Sinusoidal Waveform Construction
Instantaneous values, angular velocity
Sinusoidal alternating variables can be displayed both in a line chart or as a pointer diagram. In the pointer diagram, the pointer rotates counterclockwise. The pointer length corresponds to the peak value of the alternating quantity.
Sinusoidal Waveform Construction
For a given peak value up , the instantaneous value u(t) of a sinusoidal alternating voltage can be calculated for any point in time or angle:
u(t) = f(α) = up sin(α) (1)
In the pointer diagram, the pointer with the length r travels the distance s within one period, which corresponds to the circumference :
s = 2 π r (2)
The period duration is directly related to the rotational speed:
v = (2 π r) because t = T (3)
. T
In order to be independent of the radius length, we now refer to the unit circle with r = 1 and thus get the angular velocity ω with the unit s-1:
ω = v = ( 2 π ) f = 1 / T
. r T
=> ω = 2 π f (4)
The angular velocity ω is the angular change per time.
The angle passed in a given time is determined by the following equation:
α = ω t = 2 π f t (5)
If the peak value and frequency also are known, then the instantaneous value can be determined for each point in time:
u = f(t) = up * sin(ω t) = up * sin(2 π f t) (6)
Exercise
Our mains voltage has an RMS value of 230 V and a mains frequency of 50 Hz. Determine the times for the values of the instantaneous voltage + 100 V and - 100 V starting from the zero.
