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-====== 6. Inductances in Circuits ======+====== 6 Inductances in Circuits ======
  
 ===== 6.1 Basic Circuits ===== ===== 6.1 Basic Circuits =====
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 ==== 6.1.1 Series Circuits ==== ==== 6.1.1 Series Circuits ====
  
-Based on $L = {{ \Psi(t)}\over{i}}$ and Kirchhoff's mesh law ($i=const$) the series circuit of inductions can be interpreted as a single current $i$ which generates multiple linked fluxes $\Psi$. Since the current must stay constant in the series circuit, the following applies for the equivalent inductor of a series connection of signle ones:+Based on $L = {{ \Psi(t)}\over{i}}$ and Kirchhoff's mesh law ($i=\rm const$) the series circuit of inductions can be interpreted as a single current $i$ which generates multiple linked fluxes $\Psi$. Since the current must stay constant in the series circuit, the following applies for the equivalent inductor of a series connection of single ones:
  
-\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}                                       & = &u_1                                    & + &u_2                        &+ ... \\  
 +& L_{\rm eq} {{{\rm d}i_{\rm eq} }\over{{\rm d}t}} & = &L_{1} {{{\rm d}i_{1} }\over{{\rm d}t}} & + &L_{2} {{di_{2} }\over{dt}} &+ ... \\  
 +& L_{\rm eq} {{{\rm d}i }\over{{\rm d}t}}          & = &L_{1} {{{\rm d}i     }\over{{\rm d}t}} & + &L_{2} {{di     }\over{dt}} &+ ... \\  
 +& L_{\rm eq}                                       & = &L_{1}                                  & + &L_{2}                      &+ ... \\  
 +\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's mesh law:+For parallel circuits, one can also start with the principles based on Kirchhoff's mesh law:
  
-\begin{align*} u_{eq}= u_1 = u_2 = ... \\ \end{align*}+\begin{align*} u_{\rm eq}= u_1 = u_2 = ... \\ \end{align*}
  
 and Kirchhoff's nodal law: and Kirchhoff's nodal law:
  
-\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} dt &= L \cdot i \\ i &= {{1}\over{L}} \cdot \int u_{ind} dt \\ \end{align*}+\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 \\  
 +                        i &= {{1}\over{L}} \cdot \int u_{\rm ind} {\rm d}t \\  
 +\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} dt &=& {{1}\over{L_1}} \cdot \int u_{1} dt &+& {{1}\over{L_2}} \cdot \int u_{2} dt &+& ... \\ {{1}\over{L_{eq}}} \cdot \int u dt &=& {{1}\over{L_1}} \cdot \int u dt &+& {{1}\over{L_2}} \cdot \int u dt &+& ... \\ {{1}\over{L_{eq}}} &=& {{1}\over{L_1}} &+& {{1}\over{L_2}} &+& ... \\ \end{align*}+\begin{align*}  
 +                                  i_{\rm eq}          &=& i_1                                       &+& i_2                                       &+& ... \\  
 +{{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}}                           &+& {{1}\over{L_2}}                           &+& ... \\  
 +\end{align*}
  
 <callout icon="fa fa-exclamation" color="red" title="Notice:"> The inductor behaves in the parallel and series circuit similar to the resistor. </callout> <callout icon="fa fa-exclamation" color="red" title="Notice:"> The inductor behaves in the parallel and series circuit similar to the resistor. </callout>
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 ==== 6.1.3 in AC Circuits ==== ==== 6.1.3 in AC Circuits ====
  
-For AC circuits (i.e. with sinosidal signals) the impedance $Z$ based on the real part $R$ and imaginary part $X$ has to be considered. In order to do so, one has to solve:+For AC circuits (i.e. with sinusoidal signals) the impedance $Z$ based on the real part $R$ and imaginary part $X$ has to be considered. To do so, one has to solve:
  
