DRE1113 Chapter 1 — Basic Concept

Welcome to Chapter 1: Basic Concept

This lab follows the syllabus in order, 1.1 → 1.8. Open one subtopic at a time. Each subtopic has the same four stages. Finish all four, then press Mark complete.

0 of 8 subtopics complete

Course alignment

CO1 (C3, C4) Bloom's Taxonomy: C3 Apply, C4 Analyse PO1: Engineering Knowledge
Attribute the basic concepts of electrical quantities by using basic circuit laws (Ohm's law and Kirchhoff's law) and simplification of resistive circuits.

Week	Subtopic	Title
W1	1.1	Introduction of Circuit Analysis
W1	1.2	Electrical Quantities
W2	1.3	Circuit Elements
W2	1.4	Ohm's Law and Kirchhoff's Law
W3	1.5	Nodes, Branches and Loops
W3	1.6	Resistive Circuits
W4	1.7	Voltage Division and Current Division
W4	1.8	Delta–Wye Transformation

Main reference: C. Alexander and M. Sadiku, Fundamentals of Electric Circuits, 6th ed., McGraw-Hill, 2016. Further reading: J. Nilsson and S. Riedel, Electric Circuits, 10th ed., Prentice Hall, 2014.

How each subtopic works

1 · LEARNShort notes, formulas and worked examples from the slides.

2 · SIMULATEMove the sliders and watch the circuit respond.

3 · PRACTISEStep-by-step problems with new numbers every time.

4 · CHECKConcept quiz with instant feedback.

Tip: if you get a quiz question wrong, go back to Learn or Simulate, then try again.

1.1 Introduction of Circuit Analysis

Week 1CO1 (C3, C4)PO1

AWhy circuit theory?

Electrical engineering studies, designs and applies equipment, devices and systems that use electricity, electronics and electromagnetism.

Electric circuit theory and electromagnetic theory are the two fundamental theories of electrical engineering. Power, control, machines, electronics, communications and instrumentation are all built on circuit theory.

Circuit theory gives you the basic principles (voltage, current, resistance, Ohm's Law, Kirchhoff's Laws) used to design, analyse and optimise everything from a phone charger to the national grid.

BWhat do electrical engineers do?

Design new ways to use electrical power
Perform calculations for construction, installation and manufacturing specifications
Direct manufacturing, installation and testing so products meet codes
Investigate complaints, evaluate problems and recommend solutions
Design, develop and test equipment such as motors, radar, generators and communication systems

CWhat is an electric circuit?

An electric circuit is an interconnection of electrical elements linked together in a closed path so that an electric current can flow continuously.

Left: closed circuit (switch ON), current flows and the lamp lights. Right: open circuit (switch OFF), the path is broken, no current.

Element	What it does
Voltage source	The "battery" of the circuit, provides the energy that pushes current around
Resistor	Opposes (limits) the flow of current
Capacitor	Stores and releases energy in an electric field, like a small rechargeable battery
Inductor	Resists changes in current by storing energy in a magnetic field
Conductor (wire)	Connects all the components so current can flow
Switch	Opens or closes the path

DFrom physical layout to circuit diagram

A car headlight system has a battery, a switch, wires and two headlamps. In a circuit diagram we replace each part with a standard symbol: the battery becomes a voltage source, and each headlamp becomes a resistor. The diagram is easier to analyse than the physical layout.

Car headlight system: battery, switch, wires and two headlamps

(b) Circuit diagram

Figure (a) from the Chapter 1 lecture notes, Part 1, slide 10.

Tip: a circuit diagram is a map of the circuit. It drops the physical appearance and keeps only the symbols, so the circuit is easier to analyse.

Open or closed? Try the switch

Show electron flow instead

What to notice

Closed path → current flows → the lamp converts electrical energy into light and heat.
Open path → no current, even though the battery still has voltage across it.
Conventional current flows out of the + terminal, around the circuit, into the − terminal.
Electrons actually move the other way (− to +). In this course we always use conventional current.

