Sinusoid

Seventeenth Meeting

Sine Wave: Amplitude

The amplitude is the maximum displacement of the sine wave from its mean (average) position.

Simulation

Sine Wave: Cycle, Frequency and Period

Frequency (f) The number of cycles per second, Example: A sine wave with 5 cycles

per second is said to have a frequency of 5 Hz (hertz)

Cycle The basic shape of the waveform that

repeats indefinitely. Period (T)

The time taken to complete one cycle T = 1/f The mains electricity supply is

sinusoidal, with a frequency of 50 Hz. What is T?

1/50 = 0.02 s

One Cycle

Simulation

Sine Wave: Phase

Phase, or, more correctly, phase shift, Is how far a sine wave is shifted along

the horizontal axis relative to another sine wave taken as a reference

The blue sine wave is shifted 1/4 cycle to the right of the reference sine wave

If a sine wave to be generated by the rotating line a, then a sine wave lagging by a quarter of a cycle is generated by a line b at 90 degrees to line a. (Ninety degrees = ¼ cycle) why? because a complete revolution, 360

degrees, corresponds to one complete cycle of a sine wave.) Simulation

Sine wave: Equation

y = a sin(2πft + φ) y represents displacement at time t a represents the amplitude f is the frequency and φ is the phase

The term (2πft + φ) represents an angle that is growing as time passes.

This angle is measured in radians rather than degrees.

For the following sine wave, it is clear that the amplitude a has the value 5 volts. The values f and φ are not so obvious.

φ is a quarter of a cycle is 90 degrees, or π/2 radians.

Since the sine wave lags behind the reference sine wave, so φ = – π/2 radians. (radian = 57.3 degrees)

The equation for the sine wave is:

y = 5 sin(200πt – π/2) volts

Periodic Waves

y = 2 sin 2000 πt. y = 2 cos 2000 πt A cosine wave can be

regarded as a sine wave shifted in phase by a quarter of a cycle

sin wave

cos wave

Sine & Cosine

sine function v(t) = A sin (wt). A = amplitude w (omega) = angular

frequency in rad/s. ƒ = frequency in Hz, T = the period in seconds

of one cycle

Cosine function v(t) = A sin (wt).

Square Wave

The infinite extent of the spectrum results from the corners on the square waveform.

These corners are assumed to be perfectly sharp

The more we add, the sharper the corner

With only the first five harmonic,

Some Fourier Series

Partial sums of some Fourier series, up to and including ninth harmonic