KAHIBARO
Table of Contents
Alternating Quantities in Time
In direct current, the current keeps the same direction and usually the same magnitude. In alternating current, both the voltage and the current change with time, and they usually reverse direction repeatedly. This time variation is the central idea of AC circuits.
An alternating voltage is a voltage source whose polarity changes again and again. If such a source is connected to a circuit, the current it produces can also reverse direction. This is why AC is called alternating.
The most common AC form is sinusoidal. A sinusoidal voltage can be written as
$$
v(t) = V_0 \sin(\omega t + \phi)
$$
and a sinusoidal current can be written as
$$
i(t) = I_0 \sin(\omega t + \phi)
$$
Here, $V_0$ and $I_0$ are the maximum values, also called amplitudes or peak values. The symbol $\omega$ is the angular frequency, and $\phi$ is the phase constant, which tells us where in the cycle the oscillation begins at $t = 0$.
Important sinusoidal forms:
$$
v(t) = V_0 \sin(\omega t + \phi), \qquad i(t) = I_0 \sin(\omega t + \phi)
$$
$V_0$ and $I_0$ are peak values, not average values.
$\omega$ is measured in rad/s.
Period, Frequency, and Angular Frequency
Because AC repeats itself, it is periodic. The time required for one full repetition is called the period, $T$. The number of cycles per second is the frequency, $f$.
They are related by
$$
f = \frac{1}{T}
$$
and angular frequency is related to frequency by
$$
\omega = 2\pi f = \frac{2\pi}{T}
$$
If the frequency is $50 \, \text{Hz}$, then the voltage completes 50 cycles every second. If it is $60 \, \text{Hz}$, then it completes 60 cycles every second.
Key relations for periodic AC:
$$
f = \frac{1}{T}, \qquad \omega = 2\pi f = \frac{2\pi}{T}
$$
Meaning of Sign in AC
In AC, the sign of voltage or current matters. A positive voltage means one polarity, and a negative voltage means the opposite polarity. A positive current means flow in the chosen reference direction, and a negative current means flow opposite to that direction.
This does not mean something is wrong when the value becomes negative. It simply reflects reversal of direction or polarity during the cycle.
One Full Cycle of a Sine Wave
Consider the simple voltage
$$
v(t) = V_0 \sin(\omega t)
$$
At different moments in one cycle:
| Time | Value of $v(t)$ | Meaning |
|---|---|---|
| $t=0$ | $0$ | starts at zero |
| $t=T/4$ | $V_0$ | positive maximum |
| $t=T/2$ | $0$ | crosses zero |
| $t=3T/4$ | $-V_0$ | negative maximum |
| $t=T$ | $0$ | cycle repeats |
This repeating rise and fall is the basic pattern of AC voltage and current.
Peak Value and Average Over a Cycle
For a sinusoidal AC quantity, the peak value is simply the maximum magnitude, such as $V_0$ or $I_0$.
If we average a perfect sine wave over one complete cycle, the result is zero:
$$
\langle \sin(\omega t) \rangle_{\text{one cycle}} = 0
$$
So the average voltage over a full cycle is
$$
\langle v(t) \rangle = 0
$$
and the average current over a full cycle is
$$
\langle i(t) \rangle = 0
$$
This happens because the positive half-cycle and the negative half-cycle cancel each other.
For a pure sinusoidal AC signal, the average over one complete cycle is zero:
$$
\langle v \rangle = 0, \qquad \langle i \rangle = 0
$$
This does not mean AC has no effect. Even though the average over a full cycle is zero, AC can still deliver energy. That is why other measures, especially RMS values, are important. RMS quantities belong to a separate topic.
Phase and Time Shift
Two AC quantities may not reach their maximum values at the same time. If one wave is shifted relative to another, they differ in phase.
For example,
$$
v(t) = V_0 \sin(\omega t)
$$
and
$$
i(t) = I_0 \sin(\omega t + \phi)
$$
If $\phi > 0$, the current is shifted forward in time relative to the voltage. If $\phi < 0$, it is shifted backward. This phase difference becomes very important in AC circuits with capacitors and inductors, but here the main point is simply that AC quantities can oscillate together or with a shift.
AC Voltage from a Rotating Generator
A common physical source of AC voltage is a generator. In a simple generator, a coil rotates in a magnetic field. As it rotates, the induced voltage changes continuously and reverses sign every half turn. This naturally produces a sinusoidal voltage.
The details of electromagnetic induction are covered elsewhere, but this gives the main physical picture for why AC is so common in practice.
Common Forms and Units
Voltage is measured in volts, $\text{V}$. Current is measured in amperes, $\text{A}$. Frequency is measured in hertz, $\text{Hz}$.
Some common sinusoidal forms are shown below.
| Quantity | Mathematical form | Unit |
|---|---|---|
| AC voltage | $v(t) = V_0 \sin(\omega t + \phi)$ | volt, $\text{V}$ |
| AC current | $i(t) = I_0 \sin(\omega t + \phi)$ | ampere, $\text{A}$ |
| Frequency | $f = 1/T$ | hertz, $\text{Hz}$ |
| Angular frequency | $\omega = 2\pi f$ | rad/s |
A Simple Example
Suppose an AC source has voltage
$$
v(t) = 120 \sin(100\pi t)
$$
Comparing with
$$
v(t) = V_0 \sin(\omega t)
$$
we identify
$$
V_0 = 120 \, \text{V}, \qquad \omega = 100\pi \, \text{rad/s}
$$
Then
$$
f = \frac{\omega}{2\pi} = \frac{100\pi}{2\pi} = 50 \, \text{Hz}
$$
and
$$
T = \frac{1}{f} = \frac{1}{50} = 0.02 \, \text{s}
$$
So this source has a peak voltage of $120 \, \text{V}$, a frequency of $50 \, \text{Hz}$, and a period of $0.02 \, \text{s}$.
What to Remember
Alternating voltage and current change with time and regularly reverse sign. The most important mathematical model is the sine wave. Its amplitude gives the peak value, its period gives the time for one cycle, its frequency gives the number of cycles each second, and its phase tells how it is shifted in time relative to another wave.
Core facts about AC voltage and current:
AC quantities vary with time and can reverse direction.
A sinusoidal AC quantity is commonly written as
$$
v(t) = V_0 \sin(\omega t + \phi), \qquad i(t) = I_0 \sin(\omega t + \phi)
$$
The main timing relations are
$$
f = \frac{1}{T}, \qquad \omega = 2\pi f
$$
For a pure sine wave, the average over one full cycle is zero.