Activity 1.2.3 Circuit Calculations Answer Key

Activity 1.2.3 Circuit Calculations Answer Key: A Detailed Guide for Students and Educators

Understanding how to analyze simple electrical circuits is a foundational skill in engineering, physics, and technology courses. Activity 1.2.3 typically appears in introductory engineering or electronics curricula where learners apply Ohm’s Law, series‑parallel reduction techniques, and Kirchhoff’s rules to determine unknown voltages, currents, and resistances. This article provides a thorough walk‑through of the activity’s objectives, the core concepts involved, a step‑by‑step answer key with explanations, common pitfalls to avoid, and practical tips for mastering circuit calculations. By the end, readers should feel confident not only in reproducing the correct answers but also in interpreting what each result means in a real‑world context.

Overview of Activity 1.2.3

Activity 1.2.3 is designed to reinforce the relationship between voltage (V), current (I), and resistance (R) through a series of progressively challenging circuit diagrams. Learners are presented with a mix of:

• Series circuits – where components share the same current.
• Parallel circuits – where components share the same voltage.
• Combination (series‑parallel) circuits – requiring reduction to an equivalent resistance before applying Ohm’s Law.

The answer key expects students to:

1. Identify nodes and loops correctly.
2. Calculate equivalent resistance for each section.
3. Use Ohm’s Law (V = IR) to find missing quantities.
4. Verify results with Kirchhoff’s Voltage Law (KVL) and Kirchhoff’s Current Law (KCL).

Core Concepts Reviewed

Before diving into the solutions, a quick refresher on the principles that underlie the calculations helps avoid confusion.

Ohm’s Law

[ V = I \times R ] Voltage across a resistor equals the product of the current flowing through it and its resistance. This linear relationship holds for ohmic materials at constant temperature.

Series Resistance

When resistors are connected end‑to‑end, the total resistance is the sum:
[ R_{eq(series)} = R_1 + R_2 + \dots + R_n ]
Current is identical through each resistor; voltage divides proportionally to resistance.

Parallel Resistance For resistors sharing the same two nodes:

[ \frac{1}{R_{eq(parallel)}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n} ]
Voltage is identical across each branch; current divides inversely with resistance.

Kirchhoff’s Laws

• KVL – The algebraic sum of voltages around any closed loop equals zero.
• KCL – The algebraic sum of currents entering a node equals the sum leaving it.

These laws serve as checks: after solving for unknowns, plug them back into KVL/KCL to confirm consistency.

Step‑by‑Step Answer Key

Below is a detailed solution for each circuit presented in Activity 1.2.3. The numbers correspond to the diagram labels used in the activity sheet; if your version uses different labels, match the topology rather than the label.

Circuit A – Simple Series

Given:

• Voltage source (V_s = 12\text{ V})
• Resistors: (R_1 = 4\ \Omega), (R_2 = 6\ \Omega)

Solution:

1. Equivalent resistance
   [ R_{eq} = R_1 + R_2 = 4\ \Omega + 6\ \Omega = 10\ \Omega ]

2. Total current (Ohm’s Law)
   [ I_{total} = \frac{V_s}{R_{eq}} = \frac{12\text{ V}}{10\ \Omega} = 1.2\text{ A} ]

3. Voltage across each resistor

   • (V_{R1} = I_{total} \times R_1 = 1.2\text{ A} \times 4\ \Omega = 4.8\text{ V})
   • (V_{R2} = I_{total} \times R_2 = 1.2\text{ A} \times 6\ \Omega = 7.2\text{ V})

4. Check with KVL
   [ V_s - V_{R1} - V_{R2} = 12\text{ V} - 4.8\text{ V} - 7.2\text{ V} = 0\text{ V} ]
   ✓ Satisfied.

Circuit B – Simple Parallel

Given:

• Voltage source (V_s = 9\text{ V}) * Resistors: (R_1 = 3\ \Omega), (R_2 = 6\ \Omega)

Solution:

1. Equivalent resistance
   [ \frac{1}{R_{eq}} = \frac{1}{3\ \Omega} + \frac{1}{6\ \Omega} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} ]
   [ R_{eq} = 2\ \Omega ]

2. Total current
   [ I_{total} = \frac{V_s}{R_{eq}} = \frac{9\text{ V}}{2\ \Omega} = 4.5\text{ A} ]

3. Branch currents (Ohm’s Law)

   • (I_{R1} = \frac{V_s}{R_1} = \frac{9\text{ V}}{3\ \Omega} = 3\text{ A}) * (I_{R2} = \frac{V_s}{R_2} = \frac{9\text{ V}}{6\ \Omega} = 1.5\text{ A})

4. Check with KCL
   [ I_{total} = I_{R1} + I_{R2}

Completing the ParallelExample

The missing relationship is simply the statement of Kirchhoff’s Current Law at the node where the two branches re‑join the source. Substituting the branch currents we have

[I_{\text{total}} = I_{R1}+I_{R2}=3\ \text{A}+1.5\ \text{A}=4.5\ \text{A}, ]

which matches the current obtained from the equivalent resistance calculation. This confirms that the parallel network obeys KCL.

A Mixed‑Topology Circuit

Consider the network shown in the activity sheet where a 15 V source feeds a 5 Ω resistor in series with a parallel pair of 10 Ω and 15 Ω elements.

