Serial Circuits / Voltage Divider
=================================


.. tikz:: **Serial connection** of two resistors
   :xscale: 35
   :align: center
	      
   \draw (0,0)
   to[voltage source, v=$U$] (0,4)
   to[short, i=$I$] (3,4)
   to[R=$R_1$,v<=$U_1$] (3,2)
   to[R=$R_2$,v<=$U_2$] (3,0)
   to[short] (0,0);
   \draw (5,0) to (5,0);


   
Consider a circuit with two resistors in series. Again it is a closed
circuit and current will flow. The *same* current will flow through
both resistors. Using Ohm's law we then can calculate the voltages
across the resistors:



.. math::
   U_1 = I\, R_1\\
   U_2 = I\, R_2

The total voltage across both resistors is $U_1+uU2$ and is equal to
the battery voltage $U$. So:

.. math::
   U = U_1 + U_2 = I\, R_1 + I\, R_2 = I\, (R_1 + R_2)

i.e. the two resistors in series act as one resistor with resistance
$R = R_1+R_2$.

Our analysis above can be done for an arbitrary number of resistors in
a serial circuit.

*In a serial circuit the current through all resistors is the same, the
voltage across each resistor is dependent on its resistance (relative
to all other resistors in the circuit).*


The serial circuit serves as a voltage divider. In a more systems
oriented view of the same circuit we can draw it as shown below.

.. tikz::
   :xscale: 35
	    
   \draw
   (0,0) node[left] {$U_i$} to[short,o-] ++(2,0)  to[R=$R_1$] ++(0,-2) coordinate(u2)
   to[R=$R_2$] ++(0,-2) node[ground]{} 
   (u2) to[short,*-o] ++(2,0) node[right] {$U_o$}
   ;

Note that it appears that the circuit in the figure above does not
contain a loop and thus no current can flow. However in these types of
circuits it is implicitly assumed that the indicated voltages ($U_i$
and $U_o$) are with respect to ground (the arrow like symbol at the
bottom of the circuit).

It is not hard to prove that

.. math::
   U_o = \frac{R_2}{R_1+R_2} U_i