Principle of superposition

After a basic analysis of meshes and nodes in electrical networks, increasingly complex electrical networks are considered. For this purpose, the superposition method according to Helmholtz is used. The superposition method is also referred to as the principle of superposition. In the superposition principle, all sources are evaluated individually one after the other and the results of the individual calculations are superimposed.

1 Digression Systems theory

If only the input signal and the output signal are considered in a model, this is referred to as a black box. Here, no information is given about what happens between these two signals. Figure 1 describes such a model. Here, a black box is considered, which has an input variable \(F_\mathrm {E}\) and an output variable \(F_\mathrm {A}\). In the black box, the input signal is transformed into the output signal. The image as a whole is referred to as a system.

Based on the black box presented, the output signal \(F_\mathrm {A}\) can be determined as a function of the input signal \(F_\mathrm {E}\). This transformation can be described by equation 1.

\begin {equation} F_\mathrm {A}=T(F_\mathrm {E}) \label {GleichungFunktionSystem} \end {equation}

An electrical network can also be considered such a system. The electrical network shown in Figure 2 has an input voltage \(U_\mathrm {E}\) as the input variable and an output voltage \(U_\mathrm {A}\) as the output variable. The black box, which represents the system to be transformed, consists of the two resistors \(R_1\) and \(R_2\).

The relationship between the output signal of a system and the input signal from Equation 1 still applies. The output signal can be described as a transformation of the input signal as in Equation 2. The output signal is equal to the voltage drop across resistor \(R_2\). This voltage can be calculated from the product of the input voltage and the resistance ratio. This relationship was also discussed in the section on voltage dividers in Chapter 4.

\begin {equation} U_\mathrm {A}=T(U_\mathrm {E})=U_\mathrm {E}\cdot \frac {R_2}{R_1+R_2} \label {GleichungNetzwerk} \end {equation}

Causality

If an output variable is based exclusively on the transformation of an input variable, there is a direct relationship between cause and effect. This also means that the output variable does not provide a changing system response prior to an excitation. A system in which these conditions prevail is referred to as a causal system (see Figure 3). For example, a DC excitation of a system produces a time-indifferent DC response. A system that behaves differently is referred to as a non-causal system. If the value of the input signal is zero for \(t < t_0\), the value of the output signal must also be zero for the same period of time in order to fulfil the conditions of a causal system.

Time variance

If the output signal of a system always responds to an input signal in the same way over time, this is referred to as a time-invariant system. If the timing of the input signal shifts, for example by \(t_0\), the output signal must still respond in the same way, based on the input signal, in time-invariant systems.

The stability of a system also describes a relationship between the input signal and the output signal. If an input signal with finite amplitude produces an output signal that does not grow beyond all limits, the system is stable. If the output signal continues to grow after the input signal has been deactivated, the system is unstable.

\begin {equation} T(\alpha \cdot u_\mathrm {e})=\alpha \cdot T(u_\mathrm {e}) \label {GleichungLinearität} \end {equation}

Linearity

If the response of the output signal is proportional to the excitation of the input signal, the system is described as a linear system (Equation 3). If two superimposed input signals act on a system, they can be considered separately and added together in this case. In this way, the transformations of the input signals, as in Equation 4, can also be considered separately.

\begin {equation} T(u_1+u_2)=T(u_1)+T(u_2) \label {GleichungÜberlagerteSignale} \end {equation}

Systems that are both time-invariant and linear are referred to as LTI (Linear Time Invariant) systems. The systems in this module are approximated as LTI systems. Components that exhibit non-linearities are dealt with, for example, in the chapter on periodic quantities.

2 Superposition theorem

Once the electrical networks have been checked for linearity as the systems under consideration, the superposition theorem can be applied. Here, different sources can be considered separately from one another. In this way, the superposition theorem switches off all sources and switches on the individual sources in series. When sources are switched off, current sources and voltage sources are converted differently. When an ideal voltage source is switched off, a short circuit occurs at the location of the voltage source, as shown in Figure 4.

When switched off, a power source leaves only open terminals (Figure 5). This idle state prevents further consideration of this path in the network.

If all sources except one are switched off, the electrical network for the remaining source is analysed. This is then carried out sequentially with each source. At the end, the individual effects are considered as a sum. For example, the current \(I_\mathrm {R}\) through a resistor R, which is supplied by two sources \(Q_1\) and \(Q_2\), can be explained by the sum of the partial currents of the two sources (see Equation 5).

