Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
| electrical_engineering_1:dc_circuit_transients [2021/10/30 20:12] – tfischer | electrical_engineering_1:dc_circuit_transients [2024/11/08 15:21] (aktuell) – mexleadmin | ||
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| Zeile 1: | Zeile 1: | ||
| - | ====== 5. DC Circuit Transients (ob RC elements) ====== | + | ====== 5 DC Circuit Transients (on RC elements) ====== |
| <WRAP onlyprint> | <WRAP onlyprint> | ||
| - | - Capacitor in IC' | + | - Capacitor in ICs --> MOSFET |
| - | - Charge / discharge FET capacitor. | + | - Charge/ |
| </ | </ | ||
| Zeile 13: | Zeile 13: | ||
| < | < | ||
| </ | </ | ||
| - | \\ {{drawio> | + | \\ {{drawio> |
| </ | </ | ||
| - | Here we will shortly introduce the basic idea behind a capacitor. A more detailed analysis will follow in electical | + | Here we will shortly introduce the basic idea behind a capacitor. A more detailed analysis will follow in electrical |
| A capacitor consists of two insulated conductors (electrodes) separated by an insulator (cf. <imgref imageNo01 >). \\ | A capacitor consists of two insulated conductors (electrodes) separated by an insulator (cf. <imgref imageNo01 >). \\ | ||
| The electrodes serve as " | The electrodes serve as " | ||
| Zeile 24: | Zeile 24: | ||
| - These charges form an electric field in the space between the electrodes. This field stores the supplied energy. | - These charges form an electric field in the space between the electrodes. This field stores the supplied energy. | ||
| - | 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 $U$, 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: | ||
| - | * **open switch**: If there is voltage between the two metal parts, charges can also accumulate there. \\ Since the distances are usually large and air is used as the dielectric, the capacitance of the capacitor formed in this way is very small. | + | * **open switch**: If there is a voltage between the two metal parts, charges can also accumulate there. \\ Since the distances are usually large and the air is used as the dielectric, the capacitance of the capacitor formed in this way is very small. |
| - | * **Overhead line**: An overhead line also represents a capacitor against the ground potential of the earth. The charging and discharging by the alternating current leads to the fact that polarizable molecules can align themselves. For example, the water drops near the line are rolled through the field and hum with $100Hz$ and many times that (harmonics). Peak discharge results in the high frequency crackle. | + | * **Overhead line**: An overhead line also represents a capacitor against the ground potential of the earth. The charging and discharging by the alternating current leads to the fact that polarizable molecules can align themselves. For example, the water drops near the line are rolled through the field and hum with $100~\rm Hz$ and many times that (harmonics). Peak discharge results in a high-frequency crackle. |
| - | * **Conductor trace**: A trace on a PCB can also be a capacitor against a nearby ground plane. This can be a problem for digital signals (see charge and discharge curves below). | + | * **Conductor trace**: A trace on a PCB can also be a capacitor against a nearby ground plane. This can be a problem for digital signals (see the charge and discharge curves below). |
| * **Human body**: The human body can likewise pick up charge. The charge thus absorbed forms a capacitor with respect to other objects. This can be charged up to some $kV$. This is a particular problem in electrical laboratories, | * **Human body**: The human body can likewise pick up charge. The charge thus absorbed forms a capacitor with respect to other objects. This can be charged up to some $kV$. This is a particular problem in electrical laboratories, | ||
