Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung Nächste Überarbeitung Beide Seiten der Revision | ||
electrical_engineering_1:dc_circuit_transients [2021/10/31 20:51] tfischer |
electrical_engineering_1:dc_circuit_transients [2021/11/22 04:50] tfischer |
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Zeile 74: | Zeile 74: | ||
</ | </ | ||
- | < | + | In the simulation on the below you can see the circuit |
- | In the simulation on the right you can see the circuit mentioned above in a slightly modified form: | + | |
- | + | ||
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* But it is also possible to short-circuit the series circuit of $R$ and $C$ via the switch $S$. | * But it is also possible to short-circuit the series circuit of $R$ and $C$ via the switch $S$. | ||
* Furthermore, | * Furthermore, | ||
Zeile 87: | Zeile 85: | ||
- Become familiar with how the capacitor current $i_C$ and capacitor voltage $u_C$ depend on the given capacitance $C$ and resistance $R$. \\ To do this, use for $R=\{ 10\Omega, 100\Omega, 1k\Omega\}$ and $C=\{ 1\mu F, 10 \mu F\}$. How fast does the capacitor voltage $u_C$ increase in each case n? | - Become familiar with how the capacitor current $i_C$ and capacitor voltage $u_C$ depend on the given capacitance $C$ and resistance $R$. \\ To do this, use for $R=\{ 10\Omega, 100\Omega, 1k\Omega\}$ and $C=\{ 1\mu F, 10 \mu F\}$. How fast does the capacitor voltage $u_C$ increase in each case n? | ||
- Which quantity ($i_C$ or $u_C$) is continuous here? Why must this one be continuous? Why must the other quantity be discontinuous? | - Which quantity ($i_C$ or $u_C$) is continuous here? Why must this one be continuous? Why must the other quantity be discontinuous? | ||
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+ | < | ||
At the following, this circuit is divided into two separate circuits, which consider only charging and only discharging. | At the following, this circuit is divided into two separate circuits, which consider only charging and only discharging. | ||
Zeile 92: | Zeile 92: | ||
~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
- | < | + | Here a short introduction about the transient behavior of an RC element (starting at 15:07 until 24:55) |
+ | {{youtube>8nyNamrWcyE? | ||
- | To understand the charging process of a capacitor, an initially uncharged capacitor with capacitance $C$ is to be charged by a DC voltage source $U_q$ via a resistor $R$. | + | To understand the charging process of a capacitor, an initially uncharged capacitor with capacitance $C$ is to be charged by a DC voltage source $U_s$ via a resistor $R$. |
- | * In order that the voltage $U_q$ acts at a certain time $t_0 = 0 s$ the switch $S$ is closed at this time. | + | * In order that the voltage $U_s$ acts at a certain time $t_0 = 0 s$ the switch $S$ is closed at this time. |
- | * Directly after the time $t_0$ the maximum current (" | + | * Directly after the time $t_0$ the maximum current (" |
* The current causes charge carriers to flow from one electrode to the other. Thus the capacitor is charged and its voltage increases $u_C$. | * The current causes charge carriers to flow from one electrode to the other. Thus the capacitor is charged and its voltage increases $u_C$. | ||
* Thus the voltage $u_R$ across the resistor is reduced and so is the current $i_R$. | * Thus the voltage $u_R$ across the resistor is reduced and so is the current $i_R$. | ||
* With the current thus reduced, less charge flows on the capacitor. | * With the current thus reduced, less charge flows on the capacitor. | ||
