RLC Circuits, Reactance and Series and Parallel Calculations for SS 3

Class: SS 3
Term: First Term
Week: 4
Age: 16 years
Duration: 45 minutes
Subject: Radio, TV and Electronics
Curriculum Theme: RLC Circuits, Reactance and Series and Parallel Calculations
Previous Lesson: Fault-Finding Equipment, Identification, Uses and Operation
Topic: RESISTIVE INDUCTIVE CAPACITIVE (RLC) CIRCUITS.
Subject Matter: Calculation of capacitive reactance; Calculation of inductive reactance; Calculation involving RC circuits; Calculation involving RL, LC, and RLC series circuit and parallel; RLC circuits

Specific Objectives

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

Cognitive Domain

• Define capacitive and inductive reactance.
• State the formulas for calculating capacitive and inductive reactance.
• State the formulas for calculating impedance in series RC, RL, LC, and RLC circuits.
• Explain the approach to calculating total impedance or current in parallel RLC circuits.

Psychomotor Domain

• Calculate capacitive reactance given frequency and capacitance.
• Calculate inductive reactance given frequency and inductance.
• Solve problems involving impedance in series RC, RL, and LC circuits.
• Solve problems involving impedance in series RLC circuits.
• Perform calculations for parallel RLC circuits to find total current or impedance.

Reference Materials

The following resources were used in planning this lesson:

• 2025 Revised 9 Years Basic Education Curriculum
• Relevant State Unified Scheme of Work
• A suitable Radio, TV and Electronics textbook for SS 3
• The HeadTeacher Scheme of work

Instructional Materials

The teacher will teach this lesson with the aid of:

• Whiteboard and markers
• Charts showing formulas for reactance and impedance
• Calculators
• Diagrams of RC, RL, LC, and RLC circuits (series and parallel)

Rationale for the Lesson

This lesson is essential for students to understand the behavior of AC circuits containing resistors, inductors, and capacitors. Mastering these calculations provides a fundamental basis for analyzing and designing electronic systems, preparing students for advanced studies and practical applications in electronics.

Prerequisite/Previous Knowledge

Students should have prior knowledge of basic AC circuit concepts, including voltage, current, frequency, resistance, capacitance, and inductance. They should also be familiar with the concepts of phase difference and basic algebraic manipulation.

Lesson Content/Board Summary

RESISTIVE INDUCTIVE CAPACITIVE (RLC) CIRCUITS.

Introduction to Reactance

In AC circuits, capacitors and inductors oppose the flow of current, similar to how resistors do. This opposition is called reactance. There are two types of reactance:

1. Capacitive Reactance ((X_C)): The opposition offered by a capacitor to the flow of alternating current. It is inversely proportional to the frequency and capacitance.
2. Inductive Reactance ((X_L)): The opposition offered by an inductor to the flow of alternating current. It is directly proportional to the frequency and inductance.

Capacitive Reactance ((X_C))

Capacitive reactance is calculated using the formula:

(X_C = frac{1}{2pi f C})

Where:

1. (X_C) = Capacitive Reactance (measured in Ohms, (Omega))
2. (f) = Frequency of the AC supply (measured in Hertz, Hz)
3. (C) = Capacitance of the capacitor (measured in Farads, F)
4. (pi) = approximately 3.142

Example 1

Question: Calculate the capacitive reactance of a 10 (mu)F capacitor connected to a 50 Hz AC supply.

Solution:

Step 1: Write the formula.

(X_C = frac{1}{2pi f C})

Step 2: Substitute the values. (Note: 10 (mu)F = (10 times 10^{-6}) F)

(X_C = frac{1}{2 times 3.142 times 50 times 10 times 10^{-6}})

Step 3: Simplify and write the answer.

(X_C = frac{1}{0.003142} approx 318.3 Omega)

Answer: (X_C = 318.3 Omega)

Inductive Reactance ((X_L))

Inductive reactance is calculated using the formula:

(X_L = 2pi f L)

Where:

1. (X_L) = Inductive Reactance (measured in Ohms, (Omega))
2. (f) = Frequency of the AC supply (measured in Hertz, Hz)
3. (L) = Inductance of the inductor (measured in Henrys, H)
4. (pi) = approximately 3.142

Example 1

Question: Find the inductive reactance of a 200 mH inductor connected to a 50 Hz AC supply.

