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--- electrical_engineering_and_electronics_2:block06 [2026/04/11 12:22] – mexleadmin
+++ electrical_engineering_and_electronics_2:block06 [2026/04/21 03:48] (current) – mexleadmin
@@ Line 1: / Line 1: @@
-====== Block xx - xxx ======
+====== Block 06 - Complex Power ======
 ===== Learning objectives =====
@@ Line 49: / Line 49: @@
 Thus, the induced voltage $u(t)$ is given by:
 \begin{align*}
-u(t) &=               -\frac{{\rm d}                  \Psi}            {{\rm d}t} \\
+u(t) &=                -\frac{{\rm d~}                  \Psi}            {{\rm d}t} \\
-     &= -N  \cdot       \frac{{\rm d}                  \Phi}            {{\rm d}t} \\
+     &= -N  \cdot       \frac{{\rm d~}                  \Phi}            {{\rm d}t} \\
-     &= -NBA\cdot       \frac{{\rm d}       \cos \varphi(t)}            {{\rm d}t} \\
+     &= -NBA\cdot       \frac{{\rm d~}       \cos \varphi(t)}            {{\rm d}t} \\
-     &= -\hat{\Psi}\cdot\frac{{\rm {\rm d}} \cos (\omega t + \varphi_0)}{{\rm d}t} \\
+     &= -\hat{\Psi}\cdot\frac{{\rm {\rm d~}} \cos (\omega t + \varphi_0)}{{\rm d}t} \\
      &= \omega \hat{\Psi}           \cdot \sin (\omega t + \varphi_0) \\
      &= \hat{U}                     \cdot \sin (\omega t + \varphi_0) \\
@@ Line 102: / Line 102: @@
   - The use of the $\sqrt{2}$ in the definition $\color{blue}{u_R(t)} = \sqrt{2} U \sin(\omega t + \varphi_u)$ leads to the average power as $P_R = {{U^2}\over{R}}$. This formula for the power is exactly like the formula for the power in pure DC situations.
-==== Ideal Inductivity L ====
+==== Ideal Capacity C ====
-A similar approach is done for the ideal inductivity.
+Also here, we start with the basic definition of the instantaneous voltage
-We again start with the basic definition of the instantaneous voltage
-\begin{align*} \color{blue}{u_{\rm L}(t)} &= \sqrt{2}U \sin(\omega t + \varphi_u) \end{align*}
+\begin{align*} \color{blue}{u_C(t)} &= \sqrt{2}U \sin(\omega t + \varphi_u) \end{align*}
-With the defining formula for inductivity, we get:
+With the defining formula for the capacity, we get:
 \begin{align*}
-           \color{blue}{u_{\rm L}(t)} &=  L\cdot    {{{\rm d}\color{red} {i_{\rm L}(t)}}\over{{\rm d}t}} \\
+\color{red}{i_C(t)} &= C {{{\rm d}\color{blue}{u_C(t)}}\over{{\rm d}t}} \\
-\rightarrow \color{red}{i_{\rm L}(t)} &= {{1}\over{L}} \int  \color{blue}{u_{\rm L}(t)}       {\rm d}t \\
+                    &= \sqrt{2} U \omega C \cos(\omega t + \varphi_u)
-                                &= - \sqrt{2} {{U}\over{\omega L}} \cos(\omega t + \varphi_u)
 \end{align*}
-This leads to an instantaneous power $p_L(t)$ of
+This leads to an instantaneous power $p_C(t)$ of
 \begin{align*}
-p_L(t) &=   \color{blue}{u(t)}            \cdot \color{red}{i(t)} \\
+p_C(t) &= \color{blue}{u_C(t)} \cdot \color{red}{i_C(t)} \\
-       &= - 2\cdot {{U^2}\over{\omega L}} \cdot \sin(          \omega t + \varphi_u) \cos(\omega t + \varphi_u) \\
+       &= 2\cdot U^2 \omega C  \cdot \sin(         \omega t + \varphi_u) \cos(\omega t + \varphi_u) \\
-       &= - {{U^2}\over{\omega L}}        \cdot \sin( 2\cdot (\omega t + \varphi_u)) \\
+       &= +      U^2 \omega C  \cdot \sin( 2\cdot (\omega t + \varphi_u)) \\
 \end{align*}
-Again a trigonometric identity ({{https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Double-angle_formulae|Double-angle formula}}  "$\sin(2x) = 2 \sin(x)\cos(x)$") was used.
