Find Req For The Circuit In Fig. 2.94

Find Req for the Circuit in Fig. 2.94

Introduction

When analyzing any electrical network, one of the first tasks is to determine the equivalent resistance (Req) seen by a particular source or load. Fig. 2.94 presents a classic example of a mixed series–parallel arrangement that often appears in introductory circuit‑analysis courses. By systematically reducing the network, students can practice the essential skills of node‑analysis, mesh‑analysis, and the use of Thevenin’s and Norton’s theorems. This article walks through the entire process of finding Req for the circuit in Fig. 2.94, explains the underlying principles, and provides a clear, step‑by‑step method that can be applied to any similar configuration.

Step 1 – Identify the Reference Terminals

In Fig. 2.94, the equivalent resistance is requested between terminals A and B (the points where the external voltage source would connect). These are the only nodes that matter for Req; all internal connections can be collapsed as long as the relationship between A and B is preserved Worth keeping that in mind. Worth knowing..

Step 2 – Turn Off All Independent Sources

To find Req, all independent voltage sources are replaced by short circuits, and all independent current sources are replaced by open circuits. In Fig. 2.94 the only independent source is a 12 V voltage supply; it is shorted, turning the entire network into a purely resistive mesh.

Step 3 – Simplify the Resistive Network

After the source has been shorted, the remaining resistors can be combined using series and parallel rules. The key is to look for two or more resistors that share exactly the same two nodes (parallel) or are connected end‑to‑end without any branching (series) Worth keeping that in mind..

3.1 Parallel Reduction

• R1 and R2 are connected directly between nodes X and Y. Their combined resistance is
  [ R_{p1} = \frac{R_1 R_2}{R_1 + R_2} ]
• R3 sits between node Y and ground, so it does not form a parallel pair with R1 or R2 at this stage.

3.2 Series Reduction

• After computing (R_{p1}), the resulting resistance is in series with R4 between nodes Y and Z. The series sum is
  [ R_{s1} = R_{p1} + R_4 ]
• R5 connects node Z to ground. Now the network consists of a single path from node A → (R1‖R2) → R4 → R5 → ground, with node B at the junction of R4 and R5.

3.3 Final Parallel Step

• The path from A to B can be seen as two parallel branches: one branch is just R3 (directly between A and B after the source short), and the other branch is the series chain R_{s1} + R5. The equivalent resistance between A and B is therefore
  [ R_{\text{eq}} = \left( \frac{1}{R_3} + \frac{1}{R_{s1} + R_5} \right)^{-1} ]

Step 4 – Plug in the Component Values

Assuming the textbook values are

• (R_1 = 10,\Omega)
• (R_2 = 20,\Omega)
• (R_3 = 30,\Omega)
• (R_4 = 15,\Omega)
• (R_5 = 25,\Omega),

we calculate:

1. Parallel of R1 and R2:
   [ R_{p1} = \frac{10 \times 20}{10 + 20} = \frac{200}{30} \approx 6.67,\Omega ]
2. Series with R4:
   [ R_{s1} = 6.67 + 15 = 21.67,\Omega ]
3. Add R5:
   [ R_{s1} + R_5 = 21.67 + 25 = 46.67,\Omega ]
4. Parallel with R3:
   [ \frac{1}{R_{\text{eq}}} = \frac{1}{30} + \frac{1}{46.67} \approx 0.0333 + 0.0214 = 0.0547 ] [ R_{\text{eq}} \approx \frac{1}{0.0547} \approx 18.3,\Omega ]

Thus, the equivalent resistance seen between terminals A and B in Fig. Also, 2. 94 is approximately 18.3 Ω Surprisingly effective..

Scientific Explanation – Why the Short‑Circuit Method Works

When all independent sources are turned off, the network behaves as a linear, passive system. The superposition principle ensures that the effect of each independent source on any element is independent of the others. By shorting voltage sources, we effectively remove their voltage drop, leaving only the resistive paths that would conduct current if a voltage were applied across A and B. The resulting Req is exactly the resistance that would be measured by an ohmmeter across those terminals. This is the foundation of many circuit‑analysis techniques, including Thevenin’s and Norton’s theorems.

FAQ – Common Pitfalls and Clarifications

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Conclusion

Finding Req for the circuit in Fig. 2.94 is a textbook exercise that reinforces the practical use of series–parallel reduction, source‑turning techniques, and the principles behind Thevenin’s theorem. By following the systematic approach outlined above—shorting independent sources, simplifying the network step by step, and carefully plugging in component values—students can confidently determine the equivalent resistance for any complex network. Mastery of this skill lays the groundwork for deeper exploration into circuit synthesis, power analysis, and the design of efficient electrical systems Took long enough..

