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electrical_engineering_and_electronics_1:block20 [2025/12/02 18:44] – mexleadminelectrical_engineering_and_electronics_1:block20 [2025/12/16 14:14] (aktuell) – mexleadmin
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-====== Block 20 — Electromagnetic Induction and Energy ======+====== Block 20 — Inductance and Energy ======
  
 ===== Learning objectives ===== ===== Learning objectives =====
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 <WRAP> <imgcaption ImgNr15 | Self-Induction of a Coil> </imgcaption> {{drawio>SelfInductionCoil.svg}} </WRAP> <WRAP> <imgcaption ImgNr15 | Self-Induction of a Coil> </imgcaption> {{drawio>SelfInductionCoil.svg}} </WRAP>
  
-The created field density of the coil can be derived from Ampere's Circuital Law +Given by the [[block16#Recap of the fieldline images]] in Block16, we know that the $H$-field is given by magnetic voltage $\theta(t) = N \cdot i$ as:
 \begin{align*}  \begin{align*} 
-\theta(t) &= \int & \vec{H}(t) \cdot {\rm d}\vec{s} \\  +   {H}(t) &=                         {{N \cdot i }\over {l}} \\ 
-          &= \int & \vec{H}_{\rm inner}(t) \cdot {\rm d}\vec{s} & + & \int \vec{H}_{\rm outer}(t) \cdot {\rm d} \vec{s} \\  +
-          &= \int & \vec{H}(t) \cdot {\rm d}\vec{s}             & + &   0 \\  +
-          &=      & {H}(t) \cdot l \\ +
 \end{align*} \end{align*}
  
-With magnetic voltage $\theta(t) = N \cdot i$ this lead to the magnetic flux density $B(t)$+This lead to the magnetic flux density $B(t)$
  
 \begin{align*}  \begin{align*} 
-N \cdot i &= {H}(t) \cdot l \\  
-   {H}(t) &=                         {{N \cdot i }\over {l}} \\  
    {B}(t) &= \mu_0 \mu_{\rm r} \cdot {{N \cdot i }\over {l}} \\     {B}(t) &= \mu_0 \mu_{\rm r} \cdot {{N \cdot i }\over {l}} \\ 
 \end{align*} \end{align*}
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 \end{align*} \end{align*}
  
-The changing flux $\Phi$ is now creating an induced electric voltage and current, which counteracts the initial change of the current. +The changing flux $\Phi$ is now creating an induced electric voltage and current, which counteracts the initial change of the current. \\
 This effect is called **Self Induction**. The induced electric voltage $u_{\rm ind}$ is given by: This effect is called **Self Induction**. The induced electric voltage $u_{\rm ind}$ is given by:
  
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 \end{align*} \end{align*}
  
-The result means that the induced electric voltage $u_{\rm ind}$ is proportional to the change of the current ${{\rm d}\over{{\rm d}t}}i$. +The result means that the induced electric voltage $u_{\rm ind}$ is proportional to the change of the current ${{\rm d}\over{{\rm d}t}}i$. \\
 The proportionality factor is also called **Self-inductance**  $L$ (or often simply called inductance). The proportionality factor is also called **Self-inductance**  $L$ (or often simply called inductance).
  
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 \end{align*} \end{align*}
  
 +
 +==== 6 Inductances in Circuits ====
 +
 +Focus here: uncoupled inductors!
 +
 +=== Series Circuits ===
 +
 +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_{\rm eq} &= {{\sum_i \Psi_i}\over{I}} = \sum_i L_i \end{align*}
 +
 +A similar result can be derived from the induced voltage $u_{ind}= L {{{\rm d}i}\over{{\rm d}t}}$, 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_{\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*}
 +
 +===Parallel Circuits ===
 +
 +For parallel circuits, one can also start with the principles based on Kirchhoff's mesh law:
 +
 +\begin{align*} u_{\rm eq}= u_1 = u_2 = ... \\ \end{align*}
 +
 +and Kirchhoff's nodal law:
 +
 +\begin{align*} i_{\rm eq}= i_1 + i_2 + ... \\ \end{align*}
 +
 +Here, the formula for the induced voltage has to be rearranged:
 +
 +\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:
 +
 +\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>
  
  
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 ===== Exercises ===== ===== Exercises =====
 +
  
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