 Also, this result is interesting:
   - The part $\sin( 2\cdot (\omega t + \varphi_u))$ has an average value of $0$.
-  - Therefore, the average value of $p_L(t)=0$
+  - Therefore, also the average value of $p_C(t)=0$
-==== Ideal Capacity C ====
+==== Ideal Inductivity L ====
-Also here, we start with the basic definition of the instantaneous voltage
+A similar approach is done for the ideal inductivity.
+We again start with the basic definition of the instantaneous voltage
-\begin{align*} \color{blue}{u_C(t)} &= \sqrt{2}U \sin(\omega t + \varphi_u) \end{align*}
+\begin{align*} \color{blue}{u_{\rm L}(t)} &= \sqrt{2}U \sin(\omega t + \varphi_u) \end{align*}
-With the defining formula for the capacity, we get:
+With the defining formula for inductivity, we get:
 \begin{align*}
-\color{red}{i_C(t)} &= C {{{\rm d}\color{blue}{u_C(t)}}\over{{\rm d}t}} \\
+           \color{blue}{u_{\rm L}(t)} &=  L\cdot    {{{\rm d}\color{red} {i_{\rm L}(t)}}\over{{\rm d}t}} \\
-                    &= \sqrt{2} U \omega C \cos(\omega t + \varphi_u)
+\rightarrow \color{red}{i_{\rm L}(t)} &= {{1}\over{L}} \int  \color{blue}{u_{\rm L}(t)}       {\rm d}t \\
+                                &= - \sqrt{2} {{U}\over{\omega L}} \cos(\omega t + \varphi_u)
 \end{align*}
-This leads to an instantaneous power $p_C(t)$ of
+Since we assume pure AC signals the integration constant has to be 0.
+\\
+This formulas lead to an instantaneous power $p_L(t)$ of
 \begin{align*}
-p_C(t) &= \color{blue}{u_C(t)} \cdot \color{red}{i_C(t)} \\
+p_L(t) &=   \color{blue}{u(t)}            \cdot \color{red}{i(t)} \\
-       &= 2\cdot U^2 \omega C  \cdot \sin(         \omega t + \varphi_u) \cos(\omega t + \varphi_u) \\
+       &= - 2\cdot {{U^2}\over{\omega L}} \cdot \sin(          \omega t + \varphi_u) \cos(\omega t + \varphi_u) \\
-       &= +      U^2 \omega C  \cdot \sin( 2\cdot (\omega t + \varphi_u)) \\
+       &= - {{U^2}\over{\omega L}}        \cdot \sin( 2\cdot (\omega t + \varphi_u)) \\
 \end{align*}
+Again a trigonometric identity ({{https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Double-angle_formulae|Double-angle formula}}  "$\sin(2x) = 2 \sin(x)\cos(x)$") was used.
 Again this result leads to:
   - The part $\sin( 2\cdot (\omega t + \varphi_u))$ has an average value of $0$.
-  - Therefore, also the average value of $p_C(t)=0$
+  - Therefore, the average value of $p_L(t)=0$
   * Instantaneous values of power at $R$, $L$, $C$
@@ Line 314: / Line 316: @@
 Alternatively, a bad power factor can be compensated with a counteracting complex impedance. This compensating impedance has to provide enough power with the opposite sign to cancel out the unwanted reactive power. The following simulation shows an uncompensated circuit and a circuit with power factor correction. In the ladder, the voltage on the load resistor is the same, but the current provided by the power supply is smaller.
