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lab_electrical_engineering:2_capacitors:capacitors [2026/03/21 22:49] – mexleadminlab_electrical_engineering:2_capacitors:capacitors [2026/03/21 23:31] (current) – mexleadmin
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 === Direct capacitance measurement === === Direct capacitance measurement ===
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 Capacitors are components that allow the storage of electrical energy. In the charged state an electrical charge is present. This charge causes a voltage at the electrical terminals of the capacitor. \\ Capacitors are components that allow the storage of electrical energy. In the charged state an electrical charge is present. This charge causes a voltage at the electrical terminals of the capacitor. \\
 Build the following circuit on the breadboard with three capacitors, s. <imgref Fig-7_V2-Capacitor>: Build the following circuit on the breadboard with three capacitors, s. <imgref Fig-7_V2-Capacitor>:
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 {{drawio>lab_electrical_engineering:2_capacitors:Fig-7_V2-Capacitor.svg}} {{drawio>lab_electrical_engineering:2_capacitors:Fig-7_V2-Capacitor.svg}}
 <imgcaption Fig-7_V2-Capacitor | Capacitors> </imgcaption> <imgcaption Fig-7_V2-Capacitor | Capacitors> </imgcaption>
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-\\ 
  
 Measure the capacitance of capacitors $C_{\rm 1}$, $~C_{\rm 2}$, $~C_{\rm 3}$ with the multimeter and enter the measured values in <tabref Table-2-Capacitor-values_V2>. Measure the capacitance of capacitors $C_{\rm 1}$, $~C_{\rm 2}$, $~C_{\rm 3}$ with the multimeter and enter the measured values in <tabref Table-2-Capacitor-values_V2>.
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 <tabcaption Table-2-Capacitor-values_V2 | Capacitors> </tabcaption> <tabcaption Table-2-Capacitor-values_V2 | Capacitors> </tabcaption>
 \\ \\
-Capacitors can be connected in series and/or in parallel. The total capacitance of two or more capacitors in parallel is calculated as: \\ +Capacitors can be connected in series and/or in parallel. The total capacitance of two or more capacitors in parallel is calculated as:  
-\\ + 
-$C_{\rm total}$ = $C_{\rm 1}$ + $C_{\rm 2}$ + ... + $C_{\rm n}$  +$$C_{\rm total} = C_{\rm 1} + C_{\rm 2} + ... + C_{\rm n}$$ 
-\\ + 
-\\ +The total capacitance of capacitors connected in series is calculated as:  
-The total capacitance of capacitors connected in series is calculated as: \\ + 
-\\ +$$ \frac{1}{C_{\rm total}} = \frac{1}{C_{\rm 1}} + \frac{1}{C_{\rm 2}} + ... + \frac{1}{C_{\rm n}}$$ 
-$$ \frac{1}{C_{\rm total}} = \frac{1}{C_{\rm 1}} + \frac{1}{C_{\rm 2}} + ... \frac{1}{C_{\rm n}}$$+
 Series connection:  Series connection: 
   * $C_{\rm 1}+ C_{\rm 2}$ (measured between terminals 1 and 3)    * $C_{\rm 1}+ C_{\rm 2}$ (measured between terminals 1 and 3) 
   * $C_{\rm 2}+ C_{\rm 3}$ (measured between terminals 3 and 4)    * $C_{\rm 2}+ C_{\rm 3}$ (measured between terminals 3 and 4) 
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 Parallel connection:  Parallel connection: 
   * $C_{\rm 1~} || ~C_{\rm 3}$ (measured between terminals 1 and 2; wire bridge between 1 and 4)    * $C_{\rm 1~} || ~C_{\rm 3}$ (measured between terminals 1 and 2; wire bridge between 1 and 4) 
   * $C_{\rm 1~} || ~C_{\rm 2}$ (measured between terminals 1 and 2; wire bridge between 1 and 3)    * $C_{\rm 1~} || ~C_{\rm 2}$ (measured between terminals 1 and 2; wire bridge between 1 and 3) 
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 Series/parallel connection:  Series/parallel connection: 
   * $C_{\rm 1~}+(C_{\rm 2~} || ~C_{\rm 3})$ (measured between terminals 1 and 3; wire bridge between 3 and 4)    * $C_{\rm 1~}+(C_{\rm 2~} || ~C_{\rm 3})$ (measured between terminals 1 and 3; wire bridge between 3 and 4) 
-\\ \\  
  
 +\\ \\
 Enter the measured and calculated values in <tabref Table-3-Capacitor-meas-calc_V2>. Enter the measured and calculated values in <tabref Table-3-Capacitor-meas-calc_V2>.
  