| - | * **Membrane of nerve cells**: Nerve cells also result in a capacitor due to the lipid bilayer (membrane of the nerve cell) and the two cellular fluids with different electrolytes (ions). The nerve cells are surrounded by a thick layer (myelin layer) for faster transmission. This lowers the capacitance and thus increases the successive charging of successive parts of the nerve cell. In diseases such as Creutzfeldt-Jakob or multiple sclerosis, this layer thins out. This leads to delayed signal transmission which characterizes the disease patterns. | + | * **Membrane of nerve cells**: Nerve cells also result in a capacitor due to the lipid bilayer (membrane of the nerve cell) and the two cellular fluids with different electrolytes (ions). The nerve cells are surrounded by a thick layer (myelin layer) for faster transmission. This lowers the capacitance and thus increases the successive charging of successive parts of the nerve cell. In diseases such as Creutzfeldt-Jakob or multiple sclerosis, this layer thins out. This leads to the delayed signal transmission which characterizes the disease patterns. |
| - | < | + | < |
| - | In the following, the charging process of a capacitor is to be considered in more detail. For this purpose, one has to realize, that during charging of the capacitor, besides the voltage source $U_q$ and the capacitor $C$, there is always a resistance $R$ in the circuit. This is composed of the internal resistance of the (non-ideal) voltage source, the internal resistance of the capacitor and the parasitic (=interfering) resistance of the line. In practical applications it is often desired that capacitors charge in a certain time range. For this purpose, another real resistor is inserted into the circuit. The resulting series of resistor | + | In the following, the charging process of a capacitor is to be considered in more detail. For this purpose, one has to realize, that during |
| * infinitely fast switching | * infinitely fast switching | ||
| - | * resistance of $0\Omega$ in closed state (" | + | * resistance of $0~\Omega$ in the closed state (" |
| - | * resistance $\rightarrow \infty$ in open state ("open line") | + | * resistance $\rightarrow \infty$ in the open state ("open line") |
| * no capacitive effect | * no capacitive effect | ||
| Zeile 50: | Zeile 52: | ||
| In this chapter also time-varying quantities are considered. These are generally marked with lowercase letters. Examples of time-varying quantities are: | In this chapter also time-varying quantities are considered. These are generally marked with lowercase letters. Examples of time-varying quantities are: | ||
| - | * A **time-varying voltage $u_C(t)$ across a capacitor** | + | * A **time-varying voltage $u_C(t)$ across a capacitor** or the **voltage $U_{\rm s}$ of an ac voltage source** |
| * 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**. | ||
| Zeile 61: | Zeile 63: | ||
| < | < | ||
| - | === Goals === | + | === Learning Objectives |
| - | After this lesson, you should: | + | By the end of this section, you will be able to: |
| - | + | - know the time constant $\tau$ and in particularly | |
| - | - know the time constant $\tau$ and in particular be able to calculate it. | + | - determine the time characteristic of the currents and voltages at the RC element for a given resistance and capacitance. |
| - | - Be able to determine the time characteristic of the currents and voltages at the RC element for a given resistance and capacitance. | + | |
| - know the continuity conditions of electrical quantities. | - know the continuity conditions of electrical quantities. | ||
| - | - know when (=according to which measure) the capacitor is considered to be fully charged / discharged, i.e. a steady state can be considered to have been reached. | + | - know when (=according to which measure) the capacitor is considered to be fully charged/ |
| </ | </ | ||
| - | < | + | In the simulation below you can see the circuit |
| - | In the simulation on the right you can see the circuit mentioned above in a slightly modified form: | + | |
| - | + | ||
| - | | + | |
| * 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 83: | Zeile 82: | ||
| Exercises: | Exercises: | ||
| - | - 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, | + | - 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 $R=\{ 10~\Omega, 100~\Omega, |
| - Which quantity ($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? | ||