1. Combine the parallel section

[\frac{1}{R_{\text{par}}}= \frac{1}{10\ \Omega}+ \frac{1}{15\ \Omega} =\frac{3}{30}+\frac{2}{30}= \frac{5}{30} ;\Longrightarrow; R_{\text{par}}=\frac{30}{5}=6\ \Omega . ]

2. Add the series resistor

[ R_{\text{eq}} = 5\ \Omega + 6\ \Omega = 11\ \Omega . ]

3. Find the circuit current

[ I_{\text{circuit}} = \frac{15\ \text{V}}{11\ \Omega}=1.36\ \text{A (approximately)} . ]

4. Determine the voltage across the parallel branch

[ V_{\text{par}} = I_{\text{circuit}}\times R_{\text{par}} = 1.36\ \text{A}\times 6\ \Omega \approx 8.16\ \text{V}. ]

5. Compute the branch currents [ I_{10\Omega}= \frac{V_{\text{par}}}{10\ \Omega}=0.816\ \text{A}, \qquad I_{15\Omega}= \frac{V_{\text{par}}}{15\ \Omega}=0.544\ \text{A}. ]

6. Verification

   • KVL around the outer loop:
     [ 15\ \text{V} - I_{\text{circuit}}\times5\ \Omega - V_{\text{par}} = 0 . ] Substituting the numbers gives (15 - 6.8 - 8.16 \approx 0), confirming the loop equation.

   • KCL at the node splitting the parallel branch:
     [ I_{\text{circuit}} = I_{10\Omega

Conclusion: These meticulous steps underscore the critical role of foundational principles in advancing both theoretical and practical understanding of electrical systems.

Thus, rigorous adherence to these methods continues to anchor the field, ensuring precision and trustworthiness in every endeavor.

Building on the mixed‑topology example, the next logical step is to examine how the power delivered to each element is distributed throughout the network.

Power allocation – The resistor that sits in series with the parallel branch dissipates energy at a rate of

[ P_{5\Omega}=I_{\text{circuit}}^{2}\times5\ \Omega\approx(1.36\ \text{A})^{2}\times5\approx9.2\ \text{W}. ]

The two parallel elements share the remaining voltage drop of roughly 8.16 V. Their individual dissipations are therefore

[ P_{10\Omega}=V_{\text{par}}\times I_{10\Omega}\approx8.16\ \text{V}\times0.816\ \text{A}\approx6.7\ \text{W}, ] [ P_{15\Omega}=V_{\text{par}}\times I_{15\Omega}\approx8.16\ \text{V}\times0.544\ \text{A}\approx4.4\ \text{W}. ]

These figures illustrate that the larger resistance in the parallel network actually handles a smaller portion of the total energy, a consequence of the voltage being common to both branches.

Verification through alternative methods – Rather than relying solely on the step‑by‑step algebraic approach, the same results can be obtained by applying Thevenin’s theorem to the source‑resistor combination. By replacing the 5 Ω element and the voltage source with an equivalent single‑source model, the resulting Thevenin resistance matches the 11 Ω calculated earlier, and the subsequent current division yields identical branch currents. This cross‑check reinforces confidence in the analytical outcome.

Practical implications for circuit design – When a designer wishes to adjust the load behavior, modifying a single resistor in the parallel section has a disproportionate effect on the overall current distribution. For instance, increasing the 15 Ω element to 20 Ω raises its share of the total resistance, which in turn reduces its branch current and shifts more voltage toward the 10 Ω path. Such sensitivity analysis is essential when specifying components for precision applications, such as sensor interfaces or power‑management modules.

Extension to more complex networks – The techniques demonstrated—series‑parallel reduction, KVL, KCL, and power accounting—form the backbone for tackling circuits that incorporate multiple loops and nodes. By systematically simplifying portions of the network, applying the same set

of fundamental laws, and iteratively solving for unknown variables, even intricate circuits can be brought under control. Consider a circuit with multiple parallel branches, each containing a series combination of resistors and potentially other components like capacitors or inductors. The initial step would involve reducing each series combination to an equivalent resistance. Subsequently, the parallel branches could be simplified using the reciprocal relationship. This process continues until a single equivalent resistance is obtained, allowing for the calculation of the total current. From there, the branch currents and individual component voltages can be determined using KCL and KVL.

The role of simulation software – While analytical methods provide a deep understanding of circuit behavior, they can become unwieldy for complex topologies. Circuit simulation software, such as SPICE (Simulation Program with Integrated Circuit Emphasis), offers a powerful alternative. These tools allow designers to model circuits with arbitrary complexity, including non-linear components and time-varying signals. By entering the circuit schematic and component values, the software can automatically solve for voltages and currents at every node, providing a comprehensive picture of circuit performance. However, it’s crucial to remember that simulations are only as accurate as the models used. A thorough understanding of the underlying analytical principles remains essential for validating simulation results and interpreting their meaning. Furthermore, simulations can quickly explore "what-if" scenarios, allowing designers to optimize circuit performance and identify potential issues before committing to physical prototypes.

Beyond DC Analysis – The principles discussed here primarily focus on DC (Direct Current) circuit analysis. However, the same foundational concepts extend to AC (Alternating Current) circuits. In AC analysis, we introduce the concept of impedance, which is the AC equivalent of resistance and accounts for the effects of capacitors and inductors. Kirchhoff's laws remain valid, but the voltages and currents become complex numbers, representing both magnitude and phase. Techniques like phasor analysis and superposition are employed to solve for AC circuit behavior, building directly upon the principles of DC analysis. Understanding the transition from DC to AC is a critical step in mastering electrical circuit analysis.

In conclusion, the systematic application of Kirchhoff's laws, series-parallel reduction techniques, and power analysis provides a robust framework for understanding and predicting the behavior of electrical circuits. While modern simulation tools offer powerful alternatives, a firm grasp of these fundamental principles remains indispensable for effective circuit design, troubleshooting, and a deeper appreciation of the underlying physics. The ability to analyze circuits, both analytically and through simulation, empowers engineers to create reliable, efficient, and innovative electrical systems that underpin countless technologies we rely on daily.