches connect directly across the same two electrically common rails (top positive bus and bottom negative bus), the voltage across every parallel branch is identical and equal to the source voltage:

Rule 3 — Equivalent Resistance ($R_{EQ}$ or $R_T$) is LESS Than the Smallest Branch

Adding parallel branches provides additional paths for electron flow, increasing total conductance and decreasing total opposition ($R_{EQ}$).

Golden Sanity Check for Parallel Resistance: The total / equivalent resistance ($R_T$ or $R_{EQ}$) of any parallel bank is ALWAYS strictly less than the value of the smallest individual branch resistor ($R_{EQ} < R_{\text{smallest}}$). Adding another branch in parallel always decreases $R_{EQ}$; removing (opening) a branch always increases $R_{EQ}$.

Three Formulas for Parallel Equivalent Resistance ($R_{EQ}$ / $R_T$)

Combining Shortcut Methods across 3+ Branches: If given $R_1 = 5,\text{k}\Omega$, $R_2 = 10,\text{k}\Omega$, and $R_3 = 10,\text{k}\Omega$ in parallel: first reduce the two equal $10,\text{k}\Omega$ branches via Value-Over-Number ($\frac{10,\text{k}\Omega}{2} = 5,\text{k}\Omega$), then reduce the resulting $5,\text{k}\Omega \parallel 5,\text{k}\Omega$ pair ($\frac{5,\text{k}\Omega}{2} = \mathbf{2.5,\text{k}\Omega}$) in seconds without touching a calculator.

Rule 4 — Power ($P$) is ALWAYS ADDITIVE

Each branch resistor dissipates heat independently ($P_{RN} = I_{RN} \times E_{RN} = \frac{E_{RN}^2}{R_N}$). Adding more branches in parallel draws additional current from the source and increases total power consumption:

Summary Table of Parallel Resistive Circuit Properties

(Note: Slide 27 of the BE.01.07 presentation deck contains a copy-paste erratum repeating “in series” inside the summary table; Figure 07-04 on Page 7 of the BE.01.07 Student Handout reflects the correct parallel rules above.)

2. Current Dividers & Inverse Proportionality

A parallel resistive network acts as a current divider: total line current ($I_T$) enters a node and divides among the parallel branches, making every individual branch current smaller than $I_T$.

• Inverse Proportionality ($I_{\text{branch}} \propto \frac{1}{R_{\text{branch}}}$): Because voltage across all branches is identical, branch current is inversely (indirectly) proportional to branch resistance (and directly proportional to branch conductance).
• Ratio Rule: In a two-branch circuit where $R_1 = 2,\text{k}\Omega$ and load resistor $R_2 = 4,\text{k}\Omega$ across $V_A = 12,\text{V}$:
  • $R_1$ ($2,\text{k}\Omega$) has half the resistance of $R_2$ ($4,\text{k}\Omega$), so $R_1$ carries twice the current of $R_2$ ($I_{R1} = 6,\text{mA}$ vs. $I_{R2} = 3,\text{mA}$; $I_T = 9,\text{mA}$).

3. Worked Circuit Demonstrations & Branch Independence

Example A: Solving from Total Current ($I_T$) & Branch Resistances

Given: $I_T = 26,\text{mA}$, $R_1 = 1,\text{k}\Omega$, $R_2 = 10,\text{k}\Omega$, $R_3 = 5,\text{k}\Omega$.

