Module 2/5 · Weeks 4–6 · 30 h

Series, parallel circuits and Kirchhoff's laws

DRT 113 Fundamentals of Electrical and Electronics

About 85 minDraft, awaiting reviewLast updated 26 September 2026

By the end of this module you will be able to

Calculate total resistance of series, parallel and combined circuits
Use Kirchhoff's current and voltage laws to analyse circuits
Design a voltage divider for battery sensing with an ADC
Explain S and P battery cell arrangements and calculate voltage, capacity and energy
Explain power distribution through a drone's distribution board

Prerequisites: DRT 113 Module 1 (Ohm's law and power)

From one resistor to real circuits

Drone circuits never have just one component. The battery feeds several motors at once, cells combine into packs, and the flight controller measures battery voltage through two resistors. This module shows how to analyse multi-component circuits with Kirchhoff’s two basic laws.

Series and parallel circuits

Left: two resistors connected one after another in series. Right: two resistors connected across the same supply in parallel — Figure 1. Series and parallel circuits

In series, components form a single path. The current is the same through each, and voltage divides in proportion to resistance:

R_{s} = R_{1} + R_{2} + \dots

In parallel, components connect across the same two points. The voltage is the same across each, and current divides:

\frac{1}{R _{p}} = \frac{1}{R _{1}} + \frac{1}{R _{2}} + \dots R_{p} = \frac{R _{1} R _{2}}{R _{1} + R _{2}} (two resistors)

A parallel combination is always smaller than its smallest resistor, because it adds paths for current.

Example 1

100 Ω and 200 Ω resistors:

In series $R_{s} = 300 Ω$
In parallel $R_{p} = \frac{100 \times 200}{300} \approx 66.7 Ω$

Kirchhoff’s laws

Current law (KCL): at any junction, the total current flowing in equals the total flowing out, because charge does not disappear.

\sum I_{in} = \sum I_{out}

Voltage law (KVL): around any closed loop, the voltage rises from sources equal the voltage drops across components, because energy must balance.

\sum V = 0 around a loop

Example 2. Using KVL

A 12 V battery is in series with 4 Ω and 2 Ω resistors.

Total resistance 6 Ω, current $I = 12/6 = 2$ A
Voltage across 4 Ω is 8 V; across 2 Ω is 4 V
Check with KVL: $12 - 8 - 4 = 0$

Example 3. Using KCL on a distribution board

A power distribution board (PDB) connects four ESCs in parallel with the battery. If each draws 12 A, the battery must supply $4 \times 12 = 48$ A. The main leads and battery connector must handle more than 48 A, while each ESC lead carries only 12 A.

The voltage divider

The flight controller needs to know battery voltage, but a microcontroller ADC pin accepts only about 3.3 V. Two resistors in series divide the voltage down to a safe fraction.

Battery voltage through R1 100 kilohm and R2 13 kilohm to ground; the midpoint connects to the flight controller ADC — Figure 2. A voltage divider for battery sensing

V_{out} = V_{in} \frac{R _{2}}{R _{1} + R _{2}}

Example 4. Sensing a 6S battery

A fully charged 6S battery is at $6 \times 4.2 = 25.2$ V. With $R_{1} = 100$ kΩ and $R_{2} = 13$ kΩ:

V_{out} = 25.2 \times \frac{13}{113} \approx 2.90 V

That is below 3.3 V, so it is safe. Software multiplies back by $113/13 \approx 8.69$ to display the real voltage. In ArduPilot this is the battery monitor’s voltage-multiplier parameter; set it wrongly and low-battery warnings fire at the wrong time.

Battery cell arrangements

Lithium packs are built from cells and labelled with S (series) and P (parallel):

In series, voltages add and capacity stays the same.
In parallel, capacities add and voltage stays the same.

Eight 3.7 volt 5 amp-hour cells arranged as four in series, two strings in parallel, giving 14.8 volts, 10 amp-hours, 148 watt-hours — Figure 3. A 4S2P cell arrangement

Example 5. A 4S2P battery

Eight 3.7 V 5 Ah cells arranged 4S2P:

Voltage $4 \times 3.7 = 14.8$ V
Capacity $2 \times 5 = 10$ Ah
Energy $14.8 \times 10 = 148$ Wh

In series, a cell with lower voltage than the others empties first and can be over-discharged and damaged. A balance charger therefore measures and equalises each cell through the balance lead.

