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| Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
| electrical_engineering_2:inductances_in_circuits [2022/10/13 18:48] – [6.5 Examples] tfischer | electrical_engineering_2:inductances_in_circuits [2024/06/06 11:08] (aktuell) – [Exercises] mexleadmin | ||
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| Zeile 1: | Zeile 1: | ||
| - | ====== 6. Inductances in Circuits ====== | + | ====== 6 Inductances in Circuits ====== |
| ===== 6.1 Basic Circuits ===== | ===== 6.1 Basic Circuits ===== | ||
| Zeile 7: | Zeile 7: | ||
| ==== 6.1.1 Series Circuits ==== | ==== 6.1.1 Series Circuits ==== | ||
| - | Based on $L = {{ \Psi(t)}\over{i}}$ and Kirchhoff' | + | Based on $L = {{ \Psi(t)}\over{i}}$ and Kirchhoff' |
| - | \begin{align*} L_{eq} &= {{\sum_i \Psi_i}\over{I}} = \sum_i L_i \end{align*} | + | \begin{align*} L_{\rm eq} &= {{\sum_i \Psi_i}\over{I}} = \sum_i L_i \end{align*} |
| - | A similar result can be derived from the induced voltage $u_{ind}= L {{di}\over{dt}}$, when taking the situation of a series circuit (i.e. $i_1 = i_2 = i_1 = ... = i_{eq}$ and $u_{eq}= u_1 + u_2 + ...$): | + | A similar result can be derived from the induced voltage $u_{ind}= L {{{\rm d}i}\over{{\rm d}t}}$, when taking the situation of a series circuit (i.e. $i_1 = i_2 = i_1 = ... = i_{\rm eq}$ and $u_{\rm eq}= u_1 + u_2 + ...$): |
| - | \begin{align*} & u_{eq} & = &u_1 &+ &u_2 &+ ... \\ & L_{eq} {{di_{eq} }\over{dt}} & = &L_{1} {{di_{1} }\over{dt}} &+ &L_{2} {{di_{2} }\over{dt}} &+ ... \\ & L_{eq} {{di }\over{dt}} & = &L_{1} {{di }\over{dt}} &+ &L_{2} {{di }\over{dt}} &+ ... \\ & L_{eq} & = &L_{1} &+ &L_{2} &+ ... \\ \end{align*} | + | \begin{align*} |
| + | & u_{\rm eq} | ||
| + | & L_{\rm eq} {{{\rm d}i_{\rm | ||
| + | & L_{\rm eq} {{{\rm d}i }\over{{\rm d}t}} & = &L_{1} {{{\rm d}i }\over{{\rm d}t}} & + &L_{2} {{di | ||
| + | & L_{\rm eq} | ||
| + | \end{align*} | ||
| ==== 6.1.2 Parallel Circuits ==== | ==== 6.1.2 Parallel Circuits ==== | ||
| - | For parallel circuits one can also start with the principles based on Kirchhoff' | + | For parallel circuits, one can also start with the principles based on Kirchhoff' |
| - | \begin{align*} u_{eq}= u_1 = u_2 = ... \\ \end{align*} | + | \begin{align*} u_{\rm eq}= u_1 = u_2 = ... \\ \end{align*} |
| and Kirchhoff' | and Kirchhoff' | ||
| - | \begin{align*} i_{eq}= i_1 + i_2 + ... \\ \end{align*} | + | \begin{align*} i_{\rm eq}= i_1 + i_2 + ... \\ \end{align*} |
| Here, the formula for the induced voltage has to be rearranged: | Here, the formula for the induced voltage has to be rearranged: | ||
| - | \begin{align*} u_{ind} &= L {{di}\over{dt}} \quad \quad \quad \quad \bigg| \int()dt \\ \int u_{ind} | + | \begin{align*} |
| + | u_{\rm ind} &= L {{{\rm d}i}\over{{\rm d}t}} \quad \quad \quad \quad \bigg| \int(){\rm d}t \\ | ||
| + | \int u_{\rm ind} {\rm d}t &= L \cdot i \\ | ||
| + | | ||
| + | \end{align*} | ||
| By this, we get: | By this, we get: | ||
| - | \begin{align*} i_{eq} &=& i_1 &+& i_2 &+& ... \\ {{1}\over{L_{eq}}} \cdot \int u_{eq} | + | \begin{align*} |
| + | | ||
