Aufgabe 4.5.2: open circuit voltage via superposition (exam task, approx. 12 % of a 60-minute exam, WS2020)

schaltung_klws2020_2_3_1.jpg

A circuit is given with the following parameters
$R_1=5 ~\Omega$
$U_1=2 ~\rm V$
$I_2=1 ~\rm A$
$R_3=20 ~\Omega$
$U_3=8 ~\rm V$
$R_4=10 ~\Omega$

Determine the open circuit voltage between A and B using the principle of superposition.

Tips

  • What do the individual circuits look like, by which the effects of the individual sources can be calculated?
    Which equivalent resistor must be used to replace a current or voltage source when calculating the individual effects?
  • Where are the open-circuit voltages applied when looking at the individual components?

Solution

First, the individual circuits must be created, from which the effect of the individual sources between points A and B can be determined.

(Voltage) source $U_1$
  • substitute the current source $I_2$ with a short-circuit
  • substitute the voltage source $U_3$ with an open circuit

schaltung_klws2020_2_3_1_q1.jpg

The components can be moved in order to understand the circuit s bit better.

schaltung_klws2020_2_3_1_q1_1.jpg

For the open circuit, no current is flowing through any resistor. Therefore, the effect is: $U_{AB,1} = U_1$

(current) source $I_2$

  • substitute the voltage source $U_1$ with an open circuit
  • substitute the voltage source $U_3$ with an open circuit

schaltung_klws2020_2_3_1_q2.jpg

Also here, the components can be shifted for a better understanding:

schaltung_klws2020_2_3_1_q2_1.jpg

Here, the current source $I_2$ creates a voltage drop $U_{AB_2}$ on the resistor $R_2$ : $U_{\rm AB,2} = - R_1 \cdot I_2$

(Voltage) source $U_3$

  • substitute the voltage source $U_1$ with an open circuit
  • substitute the current source $I_2$ with a short-circuit

schaltung_klws2020_2_3_1_q3.jpg

Again, rearranging the circuit might help for an understanding:

schaltung_klws2020_2_3_1_q3_1.jpg

In this case, between the unloaded outputs $\rm A$ and $\rm B$ there will be an unloaded voltage divider given by $R_3$ and $R_4$. On $R_1$ there is no voltage drop since there is no current flow out of the unloaded outputs.
Therefore:

\begin{align*} U_{\rm AB,3} = \frac{R_4}{R_3 + R_4} \cdot U_3 \end{align*}



resulting voltage

\begin{align*} U_{\rm AB} &= U_1 - R_1 \cdot I_2 + \frac{R_4}{R_3 + R_4} \cdot U_3 \\ \end{align*}

Final value

\begin{align*} U_{\rm AB} &= 2 ~{\rm V} - 5 ~\Omega \cdot 1 ~{\rm A} + \frac{10 ~\Omega}{20 ~\Omega + 10 ~\Omega} \cdot 8 ~{\rm V} \\ \\ U_{\rm AB} & = 0.333... ~{\rm V} \rightarrow 0.3 ~{\rm V} \\ \end{align*}