In order to get a constant (lower) voltage from an higher voltage input or a source with a broader spread of the voltage (e.g. a battery) often linear regulators are used. One example could be to get $5V$ from the car battery voltage (between $11V...14V$) for a microcontroller in a control unit e.g. the brake control unit. Linear regulator here means that a transistor as a variable resistor is used to drop the unwanted voltage.

Below, two types of such a linear regulator are shown

1. The first simuation shows a simple series regulator with a FET. „Series“ here marks the fact that the transistor is in series to the load resistor $R_L$. The zener diode $D$ has a current limiting series resistors $R_D$ ahead. By the voltage divider of $R_D$ and $D$, a relatively constant voltage will be created.
2. The second simulation shows a more sophisticated circuit. Here there is a feedback from the output of the transistor back to the transistor controlling voltage is given by $R1$, $R2$ and the operational amplifier.

You work in the company „HHN Mechatronics & Robotics“ and are supposed to generate a bipolar signal ($-10V ... + 10V$) from a unipolar signal of a digital-to-analog converter ($0...5V$) in a project. A colleague recommended the circuit shown on the right.

1. First, analyze what change is made by pressing the switch $S$. How does the output signal change?
2. Try to determine mathematically the relationship of $U_O$ and $U_I$ as $U_O(U_I)$ by superposition.
3. The circuit still has the problem that for an positive positive half-wave the output is still negative. Which additional circuit must be provided so that this problem can be solved?