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====== Experiment 1 ======
===== DC circuit theory =====
==== Linear and non-linear resistors ====

^ Name ^ <wrap onlyprint> \\ \\ </wrap> ^
^ Student ID number ^ <wrap onlyprint> \\ \\ </wrap> ^

=== Equipment used ===

  * Bench power supply GPS 3303
  * Digital multimeter Agilent U1241A
  * Breadboard GL-36
  * Decade resistance box RD-1000, $\pm 1 \%$

The aim of this experiment is to become familiar with and investigate the following:

  * assembling simple circuits on the GL-36 breadboard
  * carrying out measurements with the Agilent U1241A digital multimeter
  * using resistor standard series and the associated colour codes
  * measuring resistances, voltages and currents

==== General measurement techniques ====

=== Voltage measurement ===

Procedure for voltage measurement:

  - Set the meter to the largest voltage range (check whether direct voltage or alternating voltage is to be measured; not necessary in auto range).
  - Connect the test leads to the correct meter sockets (the sockets marked COM and V).
  - Connect the test leads to the component under test with the correct polarity, so that the meter is connected in parallel with the component.
  - Read the measured value.

=== Current measurement ===

Procedure for current measurement:

  - Set the meter to the largest current range (check whether direct current or alternating current is to be measured; not necessary in auto range).
  - Connect the test leads to the correct meter sockets (the sockets marked COM and $\mu{\rm A}.{\rm mA}$).
  - Connect the test leads to the component under test with the correct polarity, so that the meter is connected in series with the component.
  - Read the measured value.

=== Resistance measurement ===

Procedure for resistance measurement:

  - Set the meter to resistance measurement.
  - Connect the resistor to be measured to the corresponding sockets on the meter (the sockets marked COM and $\Omega$).
  - Read the measured value.

=== Digital multimeter Agilent U1241A ===

The Agilent U1241A multimeter has automatic range selection. The following measuring ranges are available:

^ Function ^ Range ^ Accuracy ^
| DC voltage | $0 \ldots 1000 ~{\rm V}$ | $\pm 0.1 \%$ |
| AC voltage | $0 \ldots 1000 ~{\rm V}$ | $\pm 1 \%$ |
| DC current | $0 \ldots 10 ~{\rm A}$ | $\pm 0.2 \%$ |
| AC current | $0 \ldots 10 ~{\rm A}$ | $\pm 1 \%$ |
| Resistance | $0 \ldots 100 ~{\rm M}\Omega$ | $\pm 0.3 \%$ |
| Capacitance | $0 \ldots 10 ~{\rm mF}$ | $\pm 1.2 \%$ |
| Frequency | $30 ~{\rm Hz} \ldots 100 ~{\rm kHz}$ | $\pm 0.3 \%$ |

=== Physical quantities and units used ===

^ Quantity ^ Symbol ^ Unit ^ Unit symbol ^
| Voltage, potential difference | $U$ | volt $= {\rm W}\cdot{\rm A}^{-1} = {\rm kg}\cdot{\rm m}^2\cdot{\rm s}^{-3}\cdot{\rm A}^{-1}$ | ${\rm V}$ |
| Current | $I$ | ampere (base unit) | ${\rm A}$ |
| Resistance | $R$ | ohm $= {\rm V}\cdot{\rm A}^{-1} = {\rm kg}\cdot{\rm m}^2\cdot{\rm s}^{-3}\cdot{\rm A}^{-2}$ | $\Omega$ |

Conventional current direction: current flows from positive to negative.

==== Direct resistance measurement ====

Determine the nominal value and the measured value of the resistance of $R_1$ (brown, green, orange), $R_2$ (yellow, violet, red), $R_3$ (red, violet, red) and the incandescent lamp $R_{\rm L}$. Also measure the approximate resistance $R_{\rm K}$ of your body from your right hand to your left hand.

^  ^ $R_1$ ^ $R_2$ ^ $R_3$ ^ $R_{\rm L}$ ^ $R_{\rm K}$ ^
| Nominal value |  |  |  |  |  |
| Measured value |  |  |  |  |  |

How do you explain the deviation between $R_{\rm L,nom}$ and $R_{\rm L,meas}$?
<wrap onlyprint> \\ \\ \\ \\ </wrap>

What consequences can $R_{\rm K}$ have?
<wrap onlyprint> \\ \\ \\ \\ </wrap>

Now also determine the series and parallel combinations of resistors $R_1$, $R_2$ and $R_3$. State the formulae used:

$R_{\rm series} = R_{\rm a} + R_{\rm b}$

$R_{\rm parallel} = (R_{\rm a} \parallel R_{\rm b}) = \frac{R_{\rm a} \cdot R_{\rm b}}{R_{\rm a} + R_{\rm b}}$

^  ^ $R_1 + R_2$ ^ $R_1 + R_3$ ^ $R_2 + R_3$ ^ $R_1 \parallel R_2$ ^ $R_1 \parallel R_3$ ^ $R_2 \parallel R_3$ ^
| Calculated |  |  |  |  |  |  |
| Measured |  |  |  |  |  |  |

==== Indirect resistance measurement ====

Resistance can also be determined by a current/voltage measurement.