- | * Ideally, the capacitor is not fully charged to the specified voltage $U_q$ until $t \rightarrow \infty$. It then carries the charge: $q(t \rightarrow \infty)=Q = C \cdot U_q$ | + | * Ideally, the capacitor is not fully charged to the specified voltage $U_s$ until $t \rightarrow \infty$. It then carries the charge: $q(t \rightarrow \infty)=Q = C \cdot U_s$ |
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+ | < | ||
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: | ||
Zeile 118: | Zeile 121: | ||
\begin{align*} | \begin{align*} | ||
- | U_q =u_R + u_C = R \cdot i_C + u_C \tag{5.1.2} | + | U_s =u_R + u_C = R \cdot i_C + u_C \tag{5.1.2} |
\end{align*} | \end{align*} | ||
Zeile 136: | Zeile 139: | ||
\begin{align*} | \begin{align*} | ||
- | U_q &= u_R + u_C \\ | + | U_s &= u_R + u_C \\ |
&= R \cdot C \cdot {{du_C}\over{dt}} + u_C | &= R \cdot C \cdot {{du_C}\over{dt}} + u_C | ||
\end{align*} | \end{align*} | ||
Zeile 149: | Zeile 152: | ||
\begin{align*} | \begin{align*} | ||
- | U_q &= R \cdot C \cdot {{d}\over{dt}}(\mathcal{A} \cdot e^{\mathcal{B}\cdot t} + \mathcal{C}) + \mathcal{A} \cdot e^{\mathcal{B}\cdot t} + \mathcal{C} \\ | + | U_s &= R \cdot C \cdot {{d}\over{dt}}(\mathcal{A} \cdot e^{\mathcal{B}\cdot t} + \mathcal{C}) + \mathcal{A} \cdot e^{\mathcal{B}\cdot t} + \mathcal{C} \\ |
&= R \cdot C \cdot \mathcal{AB} \cdot e^{\mathcal{B}\cdot t} + \mathcal{A} \cdot e^{\mathcal{B}\cdot t} + \mathcal{C} \\ | &= R \cdot C \cdot \mathcal{AB} \cdot e^{\mathcal{B}\cdot t} + \mathcal{A} \cdot e^{\mathcal{B}\cdot t} + \mathcal{C} \\ | ||
- | U_q - \mathcal{C} & | + | U_s - \mathcal{C} & |
\end{align*} | \end{align*} | ||
Zeile 167: | Zeile 170: | ||
\begin{align*} | \begin{align*} | ||
- | u_C(t) = \mathcal{A} \cdot e^{\large{- {{t}\over{R C}} }} + U_q | + | u_C(t) = \mathcal{A} \cdot e^{\large{- {{t}\over{R C}} }} + U_s |
\end{align*} | \end{align*} | ||
Zeile 173: | Zeile 176: | ||
\begin{align*} | \begin{align*} | ||
- | 0 &= \mathcal{A} \cdot e^{\large{0}} + U_q \\ | + | 0 &= \mathcal{A} \cdot e^{\large{0}} + U_s \\ |
- | 0 &= \mathcal{A} | + | 0 &= \mathcal{A} |
- | \mathcal{A} &= - U_q | + | \mathcal{A} &= - U_s |
\end{align*} | \end{align*} | ||
Zeile 182: | Zeile 185: | ||
\begin{align*} | \begin{align*} | ||
- | u_C(t) &= - U_q \cdot e^{\large{- {{t}\over{R C}}}} + U_q | + | u_C(t) &= - U_s \cdot e^{\large{- {{t}\over{R C}}}} + U_s |
\end{align*} | \end{align*} | ||
Zeile 189: | Zeile 192: | ||
And this results in: | And this results in: | ||
\begin{align*} | \begin{align*} | ||
- | u_C(t) & | + | u_C(t) & |
\end{align*} | \end{align*} | ||
And with $(5.1.3)$, $i_C$ becomes: | And with $(5.1.3)$, $i_C$ becomes: | ||
\begin{align*} | \begin{align*} | ||
- | i_C(t) &= {{U_q}\over{R}} \cdot e^{\large{- {{t}\over{R C}} } } | + | i_C(t) &= {{U_s}\over{R}} \cdot e^{\large{- {{t}\over{R C}} } } |
\end{align*} | \end{align*} | ||
Zeile 209: | Zeile 212: | ||
* There must be a unitless term in the exponent. So $RC$ must also represent a time. This time is called **time constant** | * There must be a unitless term in the exponent. So $RC$ must also represent a time. This time is called **time constant** | ||
- | * At time $t=\tau$, we get: $u_C(t) = U_q \cdot (1 - e^{- 1}) = U_q \cdot (1 - {{1}\over{e}}) = U_q \cdot ({{e-1}\over{e}}) = 0.63 \cdot U_q = 63\% \cdot U_q $. \\ So, **the capacitor is charged to $63\%$ after one $\tau$.** | + | * At time $t=\tau$, we get: $u_C(t) = U_s \cdot (1 - e^{- 1}) = U_s \cdot (1 - {{1}\over{e}}) = U_s \cdot ({{e-1}\over{e}}) = 0.63 \cdot U_s = 63\% \cdot U_s $. \\ So, **the capacitor is charged to $63\%$ after one $\tau$.** |