-\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}  \end{align*}
  
-With the induction $u_{ind}= - L {{di}\over{dt}}$ we get:+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}} }\over{\underline{i}}} \\ &= L {{{{d\underline{i}}\over{dt}} }\over{\underline{i}}} \\ \end{align*}+\begin{align*}  
 +\underline{Z} &=  {{1} \over {\underline{i}}} \cdot L { { {\rm d}\underline{i} } \over {{\rm d}t} }  \\  
 +\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} &= L {{{{d \left(I \cdot \sqrt{2} \cdot e^{j \cdot \omega t + \varphi_0}\right)}\over{dt}}}\over{I \cdot \sqrt{2} \cdot e^{j \cdot \omega t + \varphi_0}}} \\[10pt] &= L{{ {{d}\over{dt}} \left( I \cdot \sqrt{2} \cdot e^{j \cdot \omega t + \varphi_0}\right)}\over{I \cdot \sqrt{2} \cdot e^{j \cdot \omega t + \varphi_0}}} \\[10pt] &= L \cdot{{ I \cdot \sqrt{2} \cdot {{d}\over{dt}}e^{j \cdot \omega t + \varphi_0}}\over{I \cdot \sqrt{2} \cdot e^{j \cdot \omega t + \varphi_0}}} \\[10pt] &= L \cdot{{ j \cdot \omega \cdot e^{j \cdot \omega t + \varphi_0}}\over{e^{j \cdot \omega t + \varphi_0}}} \\[10pt] &= L \cdot j \cdot \omega \\[10pt] \end{align*}+\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                                       I \cdot \sqrt{2} \cdot {{ {\rm d}} \over {{\rm d}t} } \left( {\rm e}^{{\rm j} \cdot \omega t + \varphi_0} \right)  \\  
 +              &= {{1} \over {\qquad\quad\; {\rm e}^{{\rm j} \cdot \omega t + \varphi_0}}} \cdot L                                       \qquad\; \cdot {{ {\rm d}} \over {{\rm d}t} } \left( {\rm e}^{{\rm j} \cdot \omega t + \varphi_0} \right)  \\  
 +              &= {{1} \over {\qquad\quad\; {\rm e}^{{\rm j} \cdot \omega t + \varphi_0}}} \cdot L                                       \qquad\; \cdot j\omega \cdot {\rm e}^{{\rm j} \cdot \omega t + \varphi_0}   \\ \\ 
 +\underline{Z} &= L \cdot {\rm j}\omega  \\  
 +\end{align*}
  
-<callout icon="fa fa-exclamation" color="red" title="Notice:"> In AC calculations the inductor is described with a complex impedance $\underline{Z} = L \cdot j \omega$ </callout>+<callout icon="fa fa-exclamation" color="red" title="Notice:"> In AC calculations the inductor is described with a complex impedance $\boxed{\underline{Z} = L \cdot j \omega}$ </callout>
  
 ===== 6.2 Charging and Discharging ===== ===== 6.2 Charging and Discharging =====
  
-Charging and discharging an $RL$ circuit is comparable to the RC-circuit in chapter [[:electrical_engineering_1:dc_circuit_transients|DC Circuit Transients]] fomr last semester. Details are not covered here; see [[https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Book%3A_University_Physics_II_-_Thermodynamics_Electricity_and_Magnetism_(OpenStax)/14%3A_Inductance/14.05%3A_RL_Circuits|OpenStax]]+Charging and discharging an $RL$ circuit is comparable to the RC-circuit in chapter [[:electrical_engineering_1:dc_circuit_transients|DC Circuit Transients]] from last semester. Details are not covered here; see [[https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Book%3A_University_Physics_II_-_Thermodynamics_Electricity_and_Magnetism_(OpenStax)/14%3A_Inductance/14.05%3A_RL_Circuits|OpenStax]]
  