| + | {{1}\over{L_{\rm eq}}} \cdot \int u_{\rm eq} {\rm d}t &=& {{1}\over{L_1}} \cdot \int u_{1} {\rm d}t &+& {{1}\over{L_2}} \cdot \int u_{2} {\rm d}t &+& ... \\ | ||
| + | {{1}\over{L_{\rm eq}}} \cdot \int u {\rm d}t &=& {{1}\over{L_1}} \cdot \int u {\rm d}t &+& {{1}\over{L_2}} \cdot \int u {\rm d}t &+& ... \\ | ||
| + | {{1}\over{L_{\rm eq}}} &=& {{1}\over{L_1}} | ||
| + | \end{align*} | ||
| <callout icon=" | <callout icon=" | ||
| Zeile 37: | Zeile 51: | ||
| ==== 6.1.3 in AC Circuits ==== | ==== 6.1.3 in AC Circuits ==== | ||
| - | For AC circuits (i.e. with sinosidal | + | For AC circuits (i.e. with sinusoidal |
| - | \begin{align*} \underline{Z} = {{\underline{u}}\over{\underline{i}}} \end{align*} | + | \begin{align*} \underline{Z} = {{\underline{u}}\over{\underline{i}}} = {{1}\over{\underline{i}}} \cdot \underline{u} |
| - | With the induction $u_{ind}= | + | With the induction $u_{\rm ind}= L {{{\rm d}i}\over{{\rm d}t}}$ we get: |
| - | \begin{align*} \underline{Z} &= {{ L {{d\underline{i}}\over{dt}} | + | \begin{align*} |
| + | \underline{Z} & | ||
| + | \end{align*} | ||
| - | Without limiting the generality, one can assume the current $i$ to be: $i = I \cdot \sqrt{2} \cdot e^{j \cdot \omega t + \varphi_0}$.\\ | + | Without limiting the generality, one can assume the current $i$ to be: $i = I \cdot \sqrt{2} \cdot {\rm e}^{{\rm j} \cdot \omega t + \varphi_0}$.\\ |
| - | Therefore: | + | Once inserted, the formula gets: |
| - | \begin{align*} \underline{Z} & | + | \begin{align*} |
| + | \underline{Z} &= {{1} \over {I \cdot \sqrt{2} \cdot {\rm e}^{{\rm j} \cdot \omega t + \varphi_0}}} \cdot L {{ {\rm d}} \over {{\rm d}t} } \left( I \cdot \sqrt{2} \cdot {\rm e}^{{\rm j} \cdot \omega t + \varphi_0} \right) | ||
| + | &= {{1} \over {I \cdot \sqrt{2} \cdot {\rm e}^{{\rm j} \cdot \omega t + \varphi_0}}} \cdot L | ||
| + | &= {{1} \over {\qquad\quad\; | ||
| + | &= {{1} \over {\qquad\quad\; | ||
| + | \underline{Z} | ||
| + | \end{align*} | ||
| - | <callout icon=" | + | <callout icon=" |
| ===== 6.2 Charging and Discharging ===== | ===== 6.2 Charging and Discharging ===== | ||
| - | Charging and discharging an $RL$ circuit is comparable to the RC-circuit in chapter [[: | + | Charging and discharging an $RL$ circuit is comparable to the RC-circuit in chapter [[: |
| < | < | ||
| Zeile 60: | Zeile 82: | ||
| ===== 6.3 Resonance Phenomena ===== | ===== 6.3 Resonance Phenomena ===== | ||
| - | Similar to the approach | + | Similar to last semester's approach, |
| ==== 6.3.1 RLC - Series Resonant Circuit ==== | ==== 6.3.1 RLC - Series Resonant Circuit ==== | ||
| Zeile 66: | Zeile 88: | ||
| As seen last semester, the circuits with complex impedances can be interpreted as four-terminal networks. There, we will again look at " | As seen last semester, the circuits with complex impedances can be interpreted as four-terminal networks. There, we will again look at " | ||
| - | In this chapter we look at combination where all three components resistor $R$, capacitor $C$ and inductance $L$ are used. | + | In this chapter, we look at a combination where all three components resistor $R$, capacitor $C$, and inductance $L$ are used. |
