Electronics Engineering · CHAPTER 02 · FOCUSED LESSON

Resistors

Resistance, Ohm’s law, the water-hose analogy and series/parallel networks.

01 · DEFINITION

A resistor controls
the flow of current.

A resistor is a passive component that opposes electric current. It lets engineers set currents, divide voltages, protect components and transform electrical energy into heat.

LIVE ANALOGY

The water hose

INTERACTIVE
PRESSURE12 V
FLOW100 mA

More hand pressure narrows the hose. More resistance reduces the electrical current.

FROM ALI’S ORIGINAL ENGINEERING NOTESPAGES 19–20

03 · WHAT SETS RESISTANCE?

Material, length and area.

A resistor is a controlled package of resistance. For a uniform conductor, the document connects its resistance to the material and geometry: a longer path increases resistance, while a larger cross-sectional area gives charge more room to move.

R=ρ·LA
R
Resistance · ohm (Ω)
ρ
Resistivity · Ω·m
L
Conductor length · metre (m)
A
Cross-sectional area · square metre (m²)

04 · SERIES RESISTORS

One path. The same current everywhere.

Series components are connected end-to-end, so charge has only one route. The source voltage is shared between the resistors, but the current cannot split.

DERIVATIONKirchhoff’s voltage law
UAB=UAC+UCB

The voltage from node A to node B equals the sum of the drops across R₁ and R₂.

1

Use Ohm’s law for every voltage:

ReqI = R₁I + R₂I
2

The same non-zero current I is present in every term. Divide the complete equation by I:

ReqII=R₁II+R₂II
3

The current cancels, leaving:

Req = R₁ + R₂
FOR n RESISTORSReq = R₁ + R₂ + ··· + Rn = ΣRk
LIVE CIRCUIT

Series voltage divider

SAME CURRENT
A12.0 V
R₁100 ΩU₁ = 3.75 V
C8.3 V
R₂220 ΩU₂ = 8.25 V
B0 V
CONVENTIONAL CURRENTI = I₁ = I₂ = 37.5 mA
Equivalent resistance320 Ω
Circuit current37.5 mA
Voltage check3.75 + 8.25 = 12.00 V

Change either resistance: the same current flows through both, while each voltage drop changes in proportion to its resistance.

05 · PARALLEL RESISTORS

Two nodes. Multiple current paths.

Every branch begins at node A and ends at node B, so every resistor has the same voltage. Current divides between the branches and recombines at the node.

LIVE CIRCUIT

Parallel current divider

SAME VOLTAGE
AI = 175 mA
R₁100 ΩI₁ = 120 mA
BRETURN
R₂220 ΩI₂ = 54.5 mA
UAB = U₁ = U₂12.0 V across every branch
Equivalent resistance68.75 Ω
Total current175 mA
Node-law check120 mA + 54.5 mA = 175 mA

Lower a branch resistance and watch that branch carry more current. The total is always I = I₁ + I₂.

DERIVATIONKirchhoff’s current law
I=I₁+I₂

At a node, the total current entering equals the total current leaving.

1

The voltage U is identical across the equivalent resistor and every branch. Replace each current using Ohm's law:

UReq=UR₁+UR₂
2

Factor the common non-zero voltage U, then divide both sides by U:

U·1Req=U·(1R₁+1R₂)
3

The conductances add:

1Req=1R₁+1R₂
FOR n RESISTORS1Req=1R₁+1R₂+···+1Rn
TWO-RESISTOR SHORTCUTReq=R₁R₂R₁ + R₂
SERIESI is the same

Voltages add · resistances add · Req is larger than either resistor.

VS
PARALLELU is the same

Currents add · conductances add · Req is smaller than the smallest branch.