Parallel resistors

Two two-terminal elements are in parallel when one terminal of each element belongs to the same first node and the other terminal of each belongs to the same second node. In Figure 2.1, R1 and R2 are both connected between the upper node and ground.

A node is the connected set of ideal-conductor points treated as having one electrical potential in the lumped-circuit model. Because both resistors terminate on exactly the same two nodes, the potential difference between their terminals is necessarily the same.

Let the upper node have potential V_A and the lower node have potential V_B. Define the branch voltage

The symbol V therefore means one specific potential difference between the same two nodes A and B. Since R1 and R2 connect to those same nodes,

This equality is not a new physical law. It follows directly from the definition of voltage between two nodes. For that reason Figure 2.2 shows one voltage reference V rather than inventing two independent voltages for the same node pair.

Let I be the total signed current entering the parallel combination at the upper node. Let I₁ be the signed current through R1 and I₂ the signed current through R2, using the arrows in Figure 2.3.

Kirchhoff's current law (KCL) is charge conservation written for a circuit node: if the ideal node has no unmodelled charge storage, the algebraic sum of currents at that node is zero. With the references just defined,

The second form is the familiar statement that the total current divides among the branches. It is not an extra “parallel formula”; it is simply KCL applied to the junction.

If one solved branch current were negative, its physical current would flow opposite to the chosen reference arrow. The node equation itself would remain valid.

The same voltage V appears across both resistors. Ohm's law therefore gives the two branch currents

Insert those element relations into the current balance:

Let R_eq mean the equivalent resistance: one resistor that reproduces the same terminal current I for the same applied terminal voltage V. Its terminal relation is

Comparing the two expressions for I gives

The quantity 1/R is the reciprocal of resistance. It is so useful that it receives its own name: conductance, denoted G.

The SI unit of conductance is the siemens, symbol S. One siemens equals one ampere per volt and also one reciprocal ohm:

In conductance form, the parallel rule becomes conceptually simple:

Parallel branches add conductance because they add independent paths for current between the same two nodes.

For exactly two resistors, start from the reciprocal relation and combine the fractions:

Invert both sides:

This is often called the product-over-sum form. It is convenient for two resistors, but the reciprocal or conductance form generalizes more naturally to many branches.

A common shorthand is

The symbol ∥ here means “the equivalent resistance of R1 in parallel with R2.” It is a circuit-notation operator, not ordinary multiplication.

Let N be the number of parallel resistors, and let k be the index used to label one branch. The general relation is

The Greek capital letter Σ means “sum.” The limits k=1 and N mean: add the conductance 1/R₁, then 1/R₂, and continue through 1/R_N.

For N identical resistors, each of resistance R,

Now inspect two useful ideal limits for a second branch:

• R2 → ∞: the branch approaches an open circuit—an ideal branch that carries no current. Its conductance 1/R2 tends to zero, so R_eq tends to R1.
• R2 → 0: the branch approaches an ideal short circuit—a zero-resistance path between the nodes. Its conductance becomes dominant and R_eq tends to zero.

These limits explain the rule physically: adding any finite positive parallel resistor adds positive conductance, so the equivalent resistance decreases.

Two resistors R1=3 kΩ and R2=6 kΩ are connected in parallel across an ideal 12 V source. The prefix k means kilo, so 1 kΩ=1000 Ω. The prefix m in mA means milli, so 1 mA=10⁻³ A.

Step 1 — branch currents.

Step 2 — total current using Kirchhoff's current law.

Step 3 — equivalent resistance.

Step 4 — independent terminal check.

The direct branch-current sum and the equivalent-resistor calculation agree.

Notice that R1 is half of R2 and therefore carries twice its current:

That inverse relationship between branch resistance and branch current is the seed of the current-divider relation developed later in this course.

Let P₁ and P₂ denote the powers absorbed in R1 and R2. Because both branches have the same voltage, using P=V²/R gives

The total absorbed power is

The equivalent resistance gives the same result:

If an ideal 12 V source supplies the network, the source current is 6 mA and, under the sign convention used in earlier chapters, its absorbed power is −72 mW: the negative sign means it delivers 72 mW to the resistors.

These distinctions matter later when a measurement instrument, transistor input or another circuit block is connected across an existing network. That added branch is a load: it changes the total conductance seen by the source.

1. Find the equivalent resistance of 2 kΩ in parallel with 3 kΩ.
2. An ideal 12 V source is applied across 4 kΩ and 6 kΩ in parallel. Find I1, I2, the total current I and Req.
3. Two identical 10 kΩ resistors are placed in parallel. Find Req. Then generalize to N identical resistors of value R.
4. A parallel pair has Req=1.5 kΩ and R1=3 kΩ. Find R2.
5. Estimate 1 kΩ in parallel with 1 MΩ, then calculate it exactly enough to see the loading effect.
6. Two resistors share only one node. A student calls them parallel because they are drawn side by side. What topological fact disproves the claim?

With series and parallel reductions established from first principles, the next lesson can derive the voltage divider as a direct consequence of one common series current—not as a formula to memorize in isolation.