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Is equivalent resistance always lower if we add a resistor to a passive electronic circuit?

Since my last question was probably badly formed and misunderstood this is a new one.

I was trying to find a lower bound solution for an equivalent resistance of an infinite resistor network. In order to do that I connected some nodes and found the equivalent resistance.

How can we mathematically prove that equivalent resistance between any 2 nodes is always lower if we connect any other 2 nodes in a network with a resistor.

Note that this is not example of a parallel circuit, this is example of a 2 port network where we connect a load on one port and measure resistance on another.

tl;dr: 2 port network. Prove that connecting a load will reduce input resistance. In terms of z-parameters we need a prof that \$ Z_{12} \cdot Z_{21} > 0 \$.

  • 1
    \$\begingroup\$ Why do you post two same questions? I think you should go and clarify the original question instead of creating a new one. \$\endgroup\$
    – Al Kepp
    Dec 14, 2012 at 0:02
  • \$\begingroup\$ I think that I already made the damage so the question is cluttered with non related answers and many edits and it is now hard to follow. I would rather like to delete my old question. Everyone has my sincere apologizes for posting the question again. \$\endgroup\$
    – Serge
    Dec 14, 2012 at 0:21
  • 1
    \$\begingroup\$ I've closed this question as duplicate, you should consider editing the old question and then accepting the answer that fits the most. Then we may clean it from irrelevant answers, if needed. \$\endgroup\$
    – clabacchio
    Dec 14, 2012 at 10:08

1 Answer 1


According to Wikipedia:

A network is said to be reciprocal if the voltage appearing at port 2 due to a current applied at port 1 is the same as the voltage appearing at port 1 when the same current is applied to port 2. ... In general, a network will be reciprocal if it consists entirely of linear passive components (that is, resistors, capacitors and inductors). In general, it will not be reciprocal if it contains active components such as generators.

Stated in terms of the impedance parameters, aka Z-parameters, this basically means that for a network of linear passive components,

Z12 = Z21

Therefore Z12 and Z21 have the same sign and Z12Z21 >= 0.


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