Comprehensive Ohm's Law and Complex Combination Circuit Analysis

Circuit Reduction and Resistor Identification Strategy

• Initial Resistor Labeling: The circuit contains resistors labeled from $a$ to $j$. The process begins by filling in all the conceptual "bubbles" or labels for the resistive loads: $a, b, c, d, e, f, g, h, i, j$.
• Identification of Simplification Routes: The initial goal is to find components in "simple parallel" or "simple series" before performing any mathematical calculations.     * Resistor $a$ is in series with $b$ and $g$, but because it connects to multiple paths, it is not a "simple series" that can be immediately collapsed.     * Resistor $b$ is in series with $c$ and the group $d, e$, meaning it cannot be simplified on its own.
• First Component Reduction: Resistors $d$ and $e$ are identified as being in a simple series. These will be combined into a single resistive load labeled $de$.
• Second Component Reduction: The combined load $de$ is in a simple parallel arrangement with resistor $c$. These are combined into load $cde$.
• Third Component Reduction (The Series Gift): A group of resistors is found to be in series with one another: resistor $b$, the combined load $cde$, and resistor $f$. They are combined into load $bcdef$. Note: $f$ was initially misidentified as $i$ due to handwriting, but corrected to $f$.
• Constraints on Resistor g: Resistor $g$ cannot be simplified early because it is in series with both $a$ and $h$. This is not a simple series because the current splits.
• Fourth Component Reduction: The load $bcdef$ is in parallel with resistor $g$. This results in a new combined resistance labeled $bcdefg$.
• Fifth Component Reduction: The combined load $bcdefg$ is in series with resistor $i$. This results in load $bcdefgi$.
• Sixth Component Reduction: The load $bcdefgi$ is in parallel with resistor $h$. This creates the combined load $bcdefghi$.
• Final Reduction: The remaining resistors $a$, $bcdefghi$, and $j$ are all in a simple series. Combining these yields the total resistance of the circuit ($R_{\text{total}}$).

Mathematical Reduction Calculations (Steps 1-8)

• Assigned Resistance Values:     * $a = 12\,\Omega$     * $b = 20\,\Omega$     * $c = 8\,\Omega$     * $d = 2\,\Omega$     * $e = 6\,\Omega$     * $f = 2\,\Omega$     * $g = 10\,\Omega$     * $h = 5\,\Omega$     * $i = 20\,\Omega$     * $j = 15\,\Omega$
• Step 1: Combining d and e (Series):     * Calculation: $R_{de} = R_d + R_e$     * $R_{de} = 2 + 6 = 8\,\Omega$
• Step 2: Combining c and de (Parallel):     * Calculation: Funky Formula (Reciprocal Method)     * Since $c = 8\,\Omega$ and $de = 8\,\Omega$, the parallel resistance is halved: $\frac{8}{2} = 4\,\Omega$     * $R_{cde} = 4\,\Omega$
• Step 3: Combining b, cde, and f (Series):     * Calculation: $R_{bcdef} = R_b + R_{cde} + R_f$     * $R_{bcdef} = 20 + 4 + 2 = 26\,\Omega$
• Step 4: Combining bcdef and g (Parallel):     * Calculation: $R_{bcdefg} = \frac{1}{\frac{1}{26} + \frac{1}{10}}$     * Result: $7.2\,\Omega$
• Step 5: Combining bcdefg and i (Series):     * Calculation: $R_{bcdefgi} = 7.2 + 20$     * $R_{bcdefgi} = 27.2\,\Omega$
• Step 6: Combining bcdefgi and h (Parallel):     * Calculation: $R_{bcdefghi} = \frac{1}{\frac{1}{27.2} + \frac{1}{5}}$     * Result: $4.22\,\Omega$
• Step 7: Final Series Combination (Total Resistance):     * Calculation: $R_T = R_a + R_{bcdefghi} + R_j$     * $R_T = 12 + 4.22 + 15 = 31.22\,\Omega$

