.. 4.1

RLC circuits, steady state response
===================================

**Objective**

Study the effect of series LCR elements in an AC circuit. Three
different combinations can be studied.

.. image:: schematics/RCsteadystate.svg
	   :width: 300px
.. image:: schematics/RLsteadystate.svg
	   :width: 300px
.. image:: schematics/RLCsteadystate.svg
	   :width: 300px

**Procedure**

-  Make connections one by one, as per the drawing
-  Note down the amplitude and phase measurements, in each case
-  Repeat the measurements by changing the frequency.
-  For RLC series circuit, the junction of L and C is monitored by A3
-  For resonance select :math:`C = 1~\mu F`, :math:`L = 10~mH` and :math:`f = 1600~Hz`, adjust f to
   make phase shift zero
-  The total voltage across L and C together goes almost to zero, the
   voltage across them are out of phase at resonance

**Discussion**

The applied AC voltage is measured on A1 and the voltage across the
resistor on A2. Subtracting the instantaneous values of A2 from A1 gives
the combined voltage across the inductor and capacitor. We need to use
an inductor with negligible resistance for good results. The phase
difference between current and voltage is given by
:math:`\Delta \Phi = \arctan((X_C − X_L)/X_R)`.

The total voltage, voltage across R and the voltage across LC are shown
in figure. The phasor diagram shows the phase angle between the current
and the voltage. The inductance used in this experiment is around :math:`10~mH`,
having a resistance of :math:`20~\Omega`.

At :math:`1600~Hz`, :math:`X_C \simeq X_L` and the voltage across LC is decided by the
resistance of the inductor. At the resonant frequency, the voltage drop
across LC will be minimum, decided by the resistance of the inductor.
The input A3 is connected between L and C, so that the individual
voltage drop across L and C can be displayed.
