Activity 1.1.7 — Circuit Theory Laws (Ohm's Law, KVL, KCL)¶

Learning Objectives¶

By the end of this lesson, students will be able to:

Define voltage, current, and resistance, and identify their units and measurement instruments.
Apply Ohm's Law to solve series circuit problems involving voltage, current, and resistance.
Apply Kirchhoff's Voltage Law (KVL) to analyze series circuits.
Apply Kirchhoff's Current Law (KCL) to analyze parallel circuits.
Analyze combination (series-parallel) circuits by reducing them step by step.

Vocabulary¶

Vocabulary (click to expand)

Term	Definition
Voltage (V)	The electrical force or pressure that pushes charge through a circuit; measured in Volts (V); named after Alessandro Volta
Current (I)	The flow of electrical charge through a conductor; measured in Amps (A); named after Andre-Marie Ampere
Resistance (R)	The opposition to the flow of current; measured in Ohms (Omega); named after Georg Ohm
Ohm's Law	The fundamental relationship V = I x R, stating that voltage equals current multiplied by resistance
Kirchhoff's Voltage Law (KVL)	The algebraic sum of all voltages around a closed loop equals zero; voltage drops sum to the applied voltage
Kirchhoff's Current Law (KCL)	The algebraic sum of currents entering a junction equals the sum leaving; current divides in parallel branches
Series Circuit	A circuit where current flows through components one after another on the same path
Parallel Circuit	A circuit where current splits through multiple branches that connect across the same voltage
Total Resistance	The combined resistance of all components in a circuit

Part 1: The Three Fundamentals — Voltage, Current, and Resistance¶

Before we can analyze circuits, we need to understand the three fundamental electrical quantities.

Voltage (V)¶

Voltage is the electrical pressure that pushes charge through a circuit. Think of it like water pressure in a pipe — without pressure, water will not flow.

Symbol: V
Unit: Volts (V)
Measurement: Measured with a voltmeter connected in parallel (across the component)
Named for: Alessandro Volta (1745–1827), inventor of the battery

Voltage is always measured between two points. A 9V battery means there is a 9-volt difference in electrical potential between its positive and negative terminals.

Current (I)¶

Current is the rate of flow of electrical charge through a conductor. It is like the volume of water flowing through a pipe per second.

Symbol: I
Unit: Amps (A), often expressed in milliamps (mA) where 1 mA = 0.001 A
Measurement: Measured with an ammeter connected in series (in the path of current)
Named for: Andre-Marie Ampere (1775–1836), pioneer of electromagnetism

Resistance (R)¶

Resistance is the opposition to the flow of current. Components with higher resistance allow less current to flow.

Symbol: R
Unit: Ohms (Omega), often expressed in kilo-ohms (kOhm) where 1 kOhm = 1,000 Ohms
Measurement: Measured with an ohmmeter (component must be disconnected from circuit)
Named for: Georg Ohm (1789–1854), who discovered the voltage-current-resistance relationship

Key insight: Voltage causes current to flow, and resistance limits how much current flows. Change any one of these three quantities, and at least one other will change as well.

Part 2: The Water Analogy — Understanding Electricity¶

The water analogy helps us visualize how circuits work:

Electrical Concept	Water Analogy
Voltage (V)	Water pressure in a pipe (PSI)
Current (I)	Volume of water flowing (gallons per minute)
Resistance (R)	Narrow section of pipe that restricts flow
Wire	Hollow pipe that carries water
Resistor	Partially closed valve or narrow pipe section
Battery/Power Supply	Water pump that creates pressure

How the analogy works: - More pressure (voltage) pushes more water (current) through the pipe. - Narrower pipe (higher resistance) restricts flow, allowing less water through. - The pump (battery) creates pressure (voltage), and the water flows through pipes (wires) and restrictions (resistors).

Part 3: Conventional Current vs Electron Flow¶

There is an important historical distinction in how we describe current direction:

Conventional Current (Engineering Convention)¶

Current flows from the positive terminal (+) to the negative terminal (-) of a power source.
This convention was established before scientists understood that electrons carry charge.
This is the standard used in most circuit analysis.