- | * At time $t=2 \cdot \tau$ we get: $u_C(t) = U_q \cdot (1 - e^{- 2}) = 86\% \cdot U_q = (63\% + (1-63\%) \cdot 63\% ) \cdot U_q$. So, **after each additional $\tau$, the uncharged remainder ($1-63\%$) is recharged to $63\%$**. | + | * At time $t=2 \cdot \tau$ we get: $u_C(t) = U_s \cdot (1 - e^{- 2}) = 86\% \cdot U_s = (63\% + (1-63\%) \cdot 63\% ) \cdot U_s$. So, **after each additional $\tau$, the uncharged remainder ($1-63\%$) is recharged to $63\%$**. |
* After about $t=5 \cdot \tau$, the result is a capacitor charged to over $99\%$. In real circuits, **a charged capacitor can be assumed after** $5 \cdot \tau$. | * After about $t=5 \cdot \tau$, the result is a capacitor charged to over $99\%$. In real circuits, **a charged capacitor can be assumed after** $5 \cdot \tau$. | ||
* The time constant $\tau$ can be determined graphically in several ways: | * The time constant $\tau$ can be determined graphically in several ways: | ||
Zeile 229: | Zeile 232: | ||
The following situation is considered for the discharge: | The following situation is considered for the discharge: | ||
- | * A capacitor charged to voltage $U_q$ with capacitance $C$ is short-circuited across a resistor $R$ at time $t=t_0$. | + | * A capacitor charged to voltage $U_s$ with capacitance $C$ is short-circuited across a resistor $R$ at time $t=t_0$. |
- | * As a result, the full voltage $U_q$ is initially applied to the resistor: $u_R(t_0)=U_q$ | + | * As a result, the full voltage $U_s$ is initially applied to the resistor: $u_R(t_0)=U_s$ |
* The initial discharge current is thus defined by the resistance: $i_C ={{u_R}\over{R}}$ | * The initial discharge current is thus defined by the resistance: $i_C ={{u_R}\over{R}}$ | ||
* The discharging charges lower the voltage of the capacitor $u_C$, since: $u_C = {{q(t)}\over{C}}$ | * The discharging charges lower the voltage of the capacitor $u_C$, since: $u_C = {{q(t)}\over{C}}$ | ||
Zeile 256: | Zeile 259: | ||
\begin{align*} | \begin{align*} | ||
- | U_q &= R \cdot C \cdot {{d}\over{dt}}(\mathcal{A} \cdot e^{\mathcal{B}\cdot t} + \mathcal{C}) + \mathcal{A} \cdot e^{\mathcal{B}\cdot t} + \mathcal{C} \\ | + | U_s &= R \cdot C \cdot {{d}\over{dt}}(\mathcal{A} \cdot e^{\mathcal{B}\cdot t} + \mathcal{C}) + \mathcal{A} \cdot e^{\mathcal{B}\cdot t} + \mathcal{C} \\ |
&= R \cdot C \cdot \mathcal{AB} \cdot e^{\mathcal{B}\cdot t} + \mathcal{A} \cdot e^{\mathcal{B}\cdot t} + \mathcal{C} \\ | &= R \cdot C \cdot \mathcal{AB} \cdot e^{\mathcal{B}\cdot t} + \mathcal{A} \cdot e^{\mathcal{B}\cdot t} + \mathcal{C} \\ | ||
- | U_q - \mathcal{C} & | + | U_s - \mathcal{C} & |
\end{align*} | \end{align*} | ||
Zeile 264: | Zeile 267: | ||
\begin{align*} | \begin{align*} | ||
- | \mathcal{C} = U_q \\ \\ | + | \mathcal{C} = U_s \\ \\ |
R \cdot C \cdot \mathcal{AB} + \mathcal{A} &= 0 \quad \quad | : \mathcal{A} \quad | -1 \\ | R \cdot C \cdot \mathcal{AB} + \mathcal{A} &= 0 \quad \quad | : \mathcal{A} \quad | -1 \\ | ||
Zeile 274: | Zeile 277: | ||
\begin{align*} | \begin{align*} | ||
- | u_C(t) = \mathcal{A} \cdot e^{\large{- {{t}\over{R C}} }} + U_q | + | u_C(t) = \mathcal{A} \cdot e^{\large{- {{t}\over{R C}} }} + U_s |
\end{align*} | \end{align*} | ||
- | For the solution it must still hold that at time $t_0=0$ $u_C(t_0) = U_q$ just holds: | + | For the solution it must still hold that at time $t_0=0$ $u_C(t_0) = U_s$ just holds: |
\begin{align*} | \begin{align*} | ||
- | 0 &= \mathcal{A} \cdot e^{\large{0}} + U_q \\ | + | 0 &= \mathcal{A} \cdot e^{\large{0}} + U_s \\ |
- | 0 &= \mathcal{A} | + | 0 &= \mathcal{A} |
- | \mathcal{A} &= - U_q | + | \mathcal{A} &= - U_s |
\end{align*} | \end{align*} | ||
Zeile 288: | Zeile 291: | ||
\begin{align*} | \begin{align*} | ||
