I'm currently reading a text on analog filters (Zverev) that discusses Foster's Reactance Theorem in relation to passive circuits and I was thrown a little by the text that says: "Physically, this signifies that the impedance or admittance of the reactive network always increases in value when the frequency is increasing" (Pg 37 for those with the book).

Then the description of the theorem on wikipedia (wikipedia entry) has a similar line: "It is easily seen that the reactances of inductors and capacitors individually increase with frequency and from that basis a proof for passive lossless networks generally can be constructed".

I'm obviously missing something, but I always thought that the reactance of a capacitor decreased with increasing frequency. What don't I understand?

  • \$\begingroup\$ Clearly that wiki page is wrong. \$\endgroup\$ – Andy aka Jun 18 '19 at 11:21
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    \$\begingroup\$ @Andyaka Foster's uses reactance in the sense of the imaginary part of the impedance, which has sign. A capacitor's reactance increases from large negative at low frequencies up closer and closer to zero as the frequency increases, even though the magnitude of that is decreasing. Interesting concept, I might try to wrap my head round that properly. \$\endgroup\$ – Neil_UK Jun 18 '19 at 11:37
  • \$\begingroup\$ @Neil_UK I'm not sure what you are driving at. \$\endgroup\$ – Andy aka Jun 18 '19 at 11:40
  • \$\begingroup\$ @Andyaka The wiki page is not necessarily wrong. Increasing absolute is not inconsistent with decreasing magnitude. \$\endgroup\$ – Neil_UK Jun 18 '19 at 11:44
  • \$\begingroup\$ Ah, quite possible but I thought the whole point was one of monotonicity and it needn't have said the confusing thing about a capacitor's reactance rising with frequency because monotonicity works just as well in either direction. \$\endgroup\$ – Andy aka Jun 18 '19 at 11:59

Foster's Theorem uses reactance in the strict mathematical sense of the imaginary part of the impedance. This is of course signed. By convention, a capacitor's reactance is negative. At low frequency, it's large negative. At higher frequencies, it increases towards zero. This is opposite to the magnitude of the reactance, which of course decreases with increasing frequency. Read the whole of that wikipedia article carefully. What's wrong with the wiki article is it says 'It is easily seen ...', it can be seen, but not easily.

enter image description here

Image from the wikipedia article

These both show reactance increasing (becoming more positive) with frequency.

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  • \$\begingroup\$ I agree. It's not easily seen! \$\endgroup\$ – Buck8pe Jun 18 '19 at 12:19

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