===== Sinusoidal quantities =====

==== Background Information ====
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<imgref Fig-1_V4-Sinusoidal-quantities> shows the waveforms of a sinusoidal voltage $u = \hat{U} \cdot \sin(\omega t + \varphi_{\rm u})$ and a sinusoidal current $i(t) = \hat{I} \cdot \sin(\omega t + \varphi_{\rm i})$:
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{{drawio>Fig-1_V4-Sinusoidal-quantities.svg}}\\
<WRAP centeralign> <imgcaption Fig-1_V4-Sinusoidal-quantities | Sinusoidal quantities> </imgcaption> </WRAP> 

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<WRAP clear></WRAP>

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  * $\omega = 2\pi f$ is defined as the angular frequency and has the unit $1/s$
  * $\hat{I}$ is referred to as the peak value or amplitude of the current
  * $I$ is the RMS (effective) value of the current
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In alternating current systems, the RMS value is defined as the value that produces the same amount of heat in a purely resistive load as a direct current of the same magnitude.
The crest factor is defined as the ratio of the peak value to the RMS value: 
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$$c = \frac{\hat{I}}{I} = \frac{\hat{U}}{U}$$
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For sinusoidal quantities, $c = \sqrt{2}$. \\
$\varphi = \varphi_{\rm u} - \varphi_{\rm i}$ is called the phase displacement angle or simply the phase shift.
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Given a two-terminal network (a) with the corresponding phasor diagram (b), s. 
<imgref Fig-2_V4-two-terminal-network>. \\
Sketch the time variation of current and voltage, clearly indicating the phase shift.
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{{drawio>Fig-2_V4-two-terminal-network.svg}}\\
<WRAP centeralign> <imgcaption Fig-2_V4-two-terminal-network | Two terminal network> </imgcaption> </WRAP> 
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In alternating current circuits, a two-terminal network is characterized by its impedance or complex resistance **$Z$**.\\ 
**$Z = R + jX$**, where $R$ is the real (active) resistance and $X$ is the reactive resistance.
The magnitude of the impedance (apparent resistance) is $Z = \sqrt{(R^2 + X^2)}$, s. <imgref Fig-3_V4-reative-resistance>. \\
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{{drawio>Fig-3_V4-reative-resistance.svg}}\\
<WRAP centeralign> <imgcaption Fig-3_V4-reative-resistance | Resistance characterization> </imgcaption> </WRAP> 
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The following relationships hold: \\
$$R = Z\cdot \cos\varphi, ~~~ X = Z\cdot \sin\varphi, ~~~ \varphi = \arctan\frac{X}{R}$$ 
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Similar to Ohm’s law, for sinusoidal quantities the following applies: $$Z = \frac{U}{I}$$