Voltage can exist without current (open circuit). Current cannot flow without a closed path.

1.2 Electrical Quantities

Week 1CO1 (C3, C4)PO1SI units · charge · current · voltage · power · energy

The big picture

→ → → → →

ASI units

The International System of Units (SI) lets engineers everywhere communicate results in the same way.

Quantity	Symbol	Unit	Abbr.
Charge	Q, q	coulomb	C
Current	I, i	ampere	A
Voltage	V, v, E	volt	V
Power	P	watt	W
Energy	W	joule	J
Resistance	R	ohm	Ω
Conductance	G	siemens	S
Capacitance	C	farad	F
Inductance	L	henry	H
Frequency	f	hertz	Hz
Time	t	second	s

BSI prefixes

Multiplier	Prefix	Symbol	Example
10¹²	tera	T	2 TB
10⁹	giga	G	12 GHz
10⁶	mega	M	25 MΩ
10³	kilo	k	13.2 kV
10⁻²	centi	c	30.2 cm
10⁻³	milli	m	5 mH
10⁻⁶	micro	µ	6.5 µW
10⁻⁹	nano	n	3 ns
10⁻¹²	pico	p	9 pF

Example: 600 000 m = 600 km = 600 000 000 mm = 0.6 Mm.
15 mA + 8000 µA = 15 mA + 8 mA = 23 mA.

CCharge

Electric charge is the property of subatomic particles that makes them feel a force in an electric or magnetic field. It can be positive or negative, and is measured in coulombs (C).

Charge of one electron	e = −1.602 × 10⁻¹⁹ C
Electrons in 1 coulomb	1 C = 6.24 × 10¹⁸ electrons

Law of conservation of charge: charge can only be transferred. It cannot be created or destroyed. Like charges repel, unlike charges attract.

Static shock from a doorknob: charge built up on your body discharges into the metal, which is at earth potential. Fuel trucks and buildings are grounded so charge flows safely to earth.

DCurrent

Current is the time rate of flow of charge, measured in amperes (A). 1 A = 1 C of charge passing a point in 1 s.

I = Qt

rearranged to find charge

Q = I t

Conventional current flows from the + terminal to the − terminal of the source.

	DC (direct current)	AC (alternating current)
Behaviour	Constant with time, one direction	Varies sinusoidally, reverses direction
Sources	Batteries, fuel cells, solar cells	Power plants, wall socket

DC: same value at all times, always one direction

AC: value changes and direction reverses (+ then −)

🇲🇾 Wall socket in Malaysia: 240 V, 50 Hz, Type G plug.

EVoltage

Voltage (potential difference) is the energy needed to move a unit charge through an element. 1 V = 1 J/C.

V = WQ

V: voltage (V)
W: work / energy (J)
Q: charge (C)

Also called potential difference, e.m.f., supply voltage or voltage source (symbol V_s or E). When current flows through an element, a voltage drop appears across its terminals.

FPower

Power is the time rate of expending or absorbing energy, measured in watts (W).

P = Wt = V I

P = W/t = (W/Q)(Q/t) = V · I

1 hp = 746 W. A 100 W bulb gives more light than a 60 W bulb.

GPassive sign convention

Current enters the + terminal → p = +vi → power absorbed (element is a load).
Current enters the − terminal → p = −vi → power supplied (element is a source).
Conservation of energy: ΣP = 0, so power supplied = power absorbed.

Focus on the direction of the current entering the element. Check which terminal the current enters first, then decide the sign.

Slide Example 2: check ΣP = 0.

P₁ = 20(−5) = −100 W (supplied)

P₂ = 12(5) = 60 W (absorbed)

P₃ = 8(6) = 48 W (absorbed)

P₄ = 8(−0.2 × 5) = −8 W (supplied)

ΣP = −100 + 60 + 48 − 8 = 0 ✓

HEnergy

Energy is the capacity to do work, measured in joules (J).