| - | At the following, this circuit is divided into two separate circuits, which consider only charging and only discharging. | + | < |
| + | |||
| + | In the following, this circuit is divided into two separate circuits, which consider only charging and only discharging. | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | < | + | Here is 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 $U_{\rm s}$ 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 $U_{\rm s}$ 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 $U_{\rm s}$ until $t \rightarrow \infty$. It then carries the charge: $q(t \rightarrow \infty) = Q = C \cdot U_{\rm s}$ |
| + | |||
| + | < | ||
| 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)}} = {{{\rm d}u_R}\over{{\rm d}i_R}} = {\rm 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 113: | Zeile 119: | ||
| 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*} | + | \begin{align*} |
| + | U_{\rm s} =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 ${\rm d}t$, 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 = {{{\rm d}q}\over{{\rm d}t}} \quad \text{and} \quad {\rm d}q = C \cdot {\rm d}u_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 {{{\rm d}u_C}\over{{\rm d}t}} \tag{5.1.3} | ||
| + | \end{align*} | ||
| Thus $(5.1.2)$ becomes: | Thus $(5.1.2)$ becomes: | ||
| - | \begin{align*} | + | \begin{align*} |
| + | U_{\rm s} &= u_R | ||
| + | | ||
| + | \end{align*} | ||
| --> here follows some mathematics: | --> here follows some mathematics: | ||
| Zeile 132: | Zeile 147: | ||
| \begin{align*} | \begin{align*} | ||
| - | u_C(t) = \mathcal{A} \cdot e^{\mathcal{B}\cdot t} + \mathcal{C} | + | u_C(t) = \mathcal{A} \cdot {\rm e}^{\mathcal{B}\cdot t} + \mathcal{C} |
| \end{align*} | \end{align*} | ||
| \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_{\rm s} &= R \cdot C \cdot {{\rm d}\over{{\rm d}t}}(\mathcal{A} \cdot {\rm e}^{\mathcal{B}\cdot t} + \mathcal{C}) + \mathcal{A} \cdot {\rm 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 {\rm e}^{\mathcal{B}\cdot t} + \mathcal{A} \cdot {\rm e}^{\mathcal{B}\cdot t} + \mathcal{C} \\ |
| - | U_q - \mathcal{C} & | + | U_{\rm s} - \mathcal{C} & |
| \end{align*} | \end{align*} | ||
| - | This equation must hold for every $t$. This is only possible if the left as well as the right term become equal to 0. \\ Thus: | + | This equation must hold for every $t$. This is only possible if the left, as well as the right term, become equal to 0. \\ Thus: |
| \begin{align*} | \begin{align*} | ||
| - | \mathcal{C} = U_q \\ \\ | + | \mathcal{C} = U_{\rm 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 154: | Zeile 169: | ||
| \begin{align*} | \begin{align*} | ||
| - | u_C(t) = \mathcal{A} \cdot e^{\large{- {{t}\over{R C}} }} + U_q | + | u_C(t) = \mathcal{A} \cdot {\rm e}^{\large{- {{t}\over{R C}} }} + U_{\rm s} |
| \end{align*} | \end{align*} | ||
| Zeile 160: | Zeile 175: | ||
| \begin{align*} | \begin{align*} | ||
| - | 0 &= \mathcal{A} \cdot e^{\large{0}} + U_q \\ | + | 0 &= \mathcal{A} \cdot {\rm e}^{\large{0}} + U_{\rm s} \\ |
| - | 0 &= \mathcal{A} | + | 0 &= \mathcal{A} |
| - | \mathcal{A} &= - U_q | + | \mathcal{A} &= - U_{\rm s} |
| \end{align*} | \end{align*} | ||
| Zeile 169: | Zeile 184: | ||
| \begin{align*} | \begin{align*} | ||
| - | u_C(t) &= - U_q \cdot e^{\large{- {{t}\over{R C}}}} + U_q | + | u_C(t) &= - U_{\rm s} \cdot {\rm e}^{\large{- {{t}\over{R C}}}} + U_{\rm s} |
| \end{align*} | \end{align*} | ||
| Zeile 176: | Zeile 191: | ||
| 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_{\rm s}}\over{R}} \cdot {\rm e}^{\large{- {{t}\over{R C}} } } |
| \end{align*} | \end{align*} | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | In <imgref imageNo04 >, the two time courses | + | In <imgref imageNo04 >, the two time course diagrams |
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| Zeile 196: | Zeile 211: | ||
| * 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_{\rm s} \cdot (1 - {\rm e}^{- 1}) = U_{\rm s} \cdot (1 - {{1}\over{\rm e}}) = U_{\rm s} \cdot ({{{\rm e}-1}\over{\rm e}}) = 0.63 \cdot U_{\rm s} = 63~\% \cdot U_{\rm 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_{\rm s} \cdot (1 - {\rm e}^{- 2}) = 86~\% \cdot U_{\rm s} = (63~\% + (100~\% |
