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    "# Average Power, Apparent Power, and Power Factor for Sinusoidal Excitation\n",
    "Ig $v(t) = V_m cos(\\omega t)$, then then because we have a non-linear impedance, $i(t) = \\sum c_n cos(n \\omega t + \\phi_n)$.  The cause of harmonics can be seen if we take a simple I-V curve that has clipping for the current.  You can write the current as a power series (using Taylor's work), and from that you can see how it is the same as a Fourier series with harmonics caused by the clipping.  The resulting waveform is periodic, but now has frequency components at $n\\omega$.  ![Cause of Harmonics](Cause_of_Harmonics.png)\n",
    "\").\n"
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    "In the linear load situation, $\\bar V = V_m$ and $\\bar I = V_m/(|Z|e^{j\\theta})$ where $\\theta$ is the angle of the impedance.  Remember that the root mean square concept allowed you to say $$P=V_{rms} I_{rms} cos(\\theta) =1/2V_m V_m/|Z| cos(-\\theta) = Re[\\mathbf S]$$  From Circuits you remember with the linear load, you have:"
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    "$$I_m = V_m/|Z|$$  The complex power was $\\mathbf S$.  The magnitude of the complex power is the apparanent power,$S$.  \n",
    "$$|\\mathbf S|=S=1/2V_m I_m = V_{rms} I_{rms}$$ \n",
    "$$P=1/2V_m I_m cos(\\theta)=V_{rms} I_{rms} cos(\\theta)$$\n",
    "$$p.f. = cos(\\theta) =1/2V_mI_m/S = P/S$$"
   ]
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   "source": [
    "For the non-linear load, we use the definition for the power factor: $$ p.f. = P/S$$"
   ]
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   "source": [
    "where $P$ is the power in the fundamental.  Note that average power in other frequencies is zero.  The voltage of the fundamental multiplied by the current of a harmonic aerages to zero, because $$\\int_T cos(2\\pi t/T) cos(2\\pi n t/T + \\phi_n) dt = 0$$"
   ]
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    "For example, ask Wolfram alpha, and get this: ![Wolfram thinks it is zero](harmonic_power_zero.png) Thus the power in the harmonics is reactive power.  \n",
    "It is reflected back and forth between the source and load, and reduces the power factor.  There is a more complete write up, see [this.](https://www.a-eberle.de/sites/default/files/media/I022-1-D-1-001-04_Infobrief%20Nr%2022-EN.pdf)"
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