**Ohm's law:** In a circuit, the current increases with increasing voltage and decreases with increasing resistance.

\\
$ I = \frac{U}{R} $
\\

Build the measurement circuit shown in Figure 2 for each of the three resistors and set the voltage on the bench power supply to $12 ~{\rm V}$.

{{drawio>lab_electrical_engineering:1_dc_circuit_theory:figure_2_indirect_resistance_measurement.svg}}

Measure $U_n$ and $I_n$. From these values calculate $R_n$ in each case.

^ $I_1 / {\rm mA}$ ^ $U_1 / {\rm V}$ ^ $R_1 / {\rm k}\Omega$ ^ $I_2 / {\rm mA}$ ^ $U_2 / {\rm V}$ ^ $R_2 / {\rm k}\Omega$ ^ $I_3 / {\rm mA}$ ^ $U_3 / {\rm V}$ ^ $R_3 / {\rm k}\Omega$ ^
|  |  |  |  |  |  |  |  |  |

==== Kirchhoff's voltage law (loop law) ====

In every closed circuit and in every supply loop, the sum of all voltages is zero.

Set the voltage on the bench power supply to $12 ~{\rm V}$ and measure this voltage accurately with a multimeter. Build the measurement circuit shown in Figure 3.

{{drawio>lab_electrical_engineering:1_dc_circuit_theory:figure_3_loop_law.svg}}

Complete the voltage arrows and measure $U$, $U_1$ and $U_2$.

^ $U$ ^ $U_1$ ^ $U_2$ ^
|  |  |  |

What is the loop equation here?
<wrap onlyprint> \\ \\ \\ \\ </wrap>

Verify the formula using the measured values:
<wrap onlyprint> \\ \\ \\ \\ </wrap>

The resistors $R_1$ and $R_2$ connected in series form a voltage divider. In what ratio are the voltages $U_1$ and $U_2$?

$U_1 / U_2 =$ <wrap onlyprint> \\ \\ </wrap> $=$ <wrap onlyprint> \\ \\ </wrap>

==== Kirchhoff's current law (node law) ====

At every branch point, the sum of all currents flowing into and out of the node is zero.

Set the voltage on the bench power supply to $12 ~{\rm V}$ and measure the voltage accurately with a multimeter. As a first step, build the measurement circuit shown in Figure 4.

{{drawio>lab_electrical_engineering:1_dc_circuit_theory:figure_4_branch_currents.svg}}

Draw the arrows for the directions of currents $I_1$ and $I_2$ in Figure 4. On both multimeters the DC current range and the polarity must be set before switching on. Then measure currents $I_1$ and $I_2$ and enter the measured values in Table 5.

{{drawio>lab_electrical_engineering:1_dc_circuit_theory:figure_4_total_current_and_node_K.svg}}

In what ratio are currents $I_1$ and $I_2$?

$I_1 / I_2 =$ <wrap onlyprint> \\ \\ </wrap> $=$ <wrap onlyprint> \\ \\ </wrap>

Switch the bench power supply on again and measure the current $I$. Enter its value in Table 5.

^ $I$ ^ $I_1$ ^ $I_2$ ^
|  |  |  |

Determine the node equation for node $K$ and verify its validity.
<wrap onlyprint> \\ \\ \\ \\ </wrap>

Using the measured values of resistors $R_1$, $R_2$ and $R_3$, calculate the total resistance $R_{\rm KP}$.
<wrap onlyprint> \\ \\ \\ \\ </wrap>

Using the calculated value of $R_{\rm KP}$, verify the measured value of the total current:

$I = \frac{U}{R_{\rm KP}} =$ <wrap onlyprint> \\ \\ </wrap> $=$ <wrap onlyprint> \\ \\ </wrap>

==== Voltage divider as a voltage source (a) ====

The voltage divider shown in Figure 6 is initially in the unloaded condition, because the entire current supplied by the bench power supply flows through the series-connected resistors $R_1$ and $R_2$. A resistor connected in parallel with $R_2$ loads the voltage divider.

Set the voltage on the bench power supply to $12 ~{\rm V}$ and measure the exact voltage with a multimeter. Build the measurement circuit shown in Figure 6.

