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--- electrical_engineering_1:dc_circuit_transients [2021/10/30 20:12]
tfischer
+++ electrical_engineering_1:dc_circuit_transients [2021/10/31 20:51]
tfischer
@@ Zeile 26: / 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 $C$:
-\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:
@@ Zeile 103: / 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:
-\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]].
@@ Zeile 113: / 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.
-\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:
-\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:
-\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:
-\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: #