Module lab

In class

Build series and parallel circuits with three resistors. Measure voltage and current everywhere and check KCL and KVL from your readings.
Design a voltage divider for a 4S battery giving at most 3.3 V. Build it on a breadboard, feed it from a supply instead of a battery and compare with your calculation.
Check a sample battery with a cell checker. Record each cell voltage and find the largest difference.

Common mistakes

Watch out

Adding parallel resistances directly. Use reciprocals.
Reversing the divider. Swapping $R_{1}$ and $R_{2}$ gives too high a voltage and can destroy the ADC.
Mixing up S and P. Series raises voltage; parallel raises capacity.
Forgetting that main leads carry the total current of everything in parallel.

Summary

Series: resistances add, current is the same. Parallel: reciprocals add, voltage is the same.
KCL: current in equals current out at a junction. KVL: voltages around a loop sum to zero.
A voltage divider $V_{out} = V_{in} R_{2} / (R_{1} + R_{2})$ lets an ADC measure high voltage.
An xSyP pack multiplies voltage by S and capacity by P, and needs balance charging.

Check your understanding

Three 30 Ω resistors are in parallel. What is the total resistance?
A 9 V battery is in series with 1 kΩ and 2 kΩ. What is the voltage across 2 kΩ?
At a junction, 5 A and 3 A flow in and 6 A flows out one branch. What flows out the other?
What energy does a 6S1P pack of 3.7 V 4 Ah cells store?
A divider with $R_{1} = 47$ kΩ and $R_{2} = 10$ kΩ has $V_{in} = 16.8$ V. What is $V_{out}$ ?

Answers

$30/3 = 10 Ω$
Current $9/3000 = 3$ mA; voltage $0.003 \times 2000 = 6$ V
$5 + 3 - 6 = 2$ A
$6 \times 3.7 \times 4 = 88.8$ Wh
$16.8 \times 10/57 \approx 2.95$ V

Key formulas

Series	$R_{s} = R_{1} + R_{2} + \dots$
Parallel	$\frac{1}{R _{p}} = \frac{1}{R _{1}} + \frac{1}{R _{2}} + \dots$
Two in parallel	$R_{p} = \frac{R _{1} R _{2}}{R _{1} + R _{2}}$
Kirchhoff's current law (KCL)	$\sum I_{in} = \sum I_{out}$
Kirchhoff's voltage law (KVL)	$\sum V = 0 around a loop$
Voltage divider	$V_{out} = V_{in} \frac{R _{2}}{R _{1} + R _{2}}$

Key references

Ling, S. J., Sanny, J., & Moebs, W. (2016). University physics volume 2. OpenStax. link
Boylestad, R. L., & Nashelsky, L. (2013). Electronic devices and circuit theory (11th ed.). Pearson.
ArduPilot Dev Team. ArduPilot documentation. link

In class / field

Lab or field practice from worksheets with a safety checklist

Learning evidence: Checked worksheets and quiz results

Module quiz

This is a formative self-check, not a graded exam

Knowledge domain: Electrical, electronics and power systems · Aircraft, structures and design · Installation, maintenance and testing

Series, parallel circuits and Kirchhoff's laws

By the end of this module you will be able to

From one resistor to real circuits

Series and parallel circuits

Kirchhoff’s laws

The voltage divider

Battery cell arrangements

Module lab

Common mistakes

Summary

Check your understanding

Key formulas

Key references

Further reading

DC and AC electrical circuits

Designing and integrating drone systems

In class / field

Module quiz

Series, parallel circuits and Kirchhoff's laws

Lesson

By the end of this module you will be able to

From one resistor to real circuits

Series and parallel circuits

Kirchhoff’s laws

The voltage divider

Battery cell arrangements

Module lab

Common mistakes

Summary

Check your understanding

Key formulas

Key references

Further reading

DC and AC electrical circuits

Designing and integrating drone systems

In class / field

Module quiz