A sine wave is a periodic signal that varies smoothly following the sine function, passing through positive, zero, and negative values continuously.
If you connect an oscilloscope to the wall outlet in your house, you won’t see a straight line. You’ll see a curve that smoothly rises and falls, going from positive to negative and back again.
Do not directly connect a common benchtop oscilloscope to a wall outlet: the probe’s ground is usually tied to earth and can create a short circuit or an electrocution hazard. This measurement requires proper equipment, probes, and training.
We call that shape a sine wave (or sinusoidal).
Why this shape and not another? Why not a square or triangular wave? We saw the answer in the previous module: Because of the generators.
Electricity is generated by rotating a coil within a magnetic field. And the mathematical projection of circular motion onto a linear axis is precisely a sine.
Alternating current is the shadow of a rotating circle.
The Cycle
Before measuring anything, let’s define the basic unit: the Cycle. A cycle is the complete path of the wave until it starts repeating:
Starts at 0.
Rises to the positive maximum (Peak).
Falls through 0 to the negative minimum (Trough).
Rises back to 0.
From there, the pattern repeats identically.
The Period (
The Period answers the question: How much time does the wave take to complete one entire cycle?
It is represented by the letter 𝑇 and, since it is time, it is measured in seconds (𝑠) (or milliseconds 𝑚𝑠).
- If the wave is very slow, the period is large.
- If the wave is very fast, the period is tiny.
The Frequency (
Frequency is the inverse of the period. It answers: How many complete cycles fit in one second?
It is represented by the letter
This relationship is fundamental.
- In Europe: The mains grid operates at 50 Hz. It completes 50 cycles and crosses zero 100 times per second. The period is
(20 ms). - In the Americas: The grid operates at 60 Hz. It is slightly faster:
.
When we say alternating current “goes back and forth,” it means the light in your room turns off and on 100 times per second (twice per cycle, when crossing zero).
Your eye is not fast enough to see it, but your brain sometimes notices it (flicker).
Angular Velocity (
Frequency (𝑓) tells us revolutions per second. It’s great for humans. But mathematical formulas (sines and cosines) don’t work with “revolutions”; they work with angles (radians).
We need to know how many radians per second our imaginary generator sweeps. We call this angular velocity or pulsation, and it is represented by the Greek letter omega (
Since one full revolution (one cycle) is 2π radians, the relationship is direct:
- Unit: Radians per second (𝑟𝑎𝑑/𝑠).
Quick calculation example: For the European mains grid at 50 Hz:
The approximate value 314 rad/s appears frequently because it corresponds to a 50 Hz signal.
The Wave Equation
Putting all this together, we can write the mathematical formula that describes the voltage at any instant in time:
: The voltage value at that precise instant. : The maximum height of the wave (Amplitude). : The shape of the wave. : The angle we are at at that moment.