| < | < | ||
| - | If a resistor $R$, a capacitor $C$ and an inductance $L$ are connected in series, the result is a **series resonant circuit**. In this case, it not clearly defined, what the output voltage is. Consequently, | + | If a resistor $R$, a capacitor $C$, and an inductance $L$ are connected in series, the result is a **series resonant circuit**. In this case, it is not clearly defined, what the output voltage is. Consequently, |
| \begin{align*} \underline{U}_I = \underline{U}_R + \underline{U}_L + \underline{U}_C \end{align*} | \begin{align*} \underline{U}_I = \underline{U}_R + \underline{U}_L + \underline{U}_C \end{align*} | ||
| Zeile 76: | Zeile 98: | ||
| Since the current in the circuit must be constant, the total impedance can be determined here in a simple way: | Since the current in the circuit must be constant, the total impedance can be determined here in a simple way: | ||
| - | \begin{align*} \underline{U}_I &= R \cdot \underline{I} + j \omega L \cdot \underline{I} + \frac {1}{j\omega C } \cdot \underline{I} \\ \underline{U}_I &= \left( R + j \omega L - j \cdot \frac {1}{\omega C } \right) \cdot \underline{I} \\ \underline{Z}_{ges} &= R + j \omega L - j \cdot \frac {1}{\omega C } \end{align*} | + | \begin{align*} |
| + | \underline{U}_I | ||
| + | \underline{U}_I | ||
| + | \underline{Z}_{\rm eq} & | ||
| + | \end{align*} | ||
| - | By this, the magnitude of the (input) voltage $U_I$, the (input or total) impedance $Z$ and the phase result to: | + | By this, the magnitude of the (input) voltage $U_I$, the (input or total) impedance $Z$, and the phase result to: |
| - | \begin{align*} U_I &= \sqrt{U_R^2 + (U_Z)^2} = \sqrt{U_R^2 + (U_L - U_C)^2} \end{align*} | + | \begin{align*} |
| + | U_I &= \sqrt{U_R^2 + (U_Z )^2} | ||
| + | = \sqrt{U_R^2 + (U_L - U_C)^2} | ||
| + | \end{align*} | ||
| - | \begin{align*} Z &= \sqrt{R^2 + X^2} = \sqrt{R^2 + (\omega L - \frac{1}{\omega C})^2} \end{align*} | + | \begin{align*} |
| + | Z &= \sqrt{R^2 + X^2} | ||
| + | = \sqrt{R^2 + (\omega L - \frac{1}{\omega C})^2} | ||
| + | \end{align*} | ||
| - | \begin{align*} \varphi_u = \varphi_Z &= arctan \frac{\omega L - \frac{1}{\omega C}}{R} \end{align*} | + | \begin{align*} |
| + | \varphi_u = \varphi_Z | ||
| + | & | ||
| + | \end{align*} | ||
| There are now 3 different situations to distinguish: | There are now 3 different situations to distinguish: | ||
| * If $U_L > U_C$ the whole setup behaves like an ohmic-inductive load. This is the case at high frequencies. | * If $U_L > U_C$ the whole setup behaves like an ohmic-inductive load. This is the case at high frequencies. | ||
| - | * If $U_L=U_C$, the total input voltage $U$ is applied to the resistor. In this case, the total resistance $Z$ is minimal and only ohmic. \\ Thus, the current $I$ is then maximal. If the current is maximum, then the responses of the capacitance and inductance - their voltages - are also maximum. This situation is the **resonance case**. | + | * If $U_L = U_C$, the total input voltage $U$ is applied to the resistor. In this case, the total resistance $Z$ is minimal and only ohmic. \\ Thus, the current $I$ is then maximal. If the current is maximum, then the responses of the capacitance and inductance - their voltages - are also maximum. This situation is the **resonance case**. |