 {{drawio>lab_electrical_engineering:2_capacitors:Table-3-Capacitor-meas-calc_V2.svg}} {{drawio>lab_electrical_engineering:2_capacitors:Table-3-Capacitor-meas-calc_V2.svg}}
 <tabcaption Table-3-Capacitor-meas-calc_V2 | Capacitor meas. vs. calc.> </tabcaption> <tabcaption Table-3-Capacitor-meas-calc_V2 | Capacitor meas. vs. calc.> </tabcaption>
-\\ +\\ \\
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 The built-in capacitors have values from the E6 series. The E6 series for capacities is shown below. For the measured capacities from <tabref Table-2-Capacitor-values_V2>, determine the matching value from the E6 series and calculate the respective measurement deviation from the nominal value in %.  The built-in capacitors have values from the E6 series. The E6 series for capacities is shown below. For the measured capacities from <tabref Table-2-Capacitor-values_V2>, determine the matching value from the E6 series and calculate the respective measurement deviation from the nominal value in %. 
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 {{drawio>lab_electrical_engineering:2_capacitors:Table-6-E6-series-Cap_V2.svg}} {{drawio>lab_electrical_engineering:2_capacitors:Table-6-E6-series-Cap_V2.svg}}
 <tabcaption Table-6-E6-series-Cap_V2 | E6 series for capacitors> </tabcaption> <tabcaption Table-6-E6-series-Cap_V2 | E6 series for capacitors> </tabcaption>
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 +\\ \\ \\ \\ \\ \\ \\  
 === RC network === === RC network ===
-The capacitance of a capacitor is defined as the quotient of charge by voltage: 
  
 +The capacitance of a capacitor is defined as the quotient of charge by voltage:
 $$ C=\frac{Q}{U} $$ $$ C=\frac{Q}{U} $$
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 Capacitors must be charged via an electrical source <imgref Fig-8_V2-Capacitor-charging>.  Capacitors must be charged via an electrical source <imgref Fig-8_V2-Capacitor-charging>. 
  
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-A capacitor charges the faster the smaller the series resistor $R$ is. During charging, the voltage $u_{\rm C}$ results from a differential equation as: \\+A capacitor charges the faster the smaller the series resistor $R$ is. During charging, the voltage $u_{\rm C}$ results from a differential equation as: 
 $$ u_{\rm C}({\rm t})=U\cdot ({\rm 1}-e^{-\frac{t}{\tau}}), {\rm with~~} \tau = R\cdot C $$ $$ u_{\rm C}({\rm t})=U\cdot ({\rm 1}-e^{-\frac{t}{\tau}}), {\rm with~~} \tau = R\cdot C $$
-\\ + 
-\\ +The constant $\tau$ is called the time constant. After this time, the capacitor is charged to approx. ${\rm 63}~{\rm\%}$. The fundamental equation for the relation between current and voltage at a capacitor is:  
-The constant $\tau$ is called the time constant. After this time, the capacitor is charged to approx. ${\rm 63}~{\rm\%}$. The fundamental equation for the relation between current and voltage at a capacitor is: \\ +
-\\+
 $$ i_{\rm C}({\rm t})=C\cdot \frac{{\rm d}u_{\rm C}}{{\rm dt}} $$  $$ i_{\rm C}({\rm t})=C\cdot \frac{{\rm d}u_{\rm C}}{{\rm dt}} $$ 
-\\ + 
-From the two equations, the current through the capacitor is:\\+From the two equations, the current through the capacitor is:
 $$ i_{\rm C}({\rm t})=\frac{U}{R}\cdot e^{-\frac{t}{\tau}} $$  $$ i_{\rm C}({\rm t})=\frac{U}{R}\cdot e^{-\frac{t}{\tau}} $$ 
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 The graphical representation of voltage and current during the charging of a capacitor over time is shown in <imgref Fig-9_V2-Capacitor-voltage-current>.  The graphical representation of voltage and current during the charging of a capacitor over time is shown in <imgref Fig-9_V2-Capacitor-voltage-current>. 
  