 <WRAP>{{url>https://www.falstad.com/circuit/circuitjs.html?running=false&ctz=CQAgjCAMB0l3BOJyWoVaYwA4wGYA2SAFjACZsCBWA4q9PCKkYvEKgUwFosAoAJ3AIyIbJBCFso8eLC8ANhIJSxSqWTLEoGOBBm8AbuADsBEBq1hT4ItvHFxzGRiq8A7ucjrNa8z8i8AMpCIqpgwjbO4gBmAIbyAM4c2mTunlJgtmRefloBHlZmFum5ULwA9uDm2gAeuJBW7CAE0HnVIrIS1eIArgD6ADIAOgmxAHYAJiMAloO8eBkgAGJ6OhBgsBA8IACSkz0AxgAu4wcc81LiK9rrDSDbAEocCdMJJ2NnvEA noborder}} </WRAP> <WRAP>{{url>https://www.falstad.com/circuit/circuitjs.html?running=false&ctz=CQAgjCAMB0l3BOJyWoVaYwA4wGYA2SAFjACZsCBWA4q9PCKkYvEKgUwFosAoAJ3AIyIbJBCFso8eLC8ANhIJSxSqWTLEoGOBBm8AbuADsBEBq1hT4ItvHFxzGRiq8A7ucjrNa8z8i8AMpCIqpgwjbO4gBmAIbyAM4c2mTunlJgtmRefloBHlZmFum5ULwA9uDm2gAeuJBW7CAE0HnVIrIS1eIArgD6ADIAOgmxAHYAJiMAloO8eBkgAGJ6OhBgsBA8IACSkz0AxgAu4wcc81LiK9rrDSDbAEocCdMJJ2NnvEA noborder}} </WRAP>
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 ===== 6.3 Resonance Phenomena ===== ===== 6.3 Resonance Phenomena =====
  
-Similar to the approach last semester we now focus on circuits with inductors $L$. For preparation, please recap the chapter [[:electrical_engineering_1:circuits_under_different_frequencies|Circuits under different Frequencies]] from last semester.+Similar to last semester's approach, we now focus on circuits with inductors $L$. For preparation, please recap the chapter [[:electrical_engineering_1:circuits_under_different_frequencies|Circuits under different Frequencies]] from last semester.
  
 ==== 6.3.1 RLC - Series Resonant Circuit ==== ==== 6.3.1 RLC - Series Resonant Circuit ====
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 As seen last semester, the circuits with complex impedances can be interpreted as four-terminal networks. There, we will again look at "output versus input", i.e: $A_V = {{U_O}\over{U_I}} \rightarrow \underline{A}_V = {{\underline{U}_O}\over{\underline{U}_I}} $. As seen last semester, the circuits with complex impedances can be interpreted as four-terminal networks. There, we will again look at "output versus input", i.e: $A_V = {{U_O}\over{U_I}} \rightarrow \underline{A}_V = {{\underline{U}_O}\over{\underline{U}_I}} $.
  
-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.
  
 <WRAP> <imgcaption imageNo01 | circuit of the series resonant circuit> </imgcaption> {{drawio>SchaltungdesSerienschwingkreises.svg}} </WRAP> <WRAP> <imgcaption imageNo01 | circuit of the series resonant circuit> </imgcaption> {{drawio>SchaltungdesSerienschwingkreises.svg}} </WRAP>
  
-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, it must be considered how the voltages behave across all the individual components in the following. The total voltage (= input voltage $U_I$) results to:+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, it must be considered how the voltages behave across all the individual components in the following. The total voltage (= input voltage $U_I$) results to:
  
 \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*}
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 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        &=        R \cdot \underline{I} + {\rm j} \omega L \cdot \underline{I} +               \frac {1}{j\omega C }         \cdot \underline{I} \\  
 +\underline{U}_I        &= \left( R                     + {\rm j} \omega L                     - {\rm j} \cdot \frac {1}{ \omega C } \right) \cdot \underline{I} \\  
 +\underline{Z}_{\rm eq} &=        R                     + {\rm j} \omega L                     - {\rm j} \cdot \frac {1}{ \omega C }  
 +\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  
 +         &= \arctan \frac{\omega L - \frac{1}{\omega C}}{R}  
 +\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              & = \frac{1}{     \sqrt{LC}} \\  
 +2\pi f_0              & = \frac{1}{     \sqrt{LC}}  
 +\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**.
  