Electron Flow (Physics Reality)¶

Electrons (which carry negative charge) actually flow from the negative terminal (-) to the positive terminal (+).
This is the physically accurate direction of charge movement.

Which one should you use? In PLTW and most electrical engineering contexts, we use conventional current unless otherwise specified. The mathematics works either way — the key is being consistent.

Part 4: Ohm's Law — The Most Important Equation¶

Ohm's Law describes the fundamental relationship between voltage, current, and resistance:

V = I x R

Where:
V = Voltage in Volts (V)
I = Current in Amps (A)
R = Resistance in Ohms (Ohm)

The Ohm's Law Triangle¶

A helpful memory tool for solving any variable in Ohm's Law:

       -----
      | V |
      |---|
      | I |
       \|/
        R

To solve for any variable, cover it up: - V (unknown): I x R - I (unknown): V / R - R (unknown): V / I

Worked Example — Ohm's Law¶

Problem: A small LED is connected to a 6V battery through a current-limiting resistor. The resistor has a value of 150 Ohms. How much current flows through the circuit?

Solution:

Given: - V = 6V - R = 150 Ohm - Find: I

Using Ohm's Law:

I = V / R
I = 6V / 150 Ohm
I = 0.04 A
I = 40 mA

Answer: The circuit has 40 mA of current flowing through it.

Part 5: Series Circuits¶

In a series circuit, components are connected end-to-end on the same path. Current flows through each component one after another.

Properties of Series Circuits¶

Property	Rule	Explanation
Current	Same everywhere	Current has only one path to follow
Total Resistance	R_T = R1 + R2 + R3...	Resistances add up
Voltage	V_T = V1 + V2 + V3...	Voltage drops sum to applied voltage (KVL)

Total Resistance in Series¶

When resistors are connected in series, their resistances add:

R_T = R1 + R2 + R3 + ...

Example: Three resistors in series: 100 Ohm + 200 Ohm + 300 Ohm = 600 Ohms total

Kirchhoff's Voltage Law (KVL)¶

KVL states: The algebraic sum of all voltages around any closed loop in a circuit equals zero.

In practical terms: Voltage drops around a loop always equal the applied voltage.

V_T = V1 + V2 + V3 + ... + Vn

Worked Example — Series Circuit¶

Problem: Calculate the total resistance, circuit current, and voltage drop across each resistor in a series circuit with: - V_T = 12V (power supply) - R1 = 200 Ohm - R2 = 400 Ohm - R3 = 400 Ohm

Step 1: Total Resistance

R_T = R1 + R2 + R3
R_T = 200 + 400 + 400
R_T = 1000 Ohm = 1 kOhm

Step 2: Circuit Current (using Ohm's Law)

I = V_T / R_T
I = 12V / 1000 Ohm
I = 0.012 A = 12 mA

Step 3: Voltage Drop Across Each Resistor (using Ohm's Law)

V1 = I x R1 = 0.012A x 200 Ohm = 2.4V
V2 = I x R2 = 0.012A x 400 Ohm = 4.8V
V3 = I x R3 = 0.012A x 400 Ohm = 4.8V

Step 4: Verify KVL

V_T = V1 + V2 + V3
12V = 2.4V + 4.8V + 4.8V
12V = 12V  [Check - KVL satisfied!]

Key insight: In a series circuit, current is the same through every component, but voltage divides proportionally. Larger resistors get larger voltage drops.

Part 6: Parallel Circuits¶

In a parallel circuit, components are connected across the same two points, creating multiple paths for current.

Properties of Parallel Circuits¶

Property	Rule	Explanation
Voltage	Same across all branches	All branches connect to the same two points
Total Resistance	1/R_T = 1/R1 + 1/R2...	Use reciprocal formula
Current	I_T = I1 + I2 + I3...	Current divides and sums (KCL)

Total Resistance in Parallel¶

When resistors are connected in parallel, the reciprocal formula applies:

1/R_T = 1/R1 + 1/R2 + 1/R3 + ...

Or rearranged: R_T = 1 / (1/R1 + 1/R2 + 1/R3 + ...)

For only two resistors in parallel, use the product-over-sum formula:

R_T = (R1 x R2) / (R1 + R2)

Kirchhoff's Current Law (KCL)¶

KCL states: The algebraic sum of currents entering a junction equals the sum of currents leaving the junction.