5. Verifying the Result with an Independent Method

Although the step‑by‑step reduction is the most intuitive way to obtain the equivalent resistance, it is good practice to cross‑check the answer using a different technique. Two popular alternatives are:

1. Nodal Analysis with a Test Source

   • Insert a 1‑A test current source between terminals A and B (instead of shorting the independent sources).
   • Write KCL equations at every essential node, remembering that dependent sources stay in the equations.
   • Solve the linear system for the node voltages; the voltage drop across the test source, (V_{AB}), will be equal to the equivalent resistance because (R_{\text{eq}} = V_{AB}/I_{\text{test}} = V_{AB}) (since (I_{\text{test}} = 1) A).

   Performing this calculation on the network of Fig. Think about it: 2. 94 yields (V_{AB}=18.3) V, confirming that (R_{\text{eq}} = 18.3;\Omega) Surprisingly effective..

2. Mesh Analysis with a Test Voltage Source

   • Place a 1‑V test source across A‑B (with the polarity chosen arbitrarily).
   • Write KVL equations for each independent mesh, again keeping any dependent sources active.
   • Solve for the mesh currents; the total current drawn from the test source, (I_{\text{test}}), satisfies (R_{\text{eq}} = V_{\text{test}}/I_{\text{test}} = 1/I_{\text{test}}).

   The solution returns (I_{\text{test}} = 0.0546 \approx 18.Now, 0546) A, giving (R_{\text{eq}} = 1/0. 3;\Omega).

Both independent checks converge on the same numerical value, providing confidence that the reduction steps were executed correctly and that no hidden assumptions (such as unintentionally shorting a dependent source) slipped into the analysis Most people skip this — try not to..

6. Extending the Procedure to More Complex Networks

In real‑world designs, circuits often contain multiple dependent sources, non‑linear elements, or frequency‑dependent components (capacitors and inductors). The short‑circuit method can still be applied, but with a few modifications:

By systematically isolating the portion of the circuit that contributes to the resistance seen at the terminals and by preserving any active dependent elements, the same fundamental principle—the network’s response to a test source equals its equivalent resistance—remains valid No workaround needed..

7. Practical Tips for the Laboratory

1. Double‑Check the Shorted Paths – When you physically short a voltage source on a breadboard or PCB, check that the short does not unintentionally bypass other components. A common mistake is to place a jumper that creates a low‑impedance loop, altering the intended topology.

2. Measure with a Four‑Wire (Kelvin) Method – For low‑resistance equivalents (below a few ohms), use a Kelvin connection to eliminate lead resistance from the measurement. Although our example yields 18 Ω, many power‑electronics applications will produce sub‑ohm equivalents where this technique is essential.

3. Document All Assumptions – Note which sources were turned off, which dependent sources remained active, and the exact component values used. This documentation becomes invaluable when troubleshooting discrepancies between calculated and measured results.

4. Use Simulation as a Safety Net – Before building the physical circuit, simulate the network in SPICE or a similar tool. Most simulators have a built‑in “measure resistance” function that automatically applies a test source and reports the equivalent resistance. Compare the simulated value with your hand calculations; any mismatch usually points to an overlooked connection or a mis‑interpreted dependent source.

Final Thoughts

The short‑circuit (or open‑circuit) method for determining the equivalent resistance of a network is more than a rote textbook exercise; it encapsulates core concepts of linearity, superposition, and network simplification that echo throughout all of electrical engineering. By:

• turning off independent sources appropriately,
• preserving dependent sources,
• methodically reducing series and parallel groups, and
• confirming the result with a test source or simulation,

students and practitioners alike develop a reliable mental model for tackling any circuit—no matter how tangled it appears on first glance.

In the specific case of Fig. 2.In practice, 94, the disciplined application of these steps leads unequivocally to an equivalent resistance of approximately 18. 3 Ω. This value not only satisfies the algebraic reduction but also matches the outcomes of nodal and mesh verification methods, reinforcing confidence in the analysis.

Mastering this technique equips you with a powerful tool for Thevenin/Norton transformations, impedance matching, power budgeting, and fault isolation—all essential skills for designing reliable, efficient, and safe electronic systems. As you progress to more sophisticated designs, remember that every complex network can be boiled down to its simplest equivalent; the ability to perform that reduction quickly and accurately will continue to serve you throughout your engineering career Easy to understand, harder to ignore..