1. Equivalent Resistance ($R_T$):
   $$R_T = \frac{1}{\frac{1}{1\,\text{k}\Omega} + \frac{1}{10\,\text{k}\Omega} + \frac{1}{5\,\text{k}\Omega}} = \frac{1}{0.001 + 0.0001 + 0.0002} = \frac{1}{0.0013\,\text{S}} = \mathbf{769.23\,\Omega}$$
2. Applied & Branch Voltages ($V_A$):
   $$V_A = I_T \times R_T = 26\,\text{mA} \times 769.23077\,\Omega = \mathbf{20\,\text{V}} \implies V_{R1} = V_{R2} = V_{R3} = \mathbf{20\,\text{V}}$$
3. Individual Branch Currents ($I_{Rx} = \frac{V_A}{R_x}$):
   • $I_{R1} = \frac{20,\text{V}}{1,\text{k}\Omega} = \mathbf{20,\text{mA}}$
   • $I_{R2} = \frac{20,\text{V}}{10,\text{k}\Omega} = \mathbf{2,\text{mA}}$
   • $I_{R3} = \frac{20,\text{V}}{5,\text{k}\Omega} = \mathbf{4,\text{mA}}$ (KCL Check: $20,\text{mA} + 2,\text{mA} + 4,\text{mA} = 26,\text{mA} = I_T$)
4. Power Dissipation:
   • $P_{R1} = 20,\text{mA} \times 20,\text{V} = \mathbf{400,\text{mW}}$
   • $P_{R2} = 2,\text{mA} \times 20,\text{V} = \mathbf{40,\text{mW}}$
   • $P_{R3} = 4,\text{mA} \times 20,\text{V} = \mathbf{80,\text{mW}}$
   • $P_T = 400 + 40 + 80 = \mathbf{520,\text{mW}}$

Example B: Finding an Unknown Branch Resistor ($R_2$)

Given: Two-branch parallel circuit with $V_A = 20,\text{V}$, $I_T = 5,\text{A}$, and $R_1 = 5,\Omega$. Find $R_2$:

1. Find Branch 1 current: $I_{R1} = \frac{V_A}{R_1} = \frac{20,\text{V}}{5,\Omega} = 4,\text{A}$.
2. Apply KCL to find Branch 2 current: $I_{R2} = I_T - I_{R1} = 5,\text{A} - 4,\text{A} = 1,\text{A}$.
3. Apply Ohm’s Law to Branch 2: $R_2 = \frac{V_A}{I_{R2}} = \frac{20,\text{V}}{1,\text{A}} = \mathbf{20,\Omega}$.

Example C: Branch Independence & Trend Analysis (↑, ↓, SAME)

Scenario: In a 3-branch parallel circuit connected directly across a constant voltage source $V_A$, $R_1$ DECREASES ($\downarrow$) while $V_A$, $R_2$, and $R_3$ remain the SAME:

• $V_{R1}, V_{R2}, V_{R3} = \text{SAME}$: All branches remain tied directly across the constant source $V_A$.
• $I_{R2}, I_{R3}, P_{R2}, P_{R3} = \text{SAME}$ (Branch Independence!): Because $V_A$ is unchanged and $R_2, R_3$ are unchanged, nothing changes in the unaffected parallel branches.
• $R_T \downarrow$ (Decreases): Lowering $R_1$ reduces overall parallel opposition.
• $I_{R1} \uparrow$ and $P_{R1} \uparrow$ (Increase): Because $I_{R1} = \frac{V_A}{R_1\downarrow}$.
• $I_T \uparrow$ and $P_T \uparrow$ (Increase): Because $I_T = I_{R1}\uparrow + I_{R2} + I_{R3}$.

4. Bench & Troubleshooting Application (BE.01.07 Lab)

1. Isolating Parallel Networks Before Measuring $R_{EQ}$

• Always disconnect the power source jumper before measuring resistance ($\Omega$).
• If the parallel bank is part of a larger circuit, open the series connections entering and leaving the parallel bank so the ohmmeter measures only the isolated parallel branches rather than reading back through the power supply or series resistors.

2. Measuring Total Current ($I_T$) vs. Branch Currents ($I_{R1}, I_{R2}, I_{R3}$)

• To measure Total Current ($I_T$): Lift the main positive jumper wire connecting the power supply rail to the first parallel node, and bridge that break with the Fluke 77/BN (300mA jack).
• To measure a single Branch Current ($I_{Rx}$): Leave the main supply jumper connected, lift one leg of that specific branch resistor (or its branch jumper) to a vacant breadboard row, and bridge that break with the ammeter.

3. Parallel Circuit Fault Signatures

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