Repetition

In order to describe the frequency- and time-dependent behaviour of electrical networks, this chapter briefly summarises some important fundamentals.

To work through this chapter, you will need knowledge of basic electrical quantities, linear passive components \(R\), \(L\) and \(C\), network calculations, and complex alternating current calculations.

1 Frequency dependence of electrical components

Purely ohmic resistors \(R\) cannot store electrical energy. Voltages and currents are proportional to each other and in phase at all times.

This behaviour is independent of time and frequency. Inductances \(L\) and capacitances \(C\), on the other hand, can store and release energy. This process is inert (sluggish), meaning it takes a certain amount of time.

As a result, their behaviour is frequency-dependent. Currents and voltages in inductors and capacitors are in a (temporal) differential linear relationship to each other. This results in a phase shift between voltage and current of \(90\ ^\circ \) for both types of components.

Figure 1 shows the voltage and current curves for \(R\), \(L\) and \(C\) when excited with an alternating voltage \(u_q = U \cdot \sin (\omega t)\) for comparison.

Due to the phase shift in \(L\) and \(C\), their power (energy absorption and release) oscillates, but averages out to zero over a period. Inductances and capacitances cannot therefore perform active power, only reactive power, which is why they are also called reactive resistances.

1.1 Behaviour of inductances and capacitances

Inductors, as ideal components, store energy in the magnetic field through the effect of (self-)induction. Capacitors, as ideal components, store energy in the electric field. Both effects are described by the law of induction and Gauss’s law, respectively.

Table 1 lists the most important differences in the behaviour of inductors and capacitors in qualitative terms as an overview.

The (self-)inductance \(L\) as a property can be simplified as „inertia “of the current. Currents in inductors are continuous and lag behind the voltage. At high frequencies, the inductance blocks; at DC voltage, it behaves like a short circuit.

In contrast, capacitance \(C\) can be simplified as „inertia “of the voltage. Voltages in capacitors are constant and lag behind the current. With direct current, capacitance blocks the current; at high frequencies, it behaves like a short circuit.

Figure 2 shows real components that implement inductance and capacitance. Shown are a coil (inductance) and a capacitor (capacitance).

1.2 Comparison of linear two-pole elements \(R\), \(L\) and \(C\)

2 shows a comparison of the quantities \(R\), \(L\) and \(C\) and summarises their most important properties.

Size	Generally	El. resistance	Inductance	Capacity
Unit	\( \left [\textit {Form.s.}\right ] = \mathrm {Unit} \)	\( \left [R\right ] = \Omega \ \text {(Ohm)} \)	\( \left [L\right ] = \mathrm {H}\ \text {(Henry)} \)	\( \left [C\right ] = \mathrm {F}\ \text {(Farad)} \)
Time domain	\( \frac {\mathrm{d} }{\mathrm {d}t} \) or \( \int \mathrm {d}t \)	\( u_{\mathrm {R}} = R \cdot i_{\mathrm R} \)	\( u_{\mathrm {L}} = L \cdot \frac {\mathrm{d} }{\mathrm {d}t}\, i_{\mathrm {L}} \)	\( i_{\mathrm {C}} = C \cdot \frac {\mathrm{d} }{\mathrm {d}t}\, u_{\mathrm {C}} \)
frequency range	\( \mathrm {j}\omega \) or \( \frac {1}{\mathrm {j}\omega } \)	\( \underline {U_{\mathrm R}} = R \cdot \underline {I_{\mathrm R}} \)	\( \underline {U_{\mathrm L}} = \mathrm {j}\omega L \cdot \underline {I_{\mathrm L}} \)	\( \underline {I_{\mathrm C}} = \mathrm {j}\omega C \cdot \underline {U_{\mathrm C}} \)
Impedance	\( \underline {Z} = \frac {\underline {U}}{\underline {I}} \)	\( \underline {Z}_{\mathrm R} = R \)	\( \underline {Z}_{\mathrm L} = \mathrm {j}\omega L \)	\( \underline {Z}_{\mathrm C} = -\mathrm {j}\frac {1}{\omega C} \)
Active part	\( R = \Re \{\underline {Z}\} \)	\( R = \frac {U}{I} \)	\( 0 \)	\( 0 \)
Blind part	\( X = \Im \{\underline {Z}\} \)	\( 0 \)	\( X_{\mathrm L} = \omega L \)	\( X_{\mathrm C} = -\frac {1}{\omega C}\)

2 Double-pole (four-pole)

We already examined the frequency dependence of single ports (dipoles) in Module 3. Using complex AC calculations, we were able to determine the frequency-dependent AC resistance (impedance) \(\underline {Z}\) of individual single ports. By applying the complex current and voltage divider rules, we were also able to determine the total impedance of linear two-terminal networks. This applies to networks that consist only of linear components such as \(R\), \(L\) and \(C\).

Two-port (four-terminal) networks are an extension of single-port (two-terminal) networks. They have an input and an output side with respective input and output variables. The two-port model is well suited for describing (frequency-dependent) transmission characteristics of electrical networks.

In the simplest case, as shown in 4, a source (single-port) is connected to a load (single-port) via a two-port. Typical two-ports are, for example, amplifiers, filters or transformers.