Reverse Analysis: Solving for Voltage and Current

• Total Circuit Parameters:     * Source Voltage ($V_T$): $120\,V$ ($AC$).     * Total Resistance ($R_T$): $31.22\,\Omega$.     * Total Current ($I_T$): $I_T = \frac{V_T}{R_T} = \frac{120}{31.22} = 3.84\,A$.
• Backtracking to Step 7 (Series Rules):     * In a series circuit, current ($I$) stays the same.     * $I_a = 3.84\,A$     * $I_j = 3.84\,A$     * $I_{bcdefghi} = 3.84\,A$     * Voltage Drops:         * $V_a = 12\,\Omega \times 3.84\,A = 46.08\,V$         * $V_j = 15\,\Omega \times 3.84\,A = 57.6\,V$         * $V_{bcdefghi} = 4.22\,\Omega \times 3.84\,A = 16.2\,V$
• Backtracking to Step 6 (Parallel Rules):     * In a parallel circuit, voltage ($V$) stays the same.     * $V_h = 16.2\,V$     * $V_{bcdefgi} = 16.2\,V$     * Currents:         * $I_h = \frac{16.2\,V}{5\,\Omega} = 3.24\,A$         * $I_{bcdefgi} = \frac{16.2\,V}{27.2\,\Omega} = 0.6\,A$ (rounded from $0.599$).
• Backtracking to Step 5 (Series Rules):     * $I_{bcdefg} = 0.6\,A$     * $I_i = 0.6\,A$     * Voltage Drops:         * $V_i = 0.6\,A \times 20\,\Omega = 12\,V$         * $V_{bcdefg} = 0.6\,A \times 7.2\,\Omega = 4.32\,V$
• Backtracking to Step 4 (Parallel Rules):     * $V_g = 4.32\,V$     * $V_{bcdef} = 4.32\,V$     * Currents:         * $I_g = \frac{4.32\,V}{10\,\Omega} = 0.432\,A$         * $I_{bcdef} = \frac{4.32\,V}{26\,\Omega} \approx 0.17\,A$
• Backtracking to Step 3 (Series Rules):     * $I_b = 0.17\,A, I_{cde} = 0.17\,A, I_f = 0.17\,A$     * Voltage Drops:         * $V_b = 0.17\,A \times 20\,\Omega = 3.4\,V$         * $V_f = 0.17\,A \times 2\,\Omega = 0.34\,V$         * $V_{cde} = 0.17\,A \times 4\,\Omega = 0.68\,V$
• Backtracking to Step 2 (Parallel Rules):     * $V_c = 0.68\,V, V_{de} = 0.68\,V$     * Currents:         * $I_c = \frac{0.68\,V}{8\,\Omega} = 0.085\,A$         * $I_{de} = \frac{0.68\,V}{8\,\Omega} = 0.085\,A$
• Backtracking to Step 1 (Series Rules):     * $I_d = 0.085\,A, I_e = 0.085\,A$     * Voltage Drops:         * $V_d = 0.085\,A \times 2\,\Omega = 0.17\,V$         * $V_e = 0.085\,A \times 6\,\Omega = 0.51\,V$

Practical Engineering and Calculation Tips

• The Leading Zero Rule: Always use a leading zero for decimals (e.g., $0.6$ instead of $.6$). This prevents the decimal point from being overlooked, which would cause a massive magnitude error in calculations.
• Scientific Calculator Orientation: On smartphone calculators, users must typically turn the device sideways (landscape mode) to reveal the scientific functions, including the reciprocal button ($1/x$ or $x^{-1}$) necessary for the "funky formula".
• The Funky Formula Utility: While mathematicians may require long-form fractional addition, technicians use the reciprocal keys ($x^{-1}$) on a calculator for speed and accuracy in parallel resistance calculations.
• Precision and Error Reduction: To reduce rounding errors, it is recommended to keep at least three digits after the decimal point during intermediate calculation steps until the final result is reached.

Questions & Discussion

• Siri Interruption: During the math portion, a student's or the instructor's device activated Siri. The instructor discussed the "Hey Siri" feature versus the new direct "Siri" prompt, expressing a preference for the older "Hey Siri" activation to avoid accidental triggers.
• Aussie Space Jump Anecdote: The instructor discussed a televised event where a person (sponsored by Red Bull) rode a balloon into space and skydived back to Earth. The jumper began spinning uncontrollably and reportedly felt ill inside the suit, though the footage was cleaned up for the broadcast. This was used as a metaphor for "going down" with a student's incorrect calculation.
• Calculator Struggles: A student struggled with finding scientific functions on their phone. The instructor explained the need to toggle the portrait orientation lock or switch the layout to find the reciprocal button ($1/x$).
• Kings Way Method vs. Instructor Method: A student expressed confusion between a method taught previously (Kings Way) and the current methodology. The instructor clarified that both methods are mathematically identical and use the same six basic Ohm's Law rules; the only difference is the visual breakdown and order of documentation.
• Clarification on R_de Calculation: A student noticed a notation error where the instructor had written "18" instead of "8". This was corrected: it was a simple series of resistors $d$ ($2\,\Omega$) and $e$ ($6\,\Omega$), totaling $8\,\Omega$.

Course Logistics and Mandatory Deadlines

• Attendance Policy: Attendance for the following Wednesday lab is optional for those who have mastered the material. However, the instructor will check the quiz scores of those not in attendance to ensure they are not being "lazy".
• Ohm's Law Quiz 1: Due Wednesday night. Students are allowed three attempts, and the highest score is recorded. Even if a student expects to fail, they are encouraged to attempt it to identify mistakes.
• Ohm's Law Quiz 2: Due Thursday. A $100\%$ score is expected if a student chooses to skip Thursday's optional session.
• Friday Exam: The final exam is scheduled for Friday. It is an online exam and will not require physical attendance in the classroom. This is to accommodate the length of the exam, which exceeds the standard one-hour class block.
• Exam Prerequisites: Students must attempt the "Interplay Interactive Assignment" (a review of meters and basic concepts) before the quizzes and the exam will unlock. The grade for this specific interactive assignment does not count toward the final course grade but is a mandatory gateway task.
• Exam Deadline: The final deadline for the exam is Saturday at midnight.