ALTERNATING CURRENT

Notes on Alternating Current

7.1 Introduction to Alternating Current (AC)

• Definition: Alternating current (AC) is an electric current that reverses direction periodically, unlike direct current (DC), which flows only in one direction.
• Sine Wave: AC voltage is commonly represented by a sine wave function, such as v(t) = v_m sin(ωt), where:
  • v_m is the peak voltage.
  • ω is the angular frequency.
• Importance of AC: Most electrical devices operate on AC due to its efficient transmission over long distances and ease of voltage transformation using transformers.

7.2 AC Voltage Applied to a Resistor

• Current through Resistor: When an AC voltage is applied across a resistor, the current also varies sinusoidally: [ i(t) = i_m ext{sin}(ωt) ] Where i_m = v_m/R, following Ohm’s Law (which holds for both AC and DC).
• In-Phase Relationship: The voltage and current in a pure resistor are in phase: their peaks, zeros, and minima occur simultaneously.
• Average Current: The average current over a complete cycle is zero, but the average power is not:
  • Instantaneous power calculation: [ p = i^2 R ] leads to average power being positive due to averaging ( i^2 ) (it’s non-negative).

7.3 Phasor Representation

• Phasors: Phasors are rotating vectors used to represent AC voltage and current, making it easier to analyze phase relationships and magnitudes in AC circuits.
• Voltage and Current Phasors: In resistive circuits, voltage and current phasors are aligned, illustrating their in-phase nature.

7.4 AC Voltage Applied to an Inductor

• Inductive Circuit Behavior: For a purely inductive circuit:
  • The current lags the voltage by 90 degrees (π/2).
  • Inductive reactance (X_L) is defined as X_L = ωL.
• Zero Average Power: The average power consumed in a purely inductive circuit over one cycle is zero, as the energy oscillates back and forth between the source and the inductor.

7.5 AC Voltage Applied to a Capacitor

• Capacitive Circuit Behavior: In a pure capacitive circuit:
  • The current leads the voltage by 90 degrees (π/2).
  • The capacitive reactance is defined as X_C = 1/(ωC).
• Average Power: Similar to inductors, the average power consumed over a complete cycle is zero.

7.6 Series LCR Circuit Analysis

• Impedance: Defined as Z = R^2 + (X_L - X_C)^2.
  • The phase angle θ between the voltage across the source and current in the circuit can be expressed as [ an(θ) = \frac{X_L - X_C}{R} ].
• Power Factor: The power factor is cos(θ) and defines how effectively the circuit converts electric energy into work.
• Resonance: A series RLC circuit reaches maximum current amplitude at the resonant frequency where X_L = X_C.

7.7 Transformers

• Basic Operation: A transformer consists of two coils (primary and secondary) wrapped around a magnetic core. By electromagnetic induction, voltage can be stepped up or down between the coils.
• Voltage and Current Relationships: The relationships between primary and secondary voltages and currents are given by:
  • [ \frac{V_s}{V_p} = \frac{N_s}{N_p} ]
  • [ \frac{I_s}{I_p} = \frac{N_p}{N_s} ] Where N_p and N_s are the number of turns in the primary and secondary coils, respectively.
• Real-World Applications: In power distribution, transformers step up voltages for transmission to reduce energy losses over long distances.

Summary Points

1. AC vs. DC: AC changes direction periodically, while DC flows in one direction.
2. Sine Wave Voltage: AC voltage follows a sine wave pattern, essential for practical applications.
3. RMS Values: The root mean square values are used to represent AC voltages and currents effectively.
4. Phasor Diagrams: Employed to visualize phase relationships in AC circuits.
5. Inductive Reactance: Defined as X_L = ωL, with current lagging behind the voltage by 90 degrees.
6. Capacitive Reactance: Defined as X_C = 1/(ωC), with current leading the voltage by 90 degrees.
7. Impedance in LCR Circuits: Z = R^2 + (X_L - X_C)^2 calculates the total opposition in AC circuits with resistive, inductive, and capacitive elements.
8. Power Factor: Measures the efficiency of the circuit (cos(θ)).
9. Resonance: Achieved when the inductive reactance equals the capacitive reactance, maximizing current flow.
10. Transformers: Used to change AC voltage levels efficiently for power distribution without violating energy conservation laws.

1. AC vs. DC: AC fluctuates and can change direction, while DC is constant. 2. Voltage Representation: AC voltage is often depicted by a sine wave function. 3. RMS Values: The root mean square (RMS) allows comparison of AC with DC. 4. Phasors: Phasor diagrams facilitate analysis of AC circuits' phase relationships. 5. Inductive Behavior: Current in an inductor lags voltage by π/2; average power over a cycle is zero. 6. Capacitive Behavior: Current in a capacitor leads voltage by π/2; average power over a cycle is zero. 7. Impedance Formula: Total impedance in an LCR circuit is represented by: Z = √(R² + (X_L - X_C)²). 8. Power Factor: Cosine of the phase angle indicates how effectively the circuit uses electric power. 9. Resonance in RLC Circuits: Occurs when inductive and capacitive reactances are equal, maximizing current. 10. Transformers: Critical for voltage adjustments in power distribution, allowing efficient transmission.