Alternating Current Fundamentals

While Direct Current (DC) remains constant over time, Alternating Current (AC) reverses its direction periodically. The most common and mathematically pure form of AC is the sinusoidal waveform, which is produced naturally by rotating AC generators (alternators) in power plants.

Sinusoidal Waveforms

A sinusoidal voltage waveform can be mathematically described as a function of time ($t$):

$v(t) = V_m \sin(\omega t + \phi)$

where:

• $v(t)$ = instantaneous voltage at time $t$ [Volts, V]
• $V_m$ = amplitude (peak voltage) [Volts, V]
• $\omega$ = angular frequency [radians per second, rad/s]
• $t$ = time [seconds, s]
• $\phi$ = phase angle [radians or degrees]

A similar equation applies to alternating current, $i(t) = I_m \sin(\omega t + \theta)$.

Frequency and Period

The signal repeats itself over a specific time interval called the period ($T$). The frequency ($f$) is the number of cycles completed in one second.

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

where:

• $T$ = period [seconds, s]
• $f$ = frequency [Hertz, Hz]

Standard power grid frequencies are typically 50 Hz (most of the world) or 60 Hz (North America).

Root Mean Square (RMS) Value

Because an AC signal is constantly changing and its average over a full cycle is zero, we cannot use the average value to calculate useful power. Instead, we use the Root Mean Square (RMS) value, which is the equivalent DC value that would deliver the same average power to a resistor.

For a pure sine wave, the relationship between the peak amplitude ($V_m$) and the RMS value ($V_{rms}$) is:

$V_{rms} = \frac{V_m}{\sqrt{2}} \approx 0.707 V_m$

Worked Example

Problem: A household electrical outlet provides an AC voltage specified as $220 \text{ V}_{rms}$ at $50 \text{ Hz}$. Write the time-domain equation for this voltage, assuming a phase angle of $0^\circ$.

Solution:

1. Calculate the peak amplitude $V_m$: $V_m = V_{rms} \times \sqrt{2} = 220 \times 1.414 \approx 311.1 \text{ V}$
2. Calculate the angular frequency $\omega$: $\omega = 2 \pi f = 2 \pi (50) = 100\pi \approx 314.16 \text{ rad/s}$
3. Construct the time-domain equation: $v(t) = 311.1 \sin(314.16 t) \text{ V}$

Engineering Check

When consumer electronics or power supplies are rated for "220 V" or "120 V", these are almost always RMS values, not peak values. A 220 V AC system actually hits peaks over 311 Volts! Engineers must design insulation and components to withstand the peak voltage, not just the RMS voltage.\n