| - | * 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: | ||
| - | * Plotting the voltage value corresponding to $63\%$ on the y-axis. Finding the point of intersection with the graph. Reading the time (see green lines in <imgref imageNo04> | + | * Plotting the voltage value corresponding to $63~\%$ on the y-axis. Finding the point of intersection with the graph. Reading the time (see green lines in <imgref imageNo04> |
| * Plotting the tangent to the (voltage) charge curve at the time of the discharged capacitor. This intersects a horizontal line at the level of the charging voltage at the point $t=\tau$ (see black and light blue lines in <imgref imageNo04> | * Plotting the tangent to the (voltage) charge curve at the time of the discharged capacitor. This intersects a horizontal line at the level of the charging voltage at the point $t=\tau$ (see black and light blue lines in <imgref imageNo04> | ||
| Zeile 211: | Zeile 226: | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| 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 $U_{\rm s}$ 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 $U_{\rm s}$ is initially applied to the resistor: $u_R(t_0)=U_{\rm s}$ |
| * 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: $u_C = {{q(t)}\over{C}}$ | ||
| * Ideally, the capacitor is not fully discharged before $t \rightarrow \infty$. | * Ideally, the capacitor is not fully discharged before $t \rightarrow \infty$. | ||
| - | Also this process now is to put into formula in detail. By looking at the loop, the general result is: the sum of the two voltages across the resistor and capacitor | + | Also, this process now is to put into a formula in detail. By looking at the loop, the general result is: the sum of the two voltages across the resistor and capacitor |
| \begin{align*} | \begin{align*} | ||
| Zeile 231: | Zeile 246: | ||
| \begin{align*} | \begin{align*} | ||
| - | 0 =u_R + u_C = R \cdot C \cdot {{du_C}\over{dt}} + u_C | + | 0 =u_R + u_C = R \cdot C \cdot {{{\rm d}u_C}\over{{\rm d}t}} + u_C |
| \end{align*} | \end{align*} | ||
| Zeile 239: | Zeile 254: | ||
| \begin{align*} | \begin{align*} | ||
| - | u_C(t) = \mathcal{A} \cdot e^{\mathcal{B}\cdot t} + \mathcal{C} | + | u_C(t) = \mathcal{A} \cdot {\rm e}^{\mathcal{B}\cdot t} + \mathcal{C} |
| \end{align*} | \end{align*} | ||
| \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} \\ | + | 0 &= R \cdot C \cdot {{\rm d}\over{{\rm d}t}}(\mathcal{A} \cdot {\rm e}^{\mathcal{B}\cdot t} + \mathcal{C}) + \mathcal{A} |
| - | &= 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 {\rm e}^{\mathcal{B}\cdot t} + \mathcal{A} |
| - | U_q - \mathcal{C} & | + | - \mathcal{C} & |
| \end{align*} | \end{align*} | ||
| - | This equation must hold for every $t$. This is only possible if the left as well as the right term become equal to 0. Thus: | + | This equation must hold for every $t$. This is only possible if the left, as well as the right term, become equal to 0. Thus: |
| \begin{align*} | \begin{align*} | ||
| - | \mathcal{C} = U_q \\ \\ | + | \mathcal{C} = 0 \\ \\ |
| 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 261: | Zeile 276: | ||
| \begin{align*} | \begin{align*} | ||
| - | u_C(t) = \mathcal{A} \cdot e^{\large{- {{t}\over{R C}} }} + U_q | + | u_C(t) = \mathcal{A} \cdot {\rm e}^{\large{- {{t}\over{R C}} }} + 0 |
| \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 $t_0=0$ $u_C(t_0) = U_{\rm s}$ just holds: |
| \begin{align*} | \begin{align*} | ||
| - | 0 &= \mathcal{A} \cdot e^{\large{0}} + U_q \\ | + | U_{\rm s} &= \mathcal{A} \cdot {\rm e}^{\large{0}} |
| - | 0 &= \mathcal{A} | + | \mathcal{A} & |
| - | \mathcal{A} & | + | |
| \end{align*} | \end{align*} | ||
| - | Therfore, the result is: | + | Therefore, the result is: |
| \begin{align*} | \begin{align*} | ||
| - | u_C(t) & | + | u_C(t) & |
| \end{align*} | \end{align*} | ||
| Zeile 283: | Zeile 297: | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| 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_{\rm s}}\over{R}} \cdot {\rm e}^{\large{- {{t}\over{R C}} } } |
| \end{align*} | \end{align*} | ||
