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**Home ⇒ Overview Courses ⇒ Electrical engineering ⇒ Alternating Current ⇒ Sinusoidal Waveform Construction**

### 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.

For a given peak value u_{p }, the instantaneous value u(t) of a sinusoidal alternating voltage can be calculated for any point in time or angle:

u(t) = f(α) = u_{p }_{ }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) = u_{p} * sin(ω t) = u_{p} * 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.