# Circuit Theory/Active Filters

Active filters have op amps or amplifiers in them. They usually don\'t
have inductors because they are expensive, heavy and large when compared
to capacitors.

## Minimal Filter Transfer Functions

Each filter type has a characteristic transfer function:

  Filter Type   Transfer Function
  ------------- ---------------------------------
  High Pass     $\frac{s^2}{aS^2 + bs + c}$
  Low Pass      $\frac{1}{aS^2 + bs + c}$
  Band Pass     $\frac{s}{aS^2 + bs + c}$
  Notch         $\frac{s^2 + d}{aS^2 + bs + c}$

First the goal is to build a transfer function from the op-amp circuit
so that a bode diagram can be plotted. This is a simple matter of
substituting s for derivatives in the terminal relations.

## Active Filter Components

Rather than draw an example op amp circuit for each of the above
filters, it is easier to draw four op amp circuits that are more related
to the desired transfer function:

+-------------------------+----------------+-------------------------+
| Circuit                 | Generates      | Transfer Function       |
+=========================+================+=========================+
| ![](Activ               | Complex Pole   | derivation |                | Example90/ "wikilink")\ |
|                         |                | $-\frac{R_              |
|                         |                | 3}{R_1 + (C_2R_1R_2 + C |
|                         |                | _2R_1R_3 + C_2R_2R_3)s  |
|                         |                |  + C_1C_2R_1R_2R_3s^2}$ |
+-------------------------+----------------+-------------------------+
| ![](Acti                | Zero at Origin | $-sRC$                  |
| veFilterOriginZero.png) |                |                         |
+-------------------------+----------------+-------------------------+
| ![](Ac                  | Real Pole      | $\frac{-\frac{1}{R_     |
| tiveFilterRealPole.png) |                | 1C}}{s+\frac{1}{R_2C}}$ |
+-------------------------+----------------+-------------------------+
| ![](Ac                  | Real Zero      | $-R_                    |
| tiveFilterRealZero.png) |                | 2C(s + \frac{1}{R_1C})$ |
+-------------------------+----------------+-------------------------+

Most filter design starts with the desired
transfer function. Then the circuit is built. With the above four
configurations, any transfer function can be created.

Better filters are have stepper slopes and narrower bands in their bode
diagrams. This is measured with a factor called Q for
quality. This is why
there are so many different types of filter designs.

## Under Constrained Design

Suppose you wanted this transfer function:

$$\frac{s+1}{(s+2)(s+3)(s+4)}$$ Then an active filter could be built
like this:

<File:ActiveFilterRealZero.png%7Cstage> A\<br
\\\>$-R_2C(s + \frac{1}{R_1C})$ <File:ActiveFilterRealPole.png%7Cstage>
B\<br \\\>$\frac{-\frac{1}{R_1C}}{s+\frac{1}{R_2C}}$
<File:ActiveFilterRealPole.png%7Cstage> C\<br
\\\>$\frac{-\frac{1}{R_1C}}{s+\frac{1}{R_2C}}$
<File:ActiveFilterRealPole.png%7Cstage> D\<br
\\\>$\frac{-\frac{1}{R_1C}}{s+\frac{1}{R_2C}}$

Choosing values to match the above transfer function results in a table
like this:

  Input Stage Variables        Feedback stage variables
  ---------------------------- --------------------------
  $R_{1_A}C_A = 1$             $R_{2_A}C_A = 1$
  $R_{1_B}C_B = \frac{1}{2}$   $R_{2_B}C_B = 1$
  $R_{1_C}C_C = \frac{1}{3}$   $R_{2_C}C_C = 1$
  $R_{1_D}C_D = \frac{1}{4}$   $R_{2_D}C_D = 1$

There are 12 unknowns and 8 equations. This is called an under
constrained problem. It means that one can apply other design rules. For
example, capacitors are more expensive than resistors. Buying in bulk is
less expensive. So \... choose one value for all the capacitors and then
adjust resistor values. Choose C = 1μF Then all the resistors are 1
M&Ohm; accept R~1B~, R~1C~ and R~1D~. 
