Simple A.C Circuits | Waec Physics

Waec Lesson notes on Graphical representation of e.m.f. and current in an a.c. circuit

Simple Alternating Current (AC)

1. Alternating current (AC) periodically changes direction and magnitude.
2. AC is represented mathematically by sinusoidal functions.
3. The standard equation for AC current is $I = I_0 \sin \omega t$, where $I_0$ is the peak current and $\omega t$ is the angular frequency.
4. AC voltage is represented as $E = E_0 \sin \omega t$, where $E_0$ is the peak voltage.
5. AC is commonly used in power transmission because it can be easily transformed to different voltage levels.
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Graphical Representation of Electromotive Force and Current in AC Circuits

6. AC waveforms are typically sinusoidal, represented as a graph of current or voltage against time.
7. The peak of the sine wave represents the maximum value of current or voltage.
8. The zero crossings indicate points where the current or voltage reverses direction.
9. The waveform repeats periodically, with a time period $T$ given by $T = \frac{1}{f}$, where $f$ is the frequency.
10. The area under the AC waveform represents the net work done over a cycle.
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Graphs of Equations

11. The graphs of $I = I_0 \sin \omega t$ and $E = E_0 \sin \omega t$ are sine waves.
12. The amplitude of the wave corresponds to $I_0$ or $E_0$, the peak values.
13. The angular frequency $\omega = 2\pi f$ determines the number of cycles per second.
14. A full cycle of the wave spans $0$ to $2\pi$ radians.
15. The graphs show positive and negative half-cycles, indicating the direction of current or voltage.
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Peak and RMS Values

16. The peak value ($I_0$ or $E_0$) is the maximum instantaneous value of current or voltage in a cycle.
17. Root mean square (RMS) value represents the effective value of AC for power calculations.
18. RMS current is given by $I_{rms} = \frac{I_0}{\sqrt{2}}$, and RMS voltage is $E_{rms} = \frac{E_0}{\sqrt{2}}$.
19. RMS values are used because they equate AC power to the equivalent DC power.
20. RMS values simplify calculations in AC circuits involving power and resistance.
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Phase Relationship Between Voltage and Current in Circuit Elements

21. In a purely resistive circuit, the voltage and current are in phase.
22. The phase difference in a resistive circuit is $\phi = 0^\circ$.
23. In a purely inductive circuit, the current lags the voltage by $90^\circ$.
24. The phase difference in an inductive circuit is $\phi = -90^\circ$.
25. In a purely capacitive circuit, the current leads the voltage by $90^\circ$.
26. The phase difference in a capacitive circuit is $\phi = +90^\circ$.
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Resistor in an AC Circuit

27. In a resistive circuit, the voltage and current waveforms are identical.
28. Power is dissipated as heat in the resistor and is always positive.
29. The impedance of a resistor in an AC circuit is equal to its resistance.
30. The power factor of a resistive circuit is $\cos \phi = 1$, indicating no reactive power.
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Inductor in an AC Circuit

31. An inductor opposes changes in current due to self-inductance.
32. The current lags the voltage by $90^\circ$ in an inductive circuit.
33. The impedance of an inductor is $X_L = \omega L$, where $L$ is inductance.
34. Reactive power is stored temporarily in the magnetic field of the inductor.
35. The power factor in a purely inductive circuit is $\cos \phi = 0$.
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Capacitor in an AC Circuit

36. A capacitor opposes changes in voltage by storing charge.
37. The current leads the voltage by $90^\circ$ in a capacitive circuit.
38. The impedance of a capacitor is $X_C = \frac{1}{\omega C}$, where $C$ is capacitance.
39. Reactive power is stored temporarily in the electric field of the capacitor.
40. The power factor in a purely capacitive circuit is $\cos \phi = 0$.
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General Properties of AC Circuits

41. The total impedance in a circuit depends on the combination of resistive, inductive, and capacitive components.
42. Impedance is represented as a complex number $Z = R + jX$, where $X$ is the reactance.
43. Resonance occurs when the inductive reactance equals the capacitive reactance in a circuit.
44. Resonant circuits are used in radios and filters for frequency selection.
45. The phase angle in an AC circuit is calculated using $\tan \phi = \frac{X}{R}$.
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Applications of AC Concepts

46. AC power systems supply electricity to homes and industries due to ease of voltage transformation.
47. Alternators generate sinusoidal AC for power distribution.
48. Transformers step up or step down AC voltage for efficient transmission and distribution.
49. AC circuits with inductors and capacitors are used in tuning circuits and filters.
50. Understanding phase relationships helps in designing power factor correction systems.
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Waec Lesson notes on Reactance and impedance