- | u_C(t) &= - U_q \cdot e^{\large{- {{t}\over{R C}}}} + U_q | + | u_C(t) &= - U_s \cdot e^{\large{- {{t}\over{R C}}}} + U_s |
\end{align*} | \end{align*} | ||
Zeile 301: | Zeile 304: | ||
And this results in: | And this results in: | ||
\begin{align*} | \begin{align*} | ||
- | u_C(t) & | + | u_C(t) & |
\end{align*} | \end{align*} | ||
And with $(5.1.3)$, $i_C$ becomes: | And with $(5.1.3)$, $i_C$ becomes: | ||
\begin{align*} | \begin{align*} | ||
- | i_C(t) &= {{U_q}\over{R}} \cdot e^{\large{- {{t}\over{R C}} } } | + | i_C(t) &=- {{U_s}\over{R}} \cdot e^{\large{- {{t}\over{R C}} } } |
\end{align*} | \end{align*} | ||
- | In <imgref imageNo05 > the two time histories | + | In <imgref imageNo05 > the two time courses |
~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
Zeile 343: | Zeile 346: | ||
< | < | ||
- | Now the capacitor as energy storage is to be looked at more closely. This derivation is also explained in [[https:// | + | Now the capacitor as energy storage is to be looked at more closely. This derivation is also explained in [[https:// |
\begin{align*} | \begin{align*} | ||
Zeile 365: | Zeile 368: | ||
During the charging process | During the charging process | ||
\begin{align*} | \begin{align*} | ||
- | u_C(t) = U_q\cdot (1 - e^{ - {{t}\over{\tau}} }) \\ | + | u_C(t) = U_s\cdot (1 - e^{ - {{t}\over{\tau}} }) \\ |
- | i_C(t) = {{U_q}\over{R}} \cdot e^{ -{{t}\over{\tau}} } \tag{5.2.2} | + | i_C(t) = {{U_s}\over{R}} \cdot e^{ -{{t}\over{\tau}} } \tag{5.2.2} |
\end{align*} | \end{align*} | ||
Zeile 388: | Zeile 391: | ||
\end{align*} | \end{align*} | ||
- | Thus, for a fully discharged capacitor ($U_q=0V$), the energy stored when charging to voltage $U_q$ is $\delta W_C={{1}\over{2}} C \cdot U_q^2$. | + | Thus, for a fully discharged capacitor ($U_s=0V$), the energy stored when charging to voltage $U_s$ is $\delta W_C={{1}\over{2}} C \cdot U_s^2$. |
=== Energy consideration on the resistor === | === Energy consideration on the resistor === | ||
Zeile 401: | Zeile 404: | ||
\begin{align*} | \begin{align*} | ||
- | \Delta W_R & | + | \Delta W_R & |
- | & | + | & |
- | & | + | & |
- | & | + | & |
\end{align*} | \end{align*} | ||
Zeile 410: | Zeile 413: | ||
\begin{align*} | \begin{align*} | ||
- | \Delta W_R & | + | \Delta W_R & |
- | & | + | & |
\end{align*} | \end{align*} | ||
\begin{align*} | \begin{align*} | ||
- | \boxed{ \Delta W_R = {{1}\over{2}} \cdot {U_q^2}\cdot{C}} \tag{5.2.4} | + | \boxed{ \Delta W_R = {{1}\over{2}} \cdot {U_s^2}\cdot{C}} \tag{5.2.4} |
\end{align*} | \end{align*} | ||
- | This means that the energy converted at the resistor is independent of the resistance value (for an ideal constant voltage source $U_q$ and given capacitor $C$)! At first, this doesn' | + | This means that the energy converted at the resistor is independent of the resistance value (for an ideal constant voltage source $U_s$ and given capacitor $C$)! At first, this doesn' |
In real applications, | In real applications, | ||
Zeile 423: | Zeile 426: | ||
=== Consideration of total energy turnover === | === Consideration of total energy turnover === | ||
- | In the previous considerations, | + | In the previous considerations, |
\begin{align*} | \begin{align*} | ||
- | \Delta W_0 & | + | \Delta W_0 & |
\end{align*} | \end{align*} | ||
Zeile 432: | Zeile 435: | ||
\begin{align*} | \begin{align*} | ||
- | \Delta W_0 & | + | \Delta W_0 & |
- | & | + | & |
- | & | + | & |
- | & | + | & |
- | & | + | & |
\end{align*} | \end{align*} | ||
- | This means that only half of the energy emitted by the source is stored in the capacitor! Again, this doesn' | + | This means that only half of the energy emitted by the source is stored in the capacitor! Again, this doesn' |
< | < |