-^  ^  $\quad$  ^  $f \rightarrow 0$  ^  $\quad$  ^  $f = f_0$  ^  $\quad$  ^  $f \rightarrow \infty$  | +^                                  ^  $\quad$  ^  $f \rightarrow 0$                                                   ^  $\quad$  ^  $f = f_0$  ^  $\quad$  ^  $f \rightarrow \infty$  | 
-|voltage $U_R$ \\ at the resistor |  |  $\boldsymbol{\small{0}}$  |  |  $\boldsymbol{\LARGE{U}}$ \\ since the impedances just cancel out  |  |  $ \boldsymbol{\small{0}}$  | +|voltage $U_R$ \\ at the resistor  |  |  $\boldsymbol{\small{0}}$                                                     |  |  $\boldsymbol{\LARGE{U}}$ \\ since the impedances just cancel out  |  |  $ \boldsymbol{\small{0}}$  | 
-|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  |  |  $\boldsymbol{\small{0}}$ \\ 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 $U_C$ \\ 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**  $Q_S$:+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**  $Q_{\rm S}$: 
  
-\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}  
 +                  = \left.\frac{U_L}{U} \right\vert_{\omega = \omega_0}  
 +                  = \color{blue}{\frac{1}{R}\sqrt{\frac{L}{C}}} }  
 +\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, e.g. in an RLC element as an antenna. +  * 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, e.g. in an $RLC$ element as an antenna. 
-  * 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**  $d_S$. This is specified when using the circuit as a non-overshooting filter.+The reciprocal of the $Q$ is called **attenuation**  $d_{\rm S}$. This is specified when using the circuit as a non-overshooting filter.
  
-\begin{align*} \boxed{ d_S = \frac{1}{Q_S} = R \sqrt{\frac{C}{L}} } \end{align*}+\begin{align*} \boxed{ d_{\rm S} = \frac{1}{Q_{\rm S}} = R \sqrt{\frac{C}{L}} } \end{align*}
  
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 <WRAP> <imgcaption imageNo02 | Amplitude and Phase Response of a Series Resonant Circuit> </imgcaption> {{drawio>AmplitudeandPhaseResponse.svg}} </WRAP> <WRAP> <imgcaption imageNo02 | Amplitude and Phase Response of a Series Resonant Circuit> </imgcaption> {{drawio>AmplitudeandPhaseResponse.svg}} </WRAP>
  