-{{lab_electrical_engineering:2_capacitors:drawio>Fig-9_V2-Capacitor-voltage-current.svg}} +{{drawio>lab_electrical_engineering:2_capacitors:Fig-9_V2-Capacitor-voltage-current.svg}} 
-<imgcaption Fig-9_V2-Capacitor-voltage-current | Voltage/Current in case of charging capacitor> </imgcaption> +<imgcaption Fig-9_V2-Capacitor-voltage-current | Voltage/Current in case of charging capacitor> </imgcaption> \\ \\ 
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 Because of the exponential function, charging is theoretically only complete after an infinitely long time. The capacitor voltage equals ${\rm 63}~{\rm\%~} U$ after ${\rm 1}\cdot \tau$, ${\rm 86}~{\rm\%~} U$ after ${\rm 2}\cdot \tau$, ${\rm 95}~{\rm\%~} U$ after ${\rm 3}\cdot \tau$, ${\rm 98}~{\rm\%~} U$ after ${\rm 4}\cdot \tau$, and ${\rm 99}~{\rm\%~} U$ after ${\rm 5}\cdot \tau$. It is assumed that the capacitor is fully charged after a time span $T = {\rm 5}\cdot \tau$ and the voltage across the capacitor has reached $U$. If the charged capacitor $C$ is discharged through a resistor $R$, the solution of the differential equation for the voltage is: \\ Because of the exponential function, charging is theoretically only complete after an infinitely long time. The capacitor voltage equals ${\rm 63}~{\rm\%~} U$ after ${\rm 1}\cdot \tau$, ${\rm 86}~{\rm\%~} U$ after ${\rm 2}\cdot \tau$, ${\rm 95}~{\rm\%~} U$ after ${\rm 3}\cdot \tau$, ${\rm 98}~{\rm\%~} U$ after ${\rm 4}\cdot \tau$, and ${\rm 99}~{\rm\%~} U$ after ${\rm 5}\cdot \tau$. It is assumed that the capacitor is fully charged after a time span $T = {\rm 5}\cdot \tau$ and the voltage across the capacitor has reached $U$. If the charged capacitor $C$ is discharged through a resistor $R$, the solution of the differential equation for the voltage is: \\
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 $$ u_{\rm C}({\rm t})=U\cdot e^{-\frac{t}{\tau}} $$ $$ u_{\rm C}({\rm t})=U\cdot e^{-\frac{t}{\tau}} $$
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 For the current accordingly: \\ For the current accordingly: \\
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 $$ i_{\rm C}({\rm t})=- \frac{U}{R}\cdot e^{-\frac{t}{\tau}} $$  $$ i_{\rm C}({\rm t})=- \frac{U}{R}\cdot e^{-\frac{t}{\tau}} $$ 
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 Now build the following circuit. Connect the function generator and the oscilloscope to the circuit as shown in <imgref Fig-10_V2-Osci-Function-gen>.  Now build the following circuit. Connect the function generator and the oscilloscope to the circuit as shown in <imgref Fig-10_V2-Osci-Function-gen>. 
  
