Which Of These Diagrams Represent The Same Circuit

Which of These Diagrams Represent the Same Circuit?

When you look at several schematic drawings, the first question that often arises is: do these diagrams actually describe the same electrical circuit? At first glance, two pictures may look very different—one might have a long winding wire, another a compact block—but they could be electrically identical. Understanding how to judge equivalence is a fundamental skill for anyone studying electronics, from hobbyists wiring a breadboard to engineers designing complex printed‑circuit boards. This article walks you through the concepts, procedures, and pitfalls involved in deciding whether two circuit diagrams truly represent the same circuit.

Detailed Explanation

A circuit diagram is a graphical representation of an electrical network. Its purpose is to convey how components (resistors, capacitors, sources, etc.) are interconnected, not to show physical layout or wire length. Consequently, two diagrams can look dissimilar yet be topologically identical—they have the same set of nodes and the same connections between those nodes.

The core idea behind equivalence is graph isomorphism: if you treat each component as an edge and each junction (including the terminals of sources) as a vertex, two circuits are the same when there exists a one‑to‑one mapping between their vertices that preserves adjacency. In practice, you do not need to draw the abstract graph; you can work directly with the schematic by following a systematic reduction process:

1. Identify all distinct nodes (points where two or more component leads meet).
2. Label each component with its type and value.
3. Check the connections: which nodes each component bridges.
4. Simplify using series‑parallel reduction, source transformation, or Thévenin/Norton equivalents.
5. Compare the reduced forms; if they match, the original diagrams are equivalent.

If the networks contain dependent sources or non‑linear elements, the same topological test applies, but you must also ensure that the controlling relationships (e.g., a voltage‑controlled current source’s controlling voltage) are preserved under the node mapping.

Step‑by‑Step or Concept Breakdown

Below is a concrete, step‑by‑step method you can apply to any pair of diagrams. We’ll illustrate each step with a generic example, but the procedure works for resistors, capacitors, inductors, sources, and even more complex devices.

Step 1: Enumerate Nodes

• Scan each diagram and place a dot at every junction.
• Give each dot a label (e.g., N₁, N₂, N₃ …).
• Include the terminals of independent sources as nodes.

Step 2: List Branches (Components)

• For each component, write down the two nodes it connects. - Example: a 10 Ω resistor between N₁ and N₂ → R₁₂(10 Ω).

Step 3: Build the Connection Table

Create a table that lists, for each node, which branches are incident. This table is essentially the adjacency list of the underlying graph.

Step 4: Look for Series and Parallel Patterns

• Series: two components share exactly one node and no other component is connected to that node.
• Parallel: two components share both of their nodes.

Replace series strings with a single equivalent component (e.g., R_eq = R₁ + R₂) and parallel groups with their equivalent (e.g., 1/R_eq = 1/R₁ + 1/R₂).

Step 5: Apply Source Transformations (if needed)

• A voltage source V in series with a resistance R can be turned into a current source I = V/R in parallel with the same R (Thévenin ↔ Norton).
• This step helps to bring different diagrams into a common form.

Step 6: Reduce Until No Further Simplification Is Possible

Continue applying series/parallel reductions and source transformations until each diagram is expressed as a minimal set of nodes and branches—often a single equivalent resistance between two terminals, or a simple Thévenin/Norton pair.

Step 7: Compare the Final Forms

• If the reduced node‑branch lists are identical (same nodes, same component values, same source polarities), the original diagrams represent the same circuit. - If any mismatch remains (different equivalent resistance, different source polarity, or a different number of nodes), the circuits are distinct.

Real Examples ### Example 1: Two Resistor Networks

Diagram A

   +---[10Ω]---+---[20Ω]---+
   |           |           |
  ( )         ( )         ( )
   |           |           |
   +-----------+-----------+

(A 10 Ω resistor in series with a 20 Ω resistor, both connected between the same two nodes.)

Diagram B

   +---[30Ω]---+
   |           |
  ( )         ( )
   |           |
   +-----------+

(A single 30 Ω resistor between the same two nodes.)

Analysis - Nodes: left terminal = N₁, right terminal = N₂.

• Diagram A: series branch → R_eq = 10 Ω + 20 Ω = 30 Ω between N₁ and N₂.
• Diagram B: already a 30 Ω resistor between N₁ and N₂.
• Reduced forms match → Diagrams A and B represent the same circuit.

Example 2: Voltage Source with Series Resistance vs. Norton Equivalent

Diagram C

   +---[V=12V]---[4Ω]---+
   |                    |
  ( )                  ( )
   |                    |
   +--------------------+

(12 V source in series with a 4 Ω resistor.)

Diagram D

   +---[3A]---[4Ω]---+
   |                 |
  ( )               ( )
   |                 |
   +-----------------+

(A 3 A current source in parallel with a 4 Ω resistor.)

Analysis

• Perform source transformation on Diagram C: I = V/R = 12 V/4 Ω = 3 A, placing the 3 A source in parallel with the 4 Ω resistor → exactly Diagram

Continuing the comparison, the 3A current source in Diagram D sits in parallel with the same 4 Ω element that appears in Diagram C after transformation. Because the current source’s value was derived directly from the original voltage source (12 V ÷ 4 Ω = 3 A), the two arrangements are mathematically identical. When we label the left‑hand terminal as node X and the right‑hand terminal as node Y, each diagram collapses to a single branch connecting X to Y: a 4 Ω resistor. The presence of an independent source does not alter the topological map; it only changes how the source is represented. Hence the reduced node‑branch description is indistinguishable, confirming that Diagram C and Diagram D are two faces of the same underlying circuit.

Example 3: Mixed Series‑Parallel Network with a Switch

Consider the following two schematics that look different at first glance but are functionally equivalent once the switch is in the closed position.

Schematic E

   +---[15Ω]---+---[25Ω]---+
   |           |           |
  ( )         ( )         ( )
   |           |           |
   +---[5Ω]---+---+---[10Ω]---+

A 15 Ω resistor feeds a node that splits into two parallel arms: a 5 Ω branch and a series pair of 10 Ω and 25 Ω resistors, which later recombine before returning to the source.

Schematic F

   +---[30Ω]---+
   |           |
  ( )         ( )
   |           |
   +-----------+

A single 30 Ω resistor links the same two terminals.

When the switch in Schematic E is closed, the 5 Ω branch becomes a short circuit, effectively removing that leg from the network. The remaining path consists of the 15 Ω resistor in series with the 10 Ω‑25 Ω series combination, giving a total of 15 Ω + 10 Ω + 25 Ω = 50 Ω. However, the overall network also includes a parallel shortcut created by the 5 Ω element when the switch is open, which can be re‑expressed as an equivalent resistance of 30 Ω between the same terminals. After performing the series‑parallel reduction and simplifying the resulting expression, the effective resistance collapses to exactly 30 Ω, matching the single resistor of Schematic F. Thus, with the switch closed, the two diagrams converge to the same minimal representation.

Conclusion

The method outlined — identifying each node, listing branches, applying systematic reductions, and finally matching the simplified node‑branch signatures — provides a reliable way to decide whether disparate-looking schematics truly describe the same circuit. When the reduced forms align perfectly, the circuits are interchangeable from an analytical standpoint, even if their visual layouts employ different component arrangements or source representations. This systematic approach eliminates ambiguity and ensures that designers, analysts, and reviewers can confidently treat seemingly different diagrams as equivalents, facilitating clearer communication and more efficient circuit analysis.