W = P t

t in seconds → W in joules (watt-seconds)
t in hours → W in watt-hours (Wh) or kWh. The utility company bills in kWh.

Example 3: 100 W bulb for 2 h

W = 100 × 2 × 3600 = 720 000 J = 720 kJ

or W = 100 W × 2 h = 200 Wh

IElectricity bill (tiered rate)

Example 5: 700 kWh in January. Base charge RM 12.00; first 100 kWh at 16 sen; next 200 kWh at 10 sen; above 300 kWh at 6 sen.

Base = RM 12.00

100 × 0.16 = RM 16.00

200 × 0.10 = RM 20.00

400 × 0.06 = RM 24.00

Total = RM 72.00 → average = 72/700 = 10.29 sen/kWh

▶ Try the Energy & Bill Calculator in Simulate.

🔁 Prefix converter

Type a value, choose its prefix, then choose the prefix you want.

Quantity

Value

From (prefix of the value)

To (prefix you want)

Rule: each step on the ladder is × 1000. Moving right (to a smaller prefix) → the number gets bigger. Moving left (to a bigger prefix) → the number gets smaller.

⚡ Passive sign convention explorer

Voltage across element, V

Current through element, I

🏠 Energy & Bill Calculator

Edit the appliance list (power in W, time in hours). Default values are from Exercise 11.

Flat rate (RM per kWh)

Tiered bill (Example 5 schedule). Change the monthly usage.

Monthly usage

Base charge (RM)

First 100 kWh (sen)

Next 200 kWh (sen)

Above 300 kWh (sen)

1.3 Circuit Elements

Week 2CO1 (C3, C4)PO1passive · active · independent · dependent · resistor · measurement

APassive and active elements

	Passive element	Active element
Energy	Cannot generate energy; absorbs or stores it	Capable of generating (supplying) energy
Examples	Resistor, capacitor, inductor	Voltage source, current source, generator, battery

Easy way to remember: passive elements absorb energy, active elements deliver energy.

BIndependent sources (circle symbol)

An independent source provides a specified voltage or current that does not depend on any other circuit variable.

Left: independent voltage source. Right: independent current source (arrow shows current direction).

CDependent (controlled) sources (diamond symbol)

A dependent source provides a voltage or current that is controlled by another voltage or current somewhere in the circuit. There are four types:

Type	Output	Controlled by	Form
VCVS	voltage	a voltage	v = μ v_x
CCVS	voltage	a current	v = r i_x
VCCS	current	a voltage	i = g v_x
CCCS	current	a current	i = β i_x

Read the label: if it has v_x or i_x times a constant, it is dependent. The symbol (± or arrow) tells you whether it outputs voltage or current.

DResistance and resistivity

Resistance is the ability of a material to resist the flow of current. For a uniform conductor of length l and cross-sectional area A:

R = ρ lA

ρ: resistivity of the material (Ω·m)
l: length (m)
A: cross-sectional area (m²)

Material	ρ (Ω·m)	Use
Silver	1.64 × 10⁻⁸	Conductor
Copper	1.72 × 10⁻⁸	Conductor
Aluminium	2.8 × 10⁻⁸	Conductor
Gold	2.45 × 10⁻⁸	Conductor
Carbon	4 × 10⁻⁵	Semiconductor
Germanium	47 × 10⁻²	Semiconductor
Silicon	6.4 × 10²	Semiconductor
Paper	10¹⁰	Insulator
Mica	5 × 10¹¹	Insulator
Glass	10¹²	Insulator

Table after Alexander & Sadiku, Table 2.1.

EResistors

A resistor is a passive element used to limit current or divide voltage. Unit: ohm (Ω). 1 Ω exists when 1 V across it drives 1 A through it.

