Series and Parallel Circuits
Learning Objectives
- Distinguish a series circuit from a parallel circuit by current path and shared quantities.
- Calculate total (equivalent) resistance for resistors in series and in parallel.
- Apply the voltage-divider and current-divider rules to find individual component values.
- Analyze a mixed series-parallel circuit by reducing it step by step.
- Explain why one configuration is chosen over the other in real wiring (e.g., household outlets vs. old Christmas lights).
Quick Answer
A series circuit connects components end-to-end so there's only one path for current — every component carries the same current, and the supply voltage divides across them. A parallel circuit connects components across the same two points so there are multiple paths for current — every component sees the same voltage, and the total current splits among the branches. These two configurations are the building blocks of every circuit: even complex networks can usually be simplified by identifying series and parallel groups and reducing them step by step into a single equivalent resistance.
Series Circuits
In a series circuit, components are connected one after another, so current has only one possible path.
Key characteristics:
- Current is identical through every component: I_total = I1 = I2 = I3.
- Voltage divides across components proportional to their resistance.
- Total resistance is the simple sum: R_total = R1 + R2 + R3 + ...
- If any one component breaks the connection (e.g., a burnt-out filament), the whole circuit stops — there's no alternate path.
Voltage divider rule: V_x = V_supply × (R_x / R_total)
Worked Example 1: A 12V supply drives three resistors in series: R1 = 2Ω, R2 = 4Ω, R3 = 6Ω. Find the total resistance, the current, and the voltage across R2.
R_total = 2 + 4 + 6 = 12 Ω
I = V / R_total = 12 / 12 = 1 A
V2 = I × R2 = 1 × 4 = 4 V
(check with divider rule: V2 = 12 × (4/12) = 4 V ✓)
Parallel Circuits
In a parallel circuit, components connect across the same two nodes, giving current multiple paths.
Key characteristics:
- Voltage is identical across every branch: V1 = V2 = V3 = V_supply.
- Current splits among branches, proportional to each branch's conductance (1/R).
- Total resistance is found from reciprocals: 1/R_total = 1/R1 + 1/R2 + 1/R3 + ...
- For exactly two resistors: R_total = (R1 × R2) / (R1 + R2).
- If one branch opens, the others keep working — that's why household wiring is parallel.
Current divider rule (two resistors): I1 = I_total × (R2 / (R1 + R2))
Worked Example 2: A 10V supply feeds two resistors in parallel: R1 = 20Ω, R2 = 5Ω. Find the equivalent resistance, total current, and current through each branch.
R_total = (20 × 5) / (20 + 5) = 100 / 25 = 4 Ω
I_total = V / R_total = 10 / 4 = 2.5 A
I1 = V / R1 = 10 / 20 = 0.5 A
I2 = V / R2 = 10 / 5 = 2 A
(check: I1 + I2 = 0.5 + 2 = 2.5 A = I_total ✓)
Notice R_total (4Ω) is smaller than either branch resistor — parallel combinations always reduce total resistance because you're giving current more ways to flow.
Mixed Series-Parallel Circuits
Most real circuits combine both. The technique is to reduce step by step: find any pure series or pure parallel group, replace it with its single equivalent resistance, and repeat until one resistor remains.
Worked Example 3: A 24V supply connects to R1 = 4Ω in series with a parallel combination of R2 = 6Ω and R3 = 3Ω. Find the total current drawn from the supply.
Step 1 — reduce the parallel pair:
R_parallel = (6 × 3) / (6 + 3) = 18 / 9 = 2 Ω
Step 2 — now it's a simple series circuit:
R_total = R1 + R_parallel = 4 + 2 = 6 Ω
Step 3 — apply Ohm's Law:
I_total = 24 / 6 = 4 A
That 4A flows through R1, then splits between R2 and R3 according to the current divider rule.
Key Terms
| Term | Definition |
|---|---|
| Series circuit | Components connected end-to-end forming a single current path |
| Parallel circuit | Components connected across shared nodes, forming multiple current paths |
| Equivalent resistance | A single resistance value that behaves identically to a combination of resistors |
| Voltage divider | A series arrangement where voltage splits across resistors proportional to their resistance |
| Current divider | A parallel arrangement where current splits across branches inversely proportional to their resistance |
| Branch | One of the parallel current paths in a circuit |
| Conductance | The reciprocal of resistance (1/R), measured in siemens (S); adds directly for parallel elements |
Common Mistakes
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Misconception: "Adding more resistors always increases the circuit's total resistance." Why it's wrong: This is only true for series circuits. In parallel circuits, adding another resistor gives current another path, which always decreases total resistance. Correct: Series resistances add directly (R_total increases); parallel resistances combine via reciprocals (R_total decreases and is always less than the smallest branch).
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Misconception: "In a parallel circuit, current is the same through every branch, just like voltage." Why it's wrong: Voltage is what's shared in parallel circuits, not current — each branch draws current according to its own resistance (I = V/R for that branch). Correct: Parallel branches share the same voltage; series components share the same current. Don't mix the two up.
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Misconception: "If one bulb fails in any circuit, all the bulbs go dark." Why it's wrong: This is only true for series wiring (like old Christmas lights). In parallel wiring (like household outlets or modern light strings), a failed component only breaks its own branch — the rest keep working. Correct: Whether a single failure affects the whole circuit depends entirely on whether the components are wired in series or parallel.
