Operational Amplifiers
======================


   ..
      tikz:: **Kirchhoff's Current Law.**
      laksdl kalsdk lk lkasd
      :xscale: 50

      \draw (0,0)
      to[vsource, v=$V_0$] (0,3)
      to[short, -* , i=$I_0$] (2,3)
      to[R=$R_1$, i>_=$i_1$] (2,0) -- (0,0);
      \draw (2,3) -- (4,3)
      to[R=$R_2$, i>_=$i_2$]
      (4,0) to[short, -*] (2,0);
  


In analog electronics, including signal processing, operational
amplifiers are very important components. These are *active
components* in contrast to the passive components we have seen so
far. As such they are really outside the scope of these lecture
series. We only give a very brief introduction as we need an op amp in
one of the circuits in the practical course and because opamps are a
very nice example of the use of feedback in signal processing systems.

.. sidebar:: Powering an OpAmp

   An opamp can amplify voltages and currents and thus increase the
   power. That power has to come from a powersource. The opamp is an
   **active electronic element** just like the bare transistor. A more
   formal drawing is:

   .. tikz::
      :xscale: 65

      \draw
      (0,0) node[op amp] (opamp) {}
      (opamp.+) node[left] {$u_+$}
      (opamp.-) node[left] {$u_-$}
      (opamp.out) node[right] {$u_o$}
      (opamp.up) --++(0,0.5) node[vcc]{5\,\textnormal{V}}
      (opamp.down) --++(0,-0.5) node[vee]{-5\,\textnormal{V}}
      ;

   This opamp is 'fed' by a dual rail power source supplying both $+5$
   and $-5$ Volt. The inputs $u_{+}, u_{-}$ and output $u_o$ all have to be
   within the range of the supplied voltages. No matter what the inputs
   are, the output is always clipped in the interval of supplied
   voltages.

   It is possible (and indeed used a lot) to use $0$ and $5$ Volt as
   supplied voltages but than the opamp *cannot* give negative
   voltages as output. Something to be reckoned with if you mix
   digital and analog signals.
   
      
Functionally an opamp amplifies the potential difference at the input
$A_0$ times. I.e. $u_\text{out} = A_0 (u_{+} - u_{-}) $ . The open loop
amplification factor is very high, in the order of
$10^5$. Approximately we can take it to be infinitely large. We call
it an ideal opamp.


   .. tikz:: **Operational Amplifier (OpAmp).** The open loop gain of
      the amplifier is given by $U_0 = A_0(u_{+} - u_{-})$. Typically
      $A_0\approx10^5$.
      :xscale: 30
	       
      \draw (0, 0) node[op amp] (opamp) {$A_0$}
      (opamp.+) node[left] (plus) {$u_+$}
      (opamp.-) node[left] {$u_-$}
      (opamp.out) node[right] {$u_o$};

      
As an open loop amplifier an opamp is of little use. The output will
be clipped by the power supply and that even for very small potential
difference at the input. The opamp shines in case we use it in a
*feedback* configuration.

The easiest feedback loop configuration is the opamp with
unity gain, i.e. $u_0 = u_i$.

   .. tikz:: **Unity gain negative feedback loop.**
      :xscale: 35

      \draw[scale=1]
      (2,0.5) node [op amp] (opamp) {$A_0$}
      (0,0) node [left] { $u_i$} to [short, o-] (opamp.+)
      (opamp.out)  |- (0.5, 2) to [short] (2, 2) 
      (opamp.-) -| (0.5, 2)
      (opamp.out) to [short, -o] (4,0.5) node [right] {$u_o$};


      
..
 image:: https://blog.mbedded.ninja/electronics/components/op-amps/op-amp-schematic-voltage-follower-buffer.png



To calculate the transfer function $u_o/u_i$ for this opamp
configuration note that:

.. math::
   u_o &= A_0(u_+ - u_-)\\
   u_+ &= u_i\\
   u_- &= u_o

so

.. math::
   u_0 = A_0( u_i - u_o)

i.e.