| * If $U_L < U_C$ then the whole setup behaves like a resistive-capacitive load. This is the case at low frequencies. | * If $U_L < U_C$ then the whole setup behaves like a resistive-capacitive load. This is the case at low frequencies. | ||
| - | Again, there seems to be an singular frequency, namely when $U_L = U_C$ or $Z_L = Z_C$ holds: | + | Again, there seems to be a singular frequency, namely when $U_L = U_C$ or $Z_L = Z_C$ holds: |
| - | \begin{align*} \frac{1}{\omega_0 C} & = \omega_0 L \\ \omega_0 & = \frac{1}{\sqrt{LC}} \\ 2\pi f_0 & = \frac{1}{\sqrt{LC}} \rightarrow \boxed{ f_0 = \frac{1}{2\pi \sqrt{LC}} } \end{align*} | + | \begin{align*} |
| + | \frac{1}{\omega_0 C} & = \omega_0 L \\ | ||
| + | \omega_0 | ||
| + | 2\pi f_0 & = \frac{1}{ | ||
| + | \rightarrow \boxed{ f_0 = \frac{1}{2\pi \sqrt{LC}} } | ||
| + | \end{align*} | ||
| The frequency $f_0$ is called **resonance frequency**. | The frequency $f_0$ is called **resonance frequency**. | ||
| - | ^ ^ $\quad$ | + | ^ ^ $\quad$ |
| - | |voltage $U_R$ \\ at the resistor | | $\boldsymbol{\small{0}}$ | + | |voltage $U_R$ \\ at the resistor |
| - | |voltage $U_L$ \\ at the inductor | | $\boldsymbol{\small{0}}$ \\ because $\omega L$ becomes very small | | $\boldsymbol{\omega_0 L \cdot I = \omega_0 L \cdot \frac{U}{R} = \color{blue}{\frac{1}{R}\sqrt{\frac{L}{C}}\cdot U}}$ | | $\boldsymbol{\LARGE{U}}$ \\ since $\omega L$ becomes very large | | + | |voltage $U_L$ \\ at the inductor |
| |voltage $U_C$ \\ at the capacitor | | $\boldsymbol{\LARGE{U}}$ \\ because $\frac{1}{\omega C}$ becomes very large | | $\boldsymbol{\frac{1}{\omega_0 C} \cdot I = \frac{1}{\omega_0 C} \cdot \frac{U}{R} = \color{blue}{\frac{1}{R}\sqrt{\frac{L}{C}}\cdot U}}$ | | $\boldsymbol{\small{0}}$ \\ because $\frac{1}{\omega C}$ becomes very small | | |voltage $U_C$ \\ at the capacitor | | $\boldsymbol{\LARGE{U}}$ \\ because $\frac{1}{\omega C}$ becomes very large | | $\boldsymbol{\frac{1}{\omega_0 C} \cdot I = \frac{1}{\omega_0 C} \cdot \frac{U}{R} = \color{blue}{\frac{1}{R}\sqrt{\frac{L}{C}}\cdot U}}$ | | $\boldsymbol{\small{0}}$ \\ because $\frac{1}{\omega C}$ becomes very small | | ||
| - | The calculation in the table shows that in the resonance case, the voltage across the capacitor or inductor deviates from the input voltage by a factor $\color{blue}{\frac{1}{R}\sqrt{\frac{L}{C}}}$. This quantity is called **quality or Q-factor** | + | The calculation in the table shows that in the resonance case, the voltage across the capacitor or inductor deviates from the input voltage by a factor $\color{blue}{\frac{1}{R}\sqrt{\frac{L}{C}}}$. This quantity is called **quality or Q-factor** |
| - | \begin{align*} \boxed{ \left.Q_S = \frac{U_C}{U} \right\vert_{\omega = \omega_0} = \left.\frac{U_L}{U} \right\vert_{\omega = \omega_0} = \color{blue}{\frac{1}{R}\sqrt{\frac{L}{C}}} } \end{align*} | + | \begin{align*} |
