In Module 4
Introduction
Kirchhoff’s lawsIntroduction
Electrical networks are omnipresent in today’s world. They consist of interconnections of various electrical components that are linked together by connecting lines. In reality, they come in a wide variety of functions and sizes. An electrical network can be a small electronic circuit within a microcontroller measuring just a few micrometres, or an electrical power distribution network extending over several thousand kilometres, as symbolically shown in Figures 1 and 2.
This chapter provides an introduction to basic calculations in direct current electrical networks. The aim of these calculations is usually to calculate the voltages and currents in all components of the network. Real components are usually simplified and represented as a combination of ideal two-pole circuits (examples: Table 1). The interconnection of these dipoles can be used to construct a simplified model of the real circuit, which replicates the basic behaviour of the circuit. This model forms the basis for the mathematical calculation of the circuit.
Since only linear components such as resistors or ideal DC voltage sources are used in this chapter, an analytical calculation is always possible. In advanced modules, however, non-linear components such as real operational amplifiers or diodes are introduced. In this case, an iterative or numerical approach is often necessary.
Two-terminal elements and reference direction systems
Learning objectives: Two-terminal elements and reference direction systems
The students can
- explain the concept of two poles, name common two poles and use their circuit symbols
- explain the generator and consumer arrow system and apply it in accordance with conventions
- explain the central elements of the basic circuit and recognise the relationships between them
1 Two-terminal elements
A two-pole component (Fig. 1) is a component with two external terminals. The internal structure of these two-pole components can vary greatly in terms of type and complexity. For example, a simple electrical resistor, but also a voltage source (e.g. a car battery) or a hairdryer
(provided it only has two connecting wires) can be regarded as a two-pole component. Components with more connecting wires are referred to as three-pole (e.g. refrigerator with protective earth conductor), four-pole (e.g. transformer) or even five-pole (electric cooker with three-phase connection, protective earth conductor and neutral conductor) components.
Regardless of its internal complexity, a two-terminal network in an electrical network can be fully characterised by the relationship between current and voltage at its connection points, known as terminal behaviour. It should be noted that the current \(I_1\) of a two-terminal network shown in Figure 1 is always equal to the current \(I_2\).
The practical implementation of the components, such as actual dimensions, material properties, parasitic effects or internal inhomogeneous field strength distributions, are neglected in the network calculation with two-pole elements.
Some of the two-pole switches already familiar from previous chapters are listed as examples in Table 1.
2 Reference direction systems
The choice of counting directions for current and voltage is essentially arbitrary. When calculating electrical networks, an attempt is often made to introduce the counting directions in such a way that the currents and voltages are positive. This makes perfect sense for quantities that are known from the outset. For unknown quantities, however, the counting direction must be determined arbitrarily. This does not mean that the current actually flows in the direction of the arrow or that a positive voltage is present in the direction of the arrow. The actual direction is then expressed by the sign of the voltage. For a sign-correct description of currents and voltages, dimensioning with counting arrows is therefore absolutely necessary.
For two-pole systems, a distinction is made between the two different counting arrow systems presented in Table 2:
Key point: Consumer and generator sign conventions
Passive sign convention (PSC): Current and voltage are counted in the same direction at
the two poles. To be used with passive two-pole devices (e.g. resistors).
Active sign convention (GSC): Current and voltage are counted in opposite directions at the two poles. To be used with active two-pole devices (e.g. voltage sources).
Key point: Sign convention arrows
Reference arrows are used to define current and voltage sign conventions and must not be confused with vectors.
3 The fundamental circuit
The previously introduced two-pole (or multi-pole) systems can be combined to form an electrical network.
The idealised two-pole branches, which, as shown in Fig. 2, can also consist of several two-pole elements connected directly in series. The same current flows through all elements of a branch. In addition to the branch marked in green in Figure 2, the two-pole group \(R_1\), \(R_2\) and \(U_0\) as well as the short circuit in the drawing each form another branch.
The connection points where at least three branches meet are called nodes or junctions. A flowing electric current can split up here onto the different branches. However, the electric potential is identical for all connected terminals. A node is marked in the circuit diagram by a filled circle and is referred to as \(K_n\). Closed paths of at least two branches in a row within a network are called meshes and are abbreviated as \(M_n\). In the circuit diagram, in addition to the designation of the mesh, a direction of circulation is often indicated by an arrow that shows the direction of circulation of the mesh. In the basic circuit drawn here, in addition to the drawn mesh \(M_1\) consisting of \(R_1\), \(R_2\), \(U_0\) and the short circuit between nodes \(K_1\) and \(K_2\): The branch marked in green together with the short circuit between \(K_1\) and \(K_2\) form a mesh \(M_2\). Another mesh \(M_3\) around the outside leads around the outside of the circuit and contains all the dipoles marked, but not the short circuit between \(K_1\) and \(K_2\).
