Simple Alternating Current Circuits | Jamb(UTME)

Jamb(utme) key points on explanation of a.c. current and voltage; peak and r.m.s. values; a.c. source connected to a resistor

Explanation of A.C. Current and Voltage

1. Alternating Current (A.C.) is a type of electrical current that periodically changes direction.
2. The voltage in an A.C. circuit also alternates between positive and negative values.
3. A.C. signals are typically represented as sinusoidal waves.
4. The time it takes to complete one cycle of A.C. is called the period $(T)$.
5. The number of cycles per second is the frequency $(f)$, measured in Hertz (Hz).
6. The relationship between period and frequency is:
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   $f = \frac{1}{T}$
7. A.C. is commonly used in households and industries due to efficient power transmission.
8. The voltage and current in A.C. circuits vary with time:
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   $v(t) = V_{peak} {sin}{\omega} t$ where:
   • $V_{peak}$: Peak voltage,
   • $\omega$: Angular frequency $(\omega = 2\pi f)$.
9. Angular frequency $(\omega)$ determines how quickly the current alternates.
10. The shape of an A.C. wave can also be triangular, square, or sawtooth, but sinusoidal waves are most common.
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Peak and RMS Values

11. Peak Value: The maximum voltage or current reached during a cycle $V_{peak}$ or $I_{peak}$.
12. Root Mean Square (RMS) Value: The effective value of A.C. voltage or current.
13. RMS value represents the equivalent direct current (D.C.) value that produces the same power.
14. The relationship between RMS and peak values is:
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    $V_{rms} = \frac{V_{peak}}{\sqrt{2}}$ and $I_{rms} = \frac{I_{peak}}{\sqrt{2}}$
15. The RMS value simplifies power calculations in A.C. circuits.
16. In standard household circuits, the RMS voltage is often 230V or 120V, depending on the region.
17. The peak voltage is higher than the RMS voltage:
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    $V_{peak} = V_{rms} \cdot \sqrt{2}$
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18. For example, if $V_{rms} = 230{V}$, then $V_{peak} \approx 325{V}$.
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A.C. Source Connected to a Resistor

19. When an A.C. source is connected to a resistor, the current flows through the resistor in alternating directions.
20. The resistor opposes the flow of current with a constant resistance $R$.
21. The voltage across the resistor follows Ohm’s law:
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    $v(t) = i(t) \cdot R$
22. In a purely resistive circuit, the current and voltage are in phase.
23. Being "in phase" means the peaks and zeros of current and voltage occur at the same time.
24. The instantaneous power delivered to the resistor is:
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    $P(t) = v(t) \cdot i(t)$
25. The average power consumed by the resistor is:
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    $P_{avg} = I_{rms}^2 \cdot R$
26. The resistor dissipates energy as heat.
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Key Characteristics of A.C. Circuits with a Resistor

27. The impedance of a purely resistive circuit is equal to the resistance $(Z = R)$.
28. The current in the circuit is directly proportional to the applied voltage.
29. Increasing the resistance decreases the current for the same voltage.
30. Decreasing the resistance increases the current for the same voltage.
31. The power factor in a resistive circuit is 1 (unity), meaning all supplied power is consumed as heat.
32. Resistive A.C. circuits are the simplest type of A.C. circuit.
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Applications of A.C. Circuits with Resistors

33. They are used in heating devices like electric heaters and toasters.
34. Incandescent light bulbs use resistive circuits to convert electrical energy into light and heat.
35. Resistors in A.C. circuits are also used to limit current in specific applications.
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Additional Concepts

36. A.C. circuits with resistors are useful for understanding basic circuit behavior before adding components like inductors or capacitors.
37. The concept of RMS values helps simplify power calculations in these circuits.
38. The sinusoidal nature of A.C. signals ensures continuous energy transfer.
39. Electrical grids use A.C. for efficient long-distance power transmission.
40. The ability of A.C. to alternate ensures compatibility with transformers for voltage step-up or step-down.
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Advantages of A.C. Over D.C.

41. A.C. voltage can be easily transformed to higher or lower values using transformers.
42. A.C. is more efficient for transmitting power over long distances.
43. Generating A.C. power is more cost-effective in power plants.
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Practical Calculations

44. Knowing RMS values helps calculate the safe operating limits of electrical devices.
45. Power consumption of devices connected to an A.C. source is usually specified in RMS terms.
46. Engineers design A.C. circuits considering both peak and RMS values to ensure reliability.
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Safety Considerations

47. The peak voltage in A.C. circuits is higher than the RMS value, posing a greater risk of shock.
48. Electrical insulation in A.C. circuits is designed to handle the peak voltage.
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Summary