| - | In <imgref imageNo05 > the two time histories | + | In <imgref imageNo05 > the two time course diagrams |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Zeile 302: | Zeile 316: | ||
| ==== Periodic switching operations ==== | ==== Periodic switching operations ==== | ||
| - | < | + | < |
| - | In the simulation on the right, a periodic switching operation can be seen. The capacitor is periodically charged and discharged via the switch. Three sliders are given in the simulation to change the resistance $R$ (< | + | In the simulation on the right, a periodic switching operation can be seen. The capacitor is periodically charged and discharged via the switch. Three sliders are given in the simulation to change the resistance $R$ (< |
| Exercises: | Exercises: | ||
| - | - Increase the the frequency to $f=10kHz$ using the appropriate slider. What is the change for $u_C$ and $i_C$? | + | - Increase the the frequency to $f=10~{\rm kHz}$ using the appropriate slider. What is the change for $u_C$ and $i_C$? |
| - | - Now increase the capacitance to $C=10 \mu F$ using the corresponding slider. What is the change for $u_C$ and $i_C$? | + | - Now increase the capacitance to $C=10 ~{\rm µF}$ using the corresponding slider. What is the change for $u_C$ and $i_C$? |
| - | - Now increase the resistance to $R= 1 k\Omega$ using the corresponding slider. What is the change for $u_C$ and $i_C$? | + | - Now increase the resistance to $R= 1 ~\rm k\Omega$ using the corresponding slider. What is the change for $u_C$ and $i_C$? |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Zeile 318: | Zeile 332: | ||
| < | < | ||
| - | === Goals === | + | === Learning Objectives |
| - | After this lesson, you should: | + | By the end of this section, you will be able to: |
| - | + | - calculate the energy content in a capacitor. | |
| - | - Be able to calculate the energy content in a capacitor. | + | - calculate the change in energy of a capacitor resulting from a change in voltage between the capacitor terminals. |
| - | - Be able to calculate the change in energy of a capacitor resulting from a change in voltage between the capacitor terminals. | + | - calculate (initial) current, (final) voltage, and charge when balancing the charge of several capacitors (also via resistors). |
| - | - Be able to calculate (initial) current, (final) voltage and charge when balancing the charge of several capacitors (also via resistors). | + | |
| </ | </ | ||
| - | < | + | < |
| - | 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 339: | Zeile 352: | ||
| \begin{align*} | \begin{align*} | ||
| - | p={{dw}\over{dt}} = u \cdot i | + | p={{{\rm d}w}\over{{\rm d}t}} = u \cdot i |
| \end{align*} | \end{align*} | ||
| Zeile 347: | Zeile 360: | ||
| \begin{align*} | \begin{align*} | ||
| - | \Delta W_C = \int_{t_0}^{t_1} | + | \Delta W_C = \int_{t_0}^{t_1} |
| + | = \int_{0}^t u | ||
| + | = \int_{0}^t u_C \cdot i_C {\rm d}t \tag{5.2.1} | ||
| \end{align*} | \end{align*} | ||
| 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_{\rm s} \cdot (1 - {\rm e}^{ -{{t}\over{\tau}} }) \\ |
| - | i_C(t) = {{U_q}\over{R}} \cdot e^{ -{{t}\over{\tau}} } \tag{5.2.2} | + | i_C(t) = {{U_{\rm s}}\over{R}} \cdot |
| \end{align*} | \end{align*} | ||
| Zeile 359: | Zeile 374: | ||
| \begin{align*} | \begin{align*} | ||
| - | C = {{q(t)}\over{u_C(t)}} \quad & | + | C = {{q(t)}\over{u_C(t)}} |
| - | i_C(t) = {{d q(t)}\over{dt}} \quad & | + | i_C(t) = {{{\rm d} q(t)}\over{{\rm d}t}} \quad & |
| \end{align*} | \end{align*} | ||
| Zeile 366: | Zeile 381: | ||
| \begin{align*} | \begin{align*} | ||
| - | \Delta W_C &= \int_{0}^t u_C(t) \cdot C \cdot {{d u_C(t)}\over{dt}} dt \quad & | \text{ substitution of integration variable: } t \rightarrow u_C\\ | + | \Delta W_C &= \int_{0}^t |
| - | &= \int_{U_0}^{U_1} u_C(t) \cdot C \cdot | + | |
| - | &= C \cdot \int_{U_0}^{U_1} u_C \, d u_C \\ | + | |
| - | &= C \cdot \left[{{1}\over{2}} u_C^2 \right] _{U_0}^{U_1} \\ | + | |
| \end{align*} | \end{align*} | ||
| \begin{align*} | \begin{align*} | ||
| Zeile 375: | Zeile 390: | ||
| \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 ($U_{\rm s}=0~{\rm V}$), the energy stored when charging to voltage $U_{\rm s}$ is $\Delta W_C={{1}\over{2}} C \cdot U_{\rm s}^2$. |