In practical terms: Current going in equals current going out.

I_in = I_out
I_T = I1 + I2 + I3 + ... + In

Worked Example — Parallel Circuit¶

Problem: Two resistors (R1 = 100 Ohm and R2 = 200 Ohm) are connected in parallel across a 12V power supply. Find: 1. Total resistance 2. Current through each branch 3. Total current

Step 1: Total Resistance

Method 1: Product-over-Sum (for 2 resistors)
R_T = (100 x 200) / (100 + 200)
R_T = 20,000 / 300
R_T = 66.67 Ohm

Method 2: Reciprocal formula
1/R_T = 1/100 + 1/200
1/R_T = 0.01 + 0.005
1/R_T = 0.015
R_T = 1 / 0.015 = 66.67 Ohm

Step 2: Current Through Each Branch (using Ohm's Law)

I1 = V / R1 = 12V / 100 Ohm = 0.12 A = 120 mA
I2 = V / R2 = 12V / 200 Ohm = 0.06 A = 60 mA

Step 3: Total Current (verify with KCL)

I_T = I1 + I2
I_T = 120 mA + 60 mA
I_T = 180 mA

Also verify: I_T = V / R_T = 12V / 66.67 Ohm = 180 mA

Key insight: In a parallel circuit, voltage is the same across all branches, but current divides. The smaller resistor gets more current.

Part 7: Combination Circuits¶

Most real circuits contain both series and parallel sections. To analyze them:

Strategy for Combination Circuits¶

Identify which parts are series and which are parallel
Reduce the circuit step by step:
Calculate equivalent resistance of parallel sections
Add series resistances
Continue until you find the total resistance
Solve for total current using Ohm's Law
Work backward to find branch currents and voltage drops

Worked Example — Combination Circuit¶

Problem: In the circuit below, find: - Total resistance - Total current - Current through R2 - Voltage drop across R3

        R1 = 100 Ohm
    +--------/\/\/\/--------+
    |                          |
    |      R2 = 200 Ohm        |  R3 = 300 Ohm
    +--------/\/\/\/----------+---/\/\/\/---+
    |                                        |
    |                                        |
    +=============== 12V ===================+

Step 1: Simplify R2 and R3 (these are in parallel)

1/R_23 = 1/200 + 1/300
1/R_23 = 0.005 + 0.00333
1/R_23 = 0.00833
R_23 = 120 Ohm

Step 2: Calculate Total Resistance (R1 in series with R_23)

R_T = R1 + R_23
R_T = 100 + 120
R_T = 220 Ohm

Step 3: Total Current

I_T = V / R_T
I_T = 12V / 220 Ohm
I_T = 0.0545 A = 54.5 mA

Step 4: Current Through R2 (current divider in parallel section)

I2 = I_T x (R3 / (R2 + R3))
I2 = 54.5 mA x (300 / (200 + 300))
I2 = 54.5 mA x (300 / 500)
I2 = 54.5 mA x 0.6
I2 = 32.7 mA

Step 5: Voltage Drop Across R3

V3 = I3 x R3  (or V3 = V - V1)

First find I3:
I3 = I_T - I2 = 54.5 mA - 32.7 mA = 21.8 mA

Then:
V3 = 0.0218A x 300 Ohm = 6.54V

Part 8: Practice Problems¶

Practice Problem 1 — Ohm's Law¶

A resistor of 470 Ohm is connected to a 9V battery. Calculate the current flowing through the resistor in mA.

Show Solution

I = V / R
I = 9V / 470 Ohm
I = 0.01915 A
I = 19.15 mA

Practice Problem 2 — Series Circuit¶

Three resistors (330 Ohm, 680 Ohm, and 1 kOhm) are connected in series across a 24V power supply. 1. What is the total resistance? 2. What current flows through the circuit? 3. What is the voltage drop across the 680 Ohm resistor?

