Equivalent Resistance - Symmet

Cheatsheet Content

### Introduction to Symmetry in Circuits Symmetry can greatly simplify the calculation of equivalent resistance ($R_{eq}$) in complex resistor networks. It allows us to identify points with the same potential, or currents that cancel out, effectively reducing the network to a simpler form. This cheatsheet covers two main types of symmetry: **Parallel Axis Symmetry** and **Perpendicular Axis Symmetry**. ### Parallel Axis Symmetry #### Concept If a circuit is symmetric about an axis passing through the input and output terminals, then any two points that are mirror images with respect to this axis will have the same potential. This means: 1. **Resistors connecting mirror points can be removed without affecting $R_{eq}$.** This is because no current will flow through them if their terminals have the same potential. 2. **Resistors cut by the axis can be split into two parallel resistors.** #### Identification - Draw an imaginary axis through the input (A) and output (B) terminals. - Check if the circuit structure (resistors and connections) is identical on both sides of this axis. #### Application Steps 1. **Identify the axis of symmetry** passing through the input and output terminals. 2. **Remove any resistors** that connect two mirror-image points (if present). 3. **Split resistors** that lie directly on the axis of symmetry into two parallel resistors of double their original value (e.g., $R \to R/2$ and $R/2$ in parallel, or if it's a single resistor on the axis, it carries half the current, effectively doubling its resistance from the perspective of the *other half* of the circuit - this is a more advanced interpretation often simplified by just considering the half-circuit). A simpler way to think about this is if the axis divides a resistor, you can analyze half the circuit and then combine results. 4. **Simplify the reduced circuit** using series and parallel combinations. #### Example Consider a ladder network. If the input and output are at the ends, and the network is symmetric along its length, you can often use this principle. *Initial circuit:* A bridge circuit where the axis passes through the input and output. If the bridge is balanced, the middle resistor can be removed. If it's not balanced, but symmetric, you might find points of equal potential. ### Perpendicular Axis Symmetry #### Concept If a circuit is symmetric about an axis perpendicular to the line connecting the input and output terminals, then the potentials of mirror-image points are symmetric. This often implies: 1. **Points that are mirror images with respect to this axis (and are equidistant from the axis) will have the same potential difference relative to the axis.** 2. **Any branch connecting two such mirror points can be removed if the current flowing into one is equal to the current flowing out of the other (or if they are at the same potential).** 3. **More commonly, it implies that current entering one side of the axis must have a mirror current flowing out of the other, allowing you to short-circuit mirror points.** #### Identification - Draw an imaginary axis perpendicular to the line connecting the input (A) and output (B) terminals. - Check if the circuit structure is identical on both sides of this axis. #### Application Steps 1. **Identify the axis of symmetry** perpendicular to the input-output line. 2. **Fold the circuit along this axis.** 3. **Short-circuit (connect) all points that overlap when folded.** This is because they will have the same potential. 4. **Combine any parallel resistors** that result from the short-circuiting. 5. **Simplify the reduced circuit** using series and parallel combinations. #### Example Consider a cube network where you want to find $R_{eq}$ between opposite corners. *Initial circuit:* A resistor cube. By choosing an axis perpendicular to the diagonal connecting input/output, you can identify points that can be shorted. For instance, for $R_{eq}$ between opposite corners, there are three paths from the input. Due to symmetry, the current will divide equally among these three paths. The equipotential points can then be identified and shorted. ### General Tips for Symmetry - **Draw clearly:** Always redraw the circuit, highlighting the axis of symmetry and the points you are analyzing. - **Identify equipotential points:** The core idea behind both methods is identifying points that have the same electrical potential. - **Current distribution:** Consider how current would flow through the symmetric paths. - **Combine with other methods:** Symmetry can often be used in conjunction with series/parallel reduction, star-delta transformations, or nodal analysis. - **Practice:** Symmetry can be tricky to spot and apply correctly. Practice with various examples to build intuition.