Fixed resistor: value is constant, identified by its colour code. Bigger body = higher power rating (2 W, 1 W, ½ W, ¼ W, ⅛ W).
Variable resistor (potentiometer): three terminals with a sliding wiper. Using one end and the wiper gives a rheostat.

Conductance

G = 1R (siemens, S)

Conductance measures how easily current flows. In parallel circuits, conductances simply add.

FResistor colour code

4-band: digit · digit · multiplier · tolerance
5-band: digit · digit · digit · multiplier · tolerance

4-band example: Green Blue Yellow Silver

56 × 10k ± 10% = 560 kΩ ± 10%

Min = 560k − 56k = 504 kΩ

Max = 560k + 56k = 616 kΩ

5-band example: Red Orange Violet Black Brown

237 × 1 ± 1% = 237 Ω ± 1%

Range = 234.63 Ω to 239.37 Ω

Mnemonic: Black Brown Red Orange Yellow Green Blue Violet Grey White = 0 to 9.

GMeasuring with a multimeter

Resistance (ohmmeter)

Turn OFF / disconnect the supply. Connect the meter across the resistor only.

Current (ammeter)

Break the circuit and put the ammeter in series so the same current flows through it.

Voltage (voltmeter)

Put the voltmeter in parallel across the component. Reversed leads give a negative reading.

▶ Try the Meter Placement Lab to see what happens when a meter is connected wrongly.

🎨 Resistor colour code decoder

Nominal value

Tolerance range

Reverse: type a value and get the 4-band colours

🔌 Meter Placement Lab

Your task: connect a multimeter to measure the resistor R correctly. Choose the meter, how to connect it, and whether the supply is ON or OFF. The meter screen shows what you would really see in the lab.

1. Which meter?

2. How is it connected to R?

3. DC supply

V_s

1.4 Ohm's Law and Kirchhoff's Law

Week 2CO1 (C3, C4)PO1V = IR · P = I²R · KCL · KVL

The big picture

→ → → → →

AOhm's Law

"The voltage V across a resistor is directly proportional to the current I flowing through it."

V = I R

Memory aid: cover the one you want.

V = I × R

I = V / R

R = V / I

Water analogy: voltage = pump pressure, current = how much water flows, resistance = the narrow part of the pipe.

With R fixed, V and I are linearly proportional (straight-line V–I graph). With V fixed, I and R are inversely related.

BOhm's Law with the power equation

P = VI, substitute V = IR:

P = (IR)I = I²R

P = VI, substitute I = V/R:

P = V(V/R) = V²/R

P = VI = I²R = V²R

Slide Example 1: 20 V source across 5 Ω.

I = 20/5 = 4 A, V₁ = 4 × 5 = 20 V

P_5Ω = (20)(4) = 80 W (absorbed)

P_20V = (20)(−4) = −80 W (supplied)

COpen circuit and short circuit

Short circuit: a plain wire across an element, so R = 0. Voltage across it is v = 0, but current can be large. All the current takes the wire and bypasses the load. Current always takes the easiest path.

Open circuit: a break (gap) in the path, so R = ∞. Current through it is i = 0, but voltage can appear across the gap. Here no current flows in R₁, so v = V_s. No path means no current.

	Resistance	Voltage across	Current through
Short circuit	R = 0	v = 0	any value (set by the rest of the circuit)
Open circuit	R = ∞	any value	i = 0

DKirchhoff's Current Law (KCL)

The sum of currents entering a node equals the sum of currents leaving it. Equivalently, the algebraic sum of currents at a node is zero.

ΣI_in = ΣI_out

or, taking currents entering as + and leaving as −

ΣI = 0

At node A: i₁ + i₃ + i₄ = i₂ + i₅

or i₁ − i₂ + i₃ + i₄ − i₅ = 0

KCL comes from conservation of charge: charge cannot pile up at a node.

EKirchhoff's Voltage Law (KVL)

The algebraic sum of all voltages around a closed loop is zero: sum of rises = sum of drops.

