Unterschiede

Hier werden die Unterschiede zwischen zwei Versionen angezeigt.

Link zu dieser Vergleichsansicht

Beide Seiten der vorigen Revision Vorhergehende Überarbeitung
Nächste Überarbeitung 		Vorhergehende Überarbeitung
Nächste Überarbeitung Beide Seiten der Revision
electrical_engineering_1:dc_circuit_transients [2021/10/30 20:03]
tfischer 		electrical_engineering_1:dc_circuit_transients [2021/10/31 20:51]
tfischer
Zeile 10: Zeile 10:
 <callout> <callout>
  
-<WRAP> <imgcaption imageNo01 | Capacitor in electrical circuit> </imgcaption> \\ {{drawio>KondensatorImStromkreis}} \\ </WRAP>+<WRAP>  
 +<imgcaption imageNo01 | Capacitor in electrical circuit>  
 +</imgcaption>  
 +\\ {{drawio>KondensatorImStromkreis}} \\  
 +</WRAP>
  
 Here we will shortly introduce the basic idea behind a capacitor. A more detailed analysis will follow in electical engineering II. \\  \\  Here we will shortly introduce the basic idea behind a capacitor. A more detailed analysis will follow in electical engineering II. \\  \\ 
Zeile 22: Zeile 26:
 As larger the voltage $U$, as more charges $Q$ are stored on the electrode. This relationship is directly proportional to the proportionality constant $C$: As larger the voltage $U$, as more charges $Q$ are stored on the electrode. This relationship is directly proportional to the proportionality constant $C$:
  
-\begin{align*} C = {{Q}\over{U}} \quad \text{with:} \quad [C]=1 {{As}\over{V}}= 1 F = 1\; Farad \end{align*}+\begin{align*}  
 +C = {{Q}\over{U}} \quad \text{with:} \quad [C]=1 {{As}\over{V}}= 1 F = 1\; Farad  
 +\end{align*}
  
 But it is not always directly recognizable that a structure contains a capacitor. \\ So the following examples are also capacitors: But it is not always directly recognizable that a structure contains a capacitor. \\ So the following examples are also capacitors:
Zeile 99: Zeile 105:
 The process is now to be summarized in detail in formulas. Linear components are used in the circuit, i.e. the component values for the resistor $R$ and the capacitance $C$ are independent of the current or the voltage. Then definition equations for the resistor $R$ and the capacitance $C$ are also valid for time-varying or infinitesimal quantities: The process is now to be summarized in detail in formulas. Linear components are used in the circuit, i.e. the component values for the resistor $R$ and the capacitance $C$ are independent of the current or the voltage. Then definition equations for the resistor $R$ and the capacitance $C$ are also valid for time-varying or infinitesimal quantities:
  
-\begin{align*} R = {{u_R(t)}\over{i_R(t)}} = {{du_R}\over{di_R}} = const. \\ +\begin{align*}  
-C = {{q(t)}\over{u_C(t)}} = {{dq}\over{du_C}} = const. \tag{5.1.1} \end{align*}+R = {{u_R(t)}\over{i_R(t)}} = {{du_R}\over{di_R}} = const. \\ 
 +C = {{q(t)}  \over{u_C(t)}} = {{dq}  \over{du_C}} = const. \tag{5.1.1}  
 +\end{align*}
  
 The following explanations are also well explained in these two videos on [[https://www.youtube.com/watch?v=csFh588BODY&ab_channel=MattAnderson|charging]] and [[https://www.youtube.com/watch?v=eCOLkUPSpxc&ab_channel=lasseviren1|discharging]]. The following explanations are also well explained in these two videos on [[https://www.youtube.com/watch?v=csFh588BODY&ab_channel=MattAnderson|charging]] and [[https://www.youtube.com/watch?v=eCOLkUPSpxc&ab_channel=lasseviren1|discharging]].
Zeile 109: Zeile 117:
 By considering the loop, the general result is: the voltage of the source is equal to the sum of the two voltages across the resistor and capacitor. By considering the loop, the general result is: the voltage of the source is equal to the sum of the two voltages across the resistor and capacitor.
  
-\begin{align*} U_q =u_R + u_C = R \cdot i_C + u_C \tag{5.1.2} \end{align*}+\begin{align*}  
 +U_q =u_R + u_C = R \cdot i_C + u_C \tag{5.1.2}  
 +\end{align*}
  
 At the first instant $dt$, an infinitesimally small charge "chunk" $dq$ flows through the circuit driven by the current $i_C$ from the voltage source. For this, $(5.1.1)$ gives: At the first instant $dt$, an infinitesimally small charge "chunk" $dq$ flows through the circuit driven by the current $i_C$ from the voltage source. For this, $(5.1.1)$ gives:
  
-\begin{align*} i_C = {{dq}\over{dt}} \quad \text{and} \quad dq = C \cdot du_C \end{align*}+\begin{align*}  
 +i_C = {{dq}\over{dt}} \quad \text{and} \quad dq = C \cdot du_C  
 +\end{align*}
  
 The charging current $i_C$ can be determined from the two formulas: The charging current $i_C$ can be determined from the two formulas:
  
-\begin{align*} i_C = C \cdot {{du_C}\over{dt}} \tag{5.1.3} \end{align*}+\begin{align*}  
 +i_C = C \cdot {{du_C}\over{dt}} \tag{5.1.3}  
 +\end{align*}
  
 Thus $(5.1.2)$ becomes: Thus $(5.1.2)$ becomes:
  
-\begin{align*} U_q &=u_R + u_C &= R \cdot C \cdot {{du_C}\over{dt}} + u_C \end{align*}+\begin{align*}  
 +U_q &= u_R                               + u_C \\ 
 +    &= R \cdot C \cdot {{du_C}\over{dt}} + u_C  
 +\end{align*}
  
 --> here follows some mathematics: # --> here follows some mathematics: #