[sdiy] Frequency multipliers

David G. Dixon dixon at interchange.ubc.ca
Sat Jun 5 01:21:56 CEST 2010

> By what method does one convert an equation into a circuit?

That's just about all we ever do, whether we realize it or not!  All circuit
elements have transfer functions which dictate the ratio of voltage out to
voltage in.  If you know these functions, and with Ohm's and Kirchhoff's
Laws, then you can design whole circuits just from equations.

When you apply a gain, you are multiplying by a constant.  When you use a
summing opamp, you are adding and subtracting.  A multiplier multiplies two
voltages together.  A resistive divider divides a voltage by a constant.  A
transistor generates an exponential current from a linear base-emitter
voltage.  A differential pair generates a current which is the hyperbolic
tangent of the differential voltages on the bases (approximately).  Filters
and other reactive elements have transfer functions which are ratios of
polynomials in the frequency domain -- that's a whole lot of fun, if you
understand how to use it.  Since the frequency domain is more or less the
same thing as the Laplace domain, and since Laplace transforms are used to
solve differential equations, then filter circuits can be used to solve
differential equations.  Indeed, our humble state-variable filter can be
used to solve the classic mass-spring-dashpot problem of classical physics.
In fact, just about all the circuits we use in synthesizers were originally
developed for analog computing.  The operational amplifier was invented to
make analog computing easier.

So, for this equation:

cos(3z) = -3*cos(z) + 4*cos^3(z)

presuming you already have a cos(z) from somewhere (a quadrature oscillator
or a VCO), then feed it to both inputs of a multiplier to get the square,
then feed the square and the original signal into another multiplier to get
cos^3.  Then, feed this to the + terminal of a summer with the correct
resistors to give it a gain of 4 whilst applying the original signal to the
- terminal of the summer through the correct resistor to give it a gain of
-3.  The output of the summer will be cos(3z), the cosine at three times the
frequency of the original.*  Voila!

* (I think, pending testing.)

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