ΣV_rise = ΣV_drop

or, going around the loop with signs

ΣV = 0

V_s = V₁ + V₂ + V₃ + V₄

−V_s + V₁ + V₂ + V₃ + V₄ = 0

Sign rule: travel around the loop (clockwise or counter-clockwise). If you meet the − terminal first, write −V. If you meet the + terminal first, write +V.

▶Video: Kirchhoff's Laws (KCL and KVL)

Watch this after reading KCL and KVL above, then try the KCL and KVL explorers in Simulate.

Video not playing? Open it on YouTube.

FUsing KVL and KCL together (slide KVL & KCL Example 1)

Ohm's Law

v₁ = 8i₁, v₂ = 3i₂, v₃ = 6i₃

KVL loop 1

−30 + 8i₁ + 3i₂ = 0 … [1]

KVL loop 2

−3i₂ + 6i₃ = 0 … [2]

KCL node a

i₁ − i₂ − i₃ = 0 … [3]

Solve (calculator matrix mode)

i₁ = 3 A, i₂ = 2 A, i₃ = 1 A → v₁ = 24 V, v₂ = v₃ = 6 V

📈 Ohm's Law explorer

Voltage, V

Resistance, R

V–I line for this R (slope = 1/R)operating point

🔀 KCL node explorer

Set four known currents (positive = arrow as drawn, negative = opposite). KCL finds the unknown I_x.

🔁 KVL loop explorer

V_s

R₁

R₂

R₃

🧩 Two-loop solver (KVL + KCL)

V_s

R₁

R₂

R₃

1.5 Nodes, Branches and Loops

Week 3CO1 (C3, C4)PO1branch · node · essential node · loop · mesh · series · parallel

ADefinitions

Element: a component or a source
Branch: a path that represents a single element
Node: a point of connection between two or more branches (a whole wire counts as one node)
Essential node: a node where three or more branches join
Loop: any closed path in a circuit
Mesh: a loop that does not enclose any other loop

number of meshes (independent loops) l = b − n + 1

Common mistake: a long wire with several dots is still one node. All points joined by plain wire (no component in between) form one node.

BSlide Example 1

4 elements → 4 branches (10 V, 5 Ω, 6 Ω, 2 A)
3 nodes (coloured), 2 of them essential
3 loops, 2 meshes → check: l = 4 − 3 + 1 = 2 ✓

CSeries connection

Two elements are in series if they share a single node and no other element connects to that node. Series elements carry the same current.

i_A = i_B = i_C

DParallel connection

Elements are in parallel if they are connected to the same two nodes. Parallel elements have the same voltage.

v = v_A = v_B = v_C

ESlide Example 5

5 elements → 5 branches: 10 V, 1 Ω, 2 Ω, 5 Ω, 4 Ω
3 nodes; 3 meshes (5 − 3 + 1 = 3); 6 loops in total
Parallel: 1 Ω ∥ 2 Ω, and 10 V ∥ 4 Ω
The 5 Ω is neither in series nor in parallel with a single element.

▶ Try the Circuit Explorer: turn on the node, branch and mesh overlays for each circuit.

🔍 Circuit Explorer

Colour the nodes Number the branches Shade the meshes

1.6 Resistive Circuits

Week 3CO1 (C3, C4)PO1series · parallel · series-parallel

ASeries circuit

R_eq = R₁ + R₂ + … + R_N

The same current flows everywhere: I = V_s/R_eq
Each drop: V_k = I R_k; larger R → larger drop
KVL: V_s = V₁ + V₂ + V₃ + V₄
Polarity of the drop: current enters the resistor at +

Example: 10 + 30 + 100 = 140 Ω. With 8.4 V: I = 8.4/140 = 60 mA.

