Chapter 16 – Alternating Current
Introduction to Alternating Current (AC)
Alternating Current (AC) is a type of electrical current in which the direction of the current flow reverses periodically. Unlike Direct Current (DC), where the flow of charge is unidirectional, AC constantly changes direction, typically in a sinusoidal waveform.
Characteristics of AC
- Frequency (f): The number of cycles the current completes per second, measured in Hertz (Hz). The standard frequency for AC in Pakistan is 50 Hz.
- Amplitude (I₀ or V₀): The maximum value of current or voltage in an AC waveform.
- RMS Value: The Root Mean Square (RMS) value is a measure of the effective value of AC, equivalent to a DC value that would produce the same power. For current ( I ) and voltage ( V ):
[ I_{\text{RMS}} = \frac{I_0}{\sqrt{2}}, \quad V_{\text{RMS}} = \frac{V_0}{\sqrt{2}} ] - Angular Frequency (ω): The rate at which the current oscillates, given by:
[ \omega = 2\pi f ]
AC Circuit Components
- Resistors in AC Circuits:
- The voltage and current in a purely resistive AC circuit are in phase, meaning they reach their maximum and minimum values simultaneously.
- The power dissipated in the resistor is given by:
[ P = I_{\text{RMS}}^2 R ]
- Inductors in AC Circuits:
- In a purely inductive circuit, the current lags the voltage by 90 degrees (π/2 radians).
- The inductive reactance ((X_L)) opposes the current flow and is given by:
[ X_L = \omega L = 2\pi f L ]
- Capacitors in AC Circuits:
- In a purely capacitive circuit, the current leads the voltage by 90 degrees (π/2 radians).
- The capacitive reactance ((X_C)) opposes the current flow and is given by:
[ X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C} ]
Impedance and Phase Angle
- Impedance (Z): The total opposition to AC in a circuit with resistance, inductance, and capacitance. It is given by:
[ Z = \sqrt{R^2 + (X_L – X_C)^2} ] - Phase Angle (ϕ): The angle of lag or lead between the voltage and current in the circuit, given by:
[ \tan(\phi) = \frac{X_L – X_C}{R} ]
Power in AC Circuits
- Instantaneous Power (p): The power at any instant, given by:
[ p(t) = V(t) \cdot I(t) ] - Average Power (P): The average power delivered to a circuit, given by:
[ P = V_{\text{RMS}} \cdot I_{\text{RMS}} \cdot \cos(\phi) ]
where ( \cos(\phi) ) is the power factor, indicating how much of the power is actually used.
Resonance in AC Circuits
Resonance occurs in a series RLC circuit when the inductive reactance (X_L) equals the capacitive reactance (X_C). At resonance, the impedance is purely resistive, and the circuit allows maximum current to flow. The resonant frequency is given by:
[ f_0 = \frac{1}{2\pi\sqrt{LC}} ]
Transformers
Transformers are devices used to increase or decrease the voltage of AC. They operate on the principle of electromagnetic induction and are essential for transmitting electrical energy over long distances. The ratio of the voltages in the primary and secondary coils of a transformer is given by:
[ \frac{V_s}{V_p} = \frac{N_s}{N_p} ]
where ( V_s ) and ( V_p ) are the secondary and primary voltages, and ( N_s ) and ( N_p ) are the number of turns in the secondary and primary coils.
Applications of AC
- Power Distribution: AC is the standard for power distribution because it is easily transformed to different voltages, reducing energy losses over long distances.
- Household Appliances: Most household appliances operate on AC, including lights, fans, and refrigerators.
- Motors and Generators: AC motors and generators are widely used in industrial and domestic applications due to their efficiency and durability.
Summary
The study of alternating current is fundamental to understanding how electricity is generated, transmitted, and utilized in modern society. Key concepts such as impedance, resonance, and the behavior of AC in different circuit components are crucial for designing and analyzing electrical systems.
Important Formulas
- RMS Values: ( I_{\text{RMS}} = \frac{I_0}{\sqrt{2}}, \quad V_{\text{RMS}} = \frac{V_0}{\sqrt{2}} )
- Inductive Reactance: ( X_L = \omega L )
- Capacitive Reactance: ( X_C = \frac{1}{\omega C} )
- Impedance: ( Z = \sqrt{R^2 + (X_L – X_C)^2} )
- Average Power: ( P = V_{\text{RMS}} \cdot I_{\text{RMS}} \cdot \cos(\phi) )
- Resonant Frequency: ( f_0 = \frac{1}{2\pi\sqrt{LC}} )
- Transformer Voltage Ratio: ( \frac{V_s}{V_p} = \frac{N_s}{N_p} )
This chapter provides a comprehensive understanding of alternating current, which is essential for both theoretical knowledge and practical applications in the field of electrical engineering.