Solution:

Step 1: Write the formula.

(X_L = 2pi f L)

Step 2: Substitute the values. (Note: 200 mH = 0.2 H)

(X_L = 2 times 3.142 times 50 times 0.2)

Step 3: Simplify and write the answer.

(X_L = 62.84 Omega)

Answer: (X_L = 62.84 Omega)

Impedance ((Z)) in AC Circuits

Impedance is the total opposition to current flow in an AC circuit, combining resistance and reactance. It is measured in Ohms ((Omega)).

RC Series Circuits

For a series circuit containing a resistor (R) and a capacitor (C), the impedance ((Z)) is given by:

(Z = sqrt{R^2 + X_C^2})

Where:

1. (R) = Resistance ((Omega))
2. (X_C) = Capacitive Reactance ((Omega))

Example 1

Question: A series RC circuit has a resistance of 30 (Omega) and a capacitive reactance of 40 (Omega). Calculate the total impedance.

Solution:

Step 1: Write the formula.

(Z = sqrt{R^2 + X_C^2})

Step 2: Substitute the values.

(Z = sqrt{30^2 + 40^2})

(Z = sqrt{900 + 1600})

(Z = sqrt{2500})

Step 3: Simplify and write the answer.

(Z = 50 Omega)

Answer: (Z = 50 Omega)

RL Series Circuits

For a series circuit containing a resistor (R) and an inductor (L), the impedance ((Z)) is given by:

(Z = sqrt{R^2 + X_L^2})

Where:

1. (R) = Resistance ((Omega))
2. (X_L) = Inductive Reactance ((Omega))

Example 1

Question: A series RL circuit has a resistance of 60 (Omega) and an inductive reactance of 80 (Omega). Calculate the total impedance.

Solution:

Step 1: Write the formula.

(Z = sqrt{R^2 + X_L^2})

Step 2: Substitute the values.

(Z = sqrt{60^2 + 80^2})

(Z = sqrt{3600 + 6400})

(Z = sqrt{10000})

Step 3: Simplify and write the answer.

(Z = 100 Omega)

Answer: (Z = 100 Omega)

LC Series Circuits

For a series circuit containing an inductor (L) and a capacitor (C), the impedance ((Z)) is given by:

(Z = |X_L – X_C|)

Where:

1. (X_L) = Inductive Reactance ((Omega))
2. (X_C) = Capacitive Reactance ((Omega))

Example 1

Question: In a series LC circuit, (X_L = 120 Omega) and (X_C = 70 Omega). Calculate the total impedance.

Solution:

Step 1: Write the formula.

(Z = |X_L – X_C|)

Step 2: Substitute the values.

(Z = |120 – 70|)

Step 3: Simplify and write the answer.

(Z = 50 Omega)

Answer: (Z = 50 Omega)

RLC Series Circuits

For a series circuit containing a resistor (R), an inductor (L), and a capacitor (C), the total impedance ((Z)) is given by:

(Z = sqrt{R^2 + (X_L – X_C)^2})

Where:

1. (R) = Resistance ((Omega))
2. (X_L) = Inductive Reactance ((Omega))
3. (X_C) = Capacitive Reactance ((Omega))

Example 1

Question: A series RLC circuit has (R = 20 Omega), (X_L = 100 Omega), and (X_C = 70 Omega). Calculate the total impedance.

Solution:

Step 1: Write the formula.

(Z = sqrt{R^2 + (X_L – X_C)^2})

Step 2: Substitute the values.

(Z = sqrt{20^2 + (100 – 70)^2})

(Z = sqrt{20^2 + 30^2})

(Z = sqrt{400 + 900})

(Z = sqrt{1300})

Step 3: Simplify and write the answer.