Comparison and Connections
| Property | Series | Parallel |
|---|---|---|
| Current | Same through all components | Divides among branches |
| Voltage | Divides across components | Same across all branches |
| Total resistance | R_total = R1 + R2 + ... (increases) | 1/R_total = 1/R1 + 1/R2 + ... (decreases) |
| Effect of component failure | Entire circuit stops | Other branches keep working |
| Typical real-world use | Old-style string lights, fuses | Household wiring, car headlights |
Practice Questions
Recall 1: Write the formula for total resistance in a series circuit and in a parallel circuit (two resistors). Answer guidance: Series: R_total = R1 + R2 + ...; Parallel (two resistors): R_total = (R1×R2)/(R1+R2).
Recall 2: What quantity is identical across all components in a series circuit, and what quantity is identical across all branches in a parallel circuit? Answer guidance: Series — current is identical; Parallel — voltage is identical.
Understanding 1: Explain why parallel resistance is always less than the smallest individual resistor. Answer guidance: Each parallel branch offers an additional path for current, so for a fixed voltage, total current can only increase, meaning by R=V/I, total resistance must decrease below any single branch's value.
Understanding 2: Why does one failed bulb turn off the whole string in a series light set but not in a parallel one? Answer guidance: In series, there's only one current path, so a break anywhere stops all current. In parallel, each bulb has its own path back to the supply, so a break in one branch doesn't affect the others.
Application 1: Three resistors, 10Ω, 20Ω, and 30Ω, are connected in series across a 60V supply. Find the current and the voltage across the 20Ω resistor. Answer guidance: R_total = 60Ω; I = 60/60 = 1A; V(20Ω) = 1×20 = 20V.
Application 2: Two resistors, 12Ω and 4Ω, are in parallel across an 8A current source. Use the current divider rule to find the current through the 4Ω resistor. Answer guidance: I(4Ω branch) = I_total × R(other)/(R1+R2) = 8 × 12/(12+4) = 8 × 0.75 = 6A.
Analysis 1: A circuit has R1 = 5Ω in series with a parallel pair of R2 = 10Ω and R3 = 10Ω, all fed by a 15V supply. Find the total current and the current through R2. Answer guidance: R_parallel = (10×10)/20 = 5Ω; R_total = 5+5 = 10Ω; I_total = 15/10 = 1.5A; this splits equally between R2 and R3 (equal resistances), so I(R2) = 0.75A.
Analysis 2: Compare what happens to total power dissipation if two identical resistors are rewired from series to parallel across the same supply voltage. Which configuration draws more total power, and why? Answer guidance: Parallel draws more power. In series, R_total = 2R so P = V²/(2R); in parallel, R_total = R/2 so P = V²/(R/2) = 2V²/R — four times more power than the series case, because lower resistance allows more current to flow for the same voltage.
FAQ
Q1: Why is household wiring parallel instead of series? So that every appliance receives the full supply voltage independently, and switching off or unplugging one appliance doesn't cut power to the others.
Q2: Can a circuit be both series and parallel at the same time? Yes — most practical circuits are "series-parallel" combinations. You solve them by reducing sub-groups (pure series or pure parallel sections) into single equivalent resistors, one step at a time.
Q3: Does the current divider rule work for more than two resistors? It gets more complex — for more than two branches, it's usually easier to find the total equivalent resistance, get the total current, then find the voltage across the parallel section and apply Ohm's Law to each branch individually.
Q4: Why do old Christmas lights go fully dark when one bulb burns out, but new LED strings don't? Old strings wired the bulbs in series, so one broken filament interrupts the single current path. Modern LED strings typically wire small groups in parallel (or use bypass circuitry), so a single failure only affects a small section, if anything.
Q5: If I put a very large resistor in parallel with a very small one, does it matter much? Not much — the parallel combination will be dominated by (and close to) the smaller resistor's value, since it offers a much easier path for current. A useful approximation: when one resistor is far larger than the other, the parallel total is approximately equal to the smaller resistor.
Quick Revision
- Series: same current through all components; voltages add up to the supply voltage.
- Series total resistance: R_total = R1 + R2 + R3 + ...
- Parallel: same voltage across all branches; currents add up to the total current.
- Parallel total resistance: 1/R_total = 1/R1 + 1/R2 + ...; for two resistors, R_total = (R1×R2)/(R1+R2).
- Parallel R_total is always less than the smallest branch resistance.
- Voltage divider rule: V_x = V_supply × (R_x / R_total) — for series circuits.
- Current divider rule (two branches): I_x = I_total × (R_other / (R1+R2)) — for parallel circuits.
- A break in a series circuit stops all current; a break in one parallel branch leaves the rest working.
- Mixed circuits are solved by reducing series and parallel sub-groups step by step.
- Household wiring is parallel; old-style Christmas lights were series.
Related Topics
Prerequisites: Ohm's Law, basic voltage/current/resistance concepts.
Related Topics: Kirchhoff's Laws, Power Calculations, Network Theorems.
Next Topics: Thevenin's and Norton's Theorems, Network Theorems.