| + | \boxed{ Q_{\rm S} = \left.\frac{U_C}{U} \right\vert_{\omega = \omega_0} | ||
| + | | ||
| + | | ||
| + | \end{align*} | ||
| - | The quality can be greater than, less than or equal to 1. | + | The quality can be greater than, less than, or equal to 1. The quality $Q_{\rm S}$ does not have a unit and should not be confused with the charge $Q$. |
| - | * If the quality is very high, the overshoot of the voltages at the impedances becomes very large in the resonance case. This is useful and necessary in various applications, | + | * If the quality is very high, the overshoot of the voltages at the impedances becomes very large in the resonance case. This is useful and necessary in various applications, |
| - | * If the Q is very small, overshoot is no longer seen. Depending on the impedance at which the output voltage is measured, a high-pass or low-pass is formed similar to the RC or RL element. However, this has a steeper slope in the blocking range. This means that the filter effect is better. | + | * If the $Q$ is very small, overshoot is no longer seen. Depending on the impedance at which the output voltage is measured, a high-pass or low-pass is formed similar to the $RC$ or $RL$ element. However, this has a steeper slope in the blocking range. This means that the filter effect is better. |
| - | The reciprocal of the Q is called **attenuation** | + | The reciprocal of the $Q$ is called **attenuation** |
| - | \begin{align*} \boxed{ | + | \begin{align*} \boxed{ |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Zeile 120: | Zeile 164: | ||
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| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Zeile 129: | Zeile 192: | ||
| * Decoupling | * Decoupling | ||
| * Filter | * Filter | ||
| - | * unwanted coupling and cirucit | + | * unwanted coupling and circuit |
| ===== 6.5 Examples ===== | ===== 6.5 Examples ===== | ||
| Zeile 153: | Zeile 216: | ||
| <panel type=" | <panel type=" | ||
| - | A $R$-$L$-$C$ series circuit uses a capacity of $C=1 \mu F$. The circuit is feed by a voltage source with $U_I$ at $f_1 = 50Hz$. | + | A $R$-$L$-$C$ series circuit uses a capacity of $C=100 ~\rm µF$. A voltage source with $U_I$ feeds the circuit |
| - Which values does $R$ and $L$ need to have, when the resonance voltage $|\underline{U}_L|$ and $|\underline{U}_C|$ at $f_1$ shall show the double value of the input voltage $U_I$? | - Which values does $R$ and $L$ need to have, when the resonance voltage $|\underline{U}_L|$ and $|\underline{U}_C|$ at $f_1$ shall show the double value of the input voltage $U_I$? | ||
| - | - The components of question 1. shall now be used. What would be the value of ${{|\underline{U}_C|} \over {|\underline{U}_I|}} $ for $f_2 = 60Hz$? | + | - The components of question 1. shall now be used. What would be the value of ${{|\underline{U}_C|} \over {|\underline{U}_I|}} $ for $f_2 = 60~\rm Hz$? |
| </ | </ | ||
| Zeile 162: | Zeile 225: | ||
| <panel type=" | <panel type=" | ||
