Class: Senior Secondary School 2 (SS2 / SSS2)
Term: 1st Term
Week: 8
Age: 16 years
Duration: 45 minutes
Subject: Basic Electronics
Curriculum Theme: Electrical Circuits and Components
Previous Lesson: Power in AC Circuits: Advantages, Disadvantages and Calculation of Power Factor.
Topic: POWER IN AC CIRCUITS
Subject Matter: Meaning of POWER IN AC CIRCUITS (recap meaning of power in AC circuits), Q factor (quality factor of a resonant circuit), bandwidth (FH and FL frequency limits).

Specific Objectives

By the end of the lesson, pupils should be able to:

Cognitive Domain:

Define power in AC circuits, distinguishing between active, reactive, and apparent power.
Define Q-factor (quality factor) of a resonant circuit.
Define bandwidth, identifying the upper (FH) and lower (FL) cut-off frequencies.
State the formulas for calculating Q-factor and bandwidth.

Affective Domain:

Appreciate the importance of Q-factor and bandwidth in determining the performance of resonant circuits.
Recognize the practical applications of Q-factor and bandwidth in electronic systems.

Psychomotor Domain:

Calculate the Q-factor for a given resonant circuit.
Calculate the bandwidth of a resonant circuit using given parameters.
Sketch a frequency response curve showing resonant frequency and bandwidth.

Discuss the relevance of AC power concepts in everyday electronic devices and communication systems.

Reference Materials

The following resources were used in planning this lesson:

Senior Secondary Schools Education Curriculum
State Unified Scheme of Work
Basic Electronics for Senior Secondary Schools Textbook

Instructional Materials

The teacher will teach this lesson with the aid of:

Calculator
Power triangle diagram
Circuit diagrams of series and parallel RLC resonant circuits
Whiteboard and markers

Rationale for the Lesson

This lesson enables pupils to understand how power behaves in alternating current circuits and to analyze the selectivity of resonant circuits. Understanding Q-factor and bandwidth is important for designing and troubleshooting filters, oscillators, and communication systems.

Prerequisite/Previous Knowledge

Pupils should have prior knowledge of basic AC circuit concepts, including resistance, inductance, capacitance, impedance, and the concept of resonance.

Lesson Content/Board Summary

POWER IN AC CIRCUITS

Recap: Power in AC Circuits

In AC circuits, power is categorized into three types:

Active Power (P): Also known as real power, measured in Watts (W). It is the actual power dissipated by the resistive components of the circuit.
Formula: P = V_RMS × I_RMS × cos(φ)
Reactive Power (Q): Measured in Volt-Ampere Reactive (VAR). It is the power exchanged between the source and the reactive components (inductors and capacitors), not dissipated.
Formula: Q = V_RMS × I_RMS × sin(φ)
Apparent Power (S): Measured in Volt-Ampere (VA). It is the total power delivered by the source, which is the vector sum of active and reactive power.
Formula: S = V_RMS × I_RMS = √(P² + Q²)

Power Triangle: Represents the relationship between P, Q, and S, where S is the hypotenuse, P is the adjacent side (horizontal), and Q is the opposite side (vertical), with φ being the phase angle between voltage and current.

Q-Factor (Quality Factor) of a Resonant Circuit

The Q-factor is a dimensionless parameter that describes the sharpness or selectivity of a resonant circuit’s frequency response. A higher Q-factor indicates a sharper, more selective resonance curve, meaning the circuit responds strongly to frequencies very close to its resonant frequency and rejects others.

Formula for Series RLC Circuit: Q = (ω₀L) / R = 1 / (ω₀RC)

Where:

Q = Quality Factor (dimensionless)
ω₀ = Resonant angular frequency (rad/s) = 2πf₀
L = Inductance (Henry, H)
R = Resistance (Ohms, Ω)
C = Capacitance (Farads, F)

Alternatively, Q can be expressed as: Q = f₀ / BW

Where:

f₀ = Resonant frequency (Hz)
BW = Bandwidth (Hz)

Bandwidth (BW) of a Resonant Circuit

Bandwidth is the range of frequencies over which the power delivered to the circuit is at least half of the maximum power delivered at resonance. These are often called the half-power frequencies or cut-off frequencies.

Upper Cut-off Frequency (f_H): The frequency above resonance where the power is half the maximum, or the current/voltage is 1/√2 (approximately 0.707) of the maximum.
Lower Cut-off Frequency (f_L): The frequency below resonance where the power is half the maximum, or the current/voltage is 1/√2 (approximately 0.707) of the maximum.

Formula: BW = f_H – f_L

Relationship with Q-factor: BW = f₀ / Q

Worked Examples

Example 1: Calculating Q-factor

A series RLC circuit has a resistor of 10 Ω, an inductor of 100 mH, and a capacitor of 1 μF. Calculate the Q-factor at resonance.

