Voltage Sources and Resistors in Series and Parallel

Series and parallel are descriptions of connectivity. Two components are in series when the same current must pass through both. Components are in parallel when both of their terminals connect to the same two nodes, so they have the same voltage.

Learning objectives

After this lesson, you will be able to:

identify true series and parallel connections from nodes rather than drawing shape;
calculate equivalent resistance;
calculate branch currents and resistor voltage drops;
explain ideal source combinations and the extra constraints of real batteries;
verify that an answer is physically plausible.

Resistors in series

In a series chain, the same current flows through every resistor. Equivalent resistance is the sum:

$$
R_{eq} = R_1 + R_2 + \cdots + R_n
$$

Start at the battery's + terminal and follow the upper path through R1, R2, and R3. There is no branch; the lower wire is the return path to the battery.

BT1: Device:Battery value="12 V" rotate=0
R1: Device:R value="1 kΩ" rotate=270
R2: Device:R value="2.2 kΩ" rotate=270
R3: Device:R value="3.3 kΩ" rotate=270

layout direction=LR gap=65
group LOOP label="Three resistors in series" direction=LR gap=65 {
  BT1 R1 R2 R3
}

BT1.1 --> R1.1 color=#b91c1c
R1.2 --> R2.1 color=#2563eb
R2.2 --> R3.1 color=#2563eb
R3.2 --> BT1.2 color=#334155

For the values shown:

$$
R_{eq} = 1\ \text{k}\Omega + 2.2\ \text{k}\Omega + 3.3\ \text{k}\Omega = 6.5\ \text{k}\Omega
$$

$$
I = \frac{12\ \text{V}}{6.5\ \text{k}\Omega} \approx 1.846\ \text{mA}
$$

Each voltage drop is V_n = IR_n, giving approximately 1.85 V, 4.06 V, and 6.09 V. Their sum is 12.00 V, as Kirchhoff's voltage law predicts.

Resistors in parallel

Parallel resistors share the same two nodes and therefore the same voltage. Their conductances add:

$$
\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}
$$

For two resistors:

$$
R_{eq} = \frac{R_1R_2}{R_1+R_2}
$$

Here each resistor is a separate vertical branch between the same upper and lower rails. The junction dots show that the branch terminals share the same two nodes.

BT1: Device:Battery value="12 V source" rotate=0
R1: Device:R value="1 kΩ branch" rotate=0
R2: Device:R value="2.2 kΩ branch" rotate=0
R3: Device:R value="3.3 kΩ branch" rotate=0

layout direction=LR gap=105
group PARALLEL label="Three resistors in parallel" direction=LR gap=105 {
  BT1 R1 R2 R3
}

BT1.1 --> R1.1 color=#b91c1c
R1.1 --> R2.1 color=#b91c1c
R2.1 --> R3.1 color=#b91c1c
BT1.2 --> R1.2 color=#334155
R1.2 --> R2.2 color=#334155
R2.2 --> R3.2 color=#334155

For the values shown:

$$
R_{eq} = \left(\frac{1}{1000}+\frac{1}{2200}+\frac{1}{3300}\right)^{-1}
$$

$$
R_{eq} \approx 569\ \Omega
$$

The branch currents are 12.0 mA, 5.45 mA, and 3.64 mA; their sum is approximately 21.09 mA. The same result follows from 12 V / 569 Ω.

Fast reasonableness check

A series equivalent must be greater than its largest resistor. A parallel equivalent must be smaller than its smallest resistor. If not, revisit the calculation.

Ideal voltage sources in series

Series-connected ideal sources add algebraically according to polarity:

$$
V_{eq} = V_1 + V_2 + \cdots
$$

Two 1.5 V cells with aiding polarity provide a nominal 3.0 V. Reverse one cell and the ideal voltages oppose.

Real cells in series carry the same current, so usable pack current is constrained by the weakest cell, interconnect, protection circuit, and temperature. Capacity in ampere-hours does not add in a simple series string.

Real voltage sources in parallel

Ideal voltage sources with exactly equal voltage can be placed in parallel. Real batteries are not exactly equal, so even a small open-circuit-voltage difference can drive a large circulating current through low internal resistance.

Battery packs are not breadboard exercises

Do not directly parallel unmatched cells. Rechargeable packs require compatible cells, fusing or other protection, suitable interconnects, and a battery-management system designed for the chemistry and topology. Pack assembly can create fire, burn, and stored-energy hazards even at low voltage.

For matched cells engineered into a parallel group, nominal voltage remains that of one cell while current capability and ampere-hour capacity can increase. The safe amount depends on cell ratings and pack design—not merely on counting cells.

Series-parallel notation

A 3S2P pack contains three series groups, with two parallel cells in each group: six cells total. If each cell is nominally 3.6 V and 2.5 Ah, the idealized nominal pack is 10.8 V, 5 Ah. Charging limits, discharge limits, cell matching, protection, and thermal design still govern the real pack.

Worked example: two parallel loads

A 12 V supply feeds R1 = 1 kΩ and R2 = 3 kΩ in parallel.

$$
I_1 = \frac{12}{1000} = 12\ \text{mA}
$$

$$
I_2 = \frac{12}{3000} = 4\ \text{mA}
$$

$$
I_{source} = I_1 + I_2 = 16\ \text{mA}
$$

$$
R_{eq} = \frac{12\ \text{V}}{0.016\ \text{A}} = 750\ \Omega
$$

Power checks: R1 dissipates 144 mW; R2 dissipates 48 mW; total load power is 192 mW, equal to 12 V × 16 mA.

Interactive resistor calculator

Common mistakes

Deciding series or parallel from how a drawing looks instead of which nodes connect.
Adding parallel resistances directly.
Forgetting that parallel branches increase total source current.
Treating ampere-hours as amperes.
Assuming ideal source-combination rules make arbitrary batteries safe to combine.
Checking equivalent resistance but not individual resistor power.

Summary

Series components share current; parallel components share voltage.
Series resistances add; parallel conductances add.
Equivalent-resistance bounds provide a quick error check.
Ideal source rules are only the beginning of real battery-pack design.
Verify current, voltage, power, and source limits for every branch.

Next: Capacitance and Inductance.