🚀 go-pugleaf

     I'm interested in getting a math package to help me out in both the
traditional and some "extended" EE circuit theory problems.  Most of these
problems are solvable or simplifiable, but the process of solving/simplifying
the equations usually takes (me) hours without a math package.

     Electronic circuits (well, some of the analog ones anyway) spend their
life numerically "solving" differential equations and generating the answers as
current and voltage waveforms.  Analog EEs convert circuit topology to
differential equations and differential equations back to circuit topologies.
The diffeqs and topologies don't always match to begin with; it's usually
necessary to manipulate the diffeqs and topologies to develop a circuit or
understand what the circuit is doing.

     The "challenge" problem is a simple oscillator circuit converted straight
(with minimal reduction) into raw equations.  There is a twist -- basic circuit
theory assumes linear elements everywhere, but this oscillator uses a
transistor in its nonlinear mode, so I've included a simple nonlinear model
[the e^ term ] of the transistor in the equations.  This nonlinear model
changes the problem so that I can't solve it with Laplace transforms any longer
(in fact, I'm not sure it's solvable in closed form).  Note that SPICE, a
circuit simulator, makes the numerical solution of these equations easy for me
(Vc(t) looks like a sine wave).  However, SPICE doesn't provide the
understanding that solved diffeqs would.

      I'd appreciate it if someone out there would try these equations in his/
her favorite math package and let me know if:

      a) The math package choked, and 
      b) if not, what the "simpler forms" of the output looked like

Note again that I'm not at all sure that the problem (as stated) is solvable
in closed form.  (SPICE tells me that, with reasonable values for the 
parameters, the oscillator works :-)

      Please email your responses to me, clw@hprnd.rose.hp.com, and I'll 
summarize the results if there is enough interest.

Thanks,
    Carl Wuebker * HP Roseville * clw@hprnd.rose.hp.com * (916) 785-4296

--- "Challenge" EE Problem: Simple Oscillator w/Nonlinear Model of Transistor

     The (raw) simultaneous integro-differential equations which describe the 
circuit are shown below.  L, Cr, Cf, R are all fixed, positive numbers with
"reasonable" engineering values shown below.  Ve(t) and Vc(t) are varying
functions of time.  The problem is to pick values of L, Cr, Cf and R which
make the the output Vc(t) "close to" a sine wave.  Usually, you can pick one
or two values arbitrarily, but the circuit topology constrains the rest if
you want a pure (or close to pure) sine-wave output.  I'm interested in how
you would, say, choose Cr and L (to determine output frequency) and, from
there, go about picking Cf and R.  This usually involves generating equations
for the output (Vc(t)) and picking Cf and R (in terms of Cr and L) to minimize
everything but the sinusoidal part of the Vc(t) waveform.

      Reasonable Values for Parameters
      --------------------------------
           Is     Use 10^-15 (it varies between 10^14 and 10^15)
           e      2.718...
           K      0.026

           L      10^- 6 .. 10^-3
           Cr,Cf  10^-12 .. 10^-4
           R           1 .. 10^4

           Ve(t=0)  Any initial value (your choice) in the range of 1..9 
                    [ I suggest using a number close to 1 here ]
           Vc(t=0)  Any initial value (your choice) in the range of 2..9 
                    [ I suggest using a number close to 5 here ]

      Differential Equations
      ----------------------

           d is differentiation symbol

           /
           | is integration symbol.
           /

           * is multiply.

      Equation 1:

        1  /t                      d(Vc(t)-9)        d(Vc(t)-Ve(t))
       --- |  (Vc(t)-9)dt' +  Cr * ---------- + Cf * -------------- +
        L  /0                          dt                  dt

                                      K*(1.75-Ve(t))
                              Is * ( e              -1) = 0
                      

      Equation 2:
                         d(Ve(t)-Vc(t))                 K*(1.75-Ve(t))   
        R * Ve(t) + Cf * -------------- - 1.01 * Is * (e              -1) = 0
                               dt