-<accordion> <panel title="Series Resonant Circuit in Time Domain (Voltage on Inductor)"> <WRAP>{{url>https://www.falstad.com/circuit/circuitjs.html?running=false&ctz=CQAgjCAMB0l5AWAnC1b0DYrQEw8gOyFgAcYArBpBkQlgpCOSAgMxMCmAtGGAFAAlcDhIgk9UeKjgWcJtMYxyfAE7DJWfBums+AYxBaxmktsWx4CAtgSUwGZCWrkkpGlhhx+AG0OnjfqJgBB7YXiBcMATIGDjRkE4EcZDkCFB8AOaB4CHZRop8+NYi2qzkOAFpAPYArgAufFXSEIwMSEHmLYbykHysoowAYhDsDIxg8BEQgyocAI41HAB2egCefEA 500,400 noborder}} </WRAP> </panel> <panel title="Series Resonant Circuit in Frequency Domain (Voltage on Inductor)"> <WRAP>{{url>https://falstad.com/afilter/circuitjs.html?running=false&cct=$+1+0.000005+5+50+5+50%0A%25+4+1959.030510288011%0Ac+256+80+304+80+0+0.000047+0%0Ar+192+80+256+80+0+3%0Ag+304+160+304+192+0%0A170+192+80+160+80+3+20+1000+5+0.1%0Al+304+160+304+80+0+0.01+0.46265716582988115%0AO+304+80+352+80+0%0Ao+3+16+0+34+5+0.00009765625+0+-1+in%0A 500,400 noborder}} </WRAP> </panel> <panel title="Series Resonant Circuit in Frequency Domain (Voltage on Capacitor)"> <WRAP>{{url>https://falstad.com/afilter/circuitjs.html?running=false&cct=$+1+0.000005+5+50+5+50%0A%25+4+1959.030510288011%0Ac+304+160+304+80+0+0.000047+0%0Ar+192+80+256+80+0+3%0Ag+304+160+304+192+0%0A170+192+80+160+80+3+20+1000+5+0.1%0Al+256+80+304+80+0+0.01+0.46265716582988115%0AO+304+80+352+80+0%0Ao+3+16+0+34+5+0.00009765625+0+-1+in%0A 500,400 noborder}} </WRAP> ~~PAGEBREAK~~ ~~CLEARFIX~~ </panel> <panel title="Series Resonant Circuit in Frequency Domain (Voltage on Resistor)"> <WRAP>{{url>https://falstad.com/afilter/circuitjs.html?running=false&cct=$+1+0.000005+5+50+5+50%0A%25+4+1959.030510288011%0Ac+192+80+256+80+0+0.000047+0%0Ar+304+80+304+160+0+3%0Ag+304+160+304+192+0%0A170+192+80+160+80+3+20+1000+5+0.1%0Al+256+80+304+80+0+0.01+0.46265716582988115%0AO+304+80+352+80+0%0Ao+3+16+0+34+5+0.00009765625+0+-1+in%0A 500,400 noborder}} </WRAP> </panel> <panel title="Parallel Resonant Circuit in Frequency Domain (Voltage on Resistor)"> <WRAP>{{url>https://falstad.com/afilter/circuitjs.html?running=false&cct=$+1+0.000005+5+50+5+50%0A%25+4+17425.09278334521%0Ac+176+96+176+16+0+0.00038999999999999994+0%0Ag+240+144+240+176+0%0A170+128+16+96+16+3+20+1000+5+0.1%0Al+240+96+240+16+0+0.00009999999999999999+7.7025416617456886%0AO+240+96+288+96+0%0Aw+128+16+176+16+0%0Ar+240+96+240+144+0+0.1%0Aw+176+96+240+96+0%0Aw+240+16+176+16+0%0Ao+2+16+0+34+5+0.00009765625+0+-1+in%0A 500,400 noborder}} </WRAP> </panel> </accordion>+<accordion>  
 +<panel title="Series Resonant Circuit in Time Domain (Voltage on Inductor)">  
 +<WRAP>{{url>https://www.falstad.com/circuit/circuitjs.html?running=false&ctz=CQAgjCAMB0l5AWAnC1b0DYrQEw8gOyFgAcYArBpBkQlgpCOSAgMxMCmAtGGAFAAlcDhIgk9UeKjgWcJtMYxyfAE7DJWfBums+AYxBaxmktsWx4CAtgSUwGZCWrkkpGlhhx+AG0OnjfqJgBB7YXiBcMATIGDjRkE4EcZDkCFB8AOaB4CHZRop8+NYi2qzkOAFpAPYArgAufFXSEIwMSEHmLYbykHysoowAYhDsDIxg8BEQgyocAI41HAB2egCefEA 500,400 noborder}}  
 +</WRAP> </panel>  
 + 
 +<panel title="Series Resonant Circuit in Frequency Domain (Voltage on Inductor)">  
 +<WRAP>{{url>https://falstad.com/afilter/circuitjs.html?running=false&cct=$+1+0.000005+5+50+5+50%0A%25+4+1959.030510288011%0Ac+256+80+304+80+0+0.000047+0%0Ar+192+80+256+80+0+3%0Ag+304+160+304+192+0%0A170+192+80+160+80+3+20+1000+5+0.1%0Al+304+160+304+80+0+0.01+0.46265716582988115%0AO+304+80+352+80+0%0Ao+3+16+0+34+5+0.00009765625+0+-1+in%0A 500,400 noborder}}  
 +</WRAP> </panel>  
 + 
 +<panel title="Series Resonant Circuit in Frequency Domain (Voltage on Capacitor)">  
 +<WRAP>{{url>https://falstad.com/afilter/circuitjs.html?running=false&cct=$+1+0.000005+5+50+5+50%0A%25+4+1959.030510288011%0Ac+304+160+304+80+0+0.000047+0%0Ar+192+80+256+80+0+3%0Ag+304+160+304+192+0%0A170+192+80+160+80+3+20+1000+5+0.1%0Al+256+80+304+80+0+0.01+0.46265716582988115%0AO+304+80+352+80+0%0Ao+3+16+0+34+5+0.00009765625+0+-1+in%0A 500,400 noborder}}  
 +</WRAP> ~~PAGEBREAK~~ ~~CLEARFIX~~ </panel>  
 + 
 +<panel title="Series Resonant Circuit in Frequency Domain (Voltage on Resistor)">  
 +<WRAP>{{url>https://falstad.com/afilter/circuitjs.html?running=false&cct=$+1+0.000005+5+50+5+50%0A%25+4+1959.030510288011%0Ac+192+80+256+80+0+0.000047+0%0Ar+304+80+304+160+0+3%0Ag+304+160+304+192+0%0A170+192+80+160+80+3+20+1000+5+0.1%0Al+256+80+304+80+0+0.01+0.46265716582988115%0AO+304+80+352+80+0%0Ao+3+16+0+34+5+0.00009765625+0+-1+in%0A 500,400 noborder}}  
 +</WRAP> </panel>  
 + 
 +<panel title="Parallel Resonant Circuit in Frequency Domain (Voltage on Resistor)">  
 +<WRAP>{{url>https://falstad.com/afilter/circuitjs.html?running=false&cct=$+1+0.000005+5+50+5+50%0A%25+4+17425.09278334521%0Ac+176+96+176+16+0+0.00038999999999999994+0%0Ag+240+144+240+176+0%0A170+128+16+96+16+3+20+1000+5+0.1%0Al+240+96+240+16+0+0.00009999999999999999+7.7025416617456886%0AO+240+96+288+96+0%0Aw+128+16+176+16+0%0Ar+240+96+240+144+0+0.1%0Aw+176+96+240+96+0%0Aw+240+16+176+16+0%0Ao+2+16+0+34+5+0.00009765625+0+-1+in%0A 500,400 noborder}}  
 +</WRAP> </panel> </accordion>
  