49. A.C. voltage and current alternate direction periodically, following sinusoidal patterns in most cases.
50. Understanding peak and RMS values and the behavior of resistive A.C. circuits is fundamental to analyzing and designing electrical systems.
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Jamb(utme) key points on a.c source connected to a capacitor- (capacitive reactance); ac source connected to an inductor (inductive reactance); R-L-C circuits

A.C. Source Connected to a Capacitor: Capacitive Reactance

1. When an A.C. source is connected to a capacitor, the capacitor charges and discharges as the voltage alternates.
2. Current in the circuit leads the voltage by $90^\circ$ (a quarter cycle).
3. The opposition offered by the capacitor to the flow of A.C. is called capacitive reactance.
4. Capacitive reactance $(X_C)$ depends on the frequency $(f)$ of the A.C. source and the capacitance $(C)$ of the capacitor.
5. The formula for capacitive reactance is:
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   $X_C = \frac{1}{2\pi f C}$
6. $X_C$ is measured in ohms $(\Omega)$.
7. A higher frequency results in a lower capacitive reactance.
8. A larger capacitance also reduces capacitive reactance.
9. In a purely capacitive circuit, the current is maximum when $X_C$ is minimum.
10. Capacitors do not dissipate energy; they store it in the form of an electric field.
11. The average power consumed in a purely capacitive circuit is zero because energy is alternately stored and returned to the circuit.
12. Capacitors allow high-frequency signals to pass but block low-frequency signals.
13. This property is used in high-pass filters to eliminate low-frequency noise.
14. Capacitors in A.C. circuits are used for tuning and frequency selection.
15. They are essential in radio, television, and communication circuits.
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A.C. Source Connected to an Inductor: Inductive Reactance

16. When an A.C. source is connected to an inductor, a changing current induces an opposing voltage (back emf).
17. The opposition offered by the inductor to A.C. is called inductive reactance.
18. Inductive reactance $(X_L)$ depends on the frequency $(f)$ of the A.C. source and the inductance $(L)$ of the inductor.
19. The formula for inductive reactance is:
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    $X_L = 2\pi f L$
20. $X_L$ is measured in ohms $(\Omega)$.
21. A higher frequency increases inductive reactance.
22. A larger inductance also increases inductive reactance.
23. In a purely inductive circuit, voltage leads the current by $90^\circ$ (a quarter cycle).
24. Inductors store energy in the form of a magnetic field.
25. The average power consumed in a purely inductive circuit is zero because energy is alternately stored and returned to the circuit.
26. Inductors allow low-frequency signals to pass but block high-frequency signals.
27. This property is used in low-pass filters to eliminate high-frequency noise.
28. Inductors in A.C. circuits are used for current smoothing and impedance matching.
29. They are commonly found in transformers, motors, and electromagnetic relays.
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R-L-C Circuits in Physics

Series R-L-C Circuit

30. A series R-L-C circuit has a resistor, inductor, and capacitor connected in series with an A.C. source.
31. The total opposition to current flow is called impedance $(Z)$.
32. Impedance is the combination of resistance $(R)$, inductive reactance $(X_L)$, and capacitive reactance $(X_C)$.
33. The impedance is given by:
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    $Z = \sqrt{R^2 + (X_L - X_C)^2}$
34. The current in the circuit depends on the applied voltage and the impedance:
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    $I = \frac{V}{Z}$
35. The phase angle (( \phi )) between the voltage and current is:
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    $\phi = \tan^{-1}\left(\frac{X_L - X_C}{R}\right)$
36. If $X_L > X_C$, the circuit is inductive, and the voltage leads the current.
37. If $X_C > X_L$, the circuit is capacitive, and the current leads the voltage.
38. At resonance, $X_L = X_C$, and the impedance is minimum $Z = R$.
39. The resonance frequency is given by:
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    $f_0 = \frac{1}{2\pi \sqrt{LC}}$
40. At resonance, the circuit allows maximum current to flow.
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Parallel R-L-C Circuit

41. In a parallel R-L-C circuit, the resistor, inductor, and capacitor are connected in parallel to an A.C. source.
42. The total current is the vector sum of the currents through each component.
43. The total admittance $(Y)$ of the circuit is given by:
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    $Y = \sqrt{G^2 + (B_C - B_L)^2}$ where:
    • $G = \frac{1}{R}$: Conductance,
    • $B_C = \frac{1}{X_C}$: Capacitive susceptance,
    • $B_L = \frac{1}{X_L}$: Inductive susceptance.
44. At resonance, the admittance is maximum, and the circuit draws the minimum current from the source.
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Applications of R-L-C Circuits