.. math::
   u_0(1+A_0) = A_0 u_i

and thus

.. math::
   \frac{u_0}{u_i} = \frac{A_0}{A_0+1} 

and for (very) large $A_0$ we have

.. math::
   \frac{u_0}{u_i} = 1

This circuit is also called a *buffer* or a *unity gain* amplifier. It
more or less makes the circuitry that 'makes' $u_i$ (the input) and
the circuitry that uses $u_0$ (the output) independant of each other.

Note that for $A_0\rightarrow\infty$ we have that $u_- = u_+$ and thus
that no current is flowing into (of out of) the opamp through its
inputs. These two observations can be proven to characterize circuits
with opamps that are more complex than the simple unity gain
amplifier. These observations are the principles of the **ideal
opamp**:

.. math::
   u_+ &= u_-\\
   i_- &= i_+ = 0

Consider the following opamp circuit where both $u_i$ and $u_o$ are
measured with respect to ground.


.. tikz:: **Inverting Amplifier.**
   :xscale: 50
	    
   \draw (0, 0) node[op amp, yscale=1] (opamp) {}
   (opamp.-) to[short,-* ] ++(-1, 0) coordinate(A)
   (opamp.+) -| ++(-1,-1) node[ground](B){}
   (opamp.out) to[short,*-o] ++(1, 0) coordinate(C)
   %(C |- B) node[ground]{} node[ocirc]{}
   (A) to[R,l_=$R_1$,-o] ++(-2, 0) node [left] {$u_i$}
   (A) |- ++(1,1) to[R=$R_2$] ++(2,0) -| (opamp.out) to[short,-o] ++(1,0) node [right] {$u_o$}
   ;


Applying the rules of the ideal opamp we get that $u_-=0$ (a *virtual
ground*). Given that no current is flowing into (or out of) the opamp,
i.e. $i_-=0$ we have that the current flow $i_1$ through $R_1$ equals
the current $i_1$ through $R_2$ and applying Ohm's law we have

.. math::
   i_1 &= i_2\\
   i_1 &= \frac{u_i}{R_1}\\
   u_o &= -i_2 R_2

leading to

.. math::
   \frac{u_o}{u_i} = - \frac{R_2}{R_1}


Note that again the properties of this opamp circuit are not dependent
on the opamp parameter $A_0$. This is a good thing in practice as the
open loop amplification is not constant over opamps (even of the same
type) and for any opamp is dependent on temperature and other
parameters. The use of negative feedback stabilizes the properties of
the opamp circuit.

In an exercise you are asked to calculate the transfer function for a
non inverting opamp.

Most OpAmps can amplify DC signal as well as AC signals. It depends on
the type op opamp what is *bandwidth* is. This capability can be used
to build frequency dependent opamp circuits by introducing capacitors
(and sometimes even inductors) in the feedback loop of an opamp
circuit.


Operational amplifiers can also be used to add signals. Consider the
following extension to the inverting amplifier showing a way to mix
three different audio signals.

.. Figure:: /figures/summing_audio.gif
   :align: center
   :figwidth: 70%

   **Mixing Audio Signals.** A summing opamp is used to mix three
   audio signals.
	      
Note that the US style for resistors is used (the squigly line) and
that a resistor with an arrow pointing to it is variable voltage
divider (a potentiometer). The capacitors at the input are used to
decouple the amplifier from any DC signal in the inputs.

A summation circuit using an opamp is also used in a traditional DAC
(Digital to Analog Converter). Note that logical signals ''1'' are
represented with a constant voltage (often 5 Volt) whereas a digital
''0'' is represented with a zero voltage level.

.. figure:: /figures/dac_ladder_converter.gif
   :align: center
   :figwidth: 70%

   **DAC Ladder Converter.** A summing opamp configuration used to
   convert digital into analog.