BParallel circuit

1R_eq = 1R₁ + 1R₂ + … + 1R_N

Two resistors: R_eq = R₁ × R₂R₁ + R₂

The same voltage across every branch: V_s = V₁ = V₂ = V₃
Branch current: I_k = V_s/R_k
KCL: I_s = I₁ + I₂ + I₃
R_eq is always smaller than the smallest resistor.

Examples: 3 ∥ 6 = 18/9 = 2 Ω. 6 ∥ 9 ∥ 6 ∥ 72 ∥ 6 = 1.6 Ω. 1k ∥ 2.2k ∥ 1.2k = 437.1 Ω.

CVoltage sources in series

Sources in series add algebraically. Aiding sources add; an opposing source subtracts.

E_T = 10 + 6 + 2 = 18 V (all aiding)

E_T = 9 + 3 − 4 = 8 V (one opposing)

Series vs parallel in practice

Series	Parallel
One element fails → the whole string stops working	One branch fails → the other branches still work
R_eq is large → less current	Every branch gets the full supply voltage

DSeries-parallel: the reduction method

Start from the side furthest from the source (solve from right to left).
Combine resistors that are clearly in series or clearly in parallel.
Redraw the circuit after each step. Label the nodes to see the connections.
Repeat until one R_eq is left.

5 + 1 = 6 Ω → 6 ∥ 3 = 2 Ω → 2 + 2 = 4 Ω

6 ∥ 4 = 2.4 Ω → R_eq = 4 + 2.4 + 8 = 14.4 Ω

▶ Step through this in the Reduction Stepper.

ELaboratory measurement

R_eq (ohmmeter)

Remove the supply, then connect the ohmmeter across the two ends of the whole network. It reads R_eq = R₁ + R₂.

Current (ammeter)

Break the circuit and connect the ammeter in series, so the same current flows through it.

Voltage (voltmeter)

Connect the voltmeter in parallel across the resistor. In a parallel circuit, every voltmeter reads the same value.

🔧 Series / parallel builder

Number of resistors

V_s

🪜 Reduction stepper

1.7 Principles of Voltage Division and Current Division

Week 4CO1 (C3, C4)PO1VDR (series) · CDR (parallel)

AVoltage Divider Rule (VDR), series

In series, the larger the resistance, the larger its share of the voltage. VDR finds the voltage without first finding the current.

v₁ = R₁R₁ + R₂ × v

v₂ = R₂R₁ + R₂ × v

General: v_k = R_kR_T × v for any number of series resistors.

Example 1: 2 kΩ and 8 kΩ across 10 V.

V₁ = (2k/10k)(10) = 2 V

V₂ = (8k/10k)(10) = 8 V

BCurrent Divider Rule (CDR), parallel

In parallel, the total current is shared in inverse proportion to the resistance: the smaller resistor takes more current.

i₁ = R₂R₁ + R₂ × i

i₂ = R₁R₁ + R₂ × i

Notice the opposite resistor on top. CDR uses the other resistor on top; VDR uses the resistor itself.

Example 1: 4 kΩ ∥ 8 kΩ, I_s = 6 A.

I₂ = 4k/(4k + 8k) × 6 = 2 A

CVDR and CDR in a series-parallel circuit (slide Example 2)

20 ∥ 5 = 4 Ω; 15 ∥ 15 ∥ 15 = 5 Ω; 24 ∥ 8 = 6 Ω

R_T = 4 + 4 + 5 + 5 + 6 = 24 Ω

I = 48 / 24 = 2 A

VDR: V₁ = (5/24)(48) = 10 V (check: 2 × 5 = 10 V ✓)

CDR: I_a = 5/(20 + 5) × 2 = 0.4 A

Strategy: find R_T first, then the main current. Use VDR for the voltages and CDR to split the current between branches.

➗ VDR explorer (series)

V_s

R₁

R₂

R₃ (set 0 to remove)

Resistances in kΩ.

🔱 CDR explorer (parallel)

I_s

R₁

R₂

Resistances in kΩ.

