Sinusoidal quantities

Background Information

figure 1 shows the waveforms of a sinusoidal voltage $u = \hat{U} \cdot \sin(\omega t + \varphi_{\rm u})$ and a sinusoidal current $i(t) = \hat{I} \cdot \sin(\omega t + \varphi_{\rm i})$:

• $\omega = 2\pi f$ 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

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:
$$c = \frac{\hat{I}}{I} = \frac{\hat{U}}{U}$$
For sinusoidal quantities, $c = \sqrt{2}$.
$\varphi = \varphi_{\rm u} - \varphi_{\rm i}$ is called the phase displacement angle or simply the phase shift.
Given a two-terminal network (a) with the corresponding phasor diagram (b), s. figure 2.
Sketch the time variation of current and voltage, clearly indicating the phase shift.

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. figure 3.

The following relationships hold:
$$R = Z\cdot \cos\varphi, ~~~ X = Z\cdot \sin\varphi, ~~~ \varphi = \arctan\frac{X}{R}$$
Similar to Ohm’s law, for sinusoidal quantities the following applies: $$Z = \frac{U}{I}$$