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   * Decoupling   * Decoupling
   * Filter   * Filter
-  * unwanted coupling and cirucit design+  * unwanted coupling and circuit design
  
 ===== 6.5 Examples ===== ===== 6.5 Examples =====
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 <panel type="info" title="Exercise 6.3.1 Series Resonant Circuit I"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> <panel type="info" title="Exercise 6.3.1 Series Resonant Circuit I"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
-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 $C=1 ~\rm µF$. A voltage source with $U_I$ feeds the circuit at $f_1 = 50~\rm Hz$.
  
   - 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 $f_2 = 60~\rm Hz$?
  
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 <panel type="info" title="Exercise 6.3.2 Series Resonant Circuit II"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> <panel type="info" title="Exercise 6.3.2 Series Resonant Circuit II"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
-A given $R$-$L$-$C$ series circuit is fed with a frequency, which is $20\%$ larger than the resonance frequency keeping the amplitude of the input voltage constant. In this situation, the circuit shows a current which is $30\%$ lower than the maximum current value.+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 that is $30~\%$ lower than the maximum current value.
  
   - Calculate the Quality $Q = {{1}\over{R}}\sqrt{{{L}\over{C}}}$.   - Calculate the Quality $Q = {{1}\over{R}}\sqrt{{{L}\over{C}}}$.