Sinusoidal voltage

Many alternating voltages such as the mains voltage in the public power supply network have a sinusoidal voltage curve and can then be referred to as sinusoidal voltage .

A sinusoidal alternating voltage
1 = peak value
2 = peak-valley value
3 = effective value
4 = period duration

The characteristic sine curve results from the rotary movement with which a conductor loop in the alternating current generator is moved at a constant angular speed in the magnetic field and a voltage is induced in the process.

Three interlinked alternating currents (staggered in time and distributed over three conductors) form a system called three-phase alternating current.

Temporal course

The following applies to the time dependence of a sinusoidal voltage with amplitude and frequency or angular frequency or period duration : ${\displaystyle {\hat {u}}}$ ${\displaystyle f}$ ${\displaystyle \omega }$ ${\displaystyle T}$

${\displaystyle u(t)={\hat {u}}\,\sin(\omega t)={\hat {u}}\,\sin(2\pi ft)={\hat {u}}\,\sin {\frac {2\pi t}{T}}}$with . ${\displaystyle f={\frac {1}{T}}}$

Course of the sinusoidal voltage with scale labeling on the time or angle axis

In the public network is and thus . ${\displaystyle f={\text{ 50 Hz}}}$ ${\displaystyle T={\frac {1}{f}}={\text{0,02 s = 20 ms}}}$

The instantaneous voltage can also be specified as a function of the phase angle by: ${\displaystyle \varphi (t)=\omega t}$

${\displaystyle u={\hat {u}}\cdot \sin(\varphi )\,,}$

in this illustration, 0 ° is placed as the beginning of a period in a zero crossing with a positive rise and 360 ° as the end of the period in the next zero crossing with a positive rise again.

There is a simple rule of thumb for special angles:

φ	sin φ	decimal
0 °	${\displaystyle {\tfrac {1}{2}}{\sqrt {0}}}$	0.000
30 °	${\displaystyle {\tfrac {1}{2}}{\sqrt {1}}}$	0.500
45 °	${\displaystyle {\tfrac {1}{2}}{\sqrt {2}}}$	0.707
60 °	${\displaystyle {\tfrac {1}{2}}{\sqrt {3}}}$	0.866
90 °	${\displaystyle {\tfrac {1}{2}}{\sqrt {4}}}$	1,000

Alternating voltages described with the cosine function are also called sinusoidal.

Rms value

The rms value of a voltage is referred to as, in electrical engineering also as briefly . The effective value of an alternating voltage is as large as the direct voltage with which the same power is converted to an ohmic consumer (e.g. heating) as with the alternating voltage. When specifying alternating voltages, the effective value is usually given (e.g. for a 230 V mains voltage). This value of sinusoidal alternating voltages can be calculated from the amplitude, as stated under the effective value : ${\displaystyle U_{\mathrm {eff} }}$ ${\displaystyle U}$

${\displaystyle U_{\mathrm {eff} }=U={\sqrt {{\frac {1}{T}}\int _{0}^{T}({\hat {u}}\sin(2\pi ft))^{2}\mathrm {d} t}}={\frac {\hat {u}}{\sqrt {2}}}}$

and vice versa from the rms value the peak value:

${\displaystyle {\hat {u}}={U_{\mathrm {eff} }}\cdot {\sqrt {2}}\,.}$

The ratio of the peak value to the rms value is referred to as the peak factor, which assumes this value for sinusoidal alternating voltages . ${\displaystyle {\sqrt {2}}}$

Individual evidence

1. ↑ DKE-IEV Dictionary No. 101-14-34
2. ↑ DKE-IEV Dictionary No. 103-02-03