Study Notes for Alternating Current Concepts

Alternating Current (AC) is defined as the electric current that reverses its direction periodically, alternating the flow of charges back and forth in a circuit.
The symbol used in circuits to represent AC is:

The most common waveform of AC is a sinusoidal waveform. Other forms include:
- Saw-tooth wave
- Square wave

Unit: Ampere (A)
AC can be represented with positive and negative values indicating the flow direction in a circuit:
- Positive sign: Current flows in a clockwise direction.
- Negative sign: Current flows in an anticlockwise direction.

When AC flows through a resistor, it results in an alternating potential difference (voltage) across it, characterized by sinusoidal alternating voltage.
Unit: Volt (V)

The equations for sinusoidal voltage and current can be expressed as:
  - For current:
    - At time t: $I(t) = I_0 imes ext{sin}( ext{ω}t)$ where:
      - $I_0$ : Maximum current or peak current
For voltage:
- At time t: $V(t) = V_0 imes ext{sin}( ext{ω}t)$ where:
- $V_0$ : Maximum voltage or peak voltage
Other variables include:
  - $ext{ω}$ : Angular frequency
  - $f$ : Frequency of AC voltage and current
  - $T$ : Period of one cycle
  - Phase angle: expressed in degrees (°) or radians (rad)

Root Mean Square (RMS) Current (I_rms): This is defined as the value of the steady direct current (DC) which produces the same amount of power dissipation in a resistor as the mean (average) power produced by the alternating current (AC).
Formula for RMS Current:
I_ ext{rms} = rac{I_0}{ ext{√2}}
Root Mean Square Voltage (V_rms): Similarly defined as the value of the steady direct voltage (DC) that produces the same power dissipation in a resistor as the mean (average) power produced by the AC.
Formula for RMS Voltage:
V_ ext{rms} = rac{V_0}{ ext{√2}}

The root mean square (RMS) values represent the effective value of AC. The output voltage or current can be illustrated graphically to show that bulbs powered with AC will light with the same brightness as with a corresponding DC.
For example, considered at max values:
- $I_rms$ = 0.707 $I_0$
- $V_rms$ = 0.707 $V_0$

A phasor diagram contains phasors, defined as vectors that rotate counterclockwise with a constant angular frequency, ω. They are used to represent sinusoidally varying voltages and currents and to determine the phase angle difference.
The length of a phasor corresponds to its maximum value (either voltage or current).

Pure Resistor (R)
   - When connected to a sinusoidal voltage supply, both voltage (V) and current (I) reach their maximum and minimum values at the same time. They are in phase, maintaining steady relationships as described by Ohm’s Law: $V = I imes R$
   - Instantaneous power delivered to the resistor can be calculated using:
     - $P_ ext{inst} = IV$
     - Average power dissipation in the resistor:
     - $P_ ext{avg} = I_ ext{rms}^2 imes R$
Pure Capacitor (C)
   - Connected to a sinusoidal voltage, the relationship between voltage and current is such that the current leads the voltage by 90° (or $rac{ ext{π}}{2}$ radians).
   - The voltage across the capacitor can be expressed as:
   - $V_c = -V_0 imes ext{cos}(ωt)$
   - The capacitive reactance (X_C) of a capacitor is given by X_C = rac{1}{2πfC}, where f is frequency.
   - For power dissipation in a pure capacitor:
     - Average power, $P_{avg} = 0$ (no energy is dissipated).
Pure Inductor (L)
   - In this setup, the voltage leads the current by 90° (or $rac{ ext{π}}{2}$ radians).
   - The instantaneous voltage across the inductor is expressed as:
   - $V_L = V_0 imes ext{cos}(ωt)$
   - The inductive reactance (X_L) is defined as $X_L = 2πfL$ .
   - Average power in a pure inductor is also given as:
     - $P_{avg} = 0$ (similarly, no energy is dissipated).

Impedance (Z) is the total opposition to the flow of current in an AC circuit and is given by:
- For series circuits:
- $Z = ext{√}(R^2 + (X_L - X_C)^2)$
Understanding how impedance varies with frequency explains resonance phenomena:
- At low frequencies, the capacitive reactance is high, while at high frequencies, the inductive reactance is high. Z achieves its minimum when $X_L = X_C$ , which defines
resonance frequency.

Only resistors dissipate power in an AC circuit. The average power dissipated (real power) in an RC, RL, or RLC series circuit is given by:
- $P_{avg} = I_{rms} imes V_{rms} imes ext{cos} φ$ ,
where φ is the phase difference between current and voltage.

The power factor is a dimensionless number that quantifies the efficiency of the circuit, primarily defined as:
- $ext{power factor} = ext{cos} φ$
where φ is the phase angle between the RMS current and voltage. The power dissipated in the system is related to the real power and is calculated as (real power)/(apparent power).

Solve for RMS values, reactance, impedance, power factor, and phase angles as demonstrated through detailed calculations provided in this guide (see examples 1-8 for detailed solutions).