Lesson Notes By Weeks and Term v3 - Senior Secondary 3
Resistive, Inductive and Capacitive (RLC) Circuits
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Resistive, Inductive and Capacitive (RLC) Circuits
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Subject: Radio Television And Repairs
Class: Senior Secondary 3
Term: 3rd Term
Week: 2
Theme: Electronic Devices And Circuits
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This lesson introduces Senior Secondary 3 (SS3) students to the fundamental concepts of alternating current (AC) circuits containing resistive, inductive, and capacitive components. The understanding of RLC circuits is crucial for technicians in radio and television repairs, as virtually all electronic devices operating on AC power, from simple radio receivers to complex television sets and power supply units, utilize these components. This knowledge forms the bedrock for troubleshooting, designing, and maintaining electronic systems common in Nigerian homes and industries. Performance Objectives (Learner-friendly language):
This section provides a detailed explanation of the core concepts required to understand and analyze RLC circuits. 2.
1. Alternating Current (AC) Fundamentals (Brief Review) Alternating current (AC) periodically reverses direction.
Key characteristics include: Frequency (f): The number of cycles per second, measured in Hertz (Hz). In Nigeria, the standard mains frequency is 50 Hz. Angular Frequency (ω): Related to frequency by ω = 2πf, measured in radians per second (rad/s).
Phase: The position of a point in time on a waveform cycle. When comparing two waveforms, their phase difference indicates how much one leads or lags the other. 2.
2. Resistive Circuits in AC Resistor (R): A component that opposes the flow of electric current. Its opposition is called resistance, measured in Ohms (Ω).
Behaviour in AC: Resistors offer the same opposition to AC as they do to D
C. Phase Relationship: In a purely resistive AC circuit, the voltage across the resistor and the current flowing through it are in phase. This means they reach their peak values at the same time.
Ohm's Law: V = IR (where V and I are RMS values for AC).
Power: P = VI cos(Φ), where for a purely resistive circuit, Φ = 0, so P = VI. 2.
3. Inductive Circuits in AC Inductor (L): A coil of wire that stores energy in a magnetic field when current flows through it. Its ability to store this energy is called inductance, measured in Henries (H).
Behaviour in AC: Inductors oppose changes in current. When AC flows, the continuously changing current induces a back electromotive force (EMF) that opposes the supply voltage. This opposition to AC current is called inductive reactance. Inductive Reactance (X_L): Definition: The opposition offered by an inductor to the flow of alternating current. It is frequency-dependent.
Formula: X_L = 2πfL = ωL Where: X_L = Inductive Reactance (Ohms, Ω) f = Frequency (Hertz, Hz) L = Inductance (Henries, H) ω = Angular frequency (rad/s)
Characteristics: X_L increases with increasing frequency and increasing inductance. At DC (f=0), X_L = 0, meaning an ideal inductor acts as a short circuit.
Phase Relationship: In a purely inductive AC circuit, the voltage across the inductor leads the current through it by 90 degrees (or the current lags the voltage by 90 degrees). Worked Example 2.3.1 (Inductive Reactance): An inductor of 250 mH is connected to a 240 V, 50 Hz AC supply. Calculate its inductive reactance.
Solution: Given: L = 250 mH = 250 × 10−3 H = 0.25 H f = 50 Hz Formula: X_L = 2πfL X_L = 2 × 3.142 × 50 Hz × 0.25 H X_L = 78.55 Ohms (Ω) 2.
4. Capacitive Circuits in AC Capacitor (C): A device that stores electrical energy in an electric field between two conductive plates separated by a dielectric material. Its ability to store charge is called capacitance, measured in Farads (F).
Behaviour in AC: Capacitors oppose changes in voltage. When AC is applied, the capacitor continuously charges and discharges, allowing current to flow through the circuit (not through the dielectric). This opposition to AC current is called capacitive reactance. Capacitive Reactance (X_C): Definition: The opposition offered by a capacitor to the flow of alternating current. It is frequency-dependent.
Formula: X_C = 1 / (2πfC) = 1 / (ωC)
Where: X_C = Capacitive Reactance (Ohms, Ω) f = Frequency (Hertz, Hz) C = Capacitance (Farads, F) ω = Angular frequency (rad/s)
Characteristics: X_C decreases with increasing frequency and increasing capacitance. At DC (f=0), X_C approaches infinity, meaning an ideal capacitor acts as an open circuit.
