Example: Calculating an RC Circuit
==================================

.. code:: python

    # Imports
    %precision 2
    import PhysicalQuantities as pq
    import PhysicalQuantities.numpywrapper as nw
    %load_ext PhysicalQuantities.ipython
    from PhysicalQuantities.constants import eps0
    def disp(s):
        display(HTML(s))


.. parsed-literal::

    The PhysicalQuantities.ipython extension is already loaded. To reload it, use:
      %reload_ext PhysicalQuantities.ipython


We have a simple RC circuit, consisting of a voltage source :math:`U_0`,
a resistor with a value :math:`R` and a capacitor with value :math:`C`:

.. figure:: images/rc.png
   :alt: image1

   image1

Let's start with calculating the capacitance of a simple parallel plate
capacitor:

.. code:: python

    A = 1 mm * 1 mm # Plate area
    d = 1 um        # Plate distance
    eps_r = 3       # Permittivity of dielectric material
    C  = (eps0*eps_r*A) / d
    disp("Capacitor value $C$ = %s" % C.pF)



.. raw:: html

    Capacitor value $C$ = 26.56 pF


The resistor is a thick film resistor:

.. code:: python

    ρ = 10 Ohm * 1 mm # Material resistivity
    t = 10 um           # Sheet thickness
    L = 150 um          # Structure length
    W = 10 um           # Structure width
    Rs = ρ / t 
    R = Rs * L / W
    disp("Resistor value $R$ = %s" % R.kOhm)



.. raw:: html

    Resistor value $R$ = 15.00 kOhm


Calculate transient response of circuit
---------------------------------------

.. code:: python

    u0 = 1. V
    τ = 1/(R*C)
    t = nw.linspace(0, 4/τ.base, 100).us

.. code:: python

    i0 = u0/R
    i = (i0*e**(-t*τ)).uA
    disp("Initial current is $i_0$ = %s" % i0.uA)



.. raw:: html

    Initial current is $i_0$ = 66.67 uA


Plot Current and Voltage over Time
----------------------------------

.. code:: python

    plot(t._,i._)
    grid()
    title('Circuit Current')
    xlabel('Time in %s' % t.unit)
    ylabel('Current in %s' % i.unit);


.. code:: python

    uc = u0 - i*R
    plot(t._, uc._)
    title('Capacitor Voltage')
    grid()
    xlabel('Time in %s' % t.unit)
    ylabel('Voltage in %s' % uc.unit);