| - | === Energy | + | === Energy |
| The converted energy can also be determined for the resistor: | The converted energy can also be determined for the resistor: | ||
| \begin{align*} | \begin{align*} | ||
| - | \Delta W_R = \int_{0}^t u_R \cdot i_R dt = \int_{0}^t R \cdot i_R \cdot i_R dt = R \cdot \int_{0}^t i_R^2 | + | \Delta W_R = \int_{0}^t u_R \cdot i_R {\rm d}t |
| + | = \int_{0}^t | ||
| + | = R \cdot \int_{0}^t i_R^2 | ||
| \end{align*} | \end{align*} | ||
| Zeile 388: | Zeile 405: | ||
| \begin{align*} | \begin{align*} | ||
| - | \Delta W_R & | + | \Delta W_R & |
| - | | + | |
| - | | + | |
| - | | + | |
| \end{align*} | \end{align*} | ||
| Zeile 397: | Zeile 414: | ||
| \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_{\rm 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 $U_{\rm s}$ and given capacitor $C$)! At first, this doesn' |
| - | In real applications, | + | In real applications, |
| === Consideration of total energy turnover === | === Consideration of total energy turnover === | ||
| - | In the previous considerations, | + | In the previous considerations, |
| + | So, in total, the voltage source injects the following energy: | ||
| \begin{align*} | \begin{align*} | ||
| - | \Delta W_0 & | + | \Delta W_0 & |
| \end{align*} | \end{align*} | ||
| Zeile 419: | Zeile 437: | ||
| \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 means that only half of the energy emitted by the source is stored in the capacitor! Again, |
| < | < | ||
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | This can also be tested in the following simulation. In addition to the RC element shown so far, a power meter and an integrator are also drawn in here. It is possible to display the instantaneous power and the stored energy. Via the slider < | + | This can also be tested in the following simulation. In addition to the RC element shown so far, a power meter and an integrator are also drawn here. It is possible to display the instantaneous power and the stored energy. Via the slider < |
| * left: Current $u_C$ and voltage $i_C$ at the capacitor. | * left: Current $u_C$ and voltage $i_C$ at the capacitor. | ||
| * middle: Instantaneous power $p_C = u_C \cdot i_C$ of the capacitor. | * middle: Instantaneous power $p_C = u_C \cdot i_C$ of the capacitor. | ||
| - | * right: stored energy $w_C = \int u_C \cdot i_C \; dt$ of the capacitor | + | * right: stored energy $w_C = \int u_C \cdot i_C \; {\rm d}t$ of the capacitor |
| < | < | ||
| Zeile 455: | Zeile 474: | ||
| </ | </ | ||
| - | <panel type=" | + | # |
| - | {{youtube> | + | The following circuit shows a charging/ |
| - | </ | + | The values of the components shall be the following: |
| + | * $R_1 = 1.0 \rm k\Omega$ | ||
| + | * $R_2 = 2.0 \rm k\Omega$ | ||
| + | * $R_3 = 3.0 \rm k\Omega$ | ||
| + | * $C = 1 \rm \mu F$ | ||
| + | * $S_1$ and $S_2$ are opened in the beginning (open-circuit) | ||
| - | <panel type=" | + | {{drawio>electrical_engineering_1: |
| - | {{youtube> | + | 1. For the first tasks, the switch $S_1$ gets closed at $t=t_0 = 0s$. \\ |
| - | </ | + | 1.1 What is the value of the time constant $\tau_1$? |
| - | <panel type=" | + | # |
| - | {{youtube> | + | The time constant $\tau$ is generally given as: $\tau= R\cdot C$. \\ |
| + | Now, we try to determine which $R$ and $C$ must be used here. \\ | ||
| + | To find this out, we have to look at the circuit when $S_1$ gets closed. | ||
| - | </WRAP></WRAP></panel> | + | {{drawio> |
| + | |||
| + | We see that for the time constant, we need to use $R=R_1 + R_2$. | ||
| + | |||
| + | # | ||
| + | |||
| + | # | ||
| + | \begin{align*} | ||
| + | \tau_1 &= R\cdot C \\ | ||
| + | & | ||
| + | & | ||
| + | \end{align*} | ||
| + | |||
| + | # | ||
| + | |||
| + | 1.2 What is the formula for the voltage $u_{R2}$ over the resistor $R_2$? Derive a general formula without using component values! | ||
| + | |||
| + | # | ||
| + | |||
| + | To get a general formula, we again look at the circuit, but this time with the voltage arrows. | ||
| + | |||
| + | {{drawio> | ||
| + | |||
| + | We see, that: $U_1 = u_C + u_{R2}$ and there is only one current in the loop: $i = i_C = i_{R2}$\\ | ||
