Series and Parallel Circuits: Resistor & Capacitor Examples

A few circuit values can change the whole result

A resistor network can set the current through an LED, limit the output of a voltage divider, or protect a sensor input. Before choosing components, you need to know how the parts are connected. The same resistor values can produce very different results in series and in parallel.

This guide builds on the brief electronics overview in TheOnlineToolset introduction with worked calculations for resistor networks, a loaded voltage divider, an LED, and capacitor combinations.

You can use a calculator to check each step, but the formulas are useful on their own:

Components	Equivalent value
Resistors in series	`R_total = R1 + R2 + ... + Rn`
Resistors in parallel	`1/R_total = 1/R1 + 1/R2 + ... + 1/Rn`
Capacitors in parallel	`C_total = C1 + C2 + ... + Cn`
Capacitors in series	`1/C_total = 1/C1 + 1/C2 + ... + 1/Cn`

A fast way to check your work is to simplify one clearly connected section at a time, then compare the result with the Series and Parallel Resistor Calculator.

Resistors in series: add the resistance

Resistors are in series when there is one path for current through them, with no branch between them. The same current flows through each resistor, and the source voltage is divided among them.

For resistors in series, add their values:

R_total = R1 + R2 + ... + Rn

Worked example: two series resistors

Connect a 1 kΩ resistor and a 2 kΩ resistor in series across a 9 V supply.

Find total resistance: 1 kΩ + 2 kΩ = 3 kΩ.
Find circuit current: I = V/R = 9 V / 3,000 Ω = 0.003 A, or 3 mA.
Find each voltage drop:
- Across 1 kΩ: V = IR = 0.003 A × 1,000 Ω = 3 V.
- Across 2 kΩ: V = IR = 0.003 A × 2,000 Ω = 6 V.

The drops add to the supply voltage: 3 V + 6 V = 9 V. Their power dissipation is 9 mW and 18 mW, respectively, for 27 mW total.

Use the resistor calculator to check series resistance and, when you enter a supply voltage, review the resulting electrical values.

Resistors in parallel: add conductance

Resistors are in parallel when both ends of each resistor connect to the same two electrical nodes. They share the same voltage, while the current divides between branches.

Calculate the equivalent resistance with the reciprocal formula:

1/R_total = 1/R1 + 1/R2 + ... + 1/Rn

For just two resistors, you can also use:

R_total = (R1 × R2) / (R1 + R2)

Worked example: two parallel resistors

Place 1 kΩ and 2 kΩ resistors across a 9 V supply.

R_total = (1,000 Ω × 2,000 Ω) / (1,000 Ω + 2,000 Ω) = 666.7 Ω

Each branch has 9 V across it:

1 kΩ branch current: 9 V / 1,000 Ω = 9 mA.
2 kΩ branch current: 9 V / 2,000 Ω = 4.5 mA.
Total supply current: 9 mA + 4.5 mA = 13.5 mA.

The equivalent resistance is lower than the smallest individual resistor. Adding another parallel path gives current another route through the network.

Mixed series-parallel circuits: reduce one section at a time

A circuit can contain both series and parallel groups. Its physical layout can be misleading, so identify the electrical nodes first. Components connected end to end are not necessarily in series if a branch connects at their shared point.

Then simplify a group whose connections are clear, redraw the remaining circuit, and continue until you can calculate the total current or voltage. If the original circuit needs a more complex analysis than series and parallel reductions, these formulas alone may not be enough.

Worked example: a loaded voltage divider

Suppose a 9 V supply feeds a 1 kΩ upper resistor. The lower side has a 2 kΩ resistor in parallel with a 2 kΩ load.

Combine the two lower parallel resistances: 2 kΩ || 2 kΩ = 1 kΩ.
Add the upper resistor: R_total = 1 kΩ + 1 kΩ = 2 kΩ.
Find the supply current: I = 9 V / 2,000 Ω = 4.5 mA.
The output is across the lower 1 kΩ equivalent resistance: V_out = 4.5 mA × 1,000 Ω = 4.5 V.

The two equal lower branches each carry 2.25 mA. Without the load, the divider would have 1 kΩ above 2 kΩ and produce 6 V. Connecting the load lowers the output to 4.5 V.

This is a useful example of why a divider’s output can change when you connect a device to it. The Voltage Divider Calculator supports simple and loaded divider calculations. The series and parallel resistor calculator handles a single series or parallel topology; it does not model an arbitrary mixed network as a circuit diagram.

How to choose a resistor for an LED

A resistor in series with an LED limits current. For a basic DC circuit, calculate its value with:

R = (V_supply - V_forward) / I

Here, V_forward is the LED’s forward voltage at the intended operating current, and I is the target current in amperes. Check the LED datasheet for appropriate values: forward voltage varies by LED type and operating conditions.

Worked example: a 5 V supply and one LED

For an illustrative calculation, use a 5 V supply, a red LED with an assumed 2.0 V forward voltage, and a target current of 15 mA.

Convert current to amperes: 15 mA = 0.015 A.
Find the voltage across the resistor: 5 V - 2 V = 3 V.
Calculate the resistance: R = 3 V / 0.015 A = 200 Ω.
Choose a standard value at or above that result. With 220 Ω, the nominal current is 3 V / 220 Ω ≈ 13.6 mA.
Check resistor power: P = VI = 3 V × 0.0136 A ≈ 0.041 W.