1.8 Delta–Wye Transformation

Week 4CO1 (C3, C4)PO1Δ → Y · Y → Δ · bridge network

AWhy do we need it?

In some circuits, like the bridge network, resistors are neither in series nor in parallel. We cannot reduce them with the usual rules.

The fix: replace a three-terminal Δ (delta / pi) group with an equivalent Y (wye / tee) group, or the other way round. After the swap, series and parallel rules work again.

BWye (Y) / Tee (T)

Three resistors meet at one common centre node.

CDelta (Δ) / Pi (π)

Three resistors form a triangle between three nodes.

DΔ → Y

Each Y resistor = product of the two Δ resistors that touch the same terminal ÷ sum of all three Δ resistors.

R₁ = R_b R_cR_a + R_b + R_c

↳ R₁ is at terminal 1. R_b (terminal 1 to 3) and R_c (terminal 1 to 2) are the two Δ resistors that touch terminal 1.

R₂ = R_c R_aR_a + R_b + R_c

↳ R₂ is at terminal 2. R_c (terminal 1 to 2) and R_a (terminal 2 to 3) touch terminal 2.

R₃ = R_a R_bR_a + R_b + R_c

↳ R₃ is at terminal 3. R_a (terminal 2 to 3) and R_b (terminal 1 to 3) touch terminal 3.

EY → Δ

Each Δ resistor = (R₁R₂ + R₂R₃ + R₃R₁) ÷ the Y resistor at the terminal it does NOT touch.

R_a = R₁R₂ + R₂R₃ + R₃R₁R₁

↳ R_a is connected from terminal 2 to terminal 3. It does not touch terminal 1, so divide by R₁ (the Y resistor at terminal 1).

R_b = R₁R₂ + R₂R₃ + R₃R₁R₂

↳ R_b is connected from terminal 1 to terminal 3. It does not touch terminal 2, so divide by R₂.

R_c = R₁R₂ + R₂R₃ + R₃R₁R₃

↳ R_c is connected from terminal 1 to terminal 2. It does not touch terminal 3, so divide by R₃.

GUnderstanding Y ⇄ Δ step by step

Step 1: both networks share the same three terminals

The Y and the Δ connect to the same terminals 1, 2 and 3. They are equivalent when the resistance measured between any two terminals is the same for both.

Step 2: learn the opposite pairs

Y resistor	Y resistor is connected from	Opposite Δ resistor	Δ resistor is connected from
R₁	terminal 1 to the centre node	R_a	terminal 2 to terminal 3
R₂	terminal 2 to the centre node	R_b	terminal 1 to terminal 3
R₃	terminal 3 to the centre node	R_c	terminal 1 to terminal 2

Opposite pair: R₁ goes to terminal 1, while R_a is connected from terminal 2 to terminal 3, the side of the triangle facing away from terminal 1 (it does not touch terminal 1).

Δ → Y: "product of the two neighbours ÷ sum of all three"

Each Y resistor sits at one terminal. Multiply the two Δ resistors that touch that terminal, then divide by the sum of all three.

At terminal 1 the Δ resistors touching it are R_b and R_c:

R₁ = R_b R_cR_a + R_b + R_c

Y → Δ: "sum of the products ÷ the opposite one"

First find N = R₁R₂ + R₂R₃ + R₃R₁. N is the same for all three. Each Δ resistor = N ÷ the Y resistor opposite it (see the table).

R_a is opposite R₁, so:

R_a = NR₁ = R₁R₂ + R₂R₃ + R₃R₁R₁

Worked example 1: Y → Δ

Given

R₁ = 10 Ω, R₂ = 20 Ω, R₃ = 40 Ω

Step 1: find N (only once)

N = (10)(20) + (20)(40) + (40)(10)

N = 200 + 800 + 400 = 1400

Step 2: divide N by the opposite Y resistor

R_a = NR₁ = 140010 = 140 Ω

R_b = NR₂ = 140020 = 70 Ω

R_c = NR₃ = 140040 = 35 Ω

Notice

The smallest Y resistor (R₁) gives the largest Δ resistor (R_a).