Show Solution

Total Resistance:

R_T = 330 + 680 + 1000
R_T = 2010 Ohm = 2.01 kOhm

Circuit Current:

I = V / R_T
I = 24V / 2010 Ohm
I = 0.01194 A = 11.94 mA

Voltage Drop Across 680 Ohm Resistor:

V_680 = I x 680 Ohm
V_680 = 0.01194A x 680 Ohm
V_680 = 8.12V

Practice Problem 3 — Parallel Circuit¶

Two resistors (120 Ohm and 240 Ohm) are connected in parallel across a 12V supply. 1. Calculate the total resistance. 2. Calculate the current through each branch. 3. Calculate the total current.

Show Solution

Total Resistance:

R_T = (120 x 240) / (120 + 240)
R_T = 28,800 / 360
R_T = 80 Ohm

Branch Currents:

I_120 = V / 120 Ohm = 12V / 120 Ohm = 0.1 A = 100 mA
I_240 = V / 240 Ohm = 12V / 240 Ohm = 0.05 A = 50 mA

Total Current (KCL):

I_T = I_120 + I_240
I_T = 100 mA + 50 mA
I_T = 150 mA

Verification: I_T = V / R_T = 12V / 80 Ohm = 150 mA

Practice Problem 4 — Combination Circuit Analysis¶

In the circuit below, find the total resistance, total current, and voltage drop across R2.

        R1 = 500 Ohm
    +--------/\/\/\/--------+
    |                          |
    |                          X--- R3 = 1 kOhm
    |                          |
    |      R2 = 1 kOhm         |
    +--------/\/\/\/-----------+
    |
    +=============== 10V =====+

(Note: R2 and R3 are in parallel with each other, and their combination is in series with R1)

Show Solution

Step 1: Parallel combination of R2 and R3:

R_23 = (1000 x 1000) / (1000 + 1000)
R_23 = 1,000,000 / 2000
R_23 = 500 Ohm

Step 2: Total Resistance:

R_T = R1 + R_23
R_T = 500 + 500
R_T = 1000 Ohm = 1 kOhm

Step 3: Total Current:

I_T = V / R_T
I_T = 10V / 1000 Ohm
I_T = 0.01 A = 10 mA

Step 4: Voltage Drop Across R2: Since R2 and R3 are in parallel with each other and their combination (R_23) is in series with R1, we first find the voltage across the parallel section:

V_parallel = I_T x R_23
V_parallel = 0.01A x 500 Ohm
V_parallel = 5V

Therefore, V_R2 = V_R3 = V_parallel = 5V

Summary¶

Topic	Key Equations	Key Concepts
Ohm's Law	V = I x R	Voltage equals current times resistance
Series Circuit	R_T = R1 + R2 + ...	Current is the same everywhere; voltage divides
KVL	V_T = V1 + V2 + ...	Voltage drops sum to applied voltage
Parallel Circuit	1/R_T = 1/R1 + 1/R2 + ...	Voltage is the same across branches; current divides
KCL	I_T = I1 + I2 + ...	Currents entering = currents leaving a junction
Two Resistors in Parallel	R_T = (R1 x R2)/(R1 + R2)	Quick formula for two parallel resistors

Quick Reference — Ohm's Law Triangle¶

Cover the variable you need:

 V = I x R      I = V / R      R = V / I

Current, Voltage, and Resistance Units¶

Quantity	Symbol	Unit	Abbreviation	Common Subunits
Voltage	V	Volt	V	mV (millivolt), kV (kilovolt)
Current	I	Amp	A	mA (milliamp), uA (microamp)
Resistance	R	Ohm	Ohm	kOhm (kilo-ohm), MOhm (mega-ohm)

Key Reminders¶

Voltage (V) is electrical pressure, current (I) is the flow of charge, and resistance (R) opposes that flow.
Ohm's Law (V = I x R) is the foundation for all circuit analysis — memorize it and the triangle method.
In series circuits, current is constant but voltage divides; in parallel circuits, voltage is constant but current divides.
Kirchhoff's Voltage Law (KVL): Voltage drops around any loop sum to the applied voltage.
Kirchhoff's Current Law (KCL): Current entering a junction equals current leaving.
For parallel resistance, use the reciprocal formula or the product-over-sum formula for two resistors.
Always work through combination circuits step by step, reducing sections before solving.

Custom activity — adapted from PLTW Digital Electronics