For the connected load $R_{\rm L} = 10 ~{\rm k}\Omega$, the voltage divider represents a voltage source. Like any voltage source, it has a source voltage (open-circuit voltage) $U_0$ and an internal resistance $R_{\rm i}$. The internal resistance of the voltage divider, regarded as a voltage source, results from the parallel connection of divider resistors $R_1$ and $R_2$:

\\
$R_{\rm i} = R_1 \parallel R_2 = \frac{R_1 \cdot R_2}{R_1 + R_2}$
\\

Using the measured values of resistors $R_1$ and $R_2$, calculate the internal resistance of the voltage source and determine the source voltage:

$R_{\rm i} =$ <wrap onlyprint> \\ \\ </wrap> \\
$U_0 =$ <wrap onlyprint> \\ \\ </wrap>

The power supplied by the bench power supply $P_0$ can be calculated using the following equation:

\\
$P_0 = U \cdot I_1$
\\

The power consumed by the load resistor can be determined using the following equation:

\\
$P_{\rm L} = R_{\rm L} \cdot I_2^2$
\\

{{drawio>lab_electrical_engineering:1_dc_circuit_theory:figure_6_loaded_voltage_divider.svg}}

==== Voltage divider as a voltage source (b) ====

Draw the equivalent voltage source of the voltage divider:

{{drawio>lab_electrical_engineering:1_dc_circuit_theory:figure_6b_equivalent_voltage_source.svg}}

What value would $U_2$ have without $R_{\rm L}$?
$U_{2,0} =$ <wrap onlyprint> \\ \\ </wrap>

Calculate $U_{2{\rm L}}$ and $I_2$ for $R_{\rm L} = 10 ~{\rm k}\Omega$ using the values of the equivalent voltage source. State the formulae used.

$U_{2{\rm L}}:$ <wrap onlyprint> \\ \\ \\ \\ </wrap>

$I_2:$ <wrap onlyprint> \\ \\ \\ \\ </wrap>

Verify the values by measurement:

$U_{2{\rm L},meas}:$ <wrap onlyprint> \\ \\ </wrap>

$I_{2,{\rm meas}}:$ <wrap onlyprint> \\ \\ </wrap>

Verify the values using Kirchhoff's laws. State the formulae used.

$U_{2{\rm L}}:$ <wrap onlyprint> \\ \\ \\ \\ </wrap>

$I_2:$ <wrap onlyprint> \\ \\ \\ \\ </wrap>

==== Non-linear resistors ====

All resistors investigated so far are linear resistors, for which the characteristic $I = f(U)$ is a straight line. See Figure 7. The resistance value of a linear resistor is independent of the current $I$ flowing through it or of the applied voltage $U$.

{{drawio>lab_electrical_engineering:1_dc_circuit_theory:figure_7_linear_characteristic.svg}}

For non-linear resistors there is no proportionality between current and voltage. The characteristic of such a resistor is shown in Figure 8. For these resistors one speaks of the static resistance $R$ and the dynamic (or differential) resistance $r$.

The static resistance is determined for a particular operating point: at a given voltage, the current is read from the resistance characteristic. The calculation is carried out according to Ohm's law:

\\
$R = \frac{U}{I}$
\\

The differential resistance around the operating point is calculated from the current difference caused by a change in the applied voltage:

\\
$r = \frac{\Delta U}{\Delta I}$
\\

{{drawio>lab_electrical_engineering:1_dc_circuit_theory:figure_8_non_linear_characteristic.svg}}

As an example of a non-linear resistor, an incandescent lamp is investigated. Build the measurement circuit shown in Figure 9.

{{drawio>lab_electrical_engineering:1_dc_circuit_theory:figure_9_incandescent_lamp_measurement_circuit.svg}}

Set the bench power supply to the voltage values from Table 7. Measure the corresponding current values and enter them in Table 7.

^ $U / {\rm V}$ ^ 0.5 ^ 1.0 ^ 2.0 ^ 3.0 ^ 4.0 ^ 5.0 ^ 6.0 ^ 7.0 ^ 8.0 ^
| $I / {\rm mA}$ |  |  |  |  |  |  |  |  |  |

Plot the characteristic $I = f(U)$.
<wrap onlyprint> \\ \\ \\ \\ \\ \\ </wrap>

Calculate the static resistance $R$ at the operating point $U = 7.0 ~{\rm V}$.
<wrap onlyprint> \\ \\ \\ \\ </wrap>

Calculate the dynamic resistance $r$ at the operating point $U = 7.0 ~{\rm V}$.
<wrap onlyprint> \\ \\ \\ \\ </wrap>

Compare the values with those from Section 1.2 (direct resistance measurement).
<wrap onlyprint> \\ \\ \\ \\ </wrap>