Phase Relationship: In a purely capacitive AC circuit, the current through the capacitor leads the voltage across it by 90 degrees (or the voltage lags the current by 90 degrees). Worked Example 2.4.1 (Capacitive Reactance): A 2.2 μF capacitor is connected to a 240 V, 50 Hz AC supply. Calculate its capacitive reactance.
Solution: Given: C = 2.2 μF = 2.2 × 10−6 F f = 50 Hz Formula: X_C = 1 frequency and increasing capacitance. At DC (f=0), X_C approaches infinity, meaning an ideal capacitor acts as an open circuit.
Phase Relationship: In a purely capacitive AC circuit, the current through the capacitor leads the voltage across it by 90 degrees (or the voltage lags the current by 90 degrees). Worked Example 2.4.1 (Capacitive Reactance): A 2.2 μF capacitor is connected to a 240 V, 50 Hz AC supply. Calculate its capacitive reactance.
Solution: Given: C = 2.2 μF = 2.2 × 10−6 F f = 50 Hz Formula: X_C = 1 / (2πfC) X_C = 1 / (2 × 3.142 × 50 Hz × 2.2 × 10−6 F) X_C = 1 / (0.00069128) X_C = 1446.59 Ohms (Ω) 2.
5. Impedance (Z)
Definition: The total opposition to the flow of alternating current in an AC circuit containing a combination of resistors, inductors, and capacitors. It is the AC equivalent of resistance in DC circuits.
Units: Ohms (Ω).
Concept: Unlike resistance, impedance is a complex quantity because it involves both magnitude and phase. In series RLC circuits, impedance is calculated using vector addition, not simple algebraic addition, due to the phase differences between voltage and current across R, L, and C. 2.
6. Series RL, RC, LC, and RLC Circuits The total opposition (impedance) in series AC circuits is found using the Pythagorean theorem, treating resistance and reactance as components in a right-angled triangle, often visualized with phasor diagrams. 2.6.
1. Series RL Circuit (Resistor and Inductor)
Components: A resistor (R) in series with an inductor (L).
Total Impedance (Z): Z = √(R2 + X_L2) Phase Angle (Φ): The angle by which the total voltage leads the total current. tan(Φ) = X_L / R Φ = arctan(X_L / R)
Characteristics: The circuit is inductive, meaning the current lags the voltage. Worked Example 2.6.1.1 (RL Series Circuit): A 60 Ω resistor is connected in series with an inductor of 0.4 H to a 240 V, 50 Hz AC supply. Calculate the impedance of the circuit.
Solution: Given: R = 60 Ω L = 0.4 H f = 50 Hz Step 1: Calculate inductive reactance (X_L). X_L = 2πfL = 2 × 3.142 × 50 Hz × 0.4 H X_L = 125.68 Ohms (Ω)
Step 2: Calculate the total impedance (Z). Z = √(R2 + X_L2) Z = √(602 + 125.682) Z = √(3600 + 15795.45) Z = √19395.45 Z = 139.27 Ohms (Ω) 2.6.
2. Series RC Circuit (Resistor and Capacitor)
Components: A resistor (R) in series with a capacitor (C).
Total Impedance (Z): Z = √(R2 + X_C2) Phase Angle (Φ):): The angle by which the total voltage lags the total current. tan(Φ) = -X_C / R (or tan(Φ) = X_C / R, and specify current leads voltage) Φ = arctan(-X_C / R)
Characteristics: The circuit is capacitive, meaning the current leads the voltage. Worked Example 2.6.2.1 (RC Series Circuit): A 100 Ω resistor is connected in series with a 50 μF capacitor to a 240 V, 50 Hz AC supply. Calculate the impedance of the circuit.
Solution: Given: R = 100 Ω C = 50 μF = 50 × 10−6 F f = 50 Hz Step 1: Calculate capacitive reactance (X_C). X_C = 1 / (2πfC) = 1 / (2 × 3.142 × 50 Hz × 50 × 10−6 F) X_C = 1 / (0.01571) X_C = 63.65 Ohms (Ω)
Step 2: Calculate the total impedance (Z). Z = √(R2 + X_C2) Z = √(1002 + 63.652) Z = √(10000 + 4051.32) Z = √14051.32 Z = 118.54 Ohms (Ω) 2.6.
