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electrical_engineering_2:inductances_in_circuits [2024/06/05 02:56] – [Exercises] mexleadminelectrical_engineering_2:inductances_in_circuits [2024/06/06 11:08] (aktuell) – [Exercises] mexleadmin
Zeile 238: Zeile 238:
 But first, add some more info, which is always true from resonant circuits at the resonant frequency: But first, add some more info, which is always true from resonant circuits at the resonant frequency:
   - $\omega_0 = {{1}\over{\sqrt{LC}}}$   - $\omega_0 = {{1}\over{\sqrt{LC}}}$
-  - $X_{C0} = X_{L0}$ +  - $X_{C0} = - X_{L0}$ 
-  - $Z = \sqrt{R^2 + (X_L - X_C)^2}$, based on the sum of the impedances $\underline{Z}_{\rm eq} = \underline{X}_R + \underline{X}_C + \underline{X}_L$ and the Pythagorean theorem$+  - $Z = \sqrt{R^2 + (X_L + X_C)^2}$, 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.  From the task, the following is also known. 
Zeile 247: Zeile 247:
   - The circuit shows a current $30~\%$ lower than the maximum current value:    - The circuit shows a current $30~\%$ lower than the maximum current value: 
     - 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 $Z_{\rm min} = R$      - 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: $Z = {{1}\cdot{0.7}} \cdot R$+    - Therefore: $Z = {{1}\over{0.7}} \cdot R$
 \\ \\
 <fs large>__Solution 2: The fast path__</fs> <fs large>__Solution 2: The fast path__</fs>
  
-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 $+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*} \begin{align*}
 (1): && Z_0 &= R \\ (1): && Z_0 &= R \\
-(2): && Z   &= \sqrt{R^2 + (X_L - X_C)^2} \\+(2): && Z   &= \sqrt{R^2 + (X_L + X_C)^2} \\
 \end{align*} \end{align*}
  
 In formula $(2)$ the impedance $X_L$ and $X_C$ are: In formula $(2)$ the impedance $X_L$ and $X_C$ are:
   * $X_L= \omega \cdot L$ and therefore also $X_L = 1.2 \cdot \omega_0 \cdot L = 1.2 \cdot X_{L0}$   * $X_L= \omega \cdot L$ 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 $X_C = {{1}\over {1.2 \cdot \omega \cdot C}} = {{1}\over {1.2}} \cdot X_{C0}$+  * $X_C= - {{1}\over {\omega \cdot C}}$ 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 $(1)$:  With $X_{C0} = X_{L0}$ we get for $(1)$: 
Zeile 269: Zeile 269:
 \end{align*} \end{align*}
  
-Since we know that $Z = {{1}\cdot{0.7}} \cdot R$ and $Z_0 = R$, we can start by dividing $(2)$ by $(1)$:+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*} \begin{align*}
-{{(2)}\over{(1)}} : &&  {{Z}\over{Z_0}}                 &= {{\sqrt{R^2 +                     (X_L - X_C)^2}                    }\over{R}}   & &| \text{put in the info from before}\\ +{{(2)}\over{(1)}} : &&  {{Z}\over{Z_0}}                 &= {{\sqrt{R^2 +                     (X_L + X_C)^2}                    }\over{R}}   & &| \text{put in the info from before}\\ 
-                    &&  {{1}\cdot{0.7}}                 &= {{\sqrt{R^2 + X_{L0}^2 \cdot \left(1.2 - {{1}\over{1.2}} \right)^2} }\over{R}}   & &| (...)^2 \\ +                    &&  {{1}\over{0.7}}                 &= {{\sqrt{R^2 + X_{L0}^2 \cdot \left(1.2 - {{1}\over{1.2}} \right)^2} }\over{R}}   & &| (...)^2 \\ 
-                    &&  {{1}\cdot{0.7^2}}               &= {{     {R^2 + X_{L0}^2 \cdot \left(1.2 - {{1}\over{1.2}} \right)^2} }\over{R^2}} & &| \cdot R^2 \\ +                    &&  {{1}\over{0.7^2}}               &= {{     {R^2 + X_{L0}^2 \cdot \left(1.2 - {{1}\over{1.2}} \right)^2} }\over{R^2}} & &| \cdot R^2 \\ 
-                    &&  {{1}\cdot{0.7^2}}     \cdot R^2 &=        {R^2 + X_{L0}^2 \cdot \left(1.2 - {{1}\over{1.2}} \right)^2}              & &| -R^2 \\ +                    &&  {{1}\over{0.7^2}}     \cdot R^2 &=        {R^2 + X_{L0}^2 \cdot \left(1.2 - {{1}\over{1.2}} \right)^2}              & &| -R^2 \\ 
-                    && ({{1}\cdot{0.7^2}} -1) \cdot R^2 &=        {      X_{L0}^2 \cdot \left(1.2 - {{1}\over{1.2}} \right)^2}              & &| -R^2 \\+         && \left({{1}\over{0.7^2}} -1\right) \cdot R^2 &=        {      X_{L0}^2 \cdot \left(1.2 - {{1}\over{1.2}} \right)^2}              & &| : R^2 \quad | : \left(1.2 - {{1}\over{1.2}} \right)^2 \\ 
 +                && {{X_{L0}^2}\over{R^2}}           &={ { {{1}\over{0.7^2}} -1 } \over { \left(1.2 - {{1}\over{1.2}} \right)^2 } }          & &| \sqrt{...} \\ 
 +                    && {{X_{L0}}\over{R}}           &={ \sqrt{ {{1}\over{0.7^2}} -1 } \over  { 1.2 - {{1}\over{1.2}} } }                    & &| \text{with } X_{L0} = \omega_0 \cdot L = {{1}\over{\sqrt{LC}}} \cdot L = \sqrt{ {L} \over {C} }\\ 
 +                    && {{1}\over{R}}\cdot \sqrt{ {L} \over {C} }           &={ \sqrt{ {{1}\over{0.7^2}} -1 } \over  { 1.2 - {{1}\over{1.2}} } }               
 \end{align*} \end{align*}