\begin {equation} I_\mathrm {R}=f(Q_1,Q_2) \label {GleichungZweiQuellen} \end {equation}

To illustrate the superposition theorem, Figure 6 shows an electrical network with two voltage sources and three resistors. To calculate this network with more than one source, the network is considered separately for both voltage sources.

The electrical network presented is shown again in Figure 7 for the calculation of the two voltage sources separately. For the network analysis with voltage source \(U_1\), voltage source \(U_2\) is short-circuited. The two resistors \(R_2\) and \(R_3\) are now connected in parallel with each other. The voltage of \(U_1\) is now distributed across \(R_1\) and the parallel connection \(R_{23}\). Equivalently, when considering the network for the voltage source \(U_2\), the voltage source \(U_1\) is short-circuited. Now the two resistors \(R_1\) and \(R_2\) form a parallel connection. The voltage of the voltage source \(U_2\) is distributed across the parallel connection \(R_{12}\) and the resistor \(R_3\).

In equation 6 and equation 7, the voltages for the two voltage sources \(U_1\) and \(U_2\) at resistor \(R_3\) are determined separately. In equation 8, the two voltages across the resistor \(R_3\) are then added together according to the superposition theorem.

\begin {equation} U_\mathrm {R3}(U_1) = U_1 \cdot \frac {R_\mathrm {23}}{R_1+R_\mathrm {23}} \label {GleichungU1} \end {equation} \begin {equation} U_\mathrm {R3}(U_2) = U_2 \cdot \frac {R_\mathrm {12}}{R_3+R_\mathrm {12}} \label {GleichungU2} \end {equation} \begin {equation} U_\mathrm {R3} = U_\mathrm {R3}(U_1) + U_\mathrm {R3}(U_2) \label {GleichungUR3} \end {equation}

Superposition method

The electrical network shown in Figure 8 is given. The following tasks are to be completed:

Application of the superposition principle (Superposition theorem):

a)

Plotting currents and voltages

b)

Network calculation \(I_3\) for \(U_\mathrm {g}\)

c)

Network calculation \(I_3\) for \(I_\mathrm {g}\)

d)

How big is the current? \(I_3\)

a)

Specification of currents and voltages:

b)

Network calculation \(I_3(U_\mathrm {g})\) for \(U_\mathrm {g}\):

\begin {align} K_1: I_\mathrm {tot}&=I_1+I_3 \nonumber \\ I_\mathrm {tot}&=\frac {U_\mathrm {g}}{R_\mathrm {tot}}=\frac {U_\mathrm {g}}{R_\mathrm {i}+\frac {R_1\cdot (R_2+R_3)}{R_1+R_2+R_3}} \nonumber \\ \frac {I_3}{I_\mathrm {tot}}&=\frac {R_1}{R_1+R_2+R_3} \nonumber \\ I_3(U_\mathrm {g})&=I_\mathrm {tot}\cdot \frac {R_1}{R_1+R_2+R_3}=\frac {U_\mathrm {g}}{R_\mathrm {i}+\frac {R_1\cdot (R_2+R_3)}{R_1+R_2+R_3}}\cdot \frac {R_1}{R_1+R_2+R_3}\nonumber \end {align}

c)

Network calculation \(I_3(I_\mathrm {g})\) for \(I_\mathrm {g}\):

\begin {align} K_2: I_\mathrm {tot}&=I_\mathrm {g}=I_2-I_3 \nonumber \\ \frac {-I_3}{I_\mathrm {tot}}&=\frac {R_2}{(R_1||R_\mathrm {i})+R_2+R_3} \nonumber \\ I_3(I_\mathrm {g})&=-I_\mathrm {g}\cdot \frac {R_2}{(R_1||R_\mathrm {i})+R_2+R_3}\nonumber \end {align}

d)

How strong is the current \(I_3(U_\mathrm {g},I_\mathrm {g})\)?

Superposition: \begin {align} I_3(U_\mathrm {g},I_\mathrm {g}) &= I_3(U_\mathrm {g})+I_3(I_\mathrm {g}) \nonumber \\ I_3(U_\mathrm {g},I_\mathrm {g}) &= \frac {U_\mathrm {g}}{R_\mathrm {i}+\frac {R_1\cdot (R_2+R_3)}{R_1+R_2+R_3}}\cdot \frac {R_1}{R_1+R_2+R_3}+(-I_\mathrm {g}\cdot \frac {R_2}{(R_1||R_\mathrm {i})+R_2+R_3}) \nonumber \end {align}