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 {{drawio>lab_electrical_engineering:2_capacitors:Table-4-time-constant-meas_V2.svg}} {{drawio>lab_electrical_engineering:2_capacitors:Table-4-time-constant-meas_V2.svg}}
-<tabcaption Table-4-time-constant-meas_V2 | Capacitor meas. + time constant $\tau$> </tabcaption>+<tabcaption Table-4-time-constant-meas_V2 | Capacitor meas. + time constant $\tau$> </tabcaption> \\
  
 Set the voltage $u_{\rm F}$ generated by the function generator to a unipolar square with amplitude 5 $V$ (i.e., no negative signal voltages occur!). The frequency on the function generator must be chosen so that the capacitor just fully charges and then fully discharges again. \\ Set the voltage $u_{\rm F}$ generated by the function generator to a unipolar square with amplitude 5 $V$ (i.e., no negative signal voltages occur!). The frequency on the function generator must be chosen so that the capacitor just fully charges and then fully discharges again. \\
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 \\ \\ \\ \\
 $f_{\rm 1} =~{\rm ...............}$ $f_{\rm 1} =~{\rm ...............}$
-\\ + 
-\\+\\ \\  
 Sketch the voltages measured with the oscilloscope for $u_{\rm F}$, $u_{\rm C}$, and $u_{\rm R}$ in the following screen diagram. Also enter alongside the screen drawings the set $\frac{V}{\rm DIV}$ of the channels and the $\frac{T}{\rm DIV}$ of the time base. \\ Sketch the voltages measured with the oscilloscope for $u_{\rm F}$, $u_{\rm C}$, and $u_{\rm R}$ in the following screen diagram. Also enter alongside the screen drawings the set $\frac{V}{\rm DIV}$ of the channels and the $\frac{T}{\rm DIV}$ of the time base. \\
 +
 +<wrap left>
 +{{drawio>lab_electrical_engineering:2_capacitors:Fig-10_V2-Rectangle-bipolar-5kHz-U-2V.svg}}
 +<imgcaption Fig-neu | u_F, u_C, u_R> </imgcaption>
 +</wrap>
 +
 +Channel 1:  $ \frac{V}{\rm DIV} =  $ \\ \\
 +Channel 2:  $ \frac{V}{\rm DIV} =  $ \\ \\
 +Time basis: $ \frac{T}{\rm DIV} =  $ \\ 
 +
 +~~PAGEBREAK~~ ~~CLEARFIX~~
 +
 \\ \\
  
-Draw **tangents** in the screen diagram for the start of charging and the start of discharging. What is the charging current or discharging current at the beginning?\\ +Draw **tangents** in the screen diagram for the start of charging and the start of discharging. \\  
-\\ +What is the charging current or discharging current at the beginning?\\ 
-\\ + 
-${\rm .....................................................}$+\\ \\ ${\rm ........................................................................................}$ 
 +\\ \\ ${\rm ........................................................................................}$
 \\ \\ \\ \\ \\ \\
-The circuit is now to be operated at higher frequency. \\ +The circuit is now to be operated at higher frequencies. Set the frequency to: 
-Set: +
   *$f_{\rm 2} = ~{\rm 10}\cdot f_{\rm 1}$   *$f_{\rm 2} = ~{\rm 10}\cdot f_{\rm 1}$
   *$f_{\rm 3} = ~{\rm 100}\cdot f_{\rm 1}$   *$f_{\rm 3} = ~{\rm 100}\cdot f_{\rm 1}$
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 {{drawio>lab_electrical_engineering:2_capacitors:Table-5-voltage-curves_V2.svg}} {{drawio>lab_electrical_engineering:2_capacitors:Table-5-voltage-curves_V2.svg}}
-<tabcaption Table-5-voltage-curves_V2 | Voltage curve u_F u_C> </tabcaption>+<tabcaption Table-5-voltage-curves_V2 | Voltage curve u_F u_C> </tabcaption> \\ 
  
 Explain your observations for the measurements with $f_{\rm 2}$, $f_{\rm 3}$:\\ Explain your observations for the measurements with $f_{\rm 2}$, $f_{\rm 3}$:\\
 +\\ \\ ${\rm ........................................................................................}$
 +\\ \\ ${\rm ........................................................................................}$ 
 +\\ \\ ${\rm ........................................................................................}$ 
 +\\ \\ ${\rm ........................................................................................}$ 
 +
 \\ \\  \\ \\ 
-${\rm ........................................................................................}$ \\ \\ 
-${\rm ........................................................................................}$ \\ \\ 
-${\rm ........................................................................................}$ \\ \\ 
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