Step 1: Calculate the resonant angular frequency (ω₀).

L = 100 mH = 100 × 10⁻³ H

C = 1 μF = 1 × 10⁻⁶ F

ω₀ = 1 / √(LC)

ω₀ = 1 / √((100 × 10⁻³) × (1 × 10⁻⁶))

ω₀ = 1 / √(10⁻⁷) = 1 / 0.0003162277

ω₀ ≈ 3162.28 rad/s

Step 2: Calculate the Q-factor.

Q = (ω₀L) / R

Q = (3162.28 rad/s × 100 × 10⁻³ H) / 10 Ω

Q = 316.228 / 10

Q ≈ 31.62

The Q-factor of the circuit is approximately 31.62.

Example 2: Calculating Bandwidth

If the resonant frequency (f₀) of a circuit is 10 kHz and its Q-factor is 20, calculate the bandwidth (BW) of the circuit.

Step 1: Identify the given values.

f₀ = 10 kHz = 10,000 Hz

Q = 20

Step 2: Use the formula relating bandwidth, resonant frequency, and Q-factor.

BW = f₀ / Q

BW = 10,000 Hz / 20

BW = 500 Hz

The bandwidth of the circuit is 500 Hz.

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher greets the pupils and reviews the previous lesson on AC circuit components and basic resonance.
Pupils’ Activity: Pupils respond to greetings and recall key points from the previous lesson.
Learning Point: Pupils connect new learning to prior knowledge.

Step 2: Recap of Power in AC Circuits

Time: 10 minutes
Teaching Skill: Explanation/Recall
Teacher’s Activity: The teacher leads a discussion on the three types of power in AC circuits (active, reactive, apparent) and explains the power triangle, writing the formulas on the board.
Pupils’ Activity: Pupils contribute to the discussion, recall definitions, and copy formulas from the board.
Learning Point: Pupils reinforce their understanding of AC power fundamentals.

Step 3: Introduction to Q-Factor

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher defines Q-factor, explains its significance in determining the sharpness of resonance, and presents the relevant formulas for calculating it. The teacher may use a frequency response curve diagram to illustrate.
Pupils’ Activity: Pupils listen attentively, ask questions for clarification, and copy definitions and formulas.
Learning Point: Pupils understand what Q-factor is and how it describes circuit selectivity.

Step 4: Explanation of Bandwidth (FH and FL)

Time: 8 minutes
Teaching Skill: Explanation/Illustration
Teacher’s Activity: The teacher defines bandwidth, explains the upper (FH) and lower (FL) cut-off frequencies, and shows the relationship between bandwidth, resonant frequency, and Q-factor.
Pupils’ Activity: Pupils observe the illustrations, note down definitions, and understand the concept of frequency limits.
Learning Point: Pupils grasp the concept of bandwidth and its relation to Q-factor.

Step 5: Worked Examples

Time: 7 minutes
Teaching Skill: Problem Solving/Demonstration
Teacher’s Activity: The teacher solves example problems on the board, demonstrating how to calculate Q-factor and bandwidth using the formulas.
Pupils’ Activity: Pupils follow the steps, ask questions, and ensure they understand the calculations.
Learning Point: Pupils learn the practical application of the formulas.

Step 6: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

Define Q-factor in your own words.
What is bandwidth in an AC resonant circuit?
State the formula that relates bandwidth, resonant frequency, and Q-factor.
If a circuit has a resonant frequency of 5 kHz and a Q-factor of 25, calculate its bandwidth.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 5 minutes
Teaching Skill: Summarization/Assignment
Teacher’s Activity: The teacher summarizes the key points of the lesson and assigns homework, which includes solving more problems related to Q-factor and bandwidth.
Pupils’ Activity: Pupils listen to the summary and copy down the homework assignment.
Learning Point: Pupils consolidate their learning and prepare for further practice.

Lesson Keywords

Q-factor – A measure of the sharpness of resonance in an AC circuit.
Bandwidth – The range of frequencies over which a circuit’s power output is at least half of the maximum.
Resonant Frequency – The frequency at which a circuit exhibits maximum response (minimum impedance in series, maximum in parallel).
Cut-off Frequencies – The upper (f_H) and lower (f_L) frequencies at which the power delivered to a circuit is half the maximum.
Power Factor – The ratio of active power to apparent power in an AC circuit.

Differentiation

For pupils who grasp the concepts quickly, the teacher can provide more complex RLC circuit problems or ask them to research the applications of high and low Q-factor circuits. For pupils needing more support, the teacher will provide simplified examples and offer one-on-one guidance during practice sessions.

Note for teachers using this lesson plan

Ensure pupils have a solid understanding of basic AC circuit analysis before introducing Q-factor and bandwidth. Use visual aids like frequency response curves to illustrate the concepts effectively. Encourage pupils to work through the examples step-by-step and to ask questions when they encounter difficulties.