45. Tuning Circuits: R-L-C circuits are used in radios and televisions to select specific frequencies.
46. Oscillators: R-L-C circuits help generate stable oscillating signals.
47. Filters: They are used in high-pass, low-pass, and band-pass filters for signal processing.
48. Resonant Circuits: R-L-C circuits are crucial for resonance-based applications in communication and electronics.
49. Power Systems: Used in power factor correction and harmonic filtering.
50. Medical Devices: R-L-C circuits are part of diagnostic tools like MRI machines.
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Jamb(utme) key points on vector diagram, phase angle and power factor; resistance and impedance; effective voltage in an RLC circuits; resonance and resonance frequency

Vector Diagram, Phase Angle, and Power Factor

1. A vector diagram represents the relationship between voltage and current in A.C. circuits.
2. In resistive circuits, voltage and current are in phase, so their vectors align.
3. In inductive circuits, voltage leads current by $90^\circ$.
4. In capacitive circuits, current leads voltage by $90^\circ$.
5. For R-L-C circuits, the vectors of resistance $(R)$, inductive reactance $(X_L)$, and capacitive reactance $(X_C)$ combine to form the impedance vector.
6. The angle between the voltage and current vectors is called the phase angle $(\phi)$.
7. The power factor is the cosine of the phase angle:
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   $Power Factor = \cos\phi$
8. A power factor of 1 (or 100%) means all power is being effectively used (purely resistive circuit).
9. A power factor less than 1 indicates some energy is stored in reactive components (inductors or capacitors).
10. Inductive circuits have a lagging power factor because voltage leads current.
11. Capacitive circuits have a leading power factor because current leads voltage.
12. The power factor improves energy efficiency in electrical systems.
13. Poor power factors lead to higher energy losses in transmission systems.
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Resistance and Impedance

14. Resistance $(R)$ is the opposition to current flow in a purely resistive circuit.
15. Resistance is independent of frequency and dissipates energy as heat.
16. Impedance $(Z)$ is the total opposition to current flow in an A.C. circuit.
17. Impedance combines resistance $(R)$, inductive reactance $(X_L)$, and capacitive reactance $(X_C)$.
18. The formula for impedance is:
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    $Z = \sqrt{R^2 + (X_L - X_C)^2}$
19. Impedance depends on frequency, resistance, and reactances of the circuit components.
20. In a purely resistive circuit, $Z = R$, as there is no reactance.
21. In purely inductive or capacitive circuits, $Z = X_L$ or $Z = X_C$, as there is no resistance.
22. The units of both resistance and impedance are ohms $(\Omega)$.
23. Impedance determines how much current flows in an A.C. circuit for a given voltage.
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Effective Voltage in R-L-C Circuits

24. The effective or RMS voltage is the root-mean-square value of the alternating voltage.
25. RMS voltage is calculated as:
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    $V_{rms} = \frac{V_peak}{\sqrt{2}}$
26. RMS voltage is used to measure the equivalent D.C. voltage that produces the same power.
27. In an R-L-C circuit, the total voltage $(V)$ is the vector sum of voltage drops across $R$, $L$, and $C$.
28. The relationship between the total voltage and the components is:
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    $V = \sqrt{(V_R)^2 + (V_L - V_C)^2}$
29. The voltage across the resistor is in phase with the current.
30. The voltage across the inductor leads the current by $90^\circ$.
31. The voltage across the capacitor lags the current by $90^\circ$.
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Resonance and Resonance Frequency

32. Resonance occurs in an R-L-C circuit when the inductive reactance $(X_L)$ equals the capacitive reactance $(X_C)$.
33. At resonance, the total reactance is zero, so impedance is purely resistive:
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    $Z = R$
34. The circuit draws maximum current at resonance for a given voltage.
35. The phase angle at resonance is zero, so voltage and current are in phase.
36. The resonance frequency $f_0$ is determined by the inductance $L$ and capacitance $(C)$:
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    $f_0 = \frac{1}{2\pi \sqrt{LC}}$
37. Resonance frequency depends only on $L$ and $C$, not on $R$.
38. In a series R-L-C circuit, resonance maximizes current.
39. In a parallel R-L-C circuit, resonance minimizes current drawn from the source.
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Applications of Resonance

40. Resonance is used in radio tuning circuits to select specific frequencies.
41. Television receivers use resonance to tune into specific channels.
42. Inductive heating systems operate efficiently at resonance frequencies.
43. In power systems, resonance can cause high voltages, damaging equipment if not controlled.
44. Band-pass filters allow signals near the resonance frequency to pass while blocking others.
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Additional Key Points

45. At frequencies below resonance, the circuit behaves as a capacitive circuit.
46. At frequencies above resonance, the circuit behaves as an inductive circuit.
47. The sharpness of resonance depends on the circuit’s quality factor $(Q)$.
48. The quality factor is given by:
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    $Q = \frac{1}{R} \sqrt{\frac{L}{C}}$
49. Higher $Q$ values mean sharper and more defined resonance peaks.
50. Controlling resonance ensures stable and efficient operation in electrical and communication systems.
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