(Z approx 36.06 Omega)

Answer: (Z = 36.06 Omega)

RLC Parallel Circuits

For parallel RLC circuits, the total impedance is often calculated by first finding the individual branch currents or admittances. A common approach is to calculate the total current and then use Ohm’s law to find the total impedance.

Given a parallel RLC circuit connected to a voltage source (V):

1. Current through resistor: (I_R = frac{V}{R})
2. Current through inductor: (I_L = frac{V}{X_L})
3. Current through capacitor: (I_C = frac{V}{X_C})

The total current (I_T) is the vector sum of these currents. Since (I_R) is in phase with (V), (I_L) lags (V) by 90°, and (I_C) leads (V) by 90°, the total current is:

(I_T = sqrt{I_R^2 + (I_C – I_L)^2})

The total impedance (Z_T) is then:

(Z_T = frac{V}{I_T})

Example 1

Question: A parallel RLC circuit is connected to a 100 V AC supply. (R = 25 Omega), (X_L = 50 Omega), and (X_C = 20 Omega). Calculate the total current and total impedance.

Solution:

Step 1: Calculate individual branch currents.

(I_R = frac{V}{R} = frac{100}{25} = 4 A)

(I_L = frac{V}{X_L} = frac{100}{50} = 2 A)

(I_C = frac{V}{X_C} = frac{100}{20} = 5 A)

Step 2: Calculate the total current.

(I_T = sqrt{I_R^2 + (I_C – I_L)^2})

(I_T = sqrt{4^2 + (5 – 2)^2})

(I_T = sqrt{16 + 3^2})

(I_T = sqrt{16 + 9})

(I_T = sqrt{25})

(I_T = 5 A)

Step 3: Calculate the total impedance.

(Z_T = frac{V}{I_T} = frac{100}{5} = 20 Omega)

Answer: Total current (I_T = 5 A), Total impedance (Z_T = 20 Omega)

Teaching Methods/Instructional Techniques

Discussion, Demonstration, Guided Practice, Question and Answer, Explanation, Problem Solving, Individual Practice

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Review/Questioning

Teacher’s Activity: The teacher reminds the students about the concepts of resistance, capacitance, and inductance in AC circuits. He asks questions about what reactance is and how it differs from resistance.

Pupils’ Activity: Pupils recall and answer questions about resistance and reactance, emphasizing similarities and differences between purely reactive and purely resistive circuits.

Learning Point: Reactance and resistance review

Step 2: Capacitive Reactance Calculation

Time: 8 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher introduces the formula for capacitive reactance ((X_C)), explains each variable and its unit, and demonstrates how to calculate it using a worked example on the board.

Pupils’ Activity: Pupils listen, ask questions, and follow the teacher’s demonstration, copying the formula and example.

Learning Point: Capacitive reactance calculation

Step 3: Inductive Reactance Calculation

Time: 8 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher introduces the formula for inductive reactance ((X_L)), explains each variable and its unit, and demonstrates how to calculate it using a worked example on the board.

Pupils’ Activity: Pupils listen, ask questions, and follow the teacher’s demonstration, copying the formula and example.

Learning Point: Inductive reactance calculation

Step 4: RC and RL Series Circuit Calculations

Time: 8 minutes

Teaching Skill: Demonstration/Guided Practice

Teacher’s Activity: The teacher explains the concept of impedance and demonstrates how to calculate the total impedance for series RC and RL circuits using the relevant formulas and worked examples. He guides students through a practice problem.

Pupils’ Activity: Pupils observe the demonstration, participate in guided practice, and solve a similar problem in their notebooks.

Learning Point: Series RC and RL impedance

Step 5: LC and RLC Series Circuit Calculations

Time: 8 minutes

Teaching Skill: Demonstration/Guided Practice

Teacher’s Activity: The teacher demonstrates the calculation of impedance for series LC and RLC circuits, explaining the vector nature of reactance. He provides worked examples and allows students to attempt a problem.

Pupils’ Activity: Pupils pay attention, ask clarifying questions, and attempt to solve the practice problems for LC and RLC series circuits.

