Alternating Current Fundamentals, Terminology, and Phase Relationships

Fundamentals of Direct Current (DC) and Alternating Current (AC)

• Role of Electricity in Daily Life:

  • Electricity is deeply integrated into every sector of human activity.

  • Modern daily functioning without electrical devices would be extraordinarily difficult.

• Direct Current (DC):

  • Definition: Electric current supplied by DC power sources such as batteries, where electric charges flow in a single direction (unidirectional flow).

  • Directional Flow:

  • Conventional Current: Travels from the positive terminal to the negative terminal of a battery throughout the circuit.

  • Electronic Current: Travels from the negative terminal to the positive terminal.

  • Regardless of the convention used, the motion of charges remains strictly unidirectional.

  • Waveform and Magnitude Characteristics:

  • Direct current flows with a constant magnitude in one direction over time, yielding a flat horizontal line on a current-versus-time graph.

  • Typical Example:

  • A simple light bulb connected via wires across the positive and negative terminals of a battery.

• Alternating Current (AC):

  • Definition: Electric current supplied via wall sockets in homes (e.g., K-Electric grid supply in Pakistan), where electric charges continuously oscillate back and forth rather than flowing in a fixed direction.

  • Directional Alternation:

  • For two points $A$ and $B$ in a circuit, current flows from point $A$ to point $B$, then reverses from point $B$ to point $A$, continuously alternating between positive and negative directions.

  • Frequency of Oscillation:

  • Reverses direction roughly 50 to 60 times in one second ($50\,\text{Hz}$ to $60\,\text{Hz}$).

  • The standard AC electrical frequency used in Pakistan is $50\,\text{Hz}$.

  • Generation Source:

  • Alternating current is generated using an Alternating Current Generator (AC Generator).

Sinusoidal Pattern and Mathematical Equations of Alternating Current

• Sinusoidal Pattern Characteristics:

  • Alternating current varies according to a regular, smooth, periodic sinusoidal pattern.

  • The continuous oscillation relies on the trigonometric sine function.

  • The current smoothly increases from zero to a positive maximum value, decreases back to zero, reverses direction to reach a negative maximum value, and returns to zero.

• Trigonometric Sine Values Across Key Angles:

  • At angle $\theta = 0^\circ$: $\sin(0^\circ) = 0$

  • At angle $\theta = 90^\circ$: $\sin(90^\circ) = 1$ (positive maximum peak)

  • At angle $\theta = 180^\circ$: $\sin(180^\circ) = 0$

  • At angle $\theta = 270^\circ$: $\sin(270^\circ) = -1$ (negative maximum peak)

  • At angle $\theta = 360^\circ$: $\sin(360^\circ) = 0$ (completes one full sinusoidal wave cycle)

• Mathematical Equations for AC Quantities:

  • Instantaneous Alternating Current Equation:

  • $I = I_0 \sin(\omega t)$

  • $I$ represents the instantaneous current at time $t$.

  • $I_0$ represents the peak (maximum) current amplitude.

  • $\omega$ represents the angular frequency in radians per second.

  • $t$ represents time in seconds.

  • Instantaneous Alternating Voltage Equation:

  • $V = V_0 \sin(\omega t)$

  • $V$ represents the instantaneous voltage at time $t$.

  • $V_0$ represents the peak (maximum) voltage amplitude.

• Derivation and Relationship for Angular Frequency ($\omega$):

  • Relation between time period $T$ and angular frequency $\omega$: $T = \frac{2\pi}{\omega}$

  • Relation between time period $T$ and cyclic frequency $f$: $T = \frac{1}{f}$

  • Equating both expressions:

  • $\frac{1}{f} = \frac{2\pi}{\omega}$

  • $\omega = 2\pi f$

Key AC Terminology: Cycle, Time Period, Frequency, and Peak Value

• Cycle:

  • Definition: A single complete set of positive and negative values of any alternating quantity (voltage or current), forming one full wave with one crest and one trough.

  • Composition:

  • Positive Half-Cycle ($0.5$ cycle): Quantity rises from $0$ to maximum positive peak and returns to $0$.

  • Negative Half-Cycle ($0.5$ cycle): Quantity alternates direction, rises from $0$ to negative maximum peak and returns to $0$.

  • Combined Positive and Negative Half-Cycles make $1$ complete cycle.

• Time Period ($T$):

  • Definition: The total time required to complete one full cycle (one complete wave) of alternating current or voltage.

  • Symbol: $T$

  • Practical Example:

  • If a waveform displays $2$ complete wave cycles occurring within a span of $10\,\text{ms}$, the time period for one cycle is:

  • $T = \frac{10\,\text{ms}}{2} = 5\,\text{ms}$

• Frequency ($f$):

  • Definition: The number of complete cycles or waves produced per second, measuring how many times the current reverses direction each second.

  • Unit: Hertz ($\text{Hz}$), where $1\,\text{Hz} = 1\,\text{cycle/s}$.

  • Example:

  • Grid electricity supplied by K-Electric in Pakistan operates at $50\,\text{Hz}$, meaning the current undergoes $50$ complete cycles and reverses direction $50$ times per second.

• Instantaneous Peak Value ($I_{\text{peak}}$ / $V_{\text{peak}}$):

  • Definition: The maximum amplitude (highest positive or negative value) achieved by an alternating current or voltage during any single cycle.