3. Series LC Circuit (Inductor and Capacitor)
Components: An inductor (L) in series with a capacitor (C). Net Reactance (X_Net): Since X_L and X_C are 180 degrees out of phase with each other (X_L leads voltage by 90°, X_C lags voltage by 90°), their effects cancel each other out to some extent. X_Net = X_L - X_C (or X_C - X_L, depending on which is larger)
Total Impedance (Z): Z = |X_L - X_C| * Resonance: A special condition occurs when X_L = X_
C. At this Ohms (Ω) 2.6.
3. Series LC Circuit (Inductor and Capacitor)
Components: An inductor (L) in series with a capacitor (C). Net Reactance (X_Net): Since X_L and X_C are 180 degrees out of phase with each other (X_L leads voltage by 90°, X_C lags voltage by 90°), their effects cancel each other out to some extent. X_Net = X_L - X_C (or X_C - X_L, depending on which is larger)
Total Impedance (Z): Z = |X_L - X_C| Resonance: A special condition occurs when X_L = X_C. At this resonant frequency (f_r), the net reactance is zero, and the impedance of the LC circuit is at its minimum (ideally zero). This phenomenon is crucial for tuning in radios and TVs. f_r = 1 / (2π√(LC)) Worked Example 2.6.3.1 (LC Series Circuit): An inductor of 150 mH and a capacitor of 40 μF are connected in series to a 50 Hz AC supply. Calculate the impedance of the circuit.
Solution: Given: L = 150 mH = 0.15 H C = 40 μF = 40 × 10−6 F f = 50 Hz Step 1: Calculate inductive reactance (X_L). X_L = 2πfL = 2 × 3.142 × 50 Hz × 0.15 H X_L = 47.13 Ohms (Ω)
Step 2: Calculate capacitive reactance (X_C). X_C = 1 / (2πfC) = 1 / (2 × 3.142 × 50 Hz × 40 × 10−6 F) X_C = 1 / (0.012568) X_C = 79.57 Ohms (Ω)
Step 3: Calculate the total impedance (Z). Z = |X_L - X_C| = |47.13 - 79.57| Z = |-32.44| Z = 32.44 Ohms (Ω) (Since X_C > X_L, the circuit is capacitive). 2.6.
4. Series RLC Circuit (Resistor, Inductor, and Capacitor)
Components: A resistor (R), an inductor (L), and a capacitor (C) connected in series. Net Reactance (X_Net): The difference between inductive and capacitive reactances. X_Net = X_L - X_C Total Impedance (Z): Z = √(R2 + (X_L - X_C)2) Phase Angle (Φ): The angle between the total voltage and total current. tan(Φ) = (X_L - X_C) / R Φ = arctan((X_L - X_C) / R)
Characteristics: If X_L > X_C: The circuit is inductive (current lags voltage). If X_C > X_L: The circuit is capacitive (current leads voltage). * If X_L = X_C (Resonance): The circuit is purely resistive, Z = R, and the current and voltage are in phase (Φ = 0). This is the condition for maximum current for a given voltage. Worked Example 2.6.4.1 (RLC Series Circuit): A series RLC circuit has a resistance of 40 Ω, an inductance of 0.3 H, and a capacitance of 60 μ
F. It is connected to a 240 V, 50 Hz AC supply. Calculate the impedance of the circuit.
Solution: Given: R = 40 Ω L = 0.3 H C = 60 μF = 60 × 10−6 F f = 50 Hz Step 1: Calculate inductive reactance (X_L). X_L = 2πfL = 2 × 3.142 × 50 Hz × 0.3 H X_L = 94.26 Ohms (Ω)
Step 2: Calculate capacitive reactance (X_C). X_C = 1 / (2πfC) = 1 / (2 × 3.142 × 50 Hz × 60 × 10−6 F) X_C = 1 / (0.018852) X_C = 53.04 Ohms (Ω)
Step 3: Calculate the net reactance (X_Net). X_Net = X_L - X_C = 94.26 - 53.04 X_Net = 41.22 Ohms (Ω)
Step 4: Calculate the total impedance (Z). Z = √(R2 + X_Net2) Z = √(402 + 41.222) Z = √(1600 + 1699.0884) Z = √3299.0884 Z = 57.44 Ohms (Ω) 3.