Worked example 2: Δ → Y

Given

R_a = 15 Ω, R_b = 10 Ω, R_c = 25 Ω

Step 1: sum of all three

Σ = 15 + 10 + 25 = 50 Ω

Step 2: product of the two neighbours ÷ Σ

R₁ = R_b R_cΣ = (10)(25)50 = 5 Ω

R₂ = R_c R_aΣ = (25)(15)50 = 7.5 Ω

R₃ = R_a R_bΣ = (15)(10)50 = 3 Ω

▶ Try your own values in the Δ ⇄ Y converter.

FBalanced case and naming

R_Δ = 3 R_Y

R_Y = R_Δ3

When all three resistors are equal.

Naming rule used here (Alexander & Sadiku): Y resistor R₁ sits at terminal 1. Δ resistor R_a is the side opposite terminal 1 (between terminals 2 and 3), R_b is opposite terminal 2, R_c is opposite terminal 3. Get the labels right before using the formulas.

🔺 Δ ⇄ Y converter

🌉 Bridge network solver

🏁 Chapter 1 Quiz

15 questions drawn at random from all eight subtopics. Press New quiz for a different set.

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Welcome to Chapter 1: Basic Concept

Course alignment

How each subtopic works

1.1 Introduction of Circuit Analysis

AWhy circuit theory?

BWhat do electrical engineers do?

CWhat is an electric circuit?

DFrom physical layout to circuit diagram

Open or closed? Try the switch

What to notice

1.2 Electrical Quantities

The big picture

ASI units

BSI prefixes

CCharge

DCurrent

EVoltage

FPower

GPassive sign convention

HEnergy

IElectricity bill (tiered rate)

🔁 Prefix converter

⚡ Passive sign convention explorer

🏠 Energy & Bill Calculator

1.3 Circuit Elements

APassive and active elements

BIndependent sources (circle symbol)

CDependent (controlled) sources (diamond symbol)

DResistance and resistivity

EResistors

Conductance

FResistor colour code

GMeasuring with a multimeter

Resistance (ohmmeter)

Current (ammeter)

Voltage (voltmeter)

🎨 Resistor colour code decoder

🔌 Meter Placement Lab

1.4 Ohm's Law and Kirchhoff's Law

The big picture

AOhm's Law

BOhm's Law with the power equation

COpen circuit and short circuit

DKirchhoff's Current Law (KCL)

EKirchhoff's Voltage Law (KVL)

▶Video: Kirchhoff's Laws (KCL and KVL)

FUsing KVL and KCL together (slide KVL & KCL Example 1)

📈 Ohm's Law explorer

🔀 KCL node explorer

🔁 KVL loop explorer

🧩 Two-loop solver (KVL + KCL)

1.5 Nodes, Branches and Loops

ADefinitions

BSlide Example 1

CSeries connection

DParallel connection

ESlide Example 5

🔍 Circuit Explorer

1.6 Resistive Circuits

ASeries circuit

BParallel circuit

CVoltage sources in series

Series vs parallel in practice

DSeries-parallel: the reduction method

ELaboratory measurement

Req (ohmmeter)

Current (ammeter)

Voltage (voltmeter)

🔧 Series / parallel builder

🪜 Reduction stepper

1.7 Principles of Voltage Division and Current Division

AVoltage Divider Rule (VDR), series

BCurrent Divider Rule (CDR), parallel

CVDR and CDR in a series-parallel circuit (slide Example 2)

➗ VDR explorer (series)

🔱 CDR explorer (parallel)

1.8 Delta–Wye Transformation

AWhy do we need it?

BWye (Y) / Tee (T)

R_eq (ohmmeter)