High Pass Filter
================


As for the low pass filter we design a high pass filter using just one
passive element, in this case a capacitor in series with the
driver. In the schemematic below the speaker driver is represented
with the resistor.

.. tikz::
   :xscale: 35
	    
   \draw (0,0)
   to[voltage source, v=$u_{in}$] (0,4)
   to[short, i=$i$] (2,4)
   to[C=$Z_{cap}$] (2,2)
   to[R=$R$] (2,0)
   to[short] (0,0);
   \draw (5,0) to (5,0);

   
The transfer function in this case is:

.. math::
   H(\omega) = \frac{R}{R+\frac{1}{j\omega C}} = \frac{j\omega RC}{1+j\omega RC}

The bode plot for this system with $R=8\Omega$ and $C=10\mu F$ is
given below. The cutoff frequency appears to be near 2 KHz.


.. exec_python:: highpass electronics
   :linenumbers:
   :code: shutter
   :Code_label: Show code for figure
   :results: hide

   import numpy as np
   import matplotlib.pyplot as plt
	     
   plt.clf()
	     
   R = 8
   C = 10E-6

   w = np.logspace(-1,6,num=1000)
   H = R / (R+1/(1j*w*C))

   plt.plot(w, abs(H))
   plt.xscale('log')
   plt.yscale('log')
   plt.xlabel(r'$\omega$')
   plt.ylabel(r'$|H(\omega)|$')
   plt.savefig('source/figures/bodeabsplothighpass.png')

.. figure:: /figures/bodeabsplothighpass.png
   :width: 70%
   :align: center

   **Bode plot of high pass filter.**


Let's do some quick and dirty analysis to see where the cutoff
frequency is in terms of R and C. For low frequencies we have

.. math::
   H(\omega) \approx j\omega R C

and thus

.. math::
   \log | H(\omega)| = \log(RC) + \log(\omega)

and for large frequencies:

.. math::
   H(\omega) \approx 1

and thus

.. math::
   \log | H(\omega)| = 0

The cutoff frequency is thus at $\omega_x = 1/(RC)$ and for this
choice of resistor and capacitor $f_x = \omega_x/(2\pi) = 1989 Hz$