use equations of the form x = x0 sin ωt representing a sinusoidally alternating current or voltage

⚡️ Characteristics of Alternating Currents

Alternating current (AC) is the type of electric current that flows in a sinusoidal pattern, meaning it changes direction periodically. It’s the kind of electricity that powers your home, TV, and phone charger. In physics, we describe AC with the simple equation $x = x_0 \sin(\omega t)$, where $x$ could be voltage or current, $x_0$ is the maximum value (amplitude), $\omega$ is the angular frequency, and $t$ is time.

What is Alternating Current (AC)?

Unlike direct current (DC) that flows in one direction, AC reverses direction many times per second. Think of it like a wave of water in a bathtub: the water level rises and falls smoothly, just as the voltage or current rises and falls.

Sinusoidal Waveform

The basic AC waveform is a sine wave:

$$x = x_0 \sin(\omega t)$$

• Amplitude ($x_0$) – the peak value of the wave.
• Frequency ($f$) – how many cycles per second (in hertz, Hz).
• Angular frequency ($\omega$) – related to frequency by $\omega = 2\pi f$.
• Period ($T$) – the time for one full cycle, $T = 1/f$.
• Phase shift ($\phi$) – a horizontal shift of the wave.

Key Parameters

• Amplitude ($x_0$)
• Frequency ($f$)
• Angular frequency ($\omega = 2\pi f$)
• Period ($T = 1/f$)
• Phase shift ($\phi$)
• Root‑Mean‑Square (RMS) value: $x_{\text{rms}} = \dfrac{x_0}{\sqrt{2}}$

Real‑World Examples

1. UK mains electricity: 230 V peak, 50 Hz.
2. US mains electricity: 120 V peak, 60 Hz.
3. Audio signals: 20 Hz to 20 kHz.
4. Electric motors: 50 Hz or 60 Hz AC drives the rotating magnetic field.

Analogies

Imagine a playground swing. When you push it, the swing goes forward and then backward in a smooth, repeating motion. That’s like an AC sine wave: it goes forward (positive voltage/current), then backward (negative), and repeats.

Calculations

**Example:** Calculate the RMS voltage of a 120 V peak AC supply.

Using the formula $x_{\text{rms}} = \dfrac{x_0}{\sqrt{2}}$:

$$x_{\text{rms}} = \frac{120}{\sqrt{2}} \approx 85 \text{ V}$$

So the effective voltage that would produce the same heating effect as a DC supply is about 85 V.

Table of AC Parameters