1. Teacher Activities Introduction (10 minutes): Begin by reviewing basic AC circuit concepts (frequency, voltage, current). Introduce the three fundamental components: Resistor, Inductor, Capacitor. Briefly remind students of their DC behavior. State the lesson objectives clearly, emphasizing the practical relevance in radio/TV repairs.
Concept Explanation (30 minutes): Explain the behaviour of resistors, inductors, and capacitors in AC circuits. Introduce and define inductive reactance (X_L) and capacitive reactance (X_C). Derive or present the formulas for X_L = 2πfL and X_C = 1/(2πfC), explaining each variable and unit. Use analogies (e.g., flywheel for inductor, water tank for capacitor) to illustrate the opposition to current/voltage changes. Explain the phase relationships for pure R, L, and C circuits (in phase, voltage leads current, current leads voltage). Introduce impedance (Z) as the total opposition in AC circuits. Circuit Analysis and Calculation (40 minutes): Draw and explain series RL, RC, LC, and RLC circuits. Present the impedance formulas for each circuit type: RL: Z = √(R2 + X_L2) RC: Z = √(R2 + X_C2) LC: Z = |X_L - X_C| RLC: Z = √(R2 + (X_L - X_C)2) Work through the detailed examples provided in the "Key Concepts and Explanations" section on the board, explaining each step clearly. Encourage students to follow along and ask questions. Emphasize the importance of unit conversion (e.g., mH to H, μF to F).
Guided Practice (20 minutes): Assign 2-3 guided practice problems (from section 4) for students to attempt in small groups. Circulate around the classroom, providing support, clarification, and correcting misconceptions. Facilitate group discussions and allow groups to present their solutions. Wrap-up and Real-life Connections (10 minutes): Summarize key formulas and concepts. Reiterate real-life applications in Nigerian context (radio tuning, power filters, generator stability, local repair shops). Assign independent practice questions as homework. 3.
2. Student Activities Active Listening and Note-taking: Students will listen attentively to explanations and take comprehensive notes.
Asking Questions: Students will ask clarifying questions during concept explanations and problem-solving sessions.
Group Discussion: Students will engage in discussions with peers during guided practice, sharing ideas and approaches to problem-solving.
Problem Solving: Students will actively solve problems during guided practice and independent assignments, demonstrating their understanding of reactance and impedance calculations.
Formula Recitation: Students will practice recalling and applying the formulas for X_L, X_C, and Z for different circuit types.
Unit Conversion: Students will practice converting units (mH to H, μF to F) as required in calculations.
Radio and Television Tuning: RLC circuits, especially LC circuits, are at the heart of how radios and televisions select specific stations. When the inductor or capacitor is adjusted (tuned), the circuit's resonant frequency changes. When this resonant frequency matches the frequency of a desired radio or TV station, the circuit offers minimum impedance to that signal, allowing it to be amplified and processed, while rejecting other frequencies. This is a direct application students encounter daily in Nigeria.
Power Supply Filters: In most electronic devices, AC power from the mains must be converted to smooth DC power. RLC circuits (or often RC and LC combinations) are used as filters in power supply units. Inductors and capacitors work together to smooth out the rectified AC (pulsating DC), removing ripples and ensuring a stable DC output, which is crucial for the sensitive components of radios, televisions, and other gadgets, particularly given the varying quality of power from the national grid or generators in Nigeria. Motor Control and Protection in Industry/Homes: Many electric motors, common in industrial machinery (e.g., grinding mills, textile factories) and household appliances (e.g., refrigerators, washing machines) in Nigeria, utilize inductors (motor windings) and sometimes capacitors for starting, speed control, and improving power factor. Understanding RLC interactions helps technicians troubleshoot motor faults, select appropriate start/run capacitors, and ensure efficient operation and protection of these vital machines.
Audio Crossover Networks: In high-fidelity audio systems (used in homes, churches, event centres), RLC circuits are used as crossover networks to split the audio signal into different frequency bands (lows for woofers, mids for mid-range speakers, highs for tweeters). This ensures that each speaker type receives only the frequencies it is designed to reproduce effectively, improving sound quality.