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| Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung Nächste Überarbeitung Beide Seiten der Revision | ||
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electrical_engineering_1:dc_circuit_transients [2021/10/30 19:23] tfischer |
electrical_engineering_1:dc_circuit_transients [2021/10/31 20:51] tfischer |
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| Zeile 10: | Zeile 10: | ||
| < | < | ||
| - | < | + | < |
| + | < | ||
| + | </ | ||
| + | \\ {{drawio> | ||
| + | </ | ||
| 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: | + | \begin{align*} |
| + | C = {{Q}\over{U}} \quad \text{with: | ||
| + | \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 44: | Zeile 50: | ||
| < | < | ||
| - | In this chapter also time-varying quantities are considered. These are generally marked with small letters. Examples of time-varying quantities are: | + | In this chapter also time-varying quantities are considered. These are generally marked with lowercase |
| * A **time-varying voltage $u_C(t)$ across a capacitor** | * A **time-varying voltage $u_C(t)$ across a capacitor** | ||
| * A **time-varying current $i_L(t)$ across a coil** or **time-varying current $i_C(t)$ across a capacitor**. | * A **time-varying current $i_L(t)$ across a coil** or **time-varying current $i_C(t)$ across a capacitor**. | ||
| - | Since the time dependence is already clear from the small letter, these quantities are occasionally not indicated by the trailing $(t)$. So it is $u = u(t)$. | + | Since the time dependence is already clear from the lowercase |
| </ | </ | ||
| - | ===== 5.1 Time course | + | ===== 5.1 Time Course |
| < | < | ||
| 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)} | ||
| + | \end{align*} | ||
| The following explanations are also well explained in these two videos on [[https:// | The following explanations are also well explained in these two videos on [[https:// | ||
| 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 " | At the first instant $dt$, an infinitesimally small charge " | ||
| - | \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 | ||
| + | | ||
| + | \end{align*} | ||
| --> here follows some mathematics: | --> here follows some mathematics: | ||
| Zeile 310: | Zeile 327: | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ===== 5.2 Energy | + | ===== 5.2 Energy |
| < | < | ||