| + | The current is generally given with the exponential function: $i_c = {{U}\over{R}}\cdot e^{-t/\tau}$, with $R$ given here as $R = R_1 + R_2$. | ||
| + | Therefore, $u_{R2}$ can be written as: | ||
| + | |||
| + | \begin{align*} | ||
| + | u_{R2} &= R_2 \cdot i_{R2} \\ | ||
| + | & | ||
| + | \end{align*} | ||
| + | |||
| + | # | ||
| + | |||
| + | # | ||
| + | \begin{align*} | ||
| + | u_{R2} = U_1 \cdot {{R_2}\over{R_1 + R_2}} \cdot e^{t/ \tau} | ||
| + | \end{align*} | ||
| + | # | ||
| + | |||
| + | 2. At a distinct time $t_1$, the voltage $u_C$ is charged up to $4/5 \cdot U_1$. | ||
| + | At this point, the switch $S_1$ will be opened. \\ Calculate $t_1$! | ||
| + | |||
| + | # | ||
| + | |||
| + | We can derive $u_{C}$ based on the exponential function: $u_C(t) = U_1 \cdot (1-e^{-t/ | ||
| + | Therefore, we get $t_1$ by: | ||
| + | |||
| + | \begin{align*} | ||
| + | u_C = 4/5 \cdot U_1 & | ||
| + | 4/5 & | ||
| + | e^{-t/ | ||
| + | | ||
| + | t &= -\tau \cdot \rm ln (1/5) \\ | ||
| + | \end{align*} | ||
| + | |||
| + | # | ||
| + | |||
| + | # | ||
| + | \begin{align*} | ||
| + | t & | ||
| + | \end{align*} | ||
| + | # | ||
| + | |||
| + | 3. The switch $S_2$ will get closed at the moment $t_2 = 10 ~\rm ms$. The values of the voltage sources are now: $U_1 = 5.0 ~\rm V$ and $U_2 = 10 ~\rm V$. | ||
| + | |||
| + | 3.1 What is the new time constant $\tau_2$? | ||
| + | |||
| + | # | ||
| + | |||
| + | Again, the time constant $\tau$ is given as: $\tau= R\cdot C$. \\ | ||
| + | Again, we try to determine which $R$ and $C$ must be used here. \\ | ||
| + | To find this out, we have to look at the circuit when $S_1$ is open and $S_2$ is closed. | ||
| + | |||
| + | {{drawio>electrical_engineering_1: | ||
| + | |||
| + | We see that for the time constant, we now need to use $R=R_3 + R_2$. | ||
| + | |||
| + | \begin{align*} | ||
| + | \tau_2 &= R\cdot C \\ | ||
| + | & | ||
| + | \end{align*} | ||
| + | |||
| + | # | ||
| + | |||
| + | # | ||
| + | \begin{align*} | ||
| + | \tau_2 &= 5~\rm ms \\ | ||
| + | \end{align*} | ||
| + | # | ||
| + | |||
| + | 3.2 Calculate the moment $t_3$ when $u_{R2}$ is smaller than $1/10 \cdot U_2$. | ||
| + | |||
| + | # | ||
| + | |||
| + | To calculate the moment $t_3$ when $u_{R2}$ is smaller than $1/10 \cdot U_2$, we first have to find out the value of $u_{R2}(t_2 = 10 ~\rm ms)$, when $S_2$ just got closed. \\ | ||
| + | * Starting from $t_2 = 10 ~\rm ms$, the voltage source $U_2$ charges up the capacitor $C$ further. | ||
| + | * Before at $t_1$, when $S_1$ got opened, the value of $u_c$ was: $u_c(t_1) = 4/5 \cdot U_1 = 4 ~\rm V$. | ||
| + | * This is also true for $t_2$, since between $t_1$ and $t_2$ the charge on $C$ does not change: $u_c(t_2) = 4 ~\rm V$. | ||
| + | * In the first moment after closing $S_2$ at $t_2$, the voltage drop on $R_3 + R_2$ is: $U_{R3+R2} = U_2 - u_c(t_2) = 6 ~\rm V$. | ||
| + | * So the voltage divider of $R_3 + R_2$ lead to $ \boldsymbol{u_{R2}(t_2 = 10 ~\rm ms)} = {{R_2}\over{R_3 + R2}} \cdot U_{R3+R2} = {{2 {~\rm k\Omega}}\over{3 {~\rm k\Omega} + 2 {~\rm k\Omega} }} \cdot 6 ~\rm V = \boldsymbol{2.4 ~\rm V} $ | ||
| + | |||
| + | We see that the voltage on $R_2$ has to decrease from $2.4 ~\rm V $ to $1/10 \cdot U_2 = 1 ~\rm V$. \\ | ||
| + | To calculate this, there are multiple ways. In the following, one shall be retraced: | ||
| + | * We know, that the current $i_C = i_{R2}$ subsides exponentially: | ||
| + | * So we can rearrange the task to focus on the change in current instead of the voltage. | ||
| + | * The exponential decay is true regardless of where it starts. | ||
| + | |||
| + | So from ${{i_{R2}(t)}\over{I_{R2~ 0}}} = {\rm e}^{-t/ | ||
| + | \begin{align*} | ||
| + | {{i_{R2}(t_3)}\over{i_{R2}(t_2)}} & | ||
| + | -{{t_3 - t_2}\over{\tau_2}} | ||
| + | | ||
| + | | ||
| + | \end{align*} | ||
| + | |||
| + | # | ||
| + | |||
| + | # | ||
| + | \begin{align*} | ||
| + | t_3 &= 14.4~\rm ms \\ | ||
| + | \end{align*} | ||
| + | # | ||
| + | |||
| + | 3.3 Draw the course of time of the voltage $u_C(t)$ over the capacitor. | ||
| + | |||
| + | {{drawio> | ||
| + | |||
| + | |||
| + | # | ||
| + | {{drawio> | ||
| + | # | ||
| + | |||
| + | # | ||
| + | |||
| + | {{page> | ||
| + | |||