| - | A given $R$-$L$-$C$ series circuit is fed with a frequency, | + | A given $R$-$L$-$C$ series circuit is fed with a frequency, $20~\%$ larger than the resonance frequency keeping the amplitude of the input voltage constant. In this situation, the circuit shows a current |
| + | |||
| + | Calculate the Quality $Q = {{1}\over{R}}\sqrt{{{L}\over{C}}}$. | ||
| + | |||
| + | # | ||
| + | The solution looks hard at first since no insights for the values of $R$, $C$, and $L$ are given. | ||
| + | |||
| + | However, it is possible and there are multiple ways to solve it. \\ \\ | ||
| + | |||
| + | <fs large> | ||
| + | |||
| + | But first, add some more info, which is always true from resonant circuits at the resonant frequency: | ||
| + | - $\omega_0 = {{1}\over{\sqrt{LC}}}$ | ||
| + | - $X_{C0} = - X_{L0}$ | ||
| + | - $Z = \sqrt{R^2 + (X_L + X_C)^2}$, based on the sum of the impedances $\underline{Z}_{\rm eq} = \underline{X}_R + \underline{X}_C + \underline{X}_L$ and the Pythagorean theorem$ | ||
| + | \\ | ||
| + | From the task, the following is also known. | ||
| + | - Using "a frequency, $20~\%$ larger than the resonance frequency": | ||
| + | - $f = 1.2 \cdot f_0 $ and | ||
| + | - $\omega = 1.2 \cdot \omega_0 $ | ||
| + | - The circuit shows a current | ||
| + | - The maximum current for the series resonant circuit is given for the minimum impedance $Z$. \\ The minimum impedance $Z$ is given at resonance frequency, and is $Z_{\rm min} = R$ | ||
| + | - Therefore: $Z = {{1}\over{0.7}} \cdot R$ | ||
| + | \\ | ||
| + | <fs large> | ||
| + | |||
| + | We start with $Z = \sqrt{R^2 + (X_L + X_C)^2}$ for the cases: (1) at the resonant frequency $f_0$ and (2) at the given frequency $f = 1.2 \cdot f_0 $ | ||
| + | |||
| + | \begin{align*} | ||
| + | (1): && Z_0 &= R \\ | ||
| + | (2): && Z & | ||
| + | \end{align*} | ||
| + | |||
| + | In formula $(2)$ the impedance $X_L$ and $X_C$ are: | ||
| + | * $X_L= \omega \cdot L$ and therefore also $X_L = 1.2 \cdot \omega_0 \cdot L = 1.2 \cdot X_{L0}$ | ||
| + | * $X_C= - {{1}\over {\omega \cdot C}}$ and therefore also $X_C = - {{1}\over {1.2 \cdot \omega \cdot C}} = - {{1}\over {1.2}} \cdot X_{C0}$ | ||
| + | |||
| + | With $X_{C0} = X_{L0}$ we get for $(1)$: | ||
| + | |||
| + | \begin{align*} | ||
| + | Z &= \sqrt{R^2 + \left(1.2\cdot X_{L0} - {{1}\over{1.2}} X_{L0} \right)^2} \\ | ||
| + | &= \sqrt{R^2 + X_{L0}^2 \cdot \left(1.2 - {{1}\over{1.2}} \right)^2} \\ | ||
| + | \end{align*} | ||
| + | |||
| + | Since we know that $Z = {{1}\over{0.7}} \cdot R$ and $Z_0 = R$, we can start by dividing $(2)$ by $(1)$: | ||
| + | |||
| + | \begin{align*} | ||
| + | {{(2)}\over{(1)}} : && | ||
| + | && | ||
| + | && | ||
| + | && | ||
| + | && | ||
| + | && {{X_{L0}^2}\over{R^2}} | ||
| + | && {{X_{L0}}\over{R}} | ||
| + | && {{1}\over{R}}\cdot \sqrt{ {L} \over {C} } & | ||
| + | |||
| + | \end{align*} | ||
| + | |||
| + | # | ||
| + | |||
| + | |||
| + | # | ||
| + | |||
| + | \begin{align*} | ||
| + | Q & | ||
| + | &= 2.782... \\ | ||
| + | \rightarrow Q &= 2.78 | ||
| + | \end{align*} | ||
| + | |||
| + | |||
| + | # | ||
| - | - Calculate the Quality $Q = {{1}\over{R}}\sqrt{{{L}\over{C}}}$. | ||
| </ | </ | ||