Learning Point: Series LC and RLC impedance

Step 6: RLC Parallel Circuit Calculations

Time: 8 minutes

Teaching Skill: Explanation/Problem Solving

Teacher’s Activity: The teacher explains the approach to calculating total current and impedance in parallel RLC circuits, focusing on calculating individual branch currents and then the total current. He works through an example.

Pupils’ Activity: Pupils follow the explanation and example, understanding the different approach for parallel circuits.

Learning Point: Parallel RLC circuit analysis

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

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

1. State the formula for capacitive reactance.
2. Calculate (X_L) for a 150 mH inductor at 100 Hz.
3. What is the impedance of a series RC circuit with (R = 20 Omega) and (X_C = 15 Omega)?
4. How do you calculate the total impedance of a series RLC circuit?

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Reactance and impedance calculation assessment

Step 8: Note-Taking

Time: 4 minutes

Teaching Skill: Guided Writing

Teacher’s Activity: The teacher guides pupils/students to copy the essential Board Summary notes, including formulas and key examples, into their notebooks.

Pupils’ Activity: Pupils/students copy the notes carefully into their notebooks.

Learning Point: Recording RLC circuit notes

Step 9: Conclusion

Time: 1 minute

Teaching Skill: Summarization

Teacher’s Activity: The teacher briefly summarizes the importance of understanding reactance and impedance calculations for various AC circuits and encourages students to practice more problems.

Pupils’ Activity: Pupils listen and prepare for the next lesson.

Learning Point: RLC circuit calculation summary

Continuous Assessment/Further Study

Type: Homework

Instruction: Solve the following problems in your notebooks.

1. A 22 (mu)F capacitor is connected to a 60 Hz AC supply. Calculate its capacitive reactance.
2. An inductor with an inductance of 300 mH is connected to a 50 Hz AC supply. Determine its inductive reactance.
3. A series circuit consists of a 40 (Omega) resistor and a capacitor with (X_C = 30 Omega). Find the total impedance of the circuit.
4. An RLC series circuit has (R = 50 Omega), (X_L = 120 Omega), and (X_C = 80 Omega). Calculate the total impedance.
5. For the parallel RLC circuit in Example 1 (Board Summary), if the voltage source is 120 V, calculate the total current and total impedance.

Lesson Keywords

• Reactance – Opposition to current flow in AC circuits by capacitors or inductors.
• Capacitive Reactance ((X_C)) – Opposition by a capacitor.
• Inductive Reactance ((X_L)) – Opposition by an inductor.
• Impedance ((Z)) – Total opposition to current flow in AC circuits, combining resistance and reactance.
• Series Circuit – Components connected end-to-end, current is the same through each.
• Parallel Circuit – Components connected across the same two points, voltage is the same across each.

Differentiation

For weaker learners, provide additional practice problems with simpler values and guide them step-by-step through the calculation process. For faster learners, introduce more complex problems involving phase angles or power factor calculations, or challenge them to research the applications of RLC circuits in filters or oscillators.

Suggested Lesson Videos

For further understanding and visual explanation, search on YouTube for:

• “Capacitive and Inductive Reactance Explained”
• “RLC Series Circuit Calculations”
• “RLC Parallel Circuit Analysis”

Teacher Guide for Using This Lesson Plan

Before the lesson, ensure you have all necessary instructional materials, especially charts with formulas and clear circuit diagrams. Review the concepts of reactance and impedance thoroughly to confidently guide students through the calculations. Begin by refreshing students’ memory on basic AC circuit elements. Systematically introduce each type of circuit (RC, RL, LC, RLC series, RLC parallel), presenting the relevant formulas and demonstrating calculations with clear, step-by-step examples. Encourage students to actively participate by asking questions and attempting practice problems. Pay close attention to common errors in formula application or unit conversion. The Board Summary should be copied by students after the main teaching points have been covered and understood, typically during Step 8. Check for understanding frequently through questions and short exercises. Provide individual support to students struggling with mathematical concepts and offer extension activities for those who grasp the material quickly.