-<WRAP>{{url>https://www.falstad.com/circuit/circuitjs.html?running=false&ctz=CQAgjCAMB0l3Z5OfATFaYwGZtgCwAc+ArNpAGwWGoXYjkgkj70kCmAtFgFAA2ITgE4K4fPkFExEyBjioA7KM4w4JQktpDIQ-KlL4F6SDwBOkwiH2zOUgjJBDCJHgHdBIq-htTrUN16yfuQSfiYA5uBg6H5YMSSisiYAblEx3mnSUODgSCCJgmDQJAqQCiLYtOVEIonFZgyQoRkhWbJOLgKx4o0S9tmqkIrKg+qaFNq6+iSGxjypnKiQln6Ly205iPD52dzFpeV0VbqEtRgukWsrGVdWCQMBt-3Cov0m7i+BgkvXSTwAxl8wEJ0rJ+jFoEouEImPtCBR1EIwGRsAk8HJIGAGn4cRlgcZwFxogFcWDol93plwXieiZvPRbrZLNxXgUwAB9DmQdlLIgUTnc1TREjs7g8zns7A8enfdZSFmZWSECXcgAezkxIpFgwoqBFYtQ4u5yuw7MGhrpkHocQsvXABX5XJ5TXhArNsGFoo5hv5UplNqkrRtshFTvVJE17vgXI9qGVBqj8HwQhTqbT6f53JFpvNZp4qu+5M4JBhi2YxdEfUsAFUAHb-AD2AFsAA7sWsAZwAhgAXdgAEwAOh2AMIAS1M-wArmOe-ncsySwwi-cqyBx5OZz3h65ZwALYcABQAYsORw3TKZ2P8e2OG7WeNhLLJjxAkh7wB-uCAAJK1-tTjeXb1uwPBAA noborder}} </WRAP>
+<WRAP>{{url>https://www.falstad.com/circuit/circuitjs.html?running=false&ctz=DwYwlgTgBAZgvAIgIwKgFwM6IAwDpsEFKEmmEBMqYIOuSSAzA0gCwAcLArA9gGy9tyvBqhAAjRN1QAHCQhYioANwiTUAW0ySApgFp6CAHwAoKFGAAbKAA9EugJy8orFlF3tnLFqngJsqC1psFChqHCo7PAIAdhZeeyRObAY2JN5otmjOBAB6EzNgaFsEdzYochZsNw8XH3CoVQQk-zzTcwB3GztHcsrqsor-WHDWgs7iwd6qnldBur9c-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-jSXfLse2-Rze2KS4PDxYCZNAuTJwU7JoOUpcEOCALNJCLcMOw3C9N4hjiMs1lgnYvwksk+wMsyrLsp+KiRifMzGOgxLgnIKD1HIliuPyny7IKBzPyOaw+x2XROHsNwHzcPZ5muDBmWQKC0G0RAAFUpy7KjpwwN5hszKAAGFIBATCwDQd1mrciU2o65gxJ6m4+oGpAhpGhAlpw1a0DuNa-igAAFAAxRbOwgCBtBANAwE7Kd3REwUnQAodtgjN1ZPkyClNg3x4Mk4INPXNYoFiuj4qMqNiBSgK8pogqryI29EpCCqgmx7ijXq197ME5zP1OaInA8ySQZA3iIdIwLodUkLpPC7Tkbxgyk3R2chhJjmcZ4-T4sJ2hicq-AyZq2yqfqmnSTeTMACtJ1nCNtAGoYIPOt5pDecA0DeCbewACkegBKDQFLjBwNGKBg0rIUgWHjJiNQW03zbWq2YtaPjwAgEwgA noborder}} </WRAP>
 Another explanation of the power factor can be seen here: