Alternating Voltage and Current

Classification of Voltages and Currents

• Direct Current / Voltage (DC):

  • General DC: The direction remains constant, but the magnitude is not necessarily constant.

  • Periodic DC: The direction remains constant, while the variable magnitude repeats periodically.

  • Constant DC: Both magnitude and direction remain strictly invariant over time.

• Alternating Current / Voltage (AC):

  • Alternating Voltage: Continuously changes in both sign/direction and magnitude.

  • Periodic Alternating Voltage: Continuously changes direction, with magnitude variations repeating periodically.

  • Alternating Voltage Proper (wisselspanning): A periodic alternating voltage whose average magnitude over a full period equals zero (e.g., standard sinusoidal AC).

Generation of Sinusoidal Alternating Voltage

• Alternator / Synchronous Generator: Converts mechanical energy into electrical energy by rotating a coil with $N$ turns within a uniform magnetic field $\mathbf{B}$.

• Magnetic Flux:

  • At rotation angle $\alpha$: $\phi_N = N \cdot B \cdot A \cdot \cos(\alpha)$

  • Uniform rotation at constant angular velocity $\omega$ ($\alpha = \omega t$): $\phi_N = N \cdot B \cdot A \cdot \cos(\omega t)$

• Induced Voltage (Faraday's Law of Induction):

  • $U(t) = -\frac{d\phi_N}{dt} = -\frac{d}{dt}(N \cdot B \cdot A \cdot \cos(\omega t)) = N \cdot B \cdot A \cdot \omega \cdot \sin(\omega t)$

  • General expression: $U(t) = U_{\max} \cdot \sin(\omega t)$

  • Peak voltage (amplitude): $U_{\max} = N \cdot B \cdot A \cdot \omega$

• Current through Resistive Load ($R$):

  • $I(t) = \frac{U(t)}{R} = \frac{U_{\max}}{R} \cdot \sin(\omega t) = I_{\max} \cdot \sin(\omega t)$

• Flux and Voltage Extremes:

  • At $\alpha = 0$: Magnetic flux is maximal ($\phi = B \cdot A$); the rate of change is zero, resulting in $U = 0\,V$.

  • At $\alpha = \frac{\pi}{2}$: Magnetic flux is zero ($\phi = 0$); the rate of change is maximal, resulting in peak induced voltage ($U = U_{\max}$).

Characteristic Quantities of AC Signals

• Period ($T$): Time duration in seconds ($\text{s}$) between identical consecutive points of the cycle (such as identical zero crossings or peaks).

• Frequency ($f$): Number of cycles completed per second in Hertz ($\text{Hz}$):

  • $f = \frac{1}{T}$

• Instantaneous Value ($u$, $i$, $u(t)$, $i(t)$, $U(t)$, $I(t)$): The value of the voltage or current at a specific instant in time.

• Peak Value / Amplitude ($U_{\max}$, $U_m$, $I_{\max}$, $I_m$): The maximum excursion from zero.

• Peak-to-Peak Value ($U_{tt}$, $U_{ptp}$): Total voltage difference between positive and negative peaks ($U_{ptp} = 2 \cdot U_m$ for symmetric signals).

• Effective / RMS Value ($I_f$, $U_f$, $I_{eff}$, $U_{eff}$):

  • The equivalent DC current that produces the exact same amount of heat in a given resistor $R$ over period $T$ as the alternating current.

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  • General Root-Mean-Square formula:     $I_f = \sqrt{\frac{1}{T} \int_0^T I^2(t)\,dt}$

  • Formulas for sinusoidal AC:     $I_f = \frac{I_m}{\sqrt{2}}$     $U_f = \frac{U_m}{\sqrt{2}}$

Phase, Phase Difference, and Vectorial Representation

• Phase: The smallest angle between the origin of the coordinate system and the positive zero crossing of the waveform.

  • General mathematical form: $u(t) = U_m \cdot \sin(\omega t + \alpha)$

• In Phase: Two quantities of identical frequency reach their zero crossings and peak values at the exact same moments.

• Phase Difference ($\varphi$): The angular difference between corresponding points of two alternating quantities having identical frequencies:

  • $u = U_m \cdot \sin(\omega t)$

  • $i = I_m \cdot \sin(\omega t - \varphi)$

• Leading and Lagging (Voorijlen en Naijlen):

  • A quantity leads (ijlt voor) another if it reaches its amplitude or zero crossing earlier in time.

  • A quantity lags (ijlt na) if it reaches those points later in time.

• Vectorial (Phasor) Representation:

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  • Sinusoidal quantities can be represented as rotating vectors in a vector diagram rotating counterclockwise at angular velocity $\omega$.

  • The length corresponds to the amplitude ($U_m$), or in practical circuit analysis, the RMS value ($U_f$).

  • When RMS length is used, instantaneous values equal the projection multiplied by $\sqrt{2}$.

  • Only valid for sinusoidal quantities of the same frequency.

Complex Representation of AC Quantities

• Complex Vector Representation:

  • Cartesian form: $U = a + j \cdot b$, where $a$ is the real component (horizontal axis) and $b$ is the imaginary component (vertical axis).

  • Polar form: $U = |U|\angle\alpha$

  • Modulus (vector length/magnitude): $|U| = \sqrt{a^2 + b^2}$

  • Argument (angle relative to the positive real axis): $\alpha = \operatorname{bgtan}\left(\frac{b}{a}\right)$

• Quadrants based on Signs:

  • $a > 0,\, b > 0$: First quadrant

  • $a < 0,\, b > 0$: Second quadrant

  • $a < 0,\, b < 0$: Third quadrant

  • $a > 0,\, b < 0$: Fourth quadrant

• Vector Addition in the Complex Plane:

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  • Vector addition follows the parallelogram rule, equivalent to summing real and imaginary parts:     $U = U_1 + U_2 = (a_1 + a_2) + j \cdot (b_1 + b_2)$