Reactance and Impedance

1. Reactance is the opposition offered by inductors ($X_L$) and capacitors ($X_C$) to the flow of alternating current.
2. Impedance ($Z$) is the total opposition to AC flow, combining resistance ($R$) and reactance ($X$).
3. Impedance is represented as $Z = \sqrt{R^2 + X^2}$ in simple AC circuits.
4. Reactance is frequency-dependent, with inductive reactance increasing and capacitive reactance decreasing with frequency.
5. Impedance is a complex quantity represented as $Z = R + jX$, where $j$ denotes the imaginary unit.
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Inductive Reactance

6. Inductive reactance is given by $X_L = \omega L = 2\pi f L$, where $L$ is inductance and $f$ is frequency.
7. $X_L$ increases linearly with frequency, opposing high-frequency currents.
8. In an inductor, the current lags the voltage by $90^\circ$ due to $X_L$.
9. Inductive reactance contributes to impedance in circuits with inductors.
10. High inductive reactance at high frequencies is used in filters.
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Capacitive Reactance

11. Capacitive reactance is given by $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$, where $C$ is capacitance.
12. $X_C$ decreases as frequency increases, allowing high-frequency currents to pass.
13. In a capacitor, the current leads the voltage by $90^\circ$ due to $X_C$.
14. Capacitive reactance dominates impedance in circuits with capacitors at low frequencies.
15. Low capacitive reactance enables capacitors to function as high-pass filters.
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Simple Numerical Problems on Reactance

16. Calculate $X_L$ for an inductor with $L = 0.1H$ at $f = 50Hz$: $X_L = 2\pi \cdot 50 \cdot 0.1 = 31.42 \Omega$.
17. Calculate $X_C$ for a capacitor with $C = 10 \muF$ at $f = 60Hz$: $X_C = \frac{1}{2\pi \cdot 60 \cdot 10 \times 10^{-6}} = 265.26 \Omega$.
18. Determine impedance in a circuit with $R = 50 \Omega$ and $X_L = 30 \Omega$: $Z = \sqrt{R^2 + X_L^2} = \sqrt{50^2 + 30^2} = 58.31 \Omega$.
19. Find $Z$ for a circuit with $R = 40 \Omega$ and $X_C = 20 \Omega$: $Z = \sqrt{R^2 + X_C^2} = \sqrt{40^2 + 20^2} = 44.72 \Omega$.
20. Calculate the total reactance ($X$) of a series circuit with $X_L = 40 \Omega$ and $X_C = 20 \Omega$: $X = X_L - X_C = 40 - 20 = 20 \Omega$.
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Vector Diagrams

21. Vector diagrams represent the phase relationship between voltage and current in AC circuits.
22. In a purely resistive circuit, voltage and current vectors are in phase.
23. In an inductive circuit, the voltage vector leads the current vector by $90^\circ$.
24. In a capacitive circuit, the current vector leads the voltage vector by $90^\circ$.
25. Vector diagrams help visualize the impedance and phase angle in AC circuits.
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Resonance in AC Circuits

26. Resonance occurs when the inductive reactance equals the capacitive reactance: $X_L = X_C$.
27. At resonance, the circuit impedance is purely resistive: $Z = R$.
28. The resonant frequency is given by $f_r = \frac{1}{2\pi \sqrt{LC}}$, where $L$ is inductance and $C$ is capacitance.
29. At resonance, current is maximum for a given voltage.
30. Resonance is crucial in tuning circuits for radios and filters.
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Power in AC Circuits

31. Power in AC circuits is given by $P = VI \cos \phi$, where $\phi$ is the phase angle.
32. $\cos \phi$ is the power factor, indicating the fraction of power used effectively.
33. Real power ($P$) is measured in watts (W) and represents the usable power.
34. Reactive power ($Q$) is measured in volt-amperes reactive (VAR) and represents the energy alternately stored and released.
35. Apparent power ($S$) is measured in volt-amperes (VA) and combines real and reactive power: $S = \sqrt{P^2 + Q^2}$.
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Applications in Tuning of Radio and TV

36. Resonance in LC circuits allows tuning to specific radio frequencies.
37. Adjusting $L$ or $C$ changes the resonant frequency in tuning circuits.
38. Band-pass filters in radios use resonance to select desired signals.
39. TV receivers use LC circuits to filter out unwanted frequencies.
40. Tuning circuits ensure clear reception by minimizing interference.
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Additional Insights into Reactance and Impedance

41. High $X_L$ at low frequencies blocks DC components in signals.
42. Low $X_C$ at high frequencies allows signal transmission in communication systems.
43. Impedance matching minimizes signal loss in transmission lines.
44. Variable inductors and capacitors adjust reactance for dynamic tuning.
45. Reactive components are essential in designing filters and oscillators.
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Applications of Resonance

46. Resonant circuits are used in frequency modulation and demodulation in radios.
47. Resonance in inductors and capacitors improves power transfer efficiency.
48. Tuned amplifiers in TVs enhance signal clarity.
49. Resonance is utilized in wireless power transfer technologies.
50. Precise control of resonance improves the performance of electronic devices.

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