A 1/8 W resistor is rated above that nominal dissipation, but the suitable rating for a real circuit also depends on the resistor’s datasheet, temperature, and operating conditions. The LED Resistor Calculator can compare standard values and estimate current and resistor power.

For multiple LEDs in series, add their forward-voltage drops before calculating the resistor. For example, use V_supply - (Vf1 + Vf2 + ...) as the voltage across the resistor. For parallel LED branches, use a separate current-limiting resistor for each branch; small differences between LEDs can otherwise cause uneven current sharing. This example is for a low-voltage DC circuit, not mains wiring.

Capacitors in series and parallel

Capacitors use the opposite equivalent-value rules from resistors:

In parallel, capacitances add: C_total = C1 + C2 + ... + Cn.
In series, add the reciprocals: 1/C_total = 1/C1 + 1/C2 + ... + 1/Cn.

Parallel capacitor example

A 10 µF capacitor and a 22 µF capacitor in parallel have:

C_total = 10 µF + 22 µF = 32 µF

If the ideal combination is charged to 12 V, its total charge is Q = CV = 32 µF × 12 V = 384 µC. Its stored energy is E = ½CV² = 2.304 mJ.

Series capacitor example

For 10 µF and 22 µF capacitors in series:

C_total = (10 × 22) / (10 + 22) µF = 6.875 µF

Across an ideal 12 V supply, the series pair holds 82.5 µC of charge. The ideal voltage division is 8.25 V across the 10 µF capacitor and 3.75 V across the 22 µF capacitor.

Real capacitors have tolerance, leakage, and other non-ideal behavior, so their voltage sharing may differ from this calculation. Do not treat capacitor voltage ratings as if they can simply be added to get a safe series rating. Check the parts’ specifications and the circuit design. The Series and Parallel Capacitor Calculator calculates ideal nominal combinations and, when voltage is provided, can show capacitor voltage, charge, and stored energy.

Common calculation mistakes

Choosing a formula from the drawing instead of the connections: Trace the nodes. Components that share the same two nodes are parallel; components in one unbranched path are series.
Mixing milliamperes and amperes: Convert before using formulas: 15 mA = 0.015 A.
Mixing ohms and kilohms: Use one unit consistently: 1 kΩ = 1,000 Ω.
Mixing capacitance units: For example, 1 µF = 1,000 nF. Convert values before adding or using the reciprocal formula.
Skipping the power check: A correct resistance value can still overheat if the resistor’s power rating is too low. Estimate dissipation and account for the component’s rated conditions.
Assuming every LED has the same forward voltage: Use the datasheet’s value for the specific LED and design current. A color-based preset is only a starting estimate.
Using one resistor for parallel LEDs without checking current sharing: Separate current-limiting resistors are generally needed for individual parallel branches.
Treating ideal calculations as measured guarantees: Component tolerances, temperature, supply variation, and measurement setup can all change real results.

You can also check a circuit’s voltage, current, resistance, and power relationships with the Ohm’s Law Calculator.

When an online calculator helps

A calculator is useful for checking arithmetic, comparing standard component values, or seeing how a changed input affects the result. It does not identify circuit connections for you: first decide which components are in series, which are in parallel, or whether loading changes a divider.

Choose a tool for the calculation in front of you:

Use the Series and Parallel Resistor Calculator for a series group or a parallel group.
Use the LED Resistor Calculator to estimate a limiting resistor, current, and power for a series LED setup.
Use the Series and Parallel Capacitor Calculator for nominal capacitor combinations.
Use the Voltage Divider Calculator to calculate a simple or loaded two-resistor divider.

Compare calculator results with your hand calculation and the relevant component datasheets before choosing parts.

Frequently asked questions

How do you calculate resistors in series?

Add the resistor values: R_total = R1 + R2 + ... + Rn. The same current passes through each resistor, and the supply voltage is divided across them.

What is the formula for parallel resistance?

Add the reciprocals, then take the reciprocal of the sum: 1/R_total = 1/R1 + 1/R2 + ... + 1/Rn. For two resistors, use R_total = (R1 × R2)/(R1 + R2).

Is total resistance lower in a parallel circuit?

For ordinary positive resistors, yes: the equivalent resistance is less than the smallest branch resistance. Each additional branch provides another path for current.

How do I choose a resistor for an LED?

Subtract the LED’s forward voltage from the supply voltage, then divide by the target current in amperes: R = (V_supply - Vf)/I. Choose a suitable standard resistance and check the resistor’s power rating. Use the LED datasheet rather than relying only on a typical value.

How do capacitors behave in series?

Their reciprocals add, so the equivalent capacitance is lower than the smallest capacitor in the series group. In an ideal series connection, each capacitor carries the same charge; the voltage division depends on capacitance.

Can I calculate mixed circuits online?

The series and parallel resistor calculator handles a group of resistors in one topology. For a mixed circuit, reduce simple groups step by step. The voltage divider calculator can handle a supported loaded divider, but these tools do not solve arbitrary circuit diagrams.

Why is my measured resistance different from the calculation?

The parts may have tolerance, or a measurement made in-circuit may include other current paths. Temperature and meter accuracy can also matter. For a clearer resistor measurement, disconnect power and isolate the part where practical.

Check the values before choosing components

Start by identifying the circuit connections, simplify one section at a time, and keep units consistent. Then check current, voltage, and power against the component specifications. Ideal formulas are a starting point; real parts have tolerances and operating limits.

Want to double-check your circuit calculations? Explore TheOnlineToolset’s electronics calculators for resistor networks, LED current limiting, capacitor combinations, and voltage dividers.