On heuristics of transforming unresolvable contradictions into resolvable ones

Y. B. Karasik,
Thoughts Guiding Systems Corp.,
Ottawa, Canada.
e-mail:karasik@sympatico.ca

Many problems as they are stated are unsolvable. Contradictions contained in their description are unresolvable. However, if description of such problems is rearranged they may become solvable. Unfortunately, it is not known how to rearrange the problem description to make it solvable. It is easier to propose heuristics of converting contradictions containing in the problem description into other ones, albeit not containing in the problem description, but which resolution nevertheless solves the problem.

Consider, for example, a well known method of dividing voltage with the help of resistors. When two resistors are connected in series and the voltage is tapped at the point between the two resistors this voltage is a fraction of the total voltage across both resistors:

This method does not work when we pass alternating current through the resistors. If the frequency of AC is higher than 1MHz then the resistors have parasitic capacitances and the actual circuit looks like this:

where R1, C1 and R2, C2 are resistance and parasitic capacitance of the first and the second resistors respectively.

The output voltage of this divider is

Vout = VinR2/(R2+R1((iωR2C2 + 1)/(iωR1C1+1))),

where i is the imaginary unit and ω is AC frequency.

Since parasitic capacitance depends on ω and other factors its value is uncertain. This makes Vout uncertain too.

How to design a voltage divider for AC then? (The question is not addressed to electrical engineers, which know the answer. It is addressed to those who does not know and want to find a solution by TRIZ.)

The cause of why the above voltage divider does not work is this:

  1. resistors have parasitic capacitances, and
  2. the parasitic capacitances are uncertain
So, one could be tempted to formulate the following contradictions:
  1. resistors should not have parasitic capacitances so that the divider works but they have them;
  2. the parasitic capacitance of resistors should not be uncertain but it is.
Unfortunately, both contradictions are unsolvable. We have to transform them into solvable contradictions by applying various heuristics.

One of the heuristics is to replace a contradiction "A should have X and should not have it" by a contradiction "the value of X should be negligible and should not be negligible". The other heuristics is to replace a contradiction "X should not be uncertain but it is" by a contradiction "X should be constant but it is not".

After replacing the contradictions one has to investigate why X cannot be made either negligible or constant. Maybe it could. Then the new contradictions are spurious and the problem is solved. If they indeed cannot be made neither negligible nor constant, we at least can augment the contradictions as follows:

  1. the value of X should be negligible so that to solve the problem and cannot be negligible because of Z;
  2. X should be constant so that to solve the problem but it cannot be constant because of Z.
Then we can try and resolve these contradictions. The latter is easily resolvable by separating between such opposites as "strict" and "approximate": X is made approximately constant rather than strictly constant. It is done by adding two large capacitors in parallel to the resistors. This time they are real capacitors rather than imagined ones C1 and C2 existing on the paper only to account for the parasitic capacitance of resistors.

Heuristic rules of contradictions (and problems) transformation is a missing tool in TRIZ.