| + | # | ||
| - | <panel type=" | ||
| < | < | ||
| - | On the simulation you see the two capacitors $C_1$ and $C_2$ (The two small resistors with $1\mu \Omega$ have to be there for the simulation to run). At the beginning $C_1$ is charged to $10V$ and $C_2$ to $0V$. With the switches $S_1$ and $S_2$ you can choose whether | + | In the simulation, you see the two capacitors $C_1$ and $C_2$ (The two small resistors with $1 ~\rm µ\Omega$ have to be there for the simulation to run). At the beginning, $C_1$ is charged to $10~{\rm V}$ and $C_2$ to $0~{\rm V}$. With the switches $S_1$ and $S_2$ you can choose whether |
| - the capacitances $C_1$ and $C_2$ are shorted, or | - the capacitances $C_1$ and $C_2$ are shorted, or | ||
| - the capacitors $C_1$ and $C_2$ are connected via resistor $R$. | - the capacitors $C_1$ and $C_2$ are connected via resistor $R$. | ||
| - | On the right side of the simulation there are some additional " | + | On the right side of the simulation, there are some additional " |
| - | In the following, the charging and discharging of a capacitor | + | In the following, the charging and discharging of a capacitor |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Zeile 491: | Zeile 653: | ||
| ^Voltage $u_1(C_1)$ of the first capacitor^Voltage $u_2(C_2)$ of the second capacitor^Stored energy $w_1(C_1)$^Stored energy $w_2(C_2)$^Total energy $\sum w$| | ^Voltage $u_1(C_1)$ of the first capacitor^Voltage $u_2(C_2)$ of the second capacitor^Stored energy $w_1(C_1)$^Stored energy $w_2(C_2)$^Total energy $\sum w$| | ||
| - | |Initially charged to $10V$|Initially neutrally charged ($0V$)|Initially holds: \\ $w_1(C_1)= {1 \over 2} \cdot C \cdot U^2 = {1 \over 2} \cdot 10\mu F \cdot (10V)^2 = 500\mu W$ \\ In the oscilloscope, | + | |Initially charged to $10~{\rm V}$|Initially neutrally charged ($0~{\rm V}$)|Initially holds: \\ $w_1(C_1)= {1 \over 2} \cdot C \cdot U^2 = {1 \over 2} \cdot 10~{\rm µF} \cdot (10~{\rm V})^2 = 500~{\rm µW}$ \\ In the oscilloscope, |
| - | The capacitor $C_1$ has thus initially stored the full energy and via a closing of the switch $S_2$ one would expect a balancing of the voltages and an equal distribution of the energy $w_1 + w_2 = 500\mu W$. | + | The capacitor $C_1$ has thus initially stored the full energy and via closing of the switch, $S_2$ one would expect a balancing of the voltages and an equal distribution of the energy $w_1 + w_2 = 500~\rm µW$. |
| - Close the switch $S_2$ (the toggle switch $S_1$ should point to the switch $S_2$). What do you find? | - Close the switch $S_2$ (the toggle switch $S_1$ should point to the switch $S_2$). What do you find? | ||
| - What do the voltages $u_1$ and $u_2$ do? | - What do the voltages $u_1$ and $u_2$ do? | ||
| - | - What the energies and the total energy? \\ How is this understandable with the previous total energy? | + | - What are the energies and the total energy? \\ How is this understandable with the previous total energy? |
| - Open $S_2$ - the changeover switch $S_1$ should not be changed. What do you find? | - Open $S_2$ - the changeover switch $S_1$ should not be changed. What do you find? | ||
| - What do the voltages $u_1$ and $u_2$ do? | - What do the voltages $u_1$ and $u_2$ do? | ||
| - | - What the energies and the total energy? \\ How is this understandable with the previous total energy? | + | - What are the energies and the total energy? \\ How is this understandable with the previous total energy? |
| - | - Repeat 1. and 2. several times. Can anything be deduced regarding the distribution of the energy? | + | - Repeat 1. and 2. several times. Can anything be deduced regarding the distribution of energy? |
| - Change the switch $S_2$ to the resistor. What do you observe? | - Change the switch $S_2$ to the resistor. What do you observe? | ||
| - What do the voltages $u_1$ and $u_2$ do? | - What do the voltages $u_1$ and $u_2$ do? | ||
| - | - What the energies and the total energy? \\ How is this understandable with the previous total energy? | + | - What are the energies and the total energy? \\ How is this understandable with the previous total energy? |
| + | |||
| + | # | ||
| - | </ | ||
| - | {{page> | ||