Complex AC Analysis

This part of the tutorial is only relevant to the exam for students of ETR.

So far, we have only considered resistors with alternating voltage. Next, let us look at networks of resistors and energy stores. As an example, we take a resistor and a capacitor. We calculate as in the first example in the last chapter.

RC voltage divider: source u0(t), resistor R with uR(t), capacitor C with uC(t), current i0(t)

\[ \begin{gathered} \text{Example: } u_0(t) = 325\,\mathrm{V} \cdot \sin(\omega t) \text{ with } f = 50\,\mathrm{Hz} \rightarrow \omega = 2\pi f = 314\,\frac{1}{\mathrm{s}} \\[4pt] R = 1\,\mathrm{k\Omega};\; C = 3.18\,\mathrm{\mu F} \\[4pt] \text{Mesh: } u_0(t) = u_R(t) + u_C(t) \\[4pt] \text{Component equation: } u_R(t) = R \cdot i_0(t) \\[4pt] \text{Component equation: } u_C(t) = \frac{1}{C} \cdot \int i_0(t)\,dt + U_{C0} \\[4pt] \text{Substituting into the mesh equation: } u_0(t) = R \cdot i_0(t) + \frac{1}{C} \cdot \int i_0(t)\,dt + U_{C0} \end{gathered} \]

Simulation

Within one equation, we have the current and the integral of the current. Mathematicians call this a “differential equation” (strictly speaking an integro-differential equation). You have probably not yet learned how to solve differential equations. Even if you have – the mathematics for it is tough. We have reached a point at which we cannot get any further with our mathematics.

As soon as energy stores appear in an AC network, we can no longer calculate voltages and currents “by hand”. As long as the network only contains resistors, everything is still fine. The rest of the chapter on alternating current deals with the calculation of energy stores in AC networks.

Anyone who tries to multiply or divide sinusoidal quantities quickly realises that this mathematics can no longer be done “by hand”. Integrating and differentiating sinusoidal quantities is not much fun either. Even addition only works if the phase angles of the sinusoidal quantities are the same. This is the case for the first network with the two resistors. Simulation tools calculate with sinusoidal quantities. But they use several billion arithmetic operations per second for it, so such an integration is no problem.