  • Symbol Notation: $I_{\text{peak}}$ or $I_0$ for peak current; $V_{\text{peak}}$ or $V_0$ for peak voltage.

Root Mean Square (RMS) Value and Power Equivalence

• Definition of RMS Value:

  • The Root Mean Square (RMS) value is the equivalent Direct Current (DC) voltage or current that delivers the exact same electrical power or energy as an alternating quantity.

• Mathematical Relationship for Sinusoidal AC:

  • The RMS value of a sinusoidal waveform is approximately $70\%$ ($0.707$ times) of its peak value ($V_{\text{peak}}$ or $I_{\text{peak}}$).

  • Formula for Voltage: $V_{\text{rms}} = 0.707 \times V_{\text{peak}}$

  • Formula for Current: $I_{\text{rms}} = 0.707 \times I_{\text{peak}}$

• Power Equivalence Examples:

  • Comparison of $12\,\text{V}$ AC and $12\,\text{V}$ DC:

  • A peak AC voltage of $12\,\text{V}$ does not provide the same electrical power as $12\,\text{V}$ DC.

  • The equivalent DC voltage providing equal power to $12\,\text{V}$ peak AC is calculated as $70\%$ of $12\,\text{V}$:

  • $V_{\text{DC, equiv}} = 0.707 \times 12\,\text{V} = 8.484\,\text{V}$

  • Household $220\,\text{V}$ Peak AC Voltage:

  • For an AC supply with a peak voltage of $220\,\text{V}$, the equivalent DC voltage supplying identical power is:

  • $V_{\text{DC, equiv}} = 0.707 \times 220\,\text{V} = 155.54\,\text{V}$

Phase Relationships in Alternating Quantities

• Overview of Phase Relationships:

  • Describes the timing alignment between two alternating quantities (currents or voltages) operating at equal frequencies.

  • Three distinct phase relationship conditions exist: In-Phase, Phase Lag, and Phase Lead.

• In-Phase (Same Phase):

  • Condition: Two alternating waveforms of identical frequency reach their zero values at the exact same time and attain their peak values at the exact same time.

  • Phase Difference: $\Delta\phi = 0^\circ$ (or $0\,\text{rad}$).

  • Waveform Behavior:

  • Both waveforms hit zero at angles $0^\circ, 180^\circ, 360^\circ, 540^\circ, 720^\circ$.

  • Both waveforms achieve positive maximum peaks at $90^\circ$ and negative maximum peaks at $270^\circ$.

  • Crest aligns with crest, and trough aligns with trough.

• Phase Lag and Phase Lead Definitions:

  • Phase Lag: Occurs when a waveform attains its peak or zero value after a reference waveform.

  • Phase Lead: Occurs when a waveform attains its peak or zero value before a reference waveform.

  • Identification Rule:

  • The waveform that hits its maximum peak earlier in time is leading.

  • The waveform that hits its maximum peak later in time is lagging.

• Phase Difference Examples:

  • Example 1: $90^\circ$ ($\frac{\pi}{2}$) Phase Difference:

  • Waveform 1 attains its maximum peak at $0^\circ$.

  • Waveform 2 attains its maximum peak at $90^\circ$.

  • Phase Difference: $\Delta\phi = 90^\circ - 0^\circ = 90^\circ = \frac{\pi}{2}\,\text{rad}$.

  • Waveform 1 leads Waveform 2 by $90^\circ$ ($\frac{\pi}{2}\,\text{rad}$); Waveform 2 lags Waveform 1 by $90^\circ$ ($\frac{\pi}{2}\,\text{rad}$).

  • Example 2: $180^\circ$ ($\pi$) Phase Difference:

  • Waveform 1 attains its first positive peak at $90^\circ$.

  • Waveform 2 attains its first positive peak at $270^\circ$.

  • Phase Difference: $\Delta\phi = 270^\circ - 90^\circ = 180^\circ = \pi\,\text{rad}$.

  • Waveform 1 leads Waveform 2 by $180^\circ$ ($\pi\,\text{rad}$); Waveform 2 lags Waveform 1 by $180^\circ$ ($\pi\,\text{rad}$).

Vector Representation of AC via Phasor Diagrams

• Definition of Phasor Diagram:

  • A phasor diagram is a vector representation of alternating currents or voltages operating at the same frequency, illustrating their peak amplitudes and phase relationships.

• Construction Rules for Phasor Diagrams:

  • Vector Origin: Vectors originate from a common origin point.

  • Vector Length: The length of each vector is proportional to the peak amplitude ($V_0$ or $I_0$) of the alternating quantity.

  • Vector Angle: Measured counter-clockwise with respect to the positive x-axis to indicate phase angle.

• Graphical Phasor Configurations:

  • In-Phase Configuration:

  • Both vectors lie along the exact same directional line from the origin.

  • The angle between the two vectors is $0^\circ$ ($\Delta\phi = 0$).

  • $90^\circ$ ($\frac{\pi}{2}$) Phase Lag/Lead Configuration:

  • The leading vector points counter-clockwise ahead of the lagging vector by a perpendicular angle of $90^\circ$ ($\frac{\pi}{2}\,\text{rad}$).

  • $180^\circ$ ($\pi$) Phase Lag/Lead Configuration:

  • The leading and lagging vectors point in directly opposite directions along a straight